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@@ -0,0 +1,1071 @@
+Ferroelectrically tunable magnetic and topological multistates in thin films of
+MnBi2Te4 family
+Guoliang Yu,1 Chuhan Tang,1 Zhiqiang Tian,1 Ziming Zhu,1 Anlian Pan,2 Mingxing Chen,1, ∗ and Xing-Qiu Chen3, 4
+1Key Laboratory for Matter Microstructure and Function of Hunan Province,
+Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education,
+School of Physics and Electronics, Hunan Normal University, Changsha 410081, China
+2Key Laboratory for Micro-Nano Physics and Technology of Hunan Province,
+College of Materials Science and Engineering, Hunan University, Changsha 410082, China
+3Shenyang National Laboratory for Materials Science, Institute of Metal Research,
+Chinese Academy of Sciences, 110016 Shenyang, People’s Republic of China.
+4School of Materials Science and Engineering, University of Science and
+Technology of China, 110016 Shenyang, People’s Republic of China.
+(Dated: January 3, 2023)
+Ferroelectric control of two-dimensional magnetism is promising in fabricating electronic devices
+with high speed and low energy consumption. The newly discovered layered MnBi2Te4(Bi2Te3)n
+and their Sb counterparts exhibit A-type antiferromagnetism with intriguing topological proper-
+ties. Here, we propose to obtain tunable magnetic multistates in their thin films by ferroelectrically
+manipulating the interlayer magnetic couplings (IMCs) based on the Heisenberg model and first-
+principles calculations. Our strategy relies on that interfacing the thin films with appropriate ferro-
+electric materials can switch on/off an interlayer hopping channel between Mn-eg orbitals as the po-
+larizations reversed, thus resulting in a switchable interlayer antiferromagnetism-to-ferromagnetism
+transition. On the other hand, the interface effect leads to asymmetric energy barrier heights for
+the two polarization states. These properties allow us to build ferroelectrically switchable triple
+and quadruple magnetic states with multiple Chern numbers in thin films. Our study reveals that
+ferroelectrically switchable magnetic and topological multistates in MnBi2Te4 family can be ob-
+tained by rational design for multifunctional electronic devices, which can also be applied to other
+two-dimensional magnetic materials.
+I.
+INTRODUCTION
+Two-dimensional (2D) magnetic materials provide an
+ideal platform to explore novel magnetic and electronic
+properties1–7. The delicate interlayer exchange couplings
+in these systems enable a variety of methods of manipu-
+lating their magnetism. For instance, recent studies re-
+vealed that the A-type antiferromagnetic (AFM) CrI3
+bilayer could be tuned into ferromagnetic (FM) by ex-
+ternal electric field8,9, electrostatic doping10,11, and in-
+terface engineering12–17. Twisting the bilayer may yield
+non-collinear magnetism18–20.
+Recently,
+the
+MnBi2Te4
+family,
+i.e.,
+MnBi2Te4(Bi2Te3)n and MnSb2Te4(Bi2Te3)n, which are
+hereafter referred to as MAT, have much great attention
+due to the coexistence and interesting interplay of
+intrinsic magnetism and band topology in them21–37.
+This series of materials also have a layered van der
+Waals (vdW) structure with an A-type AFM structure
+as revealed by experiments38–41, which preserves the
+combination of the time-reversal and a half lattice
+translation symmetry. As a result, many of them show
+nontrivial topological properties such as topological
+insulators21 and axion insulators22.
+Moreover, the
+systems can be turned into Weyl semimetals as the
+interlayer couplings become ferromagnetic22,23,36.
+The A-type AFM coupling in MAT yields unusual
+even-odd layer-number dependent magnetism for their
+thin films35,42,43. The even-number (even-N) systems are
+expected to have no net magnetization. Whereas those
+with odd-number (odd-N) layers have uncompensated
+magnetization. This difference can lead to distinct topo-
+logical properties for them. For instance, the thin films
+of MnBi2Te4 with odd-N septuple layers are quantum
+anomalous Hall insulators. However, those with even-N
+layers have a zero Chern number42. Instead, their topo-
+logical properties can be characterized by the so-called
+pseudospin Chern number44. Due to the weak interlayer
+interaction, small magnetic fields could induce spin-flip
+transitions, giving rise to an AFM-to-FM transition in
+the IMCs35,40.
+Chemical dopings45,46 and antisite de-
+fects47–51 can also be used to manipulate the IMCs in
+these systems, although they may complicate the nature
+of the surface states.
+First-principles calculations sug-
+gest that the AFM double-septuple MnBi2Te4 could be
+driven into a Chern insulator with a high Chern number
+under electric fields52.
+In this work, we propose to ferroelectrically tune the
+IMCs in MnBi2Te4 thin films for magnetic multistates by
+interface, which is desired for memory devices with high
+density storage, high speed, and low power consump-
+tion. We reveal that hole doping can lead to an interlayer
+AFM-to-FM transition in MAT bilayers based on the un-
+derstanding of the IMCs using the Heisenberg model. We
+provide a guideline for designing ferroelectric substrates
+that may induce transitions in the interlayer exchange
+couplings, i.e., polarization dependent IMCs, as demon-
+strated by our first-principles calculations. Moreover, we
+arXiv:2301.00515v1  [cond-mat.mtrl-sci]  2 Jan 2023
+
+2
+find that the interface effect results in symmetry breaking
+in the two polarization states of the FE substrate. This
+asymmetry allows us to design switchable magnetic mul-
+tistates in sandwich structures made of MnBi2Te4 mul-
+tilayers and 2D ferroelectric (FE) materials, which also
+exhibit distinct electronic and topological properties.
+We begin by presenting the concept of FE tuning of
+IMCs in MAT bilayers, which is shown in Fig. 1. In these
+systems, each Mn atom is coordinated with six chalcogen
+atoms, which form a distorted octahedron. The Mn-3d
+orbitals are split into triply degenerate t2g states and the
+doubly degenerate eg states due to the octahedral ligand
+field. The states are further split due to the magnetic
+exchange interaction between the Mn atoms. As a re-
+sult, the majority states of the t2g and eg orbitals are
+fully occupied by the five d electrons of the Mn2+ ions
+(see Fig. 1), resulting in a high spin state for the Mn2+
+ions.
+Like the 2D magnetic bilayers reported by Refs
+31 and 53, this type of occupation favors AFM IMCs
+between the Mn+2 ions, which are mediated by the p-
+orbitals of the nonmetallic atoms (denoted by {p . . . p}).
+Whereas FM IMCs are energetically unfavorable because
+the e↑
+g − {p . . . p} − e↑
+g hopping between the Mn-d or-
+bitals of adjacent layers is prohibited31,53,54.
+For our
+systems, reducing the occupation of the d orbitals makes
+the e↑
+g − {p . . . p} − e↑
+g hopping channel energetically fa-
+vorable, thus enhancing the stability of the FM IMCs.
+Indeed, our DFT calculations indicate that all the MAT
+bilayers undergo the AFM-to-FM transition by small hole
+dopings (see Fig. 1c and Fig. S1), which is also expected
+for their multilayers.
+The IMCs in MAT-2L can be understood using the
+following spin Hamiltonian.
+H =
+�
+i,j
+Jt
+∥Si · Sj +
+�
+m,n
+Jb
+∥Sm · Sn +
+�
+i,m
+J⊥Si · Sm, (1)
+where J∥ and J⊥ denote the intra- and interlayer ex-
+change interactions between the Mn ions, respectively.
+The intralayer ones are denoted by Jt
+∥ for the top layer
+and Jb
+∥ for the bottom layer, for which only the first
+nearest-neighbor interactions are taken into account. Jt
+∥
+are equal to Jb
+∥ for the freestanding MAT-2L. Whereas
+for the interlayer ones, the second nearest neighbors are
+included.
+For the bilayers without doping, we obtain
+positive J1st
+⊥
+and negative J2nd
+⊥
+(see Fig. 1d, Fig. S2 and
+Table S1).
+Note that the magnitude of J1st
+⊥
+is larger
+than that of J2nd
+⊥
+. So, the sum of J1st
+⊥
+and J2nd
+⊥
+, i.e.,
+¯
+J⊥ = J1st
+⊥
++ J2nd
+⊥
+, is positive, which gives rise to AFM
+IMCs. Introducing hole doping suppresses J1st
+⊥
+while en-
+hances J2nd
+⊥
+. As a result, ¯
+J⊥ decreases with increasing
+of the hole doping and eventually changes its sign across
+the AFM-to-FM transition (see Fig. 1d).
+The hole doping over the MAT bilayers can be achieved
+via interfacial charge transfer which requires suitable
+band alignments between them and the substrates.
+When their bands are in the type-I or type-II align-
+ment, interfacial charge transfer can be negligible.
+In
+(b)
+MAT-2L(FM)/P↓
+Mn1
+t2g
+eg
+EF
+t2g
+eg
+P↓
+e
+CBM
+VBM
+Mn2
+t2g
+eg
+t2g
+eg
+MAT-2L(AFM)/P↑
+(a)
+Mn1
+t2g
+eg
+EF
+t2g
+eg
+Mn2
+t2g
+eg
+t2g
+eg
+e
+P↑
+CBM
+VBM
+(c)
+0.4
+0
+-0.4
+0.2
+-0.2
+-0.6
+∆EFM-AFM (meV)
+n (hole/Mn pair)
+0
+0.02
+0.04
+0.06
+AFM
+FM
+MnBi2Te4-2L
+MnSb2Te4-2L
+(d)
+n (hole/Mn pair)
+0
+0.02
+0.04
+0.06
+AFM
+FM
+MnSb2Te4-2L
+0.08
+0
+-0.08
+0.04
+-0.04
+-0.12
+n (hole/Mn pair)
+0
+0.02
+0.04
+0.06
+AFM
+FM
+J⊥ (meV)
+MnBi2Te4-2L
+1st
+J⊥
+2nd
+J⊥
+J⊥
+FIG. 1. Interface engineering of the IMCs in MAT bilayers.
+(a) Spin states of Mn ions for the AFM interlayer coupling. In
+the presence of a substrate that has a type-I or type-II band
+alignment with the bilayer, the IMCs remain AFM. (b) The
+IMCs becomes FM when there is a type-III band alignment
+between them so that the CBM of the substrate lower than
+the VBM of the MAT bilayer. In (b), the white circles denote
+hole dopings to the Mn-eg orbital. In (a, b), arrows denote
+spins. The dark and light red curves with an arrow denote
+hopping channels.
+The one marked by a cross means that
+electron hoppings are prohibited. We use a FE material as
+the substrate, whose polarizations are labeled by P. P ↑ and
+P ↓ represent the up and down polarizations, respectively. (c)
+Energy difference between the FM and AFM states as a func-
+tion of hole doping for freestanding MnBi2Te4 and MnSb2Te4
+bilayers. ∆E = EF M − EAF M. (d) Doping dependence of J⊥
+for the two systems.
+¯
+J⊥ = J1st
+⊥
++ J2nd
+⊥
+.
+these cases, the IMCs are most likely to be AFM. In
+contrast, electrons will be transferred from the MAT bi-
+layer to the substrate when they are in the type-III band
+alignment that the valence band maximum (VBM) of
+the MAT bilayer are higher than the conduction band
+minimum (CBM) of the substrate.Namely, hole doping
+is introduced to the MAT bilayer, which is desired for
+the AFM-to-FM transition. Ferroelectrically switchable
+IMCs may be achieved if a 2D FE materials serves as the
+substrate so that reversing its polarizations gives rise to
+a switching of the band alignment from type-III to type-I
+(II) or vice versa (see Fig. 1). However, one can expect
+that the transferred electrons mainly come from the in-
+terfacial MAT layer because of the vdW-type interlayer
+bonding.Therefore, the spin-flipping most likely happen
+to the interfacial MAT layer rather than those further
+away from the substrate.
+We now come to first-principles calculations of the het-
+erostructures of MAT thin films and 2D FE materials,
+which were performed using the Vienna Ab initio Sim-
+ulation Package55. We choose In2Se3 monolayer as the
+substrate, which has been experimentally proved since
+its prediction in 201756–58.
+Their heterostructures are
+built by slightly adjusting the lattice of In2Se3 (The lat-
+tice mismatch between them is small). The pseudopoten-
+tials were constructed by the projector augmented wave
+method59,60.
+An 11 × 11 × 1 and 21 × 21 × 1 Γ-
+
+3
+centered k-mesh were used to sample the 2D Brillouin
+zone for structural relaxation and electronic structure
+calculations, respectively. The plane-wave energy cutoff
+is set to 400 eV for all calculations.
+A 20 ˚A vacuum
+region was used between adjacent plates to avoid the
+interaction between neighboring periodic images.
+Van
+der Waals (vdW) dispersion forces between the adsorbate
+and the substrate were accounted for through the DFT-
+D361. Different vdW methods/functionals such as DFT-
+D2 and optPBE-vdW were also used for comparison62–64.
+The systems were fully relaxed until the residual force on
+each atom is less than 0.01 eV/˚A. The DFT+U method65
+is used to treat electron correlations due to the partially
+filled d-orbital of Mn for which a value of 5.34 eV is
+used21. Our results on the structural properties, mag-
+netism, and band structures of the free-standing MAT
+films are consistent with previous studies21,32,36.
+The
+kinetic pathways of transitions between different polar-
+ization states are calculated using the climbing image
+nudge elastic band (CI-NEB) method66,67. The topolog-
+ical properties calculations were done using the WAN-
+NIER9068 and WannierTools package69.
+We have performed careful calculations over a number
+of stacking configurations (see Fig. S3, and Table S2).
+The lowest energy configuration is shown in Fig. 2a, in
+which the interfacial Se and Te are in the hollow sites.
+The stacking order is the same as the one for MnBi2Te4
+monolayer on In2Se370, which shows up for all MAT bi-
+layers on In2Se3. Table I summarizes the stability of the
+two magnetic states for MAT bilayers on In2Se3 mono-
+layer in different polarization states. One can see that for
+all the MAT bilayers the IMCs remain AFM when the
+polarization is pointing toward the interface, but become
+FM as the polarization is reversed. The trend is indepen-
+dent of the vdW functionals/methods (Table S3). Below
+we focus on the electronic structure of MnBi2Te4 bilayer
+on In2Se3 monolayer, i.e., MnBi2Te4-2L/In2Se3, which
+are shown in Figs. 2b-d. Those for all other MAT systems
+are shown in Figs.S4-S6 since they show pretty much the
+same trend as MnBi2Te4-2L/In2Se3. The Fermi level is
+located in the band gap for the AFM state. Whereas for
+the FM state, the conduction band of In2Se3 is shifted
+down into the valence band of MnBi2Te4-2L such that
+the Fermi level is crossing the valence band of the latter.
+This feature favors interfacial charge transfer. Fig. 2d de-
+picts the differential charge density for the two magnetic
+states, which indicates that there is almost negligible in-
+terfacial charge transfer between the MnBi2Te4-2L and
+In2Se3 for the AFM state. In contrast, the charge trans-
+fer is much more significant for the FM state than that
+for the AFM state.
+A close inspection finds that the
+charge density on the Mn atoms in the interfacial layer
+becomes positive. This confirms the picture of hole dop-
+ing over this layer and opens up the e↑
+g − {p . . . p} − e↑
+g
+hopping chanel.
+Consequently, the FM state becomes
+energetically favorable for this type of band structure.
+Thus, the FE In2Se3 monolayer fits the criterion for a
+substrate that gives switchable band alignments between
+MBT-2L(FM)/IS(↓)
+0.00
+0.04
+-0.04
+∆ρ (eÅ)
+MBT-2L(AFM)/IS(↑)
+0.00
+0.04
+-0.04
+∆ρ (eÅ)
+0
+10
+20
+30
+40
+50
+Z(Å)
+P
+d
+P
+d
+(a)
+a
+c
+(d)
+Energy (eV)
+1
+0
+-1
+Μ
+Γ
+Κ
+Μ
+Γ
+���
+���
+���
+�
+(c)
+(b)
+MnBi2Te4-2L(AFM)/In2Se3(↑)
+Μ
+Γ
+Κ
+Μ
+In2Se3
+MnBi2Te4-2L
+�
+M
+K
+K'
+Energy (eV)
+1
+0
+-1
+MnBi2Te4-2L(FM)/In2Se3(↓)
+Mn
+Bi
+Te
+In
+Se
+a
+b
+FIG. 2.
+Ferroelectric control of AFM-to-FM transition in
+MnBi2Te4 bilayers. (a) Geometric structures of MnBi2Te4-
+2L/In2Se3 heterostructures. Left panel shows the top view
+of the lowest energy configuration.
+Middle and right pan-
+els show the side view of the structures with different po-
+larizations.
+The thin purple arrows denote spins of the
+Mn ions. While the thick blue arrows denote polarizations
+of the FE substrate.
+(b, c) Band structures for the two
+states in (a), respectively, i.e., MnBi2Te4-2L(AFM)/In2Se3(↑)
+and MnBi2Te4-2L(FM)/In2Se3(↓).(d) Planar-averaged differ-
+ential charge density ∆ρ(z) for the two states shown in (b)
+and (c).
+The insets show the density contour at 0.00015
+e/˚A3. Here, abbreviations (MBT-2L(AFM)/IS(↑) and MBT-
+2L(FM)/IS(↓)) are used by incorporating the IMCs of the
+MnBi2Te4-2L and the polarization states of In2Se3 for sim-
+plicity.
+type-II and type-III with MnBi2Te4-2L. Moreover, the
+trend that the charge transfer mainly happened to the
+interfacial layer also suggests that the spin-flipping ac-
+companied by the AFM-FM transition takes place to the
+interfacial MnBi2Te4 layer. For the trilayers and quad-
+layers, our calculations find the same trend in the spin-
+flipping as the bilayers (Figs. S7 and S8).
+On the other hand, the interface has a significant im-
+pact on the polarization states of the FE In2Se3 mono-
+layer by introducing a coupling between its polarizations
+and the local dipoles of MAT. This coupling breaks the
+symmetry of the two polarization states, that is, it gives
+rise to asymmetric barrier heights for the two polariza-
+tion states. Fig. 3a shows that the state with the po-
+larizations pointing toward the MnBi2Te4 bilayer has a
+barrier height of about 152 meV (∆GT ), which is about
+68 meV lower than the one with polarizations pointing
+away from the interface (∆GA). Therefore, the critical
+electric fields needed to flip the polarizations for the for-
+mer (ET ) is smaller than that for the latter (EA), i.e.,
+ET < EA.
+The asymmetric barrier heights along with the unique
+polarization-dependent IMCs allow designing ferroelec-
+
+4
+FE1
+FE1
+FE2
+FE2
+MBT/P↑
+MBT/P↓
+(c)
+0
+200
+100
+300
+Energy (meV)
+400
+Q2
+Q3
+Q4
+Q1
+Q1
+(e)
+153
+220
+307
+220
+153
+243
+E↓
+E↑
+E↓
+E↑
+E↑
+E↓
+P
+P
+(a)
+0
+200
+100
+250
+Energy (meV)
+50
+150
+MBT-2L/IS(↓)
+MBT-2L/IS(↑)
+E↓ > E2
+A
+Q1
+P
+P
+Q2
+P
+P
+E1
+T < E↓ < E2
+A
+E↑ > E1
+A
+Q4
+P
+P
+Q3
+P
+P
+E↑ > E2
+T
+E1
+A < E↑ < E2
+T
+E↓ > E1
+T
+(d)
+ET < E↓ < EA
+(b)
+T1
+T2
+T3
+E↑ > E A
+E↓ > E A
+ET < E↑ < EA
+P
+P
+P
+P
+P
+P
+ΔGT
+(ET)
+ΔGA
+(EA)
+ΔG1
+T
+(E1
+T)
+ΔG1
+A
+(E1
+A)
+ΔG2
+T
+(E2
+T)
+ΔG2
+A
+(E2
+A)
+FIG. 3. Magnetic multistates in MnBi2Te4 thin films. (a) Kinetic pathway of the FE phase transforming in MnBi2Te4-2L/In2Se3
+(abbreviated as MBT-2L/IS). Interface effects lead to asymmetric barrier heights for the two polarization states, which are
+labelled as ∆GT and ∆GA as the polarization point toward and away from the interface, respectively. Correspondingly, the
+critical electric fields are labelled as ET and EA, respectively. (b) Triple magnetic states in In2Se3/MnBi2Te4-3L/In2Se3 and
+schematic FE transforming by controlling the external electric field. E↑ (E↓) represents the external electric fields along the z
+(-z) axis. (c) Requirement of energy barriers of the two different FE layers for quadruple magnetic states in sandwich structure
+FE1/MBT-4L/FE2, ∆GT
+1 < ∆GA
+1 < ∆GT
+2 < ∆GA
+2 . ∆GT
+1 and ∆GA
+1 are for one FE layer (FE1), which is colored in blue.
+Critical electric fields needed to overcome these barriers are denoted as ET
+1 and EA
+1 ,respectively.
+∆GT
+2 and ∆GA
+2 are for
+the other layer (FE2) colored in red, for which the critical fields are ET
+2 and EA
+2 , respectively. (d) Schematic illustration of
+quadruple magnetic states in FE1/MnBi2Te4-4L/FE2 and transforming between the states under electric fields. (e) Kinetic
+pathways of the quadruple states in In2SSe2/MBT-4L/In2Se3 during FE transforming. The convention of labeling spins of the
+Mn+2 ions and the polarizations of In2Se3 is the same as in Fig. 2.
+trically switchable magnetic multistates for MAT multi-
+layers. We illustrate the concept in MnBi2Te4 trilayers
+and quadlayers, i.e., MnBi2Te4-3L and MnBi2Te4-4L. We
+first sandwich MnBi2Te4-3L in between two In2Se3 layers
+(Fig. 3b). Suppose that both the top and bottom In2Se3
+layers have up polarizations, which can be achieved by
+applying external electric fields anyway.
+According to
+the polarization dependent IMCs discussed above, spins
+in the MnBi2Te4 layer next to the top In2Se3 layer will be
+flipped so that it will beferromagnetically coupled with
+the underneath MnBi2Te4 layer. We label this magnetic
+state as T1. Then one can apply an electrical field E↓
+antiparallel to the z axis that is larger than the critical
+field overcoming ∆GT but smaller than the critical field
+required to overcome ∆GA, i.e., ET < E↓ < EA. As a re-
+sult, the polarizations in the bottom layer will be reversed
+while those in the top layer will remain unchanged. Then,
+the magnetization of the bottom MnBi2Te4 layer will be
+flipped to be ferromagnetically coupled with the adjacent
+MnBi2Te4 layer, i.e., T2 in Fig. 3b. Further increasing
+the electric field such that E↓ > EA will also drive the
+polarizations of the top In2Se3 layer to be flipped. Cor-
+respondingly, the magnetizations of the top MnBi2Te4
+layer will be flipped, for which the magnetic state is la-
+belled as T3. Now, an electric field along the z axis, i.e.,
+E↑, will first force the polarization of the bottom In2Se3
+to be reversed when ET < E↑ < EA. As a result, the
+system will flow into T2.
+Futher enhancing E↑ to the
+level that E↑ > EA will drive the system back into T1.
+So the whole system have triple magnetic states, which
+can be ferroelectrically controlled. Likewise, sandwiching
+thicker films than triple layers by the same FE layers also
+gives rise to triple magnetic states.
+More magnetic states can be obtained by sandwiching
+the MAT thin films in between two different FE layers
+with a special combination of the barrier heights. Such a
+combination requires that the highest barrier for one FE
+monolayer should be lower than the lowest barrier for the
+other FE layer. We depict the barrier heights for the two
+different FE layers in Fig3c, ∆GT
+1 and ∆GA
+1 are for one
+
+5
+TABLE I. Energy difference between the interlayer FM and
+AFM states for freestanding MAT bilayers and their bilayers
+supported by In2Se3 monolayer in different polarization states
+(denoted by arrows). ∆E = EF M − EAF M, EF M (EAF M)
+represents the total energy of the FM (AFM) state.
+Systems
+∆E [meV/(Mn pair)]
+IMCs
+MnBi2Te4-2L
+0.21
+AFM
+MnBi2Te4-2L/In2Se3(↑)
+0.22
+AFM
+MnBi2Te4-2L/In2Se3(↓)
+-0.16
+FM
+MnSb2Te4-2L
+0.39
+AFM
+MnSb2Te4-2L/In2Se3(↑)
+0.36
+AFM
+MnSb2Te4-2L/In2Se3(↓)
+-0.40
+FM
+MnBi4Te7-2L
+0.03
+AFM
+MnBi4Te7-2L/In2Se3(↑)
+0.03
+AFM
+MnBi4Te7-2L/In2Se3(↓)
+-0.01
+FM
+MnSb4Te7-2L
+0.06
+AFM
+MnSb4Te7-2L/In2Se3(↑)
+0.09
+AFM
+MnSb4Te7-2L/In2Se3(↓)
+-0.04
+FM
+FE layer (FE1), to which the corresponding critical elec-
+tric fields are ET
+1 and EA
+1 , respectively. Whereas ∆GT
+2
+and ∆GA
+2 are for the other layer colored in red (FE2),
+for which the critical fields are ET
+2 and EA
+2 , respectively.
+In the case that ∆GT
+1 < ∆GA
+1 < ∆GT
+2 < ∆GA
+2 , i.e.,
+ET
+1 < EA
+1 < ET
+2 < EA
+2 , a layer-by-layer flipping mech-
+anism for the FE contacts can be achieved by properly
+controlling the electric field. As a result, one can have
+quadruple magnetic states based on the polarization-
+dependent IMCs in MAT heterostructures (Fig. 3d). Our
+calculations find that the barrier heights of In2Se3 and
+In2SSe2 monolayers fit the above requirement for the
+quadruple magnetic states. Specifically, we obtain 245
+meV (∆GT
+2 ) and 308 meV (∆GA
+2 ) for In2SSe2 with po-
+larizations pointing toward and away from the MnBi2Te4
+layer (see Figs.
+S9 and S10), respectively, which are
+larger than those of In2Se3 (see Fig. 3a, 152 meV for
+∆GT
+1 and 220 meV for ∆GA
+1 ). In Fig. 3e, we show the
+kinetic pathway of transforming the polarization states,
+which suggests that the quadruple states are ferroelectri-
+cally switchable.
+The ferroelectrically tunable magnetic multistates give
+rise to a variety of distinct topological properties the
+MAT thin films. We perform calculations of the Chern
+number (C) for the MnBi2Te4 multilayers with the mag-
+netic states show in Figs. 2 and 3. For the bilayer sys-
+tems, the topological properties of MnBi2Te4 remain un-
+changed upon interfacing, i.e., C = 1 for the FM state
+and C = 0 for the AFM state, which is also supported
+by the results of edge states (Figs. S11 and S12). For
+MnBi2Te4-3L, we have C = 0, -1, and 0 for T1, T2, and
+T3, respectively.
+Whereas for MnBi2Te4-4L, there are
+three Chern numbers for the quadruple states, i.e., 1,
+0, and -1. Table II summarizes the Chern numbers for
+the studied MnBi2Te4 thin films. We expect that such a
+TABLE II. Chern number (C) for MnBi2Te4 multilayers with
+different magnetic states. Arrows denote the magnetizations
+on Mn ions.
+Systems
+IMCs
+C
+MnBi2Te4-2L
+M1 (↑↓)
+0
+M2 (↓↓)
+-1
+MnBi2Te4-3L
+T1 (↓↓↑)
+0
+T2 (↓↓↓)
+-1
+T3 (↑↓↓)
+0
+MnBi2Te4-4L
+Q1 (↑↑↓↑)
+1
+Q2 (↑↑↓↓)
+0
+Q3 (↓↑↓↓)
+-1
+Q4 (↓↑↓↑)
+0
+ferroelectrically tunable multiplet for the Chern number
+may be seen in other MAT multilayers.
+In conclusion, we have proposed to ferroelectrically
+tune the magnetism of MAT thin films using model and
+first-principles calculations. The scheme is based on the
+fact that the IMCs are strongly dependent on the oc-
+cupation of d-orbitals of the Mn2+ ions. The variation
+in the occupation can be controlled by interfacing the
+films with a FE layer with appropriate band alignments.
+We have demonstrated the concept in MAT/In2Se3 het-
+erostructures by performing first-principles calculations.
+We find that the interfacing effect mainly has an im-
+pact on the interfacial MAT layer. Specifically, there is
+spin-flipping in the interfacial layer when polarizations of
+the In2Se3 are reversed, which results in ferroelectrically
+switchable IMCs and an AFM-to-FM transition. On the
+other hand, the interfacing effect leads to asymmetric en-
+ergy barrier heights, which means that different electric
+fields are needed to switch the polarizations for the two
+states. We further show that this physics can be used to
+build magnetic multistates in their sandwich structures.
+Our calculations suggest that triple and quadruple mag-
+netic states with tunable Chern number can be obtained
+for MnBi2Te4 thin films by sandwiching them in between
+appropriate FE layers. Our results will not only attract
+experimental interest in FE control of the magnetism and
+topological properties of MAT thin films, but also inspire
+designing novel magnetism in other 2D materials.
+Acknowledgments
+We thank Haijun Zhang and Zhixin Guo for useful dis-
+cussions. This work was supported by the National Nat-
+ural Science Foundation of China (Grants No. 12174098,
+No. 11774084, No. U19A2090 and No. 91833302) and
+project supported by State Key Laboratory of Powder
+Metallurgy, Central South University, Changsha, China.
+
+6
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+

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@@ -0,0 +1,611 @@
+Discrete Search in Heterogeneous Integer Spaces
+for Automated Choice of Parameters using
+Correct-by-Construction Methods
+Omar Radwan
+oradwan@usc.edu
+oradwan@alumni.usc.edu
+Viterbi School of Engineering
+University of Southern California
+Yilin Zhang
+yilinz80@usc.edu
+Viterbi School of Engineering
+University of Southern California
+Luca Geretti
+geretti@usc.edu
+Viterbi School of Engineering
+University of Southern California
+Abstract—Discrete Search of integer spaces for tool parame-
+ter values provides a powerful methodology for modeling and
+finding a heuristically optimal parameter list for a given system.
+Current tools and implementations that exist focus primarily
+on homogeneous tool parameters, and the implementations for
+heterogeneous tool parameters is lacking. In this paper we
+introduce a correct-by-construction method of heterogeneous
+parameter reachability and validity search, and further outline
+the implementation as well as a demonstration using examples
+of heterogeneous systems that this tool can be used for.
+I. INTRODUCTION
+A. Premise
+Discrete Search of integer spaces provides a powerful mech-
+anism through which to explore the reachable set of a given
+system. Current design cools work primarily for homogeneous
+parameter spaces, and mapping a heterogeneous parameter
+space into the integer domain would provide a strong backbone
+for both performance and allow for a wide range of uses in
+many hybrid systems as well as hybrid parameters that are
+contained within a single system. There are precautions that
+would need to be taken for hybrid systems, which primarily
+consist of having unsafe states, that even though they are
+reachable, they would be considered to be unsafe in a real-
+world implementation, as well dependencies between vari-
+ables, that could transcend the homogeneous dependencies that
+are trivial (i.e. comparing two integers together as compared
+to a comparison between floating point and Boolean). There
+also would exist optimal state locations of the parameter set,
+and those would be modeled using an arbitrary cost function.
+B. Related Work
+Related work consists primarily of homogeneous tool pa-
+rameter exploration implementations, and those concern them-
+selves primarily with arriving at the reachable set primarily
+for homogeneous parameter sets. This would include the tool
+Ariadne [1], which has features built-in that allow it to find
+an approximation of the given reachable set by giving by
+controlling the growth of the approximation error.
+One other concern that arises when attempting to model
+heterogeneous parameters in integer spaces is the problem
+of solvability within bounded time with close approximation,
+and as outlined in [2], there does exist a finite bound for
+finite discovery. There was a foray into unbounded analysis,
+but that is infeasible given the constraints and would be too
+computationally exhaustive. Another issue that comes up is
+discrete versus non-discrete evolution in terms of time, and
+this was a problem resolved by setting as a condition that
+there can only exist discrete time steps and discrete evolution.
+Fig. 1. From Citation [3]
+C. Our Approach
+For the implementation demonstrated in this paper, we focus
+on a number of contributions that create a fast and efficient
+method of finding the optimal set given existing constraints
+and cost. We also define a semantic format that supports the
+representation of heterogeneous parameters, which better suits
+it for discrete search along hybrid domains.
+For exploring the adjacent set space from our beginning
+iteration point(initial state), there are a number of possible
+implementation decisions that would need to be made on how
+best to explore the reachable set given the constraints. The
+path that we decided on was to create a correct by construction
+approach, that would allow the exploration tool to only explore
+the reachable set that is also valid given the constraints and
+dependencies that are supplied. Our flow is as follows: given
+a parameter list which can consist of integer, Boolean, and
+composite parameters, as well as a list of constraints and
+dependencies between variables, and a cost function, we aim
+to find a valid parameter state that satisfies all of our given
+requirements.
+For our implementation, we split our computational engine
+into two general algorithms. Our first algorithm involves com-
+arXiv:2301.13426v1  [eess.SY]  31 Jan 2023
+
+Fig. 7. Evader keeps pursuer from entering reachable set, and hence avoids collision (animation
+at [44]puting a correct-by-construction interval for a given parameter
+given our requirements, and our current state when it comes
+to other parameters that exist within our set space. The second
+algorithm is our step-by-step evolution iteration across the set
+space of the parameter list based on the computation of local
+optimal cost.
+Compared to existing and related works, our approach has
+the following contributions:
+• Developed a representation for heterogeneous parameter
+sets that allows for the discretization of all parameters
+and results in the ability for integer space exploration for
+all relevant types
+• Created a correct-by-construction approach to not only
+finding the reachable set of a given parameter set, but also
+allowing the inclusion of heterogeneous inter-parameter
+dependencies and assertions.
+• Designed a method of evolution that allows for quick
+computation of adjacent states for a given set of already
+locally-optimal parameter instances with a method of
+back-tracing and reset if arriving at an invalid location
+• Demonstrate the applicability and the versatility of our
+implementation on two examples that involve computing
+minimum cost for a computer architecture design and a
+re-programmable logic circuit with a demonstration of the
+implementation of pseudo-Boolean constraints
+II. IMPLEMENTATION
+A. Environment and Language Considerations
+We decided on implementing our design in Python [4], the
+reason for that being that Python allows a host of libraries
+and type-interfacing that would allow us to quickly prototype,
+verify, and extend during testing. We also chose Python for
+the reason of being able to interface easily with JSON [5],
+which is our input-format of choice. JSON was chosen due
+to its status as being very well-adopted and would provide an
+easy interface for other CAD tools to create tool-parameter
+sets for analysis using our program.
+We also use a number of Python libraries to do the necessary
+computations that are required for our implementation. A spe-
+cial recognition is deserved of Numpy [6], which is a library
+that allows for very quick computation of intervals, arrays,
+and sets. Since we are operating in the integer domain, integer
+arrays using Numpy libraries make the cost of computation a
+significantly smaller area of concern during implementation.
+B. Motivation for Design
+To better improve the performance of discrete search in
+heterogeneous space, there do exist a number of limitations.
+Firstly, a slight weakness exists in parsing string type asser-
+tions and evaluating them in a computationally static format
+as opposed to extensive abstract syntax trees and symbolic
+interval computation. Secondly, considering various typed
+parameters and assertion relations, it is necessary to have a
+uniform interface design such that algorithm implementation
+is isolated with complicated typed transformation, which is
+why JSON was selected, which could become unwieldy if
+enumerated or vector parameters which to be considered. In
+this case, a tool that would generate a statically-enumerated
+JSON format that is acceptable to our program would be
+required.
+C. Evolution Algorithm
+In this section, we introduce how the program will explore
+feasible set constrained by assertions. The JSON format input
+will be interpreted and loaded into our program. For the sake
+of generality, we assume that there are n parameters denoted
+as x1, · · · xn. First of all, for each parameter xi, we randomly
+generate N − 1 valid neighboring points. For the random
+sampling of these points, we experimented with a couple
+methods. One was uniform sampling from the valid interval,
+the other two where linear and square weighted sampling with
+respect of distance from the interval. After these were tested,
+we found that square weighting was the most effective, and
+we will demonstrate these findings during our examples. With
+xi itself, these N points form a list {xj
+i}N
+j=1. In total there are
+n lists.
+During the evolution process, each point will randomly
+generate a neighboring point from its valid set. Therefore, all
+n·N points will generate another n new lists. Without loss of
+generality, we denote these n new lists as {xj
+i}2N
+j=N+1. Next
+we the original list and new list with the same footnote i to get
+n new list {xj
+i}2N
+j=1. From these n list, we evaluate 2N cost
+function values as {cj = F(xj
+1, xj
+2, · · · xj
+n) | j = 1, · · · , 2N}.
+For these 2N cost values, we keep the smaller half and
+corresponding parameter values to form n new lists. Repeat
+the above steps until the ending requirements are satisfied. A
+pseudo-code for this algorithm can be found at Algorithm 1
+D. Approach for feasibility checking between heterogeneous
+parameters
+For defining the the set of parameters that would exist for a
+given system, we supply two atomic types and one composite
+type:
+1) Integer type
+2) Boolean type
+3) Composite type
+Integers exist in the Integer domain, and Boolean’s likewise
+in the Boolean domain. Composites are different in that they
+are modeled like an array, given a composite parameter C,
+C can contain any number of composites, Boolean’s, and
+integers. This allows the modeling of parameters that cannot
+be modeled as strictly scalar integer or Boolean values. Floats,
+complex numbers, and vectors are all examples of what can
+be modeled as a composite set. Furthermore, to maintain the
+desired behavior of these parameters, the constraint paradigm
+that we introduce allows us to describe the behavior of how
+these composite parameters undergo evolution.
+As an example, take Cube, which of type composite, and it
+is defined by 3-equal length sides x, y, z, such that Cube(t) =
+{x, y, z ∈ Z, x == y == z}∀t where t is time-step during
+evolution. For the case of this parameter, the instantiation of
+the of the domain of each sub-parameter would go with the
+2
+
+parameter declarations, while the instantiation of the constraint
+that is intrinsic to cubes would be added to the constraints field
+that is given.
+This paradigm of allowing composite parameters to have
+unique behaviors could lead to invalid states during evolution,
+if one sub-parameter undergoes evolution independently and
+is now not equal to the other two, that would lead to an unde-
+sirable state. For this reason correct-by-construction interval
+generation for each of the sub-parameters is done with all
+assertions and constraints in mind.
+One note on using composite parameters to model floating
+point numbers. Initially during development we had planned
+to incorporate a floating point type, however the tedious-
+ness of setting properties for floating point as an atomic
+type is redundant as all the properties of a floating point
+value(mantissa, exponent, significant figures) can be modeled
+as sub-parameters of a composite value, and the user can
+specify the desired constraints and behaviors for comparison
+and incorporation between the composite-ized floating point
+value and other parameters.
+E. Feasibility Checking given Constraints
+In this section, a detailed explanation about how to construct
+valid neighboring set is given. Suppose that there are m
+assertions {Ai}m
+i=1 on n parameters. For each parameter xi,
+assertions containing xi are selected out of m, which is
+{Ak | xi ∈ Ak}. Next, iterate through other parameters and
+apply their values into these assertions. Finally, Intersect all
+the intervals after evaluating the assertions to get the final
+interval. A new value for xi is sampled randomly from the
+final interval based on the square of their distance to xi. By
+default, values closer to xi have higher probabilities to be
+selected. More details can be found in Algorithm 2
+F. Desired Implementation Aspects that Proved Infeasible
+One initial idea that was considered well thought out and
+feasible was the incorporation of symbolic computation for
+our constraint and dependency valid interval generation. The
+Sympy [7] library in Python was going to be utilized for this
+purpose. Though the algorithm was functional, the symbolic
+computation cost was extremely prohibitive, and was not
+feasible for a general-use case. After doing much research
+to attempt to make it feasible, we discovered that even Sympy
+as an organization recognizes that the substitution and eval-
+uation is cost-prohibitive, and recommends other avenues for
+repetitive computation. For this reason we had to re-calibrate
+and find another solution. This solution was to do string
+replacement of our given parameters with their values into
+the string representation of our constraints, dependencies, and
+costs. Then these string representations would be converted
+into lambda functions that would be operated on by the
+Numpy array operations. Since Numpy on the back-end uses C
+libraries to do computation, this lessened our computation time
+by an order of magnitude, for mostly the same functionality.
+The functionality that is missing is due to the inherent
+behavioral properties of lambda functions. Symbolic compu-
+Algorithm 1: Evolution of Adjacent Optimal Cost
+Input: List of variables Lv, Iterating parameter T, List
+of assertions La, Cost function F
+Output: Optimal value of variables L∗
+v
+1 //This is for initial value selection, since we need to
+enter the set space is what we presume to be a valid
+point foreach v in Lv do
+2
+v := Sample Uniform Distribution(Lower Bound of
+v, Upper Bound of v)
+3
+Construct Vi as the set of n sample of vi
+4 end
+5 while T <= K do
+6
+foreach variable vi in Lv do
+7
+Vi is the set of n values of vi
+Svi =get intersect of all valid intervals(La, Lv, vi).
+8
+Svisorted = Arrange by incrementing closeness
+to value of ak
+9
+WeightsSvisorted = array from 0 to length of
+Svisorted
+10
+foreach w in WeightsSvisorted do
+11
+w = (length of Svisorted - index of w)2
+12
+end
+13
+Use weighted sampling of WeightsSvisorted to
+randomly sample n new values of vi from
+Svisorted.
+14
+Append these n values into Vi
+15
+Construct n new list of variables {Lj
+v}n
+j=1,
+Lj
+v[i] = Lv[i].
+16
+Pick Lk
+v with minimum F(Lk
+v) in {Lj
+v}n
+i=j.
+17
+Update Lv[i] = Lk
+v[i].
+18
+Delete vi in Vi with n highest cost values.
+19
+Update T.
+20
+end
+21 end
+22 return Lv
+tation was desired as it allowed the incorporation of very
+rigorous Boolean SAT exploration, but this is not a feature
+that is possible with the lambda paradigm. Therefore, to allow
+the extend-ability of Boolean values, fuzzy pseudo-Boolean
+logic [8] is implemented, which does allow for an adequate
+semantic representation of Boolean logic.
+III. EXAMPLES OF APPLICATION
+For an example foray to explore what our program would
+be able to handle, we decided on two different, yet related,
+domains.
+A. FPGA Synthesis
+For our first example(outlined in 2), we decided on model-
+ing our problem as an FPGA cost problem. Given a number of
+constraints on an FPGA, i.e. memory size, available memory
+ports, available input and output ports we have Routine1,2,3,
+and only two of the previously mentioned three can be
+3
+
+Algorithm 2: Get Intersect of All Valid Points in
+Bounds and Assertions
+Input: List of assertions La, List of variables Lv,
+Target variable vi
+Output: All valid set Svi of vi
+1 Initialize list of intervals Li = []
+2 foreach ak in La do
+3
+if vi appears in ak then
+4
+foreach vj in Lv do
+5
+if vj ̸= vi then
+6
+Plug in value of vj in ak.
+7
+else
+8
+continue
+9
+end
+10
+end
+11
+Append ak into Li
+12
+else
+13
+continue
+14
+end
+15 end
+16 Transform the intersection of Li into valid set Svi
+17 return Svi
+installed on the FPGA fabric, and depending on which two
+are loaded onto the fabric, we then must enable a minimum
+number of memory, I/O, and interconnection ports, as well
+as have different memory properties. We then created a
+polynomial cost function of these constraints, in an aim of
+it becoming nonlinear and make the algorithm demonstrate its
+effectiveness in traversing the set space while attempting to
+find the given most optimal cost.
+One highlight of this example is the inclusion of pseudo-
+Boolean constraints, which manifest in the requirement that
+only two of the three routines can function at any time, which
+in terms of cost, creates a piece-wise function. The parameter
+variation that is generated during random sampling is able
+to traverse this piece wise function, because even though we
+generate points using a correct-by-construction approach, in
+some cases there is no valid interval, and in that case we reset
+for that specific parameter back to the largest valid interval,
+and randomly sample that. This allows the program to exit
+any possible rut that it enters while making an early decision
+on which Routine set to choose, and so it can backtrack
+as necessary and choose another Routine set if the specific
+parameter space undergoing evolution is no longer valid. The
+results for these are demonstrated in Figure 3 with different
+weights for random sampling methods from the valid intervals
+generated.
+B. Computer Architecture Design
+Another example that we used is the creation of of a
+computer architecture system. During the creation of a new
+computer architecture, or the generation of a new implementa-
+tion of an architecture, multiple design decisions must be made
+Fig. 2. Illustration of FPGA Paradigm for Testing Our Implementation
+with respect to area, inter-connectivity, interface requirements,
+and transistor count. In this example, we model a simple
+multi-fetch, multi-execution, processor design. We drafted the
+requirements in terms of dependencies and constraints, and
+given the constraints and requirements for the interfaces and
+inter-connectivity between components, we aim to find the
+minimal transistor count. This was a more rudimentary design,
+and it aimed to find the computation limit of our implemen-
+tation. One thing that we attempted to model was having
+very large integer sets, and exploring those. Emulating design
+space exploration for computer architectures with such large
+intervals was the reason we had to refactor our computation
+engine from purely symbolic to the lambda paradigm, as
+the symbolic computation was not able to run search space
+exploration and computation in a reasonable amount of time
+with this example. The results for those example are posted in
+Figure 4, along with the variation between random sampling
+methods from the valid intervals generated.
+C. Performance and Efficacy
+As aforementioned during the discussion on the implemen-
+tation, performance was a major bottleneck in our implemen-
+tation, and there were a number of features that needed to be
+added to be able to guarantee reasonable performance. The
+first was the use of lambdas to calculate the valid interval set.
+The second, which is outlined in the algorithm, is keeping a
+short list of the least-cost neighbors that exist, and generating
+new random neighbors from that list. This allows us to have
+multiple different forays into the search space, and we could
+possibly arrive to many local minima’s, but we only choose
+the most optimal local minima. Computation time is static
+across iterations, and there are parameter options to increase
+or decrease the exhaustiveness of the search depending on the
+intended use cases.
+4
+
+L2Cache
+L1Cache
+品品
+00001
+0000
+Memory Ports
+Process 1
+Process 2
+indino
+Input
+Ports
+Ports
+Process 3We also wanted to verify the efficacy of our design and do
+the best possible effort into generating the most optimal point.
+To verify that our results where sane, we ran multiple differ-
+ent instances of both the FPGA and Computer Architecture
+description JSON files, and averaged those results out, and did
+this for three different weights for random sampling(uniform,
+linear weighted, square weighted), and what we found that in
+all cases, our results for all runs where fairly similar, but there
+are some noticeable differences worth discussion.
+Firstly, the uniform random search has better performance
+for lower iterations, and this is because during early stages
+of evolution, a majority portion of the set space has yet
+to be explored, and uniform sampling allows us to traverse
+the majority of the set space early. However after a lot of
+iterations, the square weighted random sampling from the
+interval eventually makes us arrive to a more optimal cost, and
+this is because as more and more of the set space is invalidated,
+the parameters that are undergoing evolution get much closer
+to the local optima, and square weighting allows us to more
+likely sample these local optima and arrive at them at a quicker
+rate than both uniform and linear random sampling.
+Fig. 3. Table of the Impact of Different Weights and Effect on Set Exploration
+for FPGA Example
+IV. SUMMARY
+To reiterate the major points that have been mentioned
+throughout this paper, we have created a tool that performs
+discrete search of integer spaces of mapped heterogeneous
+parameters to the integer domain, and we utilized correct-
+by-construction methods to ensure that given constraints and
+dependencies are met, while attempting to find the most opti-
+mal cost. This differs from the previous literature in that it is
+able to accommodate for heterogeneous data structures and is
+able to model hybrid systems, while comparatively the existing
+literature exists primarily for reachability and homogeneous
+parameter exploration. The main takeaways from this endeavor
+include that there is a significant divide between the tools that
+are used in industry, and the potential for tools that could be
+used to better-optimize processes and methods that are used.
+Fig. 4. Table of the Impact of Different Weights and Effect on Set Exploration
+for Architecture Example
+The main hurdle for widespread adoption of these methods
+includes a difficulty of understanding and use, as well as
+a computational cost-barrier that is evident in very complex
+systems.
+A. Wish-list of additional features
+One feature that would have been useful to incorporate
+would have been incorporating a Boolean SAT or SMT solver
+[9], which would have allowed us to bypass pseudo-Boolean
+constraints entirely, which are generated heuristically, and
+instead rigorously solve Boolean equations for all possible
+solutions. Incorporation a Boolean SAT solver such as Z3
+would’ve been time-prohibitive, but would’ve allowed for a
+greater range of expressively for constraints.
+B. Application Files
+Due to space reasons, we do not go into detail on the
+specifics of the Computer Architecture Example and the FPGA
+Example. Please contact the authors for more information.
+V. SOME THOUGHTS ON OPTIMIZATION AND USE CASES
+Optimization aims at searching for values of x which
+minimizes the objective function f bounded by constraints.
+A general formula of optimization problem is in equation (1).
+arg
+x min f(x)
+s.t. Constraints on x
+(1)
+In addition to existing gradient based methods which re-
+quires the objective function to be differentiable or even
+more smooth, discrete search algorithm proposed in this paper
+achieves a high degree of performance on all kinds of objective
+functions.
+One of the most important features of cyber-physical sys-
+tems is that they contains both continuous system components
+and discrete system components. In this case, the constraints
+may include discrete forms like SATs, and continuous forms
+like inequalities. Our discrete search algorithm can be used to
+choose optimal parameters for a cyber-physical system.
+5
+
+Different WeightTypesandCostper IterationForFPGA
+Example
+UniformWeightFPGA
+Linear Weight FPGA Square Weight FPGA
+40000000
+20000000
+10000000
+8000000
+6000000
+4000000
+1
+5
+10
+50
+100
+IterationDifferent Weight Types and Costper Iteration For Architecture
+Example
+Uniform Weight Arch
+Linear Weight Arch
+Square Weight Arch
+5000000000000000
+1000000000000000
+500000000000000
+100000000000000
+50000000000000
+10000000000000
+5
+10
+50
+100VI. FURTHER POSSIBLE WORK
+We would like to explore more about the background of
+reachability analysis. Where does this problem rise from.
+Moreover, as for existing optimization algorithms like heuristic
+algorithms, gradient based methods and interior point methods,
+what are the bottlenecks on applying these algorithms on
+hybrid system reachability analysis.
+Another topic is the connection between reachability anal-
+ysis and optimization algorithm. If the reachability problem
+can be formulated into an optimization problem, then it will
+be easier to understand the problem from the mathematical
+properties of objective function.
+REFERENCES
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+Introduction and applications. Commun. ACM, 54(9):69–77, sep 2011.
+6
+

3NFQT4oBgHgl3EQf3DZF/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf,len=227
+page_content='Discrete Search in Heterogeneous Integer Spaces  for Automated Choice of Parameters using  Correct-by-Construction Methods  Omar Radwan  oradwan@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='edu  oradwan@alumni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='edu  Viterbi School of Engineering  University of Southern California  Yilin Zhang  yilinz80@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='edu  Viterbi School of Engineering  University of Southern California  Luca Geretti  geretti@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='edu  Viterbi School of Engineering  University of Southern California  Abstract—Discrete Search of integer spaces for tool parame-  ter values provides a powerful methodology for modeling and  finding a heuristically optimal parameter list for a given system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Current tools and implementations that exist focus primarily  on homogeneous tool parameters, and the implementations for  heterogeneous tool parameters is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' In this paper we  introduce a correct-by-construction method of heterogeneous  parameter reachability and validity search, and further outline  the implementation as well as a demonstration using examples  of heterogeneous systems that this tool can be used for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' INTRODUCTION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Premise  Discrete Search of integer spaces provides a powerful mech-  anism through which to explore the reachable set of a given  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Current design cools work primarily for homogeneous  parameter spaces, and mapping a heterogeneous parameter  space into the integer domain would provide a strong backbone  for both performance and allow for a wide range of uses in  many hybrid systems as well as hybrid parameters that are  contained within a single system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' There are precautions that  would need to be taken for hybrid systems, which primarily  consist of having unsafe states, that even though they are  reachable, they would be considered to be unsafe in a real-  world implementation, as well dependencies between vari-  ables, that could transcend the homogeneous dependencies that  are trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' comparing two integers together as compared  to a comparison between floating point and Boolean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' There  also would exist optimal state locations of the parameter set,  and those would be modeled using an arbitrary cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Related Work  Related work consists primarily of homogeneous tool pa-  rameter exploration implementations, and those concern them-  selves primarily with arriving at the reachable set primarily  for homogeneous parameter sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' This would include the tool  Ariadne [1], which has features built-in that allow it to find  an approximation of the given reachable set by giving by  controlling the growth of the approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  One other concern that arises when attempting to model  heterogeneous parameters in integer spaces is the problem  of solvability within bounded time with close approximation,  and as outlined in [2], there does exist a finite bound for  finite discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' There was a foray into unbounded analysis,  but that is infeasible given the constraints and would be too  computationally exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Another issue that comes up is  discrete versus non-discrete evolution in terms of time, and  this was a problem resolved by setting as a condition that  there can only exist discrete time steps and discrete evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' From Citation [3]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Our Approach  For the implementation demonstrated in this paper, we focus  on a number of contributions that create a fast and efficient  method of finding the optimal set given existing constraints  and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' We also define a semantic format that supports the  representation of heterogeneous parameters, which better suits  it for discrete search along hybrid domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  For exploring the adjacent set space from our beginning  iteration point(initial state), there are a number of possible  implementation decisions that would need to be made on how  best to explore the reachable set given the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' The  path that we decided on was to create a correct by construction  approach, that would allow the exploration tool to only explore  the reachable set that is also valid given the constraints and  dependencies that are supplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Our flow is as follows: given  a parameter list which can consist of integer, Boolean, and  composite parameters, as well as a list of constraints and  dependencies between variables, and a cost function, we aim  to find a valid parameter state that satisfies all of our given  requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  For our implementation, we split our computational engine  into two general algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Our first algorithm involves com-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='13426v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='SY]  31 Jan 2023  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Evader keeps pursuer from entering reachable set, and hence avoids collision (animation  at [44]puting a correct-by-construction interval for a given parameter  given our requirements, and our current state when it comes  to other parameters that exist within our set space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' The second  algorithm is our step-by-step evolution iteration across the set  space of the parameter list based on the computation of local  optimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Compared to existing and related works, our approach has  the following contributions:  Developed a representation for heterogeneous parameter  sets that allows for the discretization of all parameters  and results in the ability for integer space exploration for  all relevant types  Created a correct-by-construction approach to not only  finding the reachable set of a given parameter set, but also  allowing the inclusion of heterogeneous inter-parameter  dependencies and assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='Designed a method of evolution that allows for quick  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='computation of adjacent states for a given set of already  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='locally-optimal parameter instances with a method of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='back-tracing and reset if arriving at an invalid location  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='Demonstrate the applicability and the versatility of our  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='implementation on two examples that involve computing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='minimum cost for a computer architecture design and a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='re-programmable logic circuit with a demonstration of the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='implementation of pseudo-Boolean constraints  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' IMPLEMENTATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Environment and Language Considerations  We decided on implementing our design in Python [4], the  reason for that being that Python allows a host of libraries  and type-interfacing that would allow us to quickly prototype,  verify, and extend during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' We also chose Python for  the reason of being able to interface easily with JSON [5],  which is our input-format of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' JSON was chosen due  to its status as being very well-adopted and would provide an  easy interface for other CAD tools to create tool-parameter  sets for analysis using our program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  We also use a number of Python libraries to do the necessary  computations that are required for our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' A spe-  cial recognition is deserved of Numpy [6], which is a library  that allows for very quick computation of intervals, arrays,  and sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Since we are operating in the integer domain, integer  arrays using Numpy libraries make the cost of computation a  significantly smaller area of concern during implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Motivation for Design  To better improve the performance of discrete search in  heterogeneous space, there do exist a number of limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Firstly, a slight weakness exists in parsing string type asser-  tions and evaluating them in a computationally static format  as opposed to extensive abstract syntax trees and symbolic  interval computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Secondly, considering various typed  parameters and assertion relations, it is necessary to have a  uniform interface design such that algorithm implementation  is isolated with complicated typed transformation, which is  why JSON was selected, which could become unwieldy if  enumerated or vector parameters which to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' In  this case, a tool that would generate a statically-enumerated  JSON format that is acceptable to our program would be  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Evolution Algorithm  In this section, we introduce how the program will explore  feasible set constrained by assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' The JSON format input  will be interpreted and loaded into our program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' For the sake  of generality, we assume that there are n parameters denoted  as x1, · · · xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' First of all, for each parameter xi, we randomly  generate N − 1 valid neighboring points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' For the random  sampling of these points, we experimented with a couple  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' One was uniform sampling from the valid interval,  the other two where linear and square weighted sampling with  respect of distance from the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' After these were tested,  we found that square weighting was the most effective, and  we will demonstrate these findings during our examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' With  xi itself, these N points form a list {xj  i}N  j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' In total there are  n lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  During the evolution process, each point will randomly  generate a neighboring point from its valid set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Therefore, all  n·N points will generate another n new lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Without loss of  generality, we denote these n new lists as {xj  i}2N  j=N+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Next  we the original list and new list with the same footnote i to get  n new list {xj  i}2N  j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' From these n list, we evaluate 2N cost  function values as {cj = F(xj  1, xj  2, · · · xj  n) | j = 1, · · · , 2N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  For these 2N cost values, we keep the smaller half and  corresponding parameter values to form n new lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Repeat  the above steps until the ending requirements are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' A  pseudo-code for this algorithm can be found at Algorithm 1  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Approach for feasibility checking between heterogeneous  parameters  For defining the the set of parameters that would exist for a  given system, we supply two atomic types and one composite  type:  1) Integer type  2) Boolean type  3) Composite type  Integers exist in the Integer domain, and Boolean’s likewise  in the Boolean domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Composites are different in that they  are modeled like an array, given a composite parameter C,  C can contain any number of composites, Boolean’s, and  integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' This allows the modeling of parameters that cannot  be modeled as strictly scalar integer or Boolean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Floats,  complex numbers, and vectors are all examples of what can  be modeled as a composite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Furthermore, to maintain the  desired behavior of these parameters, the constraint paradigm  that we introduce allows us to describe the behavior of how  these composite parameters undergo evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  As an example, take Cube, which of type composite, and it  is defined by 3-equal length sides x, y, z, such that Cube(t) =  {x, y, z ∈ Z, x == y == z}∀t where t is time-step during  evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' For the case of this parameter, the instantiation of  the of the domain of each sub-parameter would go with the  2  parameter declarations, while the instantiation of the constraint  that is intrinsic to cubes would be added to the constraints field  that is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  This paradigm of allowing composite parameters to have  unique behaviors could lead to invalid states during evolution,  if one sub-parameter undergoes evolution independently and  is now not equal to the other two, that would lead to an unde-  sirable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' For this reason correct-by-construction interval  generation for each of the sub-parameters is done with all  assertions and constraints in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  One note on using composite parameters to model floating  point numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Initially during development we had planned  to incorporate a floating point type, however the tedious-  ness of setting properties for floating point as an atomic  type is redundant as all the properties of a floating point  value(mantissa, exponent, significant figures) can be modeled  as sub-parameters of a composite value, and the user can  specify the desired constraints and behaviors for comparison  and incorporation between the composite-ized floating point  value and other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Feasibility Checking given Constraints  In this section, a detailed explanation about how to construct  valid neighboring set is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Suppose that there are m  assertions {Ai}m  i=1 on n parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' For each parameter xi,  assertions containing xi are selected out of m, which is  {Ak | xi ∈ Ak}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Next, iterate through other parameters and  apply their values into these assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Finally, Intersect all  the intervals after evaluating the assertions to get the final  interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' A new value for xi is sampled randomly from the  final interval based on the square of their distance to xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' By  default, values closer to xi have higher probabilities to be  selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' More details can be found in Algorithm 2  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Desired Implementation Aspects that Proved Infeasible  One initial idea that was considered well thought out and  feasible was the incorporation of symbolic computation for  our constraint and dependency valid interval generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' The  Sympy [7] library in Python was going to be utilized for this  purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Though the algorithm was functional, the symbolic  computation cost was extremely prohibitive, and was not  feasible for a general-use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' After doing much research  to attempt to make it feasible, we discovered that even Sympy  as an organization recognizes that the substitution and eval-  uation is cost-prohibitive, and recommends other avenues for  repetitive computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' For this reason we had to re-calibrate  and find another solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' This solution was to do string  replacement of our given parameters with their values into  the string representation of our constraints, dependencies, and  costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Then these string representations would be converted  into lambda functions that would be operated on by the  Numpy array operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Since Numpy on the back-end uses C  libraries to do computation, this lessened our computation time  by an order of magnitude, for mostly the same functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  The functionality that is missing is due to the inherent  behavioral properties of lambda functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Symbolic compu-  Algorithm 1: Evolution of Adjacent Optimal Cost  Input: List of variables Lv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Iterating parameter T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' List  of assertions La,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Cost function F  Output: Optimal value of variables L∗  v  1 //This is for initial value selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' since we need to  enter the set space is what we presume to be a valid  point foreach v in Lv do  2  v := Sample Uniform Distribution(Lower Bound of  v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Upper Bound of v)  3  Construct Vi as the set of n sample of vi  4 end  5 while T <= K do  6  foreach variable vi in Lv do  7  Vi is the set of n values of vi  Svi =get intersect of all valid intervals(La,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Lv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  8  Svisorted = Arrange by incrementing closeness  to value of ak  9  WeightsSvisorted = array from 0 to length of  Svisorted  10  foreach w in WeightsSvisorted do  11  w = (length of Svisorted - index of w)2  12  end  13  Use weighted sampling of WeightsSvisorted to  randomly sample n new values of vi from  Svisorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  14  Append these n values into Vi  15  Construct n new list of variables {Lj  v}n  j=1,  Lj  v[i] = Lv[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  16  Pick Lk  v with minimum F(Lk  v) in {Lj  v}n  i=j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  17  Update Lv[i] = Lk  v[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  18  Delete vi in Vi with n highest cost values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  19  Update T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  20  end  21 end  22 return Lv  tation was desired as it allowed the incorporation of very  rigorous Boolean SAT exploration, but this is not a feature  that is possible with the lambda paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Therefore, to allow  the extend-ability of Boolean values, fuzzy pseudo-Boolean  logic [8] is implemented, which does allow for an adequate  semantic representation of Boolean logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' EXAMPLES OF APPLICATION  For an example foray to explore what our program would  be able to handle, we decided on two different, yet related,  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' FPGA Synthesis  For our first example(outlined in 2), we decided on model-  ing our problem as an FPGA cost problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Given a number of  constraints on an FPGA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' memory size, available memory  ports, available input and output ports we have Routine1,2,3,  and only two of the previously mentioned three can be  3  Algorithm 2: Get Intersect of All Valid Points in  Bounds and Assertions  Input: List of assertions La, List of variables Lv,  Target variable vi  Output: All valid set Svi of vi  1 Initialize list of intervals Li = []  2 foreach ak in La do  3  if vi appears in ak then  4  foreach vj in Lv do  5  if vj ̸= vi then  6  Plug in value of vj in ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  7  else  8  continue  9  end  10  end  11  Append ak into Li  12  else  13  continue  14  end  15 end  16 Transform the intersection of Li into valid set Svi  17 return Svi  installed on the FPGA fabric, and depending on which two  are loaded onto the fabric, we then must enable a minimum  number of memory, I/O, and interconnection ports, as well  as have different memory properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' We then created a  polynomial cost function of these constraints, in an aim of  it becoming nonlinear and make the algorithm demonstrate its  effectiveness in traversing the set space while attempting to  find the given most optimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  One highlight of this example is the inclusion of pseudo-  Boolean constraints, which manifest in the requirement that  only two of the three routines can function at any time, which  in terms of cost, creates a piece-wise function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' The parameter  variation that is generated during random sampling is able  to traverse this piece wise function, because even though we  generate points using a correct-by-construction approach, in  some cases there is no valid interval, and in that case we reset  for that specific parameter back to the largest valid interval,  and randomly sample that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' This allows the program to exit  any possible rut that it enters while making an early decision  on which Routine set to choose, and so it can backtrack  as necessary and choose another Routine set if the specific  parameter space undergoing evolution is no longer valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' The  results for these are demonstrated in Figure 3 with different  weights for random sampling methods from the valid intervals  generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Computer Architecture Design  Another example that we used is the creation of of a  computer architecture system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' During the creation of a new  computer architecture, or the generation of a new implementa-  tion of an architecture, multiple design decisions must be made  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Illustration of FPGA Paradigm for Testing Our Implementation  with respect to area, inter-connectivity, interface requirements,  and transistor count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' In this example, we model a simple  multi-fetch, multi-execution, processor design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' We drafted the  requirements in terms of dependencies and constraints, and  given the constraints and requirements for the interfaces and  inter-connectivity between components, we aim to find the  minimal transistor count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' This was a more rudimentary design,  and it aimed to find the computation limit of our implemen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' One thing that we attempted to model was having  very large integer sets, and exploring those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Emulating design  space exploration for computer architectures with such large  intervals was the reason we had to refactor our computation  engine from purely symbolic to the lambda paradigm, as  the symbolic computation was not able to run search space  exploration and computation in a reasonable amount of time  with this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' The results for those example are posted in  Figure 4, along with the variation between random sampling  methods from the valid intervals generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Performance and Efficacy  As aforementioned during the discussion on the implemen-  tation, performance was a major bottleneck in our implemen-  tation, and there were a number of features that needed to be  added to be able to guarantee reasonable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' The  first was the use of lambdas to calculate the valid interval set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  The second, which is outlined in the algorithm, is keeping a  short list of the least-cost neighbors that exist, and generating  new random neighbors from that list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' This allows us to have  multiple different forays into the search space, and we could  possibly arrive to many local minima’s, but we only choose  the most optimal local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Computation time is static  across iterations, and there are parameter options to increase  or decrease the exhaustiveness of the search depending on the  intended use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  4  L2Cache  L1Cache  品品  00001  0000  Memory Ports  Process 1  Process 2  indino  Input  Ports  Ports  Process 3We also wanted to verify the efficacy of our design and do  the best possible effort into generating the most optimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  To verify that our results where sane, we ran multiple differ-  ent instances of both the FPGA and Computer Architecture  description JSON files, and averaged those results out, and did  this for three different weights for random sampling(uniform,  linear weighted, square weighted), and what we found that in  all cases, our results for all runs where fairly similar, but there  are some noticeable differences worth discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Firstly, the uniform random search has better performance  for lower iterations, and this is because during early stages  of evolution, a majority portion of the set space has yet  to be explored, and uniform sampling allows us to traverse  the majority of the set space early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' However after a lot of  iterations, the square weighted random sampling from the  interval eventually makes us arrive to a more optimal cost, and  this is because as more and more of the set space is invalidated,  the parameters that are undergoing evolution get much closer  to the local optima, and square weighting allows us to more  likely sample these local optima and arrive at them at a quicker  rate than both uniform and linear random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Table of the Impact of Different Weights and Effect on Set Exploration  for FPGA Example  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' SUMMARY  To reiterate the major points that have been mentioned  throughout this paper, we have created a tool that performs  discrete search of integer spaces of mapped heterogeneous  parameters to the integer domain, and we utilized correct-  by-construction methods to ensure that given constraints and  dependencies are met, while attempting to find the most opti-  mal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' This differs from the previous literature in that it is  able to accommodate for heterogeneous data structures and is  able to model hybrid systems, while comparatively the existing  literature exists primarily for reachability and homogeneous  parameter exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' The main takeaways from this endeavor  include that there is a significant divide between the tools that  are used in industry, and the potential for tools that could be  used to better-optimize processes and methods that are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Table of the Impact of Different Weights and Effect on Set Exploration  for Architecture Example  The main hurdle for widespread adoption of these methods  includes a difficulty of understanding and use, as well as  a computational cost-barrier that is evident in very complex  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Wish-list of additional features  One feature that would have been useful to incorporate  would have been incorporating a Boolean SAT or SMT solver  [9], which would have allowed us to bypass pseudo-Boolean  constraints entirely, which are generated heuristically, and  instead rigorously solve Boolean equations for all possible  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Incorporation a Boolean SAT solver such as Z3  would’ve been time-prohibitive, but would’ve allowed for a  greater range of expressively for constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Application Files  Due to space reasons, we do not go into detail on the  specifics of the Computer Architecture Example and the FPGA  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Please contact the authors for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' SOME THOUGHTS ON OPTIMIZATION AND USE CASES  Optimization aims at searching for values of x which  minimizes the objective function f bounded by constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  A general formula of optimization problem is in equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  arg  x min f(x)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Constraints on x  (1)  In addition to existing gradient based methods which re-  quires the objective function to be differentiable or even  more smooth, discrete search algorithm proposed in this paper  achieves a high degree of performance on all kinds of objective  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  One of the most important features of cyber-physical sys-  tems is that they contains both continuous system components  and discrete system components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' In this case, the constraints  may include discrete forms like SATs, and continuous forms  like inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Our discrete search algorithm can be used to  choose optimal parameters for a cyber-physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  5  Different WeightTypesandCostper IterationForFPGA  Example  UniformWeightFPGA  Linear Weight FPGA Square Weight FPGA  40000000  20000000  10000000  8000000  6000000  4000000  1  5  10  50  100  IterationDifferent Weight Types and Costper Iteration For Architecture  Example  Uniform Weight Arch  Linear Weight Arch  Square Weight Arch  5000000000000000  1000000000000000  500000000000000  100000000000000  50000000000000  10000000000000  5  10  50  100VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' FURTHER POSSIBLE WORK  We would like to explore more about the background of  reachability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Where does this problem rise from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Moreover, as for existing optimization algorithms like heuristic  algorithms, gradient based methods and interior point methods,  what are the bottlenecks on applying these algorithms on  hybrid system reachability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Another topic is the connection between reachability anal-  ysis and optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' If the reachability problem  can be formulated into an optimization problem, then it will  be easier to understand the problem from the mathematical  properties of objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  REFERENCES  [1] Luca Geretti, Pieter Collins, Davide Bresolin, and Tiziano Villa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Automat-  ing numerical parameters along the evolution of a nonlinear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' In  Runtime Verification: 22nd International Conference, RV 2022, Tbilisi,  Georgia, September 28–30, 2022, Proceedings, page 336–345, Berlin,  Heidelberg, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Springer-Verlag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  [2] Michele Conforti, Gerard Cornuejols, and Giacomo Zambelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Integer  Programming / Michele Conforti, G´erard Cornu´ejols, Giacomo Zambelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Springer, Cham, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  [3] Ian M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Mitchell and Claire J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Tomlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Overapproximating reachable  sets by hamilton-jacobi projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Journal of Scientific Computing,  19(1):323–346, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  [4] Guido Van Rossum and Fred L Drake Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Python reference manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  Centrum voor Wiskunde en Informatica Amsterdam, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
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+page_content=' Foundations of json schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' In Proceedings of the  25th International Conference on World Wide Web, pages 263–273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  International World Wide Web Conferences Steering Committee, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content='  [6] Charles R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Harris, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Jarrod Millman, St´efan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' van der Walt, Ralf  Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Tay-  lor, Sebastian Berg, Nathaniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Smith, Robert Kern, Matti Picus,  Stephan Hoyer, Marten H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' van Kerkwijk, Matthew Brett, Allan Haldane,  Jaime Fern´andez del R´ıo, Mark Wiebe, Pearu Peterson, Pierre G´erard-  Marchant, Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer  Abbasi, Christoph Gohlke, and Travis E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Oliphant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
+page_content=' Array programming  with NumPy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
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+page_content='  [7] Sympy Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'}
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3NFST4oBgHgl3EQfYjhQ/content/tmp_files/2301.13788v1.pdf.txt

@@ -0,0 +1,828 @@
+1 
+ 
+Synthesis and characterization of PEG-coated Zn0.3MnxFe2.7-xO4 nanoparticles as 
+the dual T1/T2-weighted MRI contrast agent  
+Bahareh Rezaei, Ahmad Kermanpur*, Sheyda Labbaf 
+Department of Materials Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran 
+ 
+ 
+ 
+ 
+ 
+Abstract 
+Super-paramagnetic nanoparticles (NPs) have been widely explored as magnetic resonance imaging 
+(MRI) contrast agents because of a combination of favorable magnetic properties, biocompability and 
+ease of fabrication. MRI using traditional T1- or T2-weighted single mode contrast-enhanced 
+techniques may yield inaccurate imaging results. In the present work, a T1/T2 dual mode contrast agent 
+based on the super-paramagnetic zinc-manganese ferrite (Zn0.3MnxFe2.7-xO4, x= 0, 0.25, 0.75 and 1) 
+NPs with small core size and a hydrophilic PEG surface coating is reported. The TEM, TGA and FTIR 
+results confirmed the formation of a uniform coating on the NPs surface. The MRI analysis revealed 
+that the Zn0.3Mn0.5Fe2.2O4 NPs had the maximum image contrast compared to other zinc-manganese 
+ferrite samples. Cell viability evaluations revealed that the coated and uncoated particles did not 
+inhibit cell growth pattern. The present PEG-coated Zn0.3Mn0.5Fe2.2O4 NPs can be utilized as a suitable 
+T1/T2-weighted MRI contrast agent for better diagnostic of abnormalities in the organs or tissues. 
+ 
+Keywords 
+Magnetic Resonance Imaging (MRI); Super-paramagnetic nanoparticles; Zn0.3MnxFe2.7-xO4 
+nanoparticles; Polyethylene Glycol (PEG) coating 
+ 
+1. Introduction 
+ 
+The most potent and painless test that gives extremely clear images of the internal organs in 
+the body is the magnetic resonance imaging (MRI) scan [1, 2]. Based on the magnetic relaxation 
+processes of water protons on soft tissue of nearly every internal structure in the human body [1, 3-5], 
+this method is a sort of diagnostic test that generates detailed images and functional information in a 
+non-invasive and real-time monitoring manner [6, 7]. It is a distinguished device since there is no 
+ionizing radiation during the imaging process and obviously reduces harmful side effects [2, 4, 8, 9]. 
+However, this test typically  provides poor anatomical details, and clinicians have some difficulties to 
+distinguish between normal and abnormal tissues due to its low sensitivity [9, 10]. Hence, the clinical 
+                                                           
+* Corresponding author; Tel. (+98)3133915738; Fax (+98)3133912752; Email: ahmad_k@iut.ac.ir 
+
+2 
+ 
+domains urgently require more reliable MR images. There is a potential to create more accurate and 
+crisper images by adding contrast agents, which enables physicians to detect organs or in-vivo systems 
+more clear. This opens up a wide range of MRI applications for therapeutic medicine in addition to 
+diagnostic radiology. Despite the fact of shorter circulation time of Gd3+ ions as a T1-weighted MRI 
+contrast agent, which renders them useless for high-resolution and/or targeted MRI [9, 11] and many 
+concerns about potential trace deposition of Gd ions in the body, known as Nephrogenic Systemic 
+Fibrosis (NSF) [12-14], which is a rare disease that frequently develops in patients with severe renal 
+failure or after liver transplantation [15], Gd-based contrast agents can shorten the T1 relaxation time 
+effectively and provide brighter images in the regions of interest [16]. Following the increased 
+awareness of this side effect, researchers have much more emphasis on alternative methods based on 
+Mn-based complexes [15]. Although no scientific relationship has been proved between the NSF side 
+effect and Mn so far, the metal is still known to pose some toxicity when inhaled. However, small 
+amounts are essential to human health, but overexposure to free Mn ions may result in the 
+neurodegenerative disorder known as ‘Manganism’ with symptoms similar Parkinson’s disease [11]. 
+Unlike Gd3+ and Mn2+ chelates, iron oxide nanoparticles (NPs) have achieved great attention 
+due to the outstanding properties they exhibit at the nano-metric scale. A large number of benefits 
+including biocompatibility, superparamagnetic behavior at room temperature, high saturation 
+magnetization that can be tailored by size, shape, composition and assembly, tunable cellular uptake, 
+biodispersibility, and large surface areas that make them a good candidate for polymer coating, 
+conjugation with targeting molecules and other probes for achieving targeting and multimodal agents 
+[17, 18] is reported for the iron oxide NPs. Super-paramagnetic NPs can be employed as T2-weighted 
+MRI contrast agents since they are more sensitive in the micro- or nano-molar range than Gd 
+complexes [17]. Clinical MR imaging applications often use iron oxide-based NPs with strong 
+magnetic moments as T2-weighted MRI contrast agents. The limited usage of iron oxide NPs as T1 
+contrast agents is due to their high transverse to longitudinal relaxivity ratio [19]. However, the use of 
+superparamagnetic NPs in MRI is constrained by a negative contrast effect and magnetic susceptibility 
+artifacts. Because the signal is frequently confused with signals from bleeding, calcification, or metal 
+deposits and the susceptibility artifacts alter the background image, the resulting dark signal in T2-
+weighted MRI may be exploited to mislead clinical diagnosis [18]. The T1-weighted MRI contrast 
+agents, however, have advantages over T2-weighted MRI contrast agents. These advantages include 
+better imaging quality, brighter images that can more effectively distinguish between normal and 
+lesion tissues, and also the ability to provide better resolution for blood imaging. Nonetheless, in T1-
+
+3 
+ 
+weighted MR imaging, some normal tissues (such as fatty tissue) may be mistaken for bright lesions 
+that have been increased by T1 contrast agents [20]. Therefore, efforts to integrate T1 and T2 imaging 
+to prevent probable MRI artifacts and produce superior clinical images have been made as a result of 
+the rising demand in the clinical diagnosis for both T1- and T2-weighted MR images. [18, 21]. 
+Additionally, when several organ scans are required, injecting one dosage offers unmatched benefits 
+to patients and doctors [16]. Super-paramagnetic NPs have the potential to exhibit significant dual 
+T1/T2 relaxation performances when their sizes are decreased to less than 10 nm, according to some 
+theoretical investigations [21-24]. Recently, super-paramagnetic iron oxide-gold composite NPs is 
+synthesized by a green method [25]. It is shown that the NPs exhibited a high relaxivities ratio (r2/r1) 
+of 13.20, indicating the potential as a T2 contrast agent.  
+Surface modification is often practical to provide better stability under physiological 
+conditions and prolong bloodstream circulation time, thereby increasing MR imaging quality [26]. 
+This surface modification is known to restrict the uptake of plasma proteins (i.e., corona proteins), 
+which lowers the likelihood that macrophages will recognize and remove them [27]. In order to 
+overcome the aforementioned difficulties, polymeric coatings on the surface of magnetic NPs are 
+recommended [28]. In a recent work [29], iron oxide ferrofluid is synthesized by thermal 
+decomposition using poly (maleic anhydride-alt-1-octadecene, noted as PMAO) as a phase 
+transferring ligand. The results have demonstrated that the magnetic particles were fully covered at 
+high coverage by long non-magnetic polymeric chains. It is shown that this ligand could improve the 
+ferrofluid stability up to as long as 6 months. The MR images in solution and in rabbit using the 
+prepared PMAO-coated magnetic NPs had the best contrast effect on T2 weighted maps.    
+Polyethylene glycol (PEG) is a highly water soluble, hydrophilic, biocompatible, non-
+antigenic, and protein-resistant polymer that is easily eliminated through the kidneys and is not 
+absorbed by humans' immune systems among all forms of polymeric coatings. PEG has also been 
+frequently employed for linking anticancer medications to proteins to prolong their half-life, as well 
+as for organ preservation [30. It also functions as an antibacterial, non-toxic lubricant and binder that 
+is frequently used in a variety of medicinal applications [31, 32]. Additionally, PEG-capped magnetic 
+NPs have demonstrated promise as effective and efficient magnetic hyperthermia candidates as well 
+as multifunctional nano-carriers for the encapsulation of hydrophobic medicines [28]. In our previous 
+work, we successfully synthesized Zn0.3MnxFe2.7-xO4 (x=0, 0.25, 0.5, 0.75 and 1) NPs by a one-step 
+citric acid-assistant hydrothermal method and reported the effect of citric acid concentration, pH of 
+the medium and the amount of Mn addition on the structure, purity, and magnetic properties of the 
+
+4 
+ 
+synthesized NPs [33]. According to the author’s knowledge, citric acid-assistant hydrothermal 
+synthesis of PEG-6000 coated Zn0.3Mn0.5Fe2.2O4 NPs as a dual mode T1/T2 imaging contrast agent 
+have not been previously reported. In the present study, PEG surface coating is applied on the surface 
+of the zinc-manganese ferrite NPs and then physiochemical properties of the optimized sample is 
+thoroughly investigated. The mono-dispersed magnetic PEG-coated and uncoated Zn-Mn ferrite NPs 
+containing different levels of Mn content is synthesized and the MR imaging of the NPs in the presence 
+of external magnetic field is investigated.  
+2. Materials and Experimental Techniques  
+2.1. Materials 
+All raw materials, including Fe (NO3)3.9 H2O, NH4OH 25%, Zn (NO3)2.4H2O, Mn (NO3)2.4H2O and 
+C6H8O7.H2O (citric acid), CH3OH, and PEG (MW=6000 g/mol) were purchased from Merck Co. with 
+minimum purity of 99%. 
+2.2. Synthesis of Mn-Zn NPs 
+In order to synthesize Zn0.3MnxF2.7-xO4 NPs, where x is the molar fraction of manganese ions (Mn2+) 
+from 0 to 1, various amounts of manganese iron nitrate, zinc nitrate and manganese nitrate were 
+dissolved in 25 ml of distilled water. A reddish brown slurry was formed after adding a solution of 
+25% NH4OH which was added for the purpose of adjusting the pH of the media to 10. The resulting 
+slurry was then washed with the deionized distilled water three times. Following the addition of the 
+citric acid (CA), the mixture was rapidly stirred for 30 minutes before being placed to a 350 ml Teflon-
+lined autoclave with a 65% fill level. The autoclave was kept at 185 °C for 15 h and then cooled to 
+room temperature [33]. Table 1 shows the experimental conditions of the synthesized samples. The 
+uncoated samples were coded as NCZMX in which X is the molar fraction of Mn2+ ions.   
+Table 1: The hydrothermal process parameters and the corresponding sample codes in the present work 
+Sample code 
+Temperature (℃) 
+Time (h) 
+Citric acid (mmol) 
+pH 
+Molar fraction of Mn2+(x) 
+NCZM 
+185 
+15 
+3.5 
+10.5 
+0 
+NCZM25 
+185 
+15 
+3.5 
+10 
+0.25 
+NCZM50 
+185 
+15 
+3.5 
+10 
+0.5 
+NCZM75 
+185 
+15 
+3.5 
+10 
+0.75 
+NCZM100 
+185 
+15 
+3.5 
+10 
+1 
+
+5 
+ 
+2.3. Coating of Mn-Zn NPs 
+15 mg of NCZM50 and NCZM25 NPs were added to 1 ml deionized distilled water and then placed 
+in an ultrasonic bath for 30 min. A polymeric solution containing 3 wt% PEG was dissolved in 1.5 ml 
+of deionized distilled water and stirred for 30 min. The prepared magnetic ferro-fluid placed on a 
+magnetic stirrer and then, the PEG solution were slowly added. This mixture was stirred at room 
+temperature for another 1 h at ambient temperature (25 °C). Finally, the coated NPs were magnetically 
+collected, washed with distilled water and dried in a vacuum oven at 40 °C for 24 h. The synthesized 
+coated NPs are named as CZM25 and CZM50. 
+2.4. Cell viability  
+The MCF-7 cells were cultured in Dulbecco’s modified Eagle’s medium DMEM (Gibco 12800, UK) 
+supplemented with 10% fetal bovine serum, 100 U/ml penicillin, 100 μg/ml streptomycin and 2 mM 
+L-glutamine at 37 °C in a humidified atmosphere of 5% CO2. The MG-63 osteoblast-like-cells were 
+seeded at a density of 10,000 cells/well in a 96 well plate and cultured with complete medium 
+containing NPs at concentrations of 50, 100 and 250 g/ml. MCF-7 cells were exposed to particles 
+for 24 h, after which Alamar Blue cytotoxicity assay was conducted and absorbance was measured at 
+450 nm using a micro-plate reader. The results represent the mean values ± SD of two individual 
+experiments each performed in quadruplicate. Differences between groups were determined by 
+student’s t test with values of p<0.05 considered significant [34, 35]. 
+2.5. Characterizations 
+Philips diffractometer, MPD-XPERT model, using CuKα radiation (λ = 1.5406 Å), was used for phase 
+identification. Estimation of the average crystallite size (L) of the samples, using the full width at half 
+maximum value (β) obtained from the spinel peaks located at every 2θ in the pattern, was  carried out 
+by the modified Scherer’s formula. According to Scherer's modified formula, Lnβ (β in radians) is 
+plotted against Ln(1/cosθ). A linear plot is obtained using the linear regression which is defined as Eq. 
+(1). The intercept of the line would be Ln(kλ/L) (k=0.9); the value of L (mean crystallite size) can be 
+obtained using all the peaks: [33, 36]. 
+𝐋𝐧𝛃 = 𝐋𝐧 ((𝟎. 𝟗𝟒𝛌
+𝐋
+) + 𝐋𝐧 ( 𝟏
+𝐜𝐨𝐬𝛉)) 
+(1)
+ 
+The miller indices of the planes were extracted from the cards in the X’Pert software. Then, the mean 
+lattice parameter was calculated based on Eq. (2) [37]: 
+
+6 
+ 
+     
+                                                                                                   (2) 
+The shape, size, and size distribution of NPs were investigated using transmission electron microscopy 
+(TEM) with energy of 200 kV at Arya Rastak company in Tehran. A droplet of diluted magnetic flux 
+was placed on a carbon coated copper mesh and placed at room temperature to allow water to 
+evaporate. The average particle size of the produced zinc-manganese ferrite NPs from the TEM and 
+SEM data was calculated by measuring the diameter of at least 100 NPs with ImageJ software. The 
+data were fitted by a log-normal distribution curve and then the mean size was obtained.  
+Fourier transform infrared spectra (FTIR) were recorded in the range of 4000-400 cm-1 to detect 
+functional groups.  
+Saturation magnetization (Ms) values were conducted from the high field part of the measured 
+magnetization curves, where the magnetization curve becomes linear and line’s slope reaches to zero.  
+Colloidal properties of the aqueous magnetic ferro-fluids were investigated using a Zeta Potential 
+Estimator to measure the surface charge of NPs, hydrodynamic size, zeta potential and poly-dispersity 
+index of NPs (in pH=7) under different conditions.  
+Thermo-gravimetric analysis (TGA) was used to investigate the presence of polymer coating on the 
+surface of NPs. 
+MRI tests were performed with a 1.5 T clinical MRI instrument with a head coil working at 37 ℃. For 
+T1 and T2-weighted MRI of in-vitro cells at 1.5 T, the following parameters were adopted: [Mat 
+(320*192), FoV (184*230), and TR (407)], [Mat (256*192), FoV (260*260), and TR (7)], [Mat 
+(320*192), FoV (184*230), TR (2570)]. In order to simulate the physiological state of the body, PBS 
+solution and water was used to create a positive and negative contrast in the images. 
+
+aj = d; × Jh,? +k;? + ?7 
+ 
+ 
+Fig. 1. Image of the prepared instrument for MRI imaging. 
+3. Results and Discussion 
+3.1. Structural properties 
+Fig. 2. shows XRD pattern of the NCZM50 NPs in which the diffraction peaks are in good agreement 
+with planes (220), (311), (222), (400), (422), (511), (440), (620), (533) and (444) representing 
+synthesis of pure spinel phase without the need for any calcination step. The crystallite size of the 
+sample was estimated as 22 nm. 
+ 
+Fig. 2. The XRD pattern of the NCZM50 sample.        
+Surface coating is important in preventing NPs from agglomeration in physiological environment 
+which also act as a barrier, effectively shielding the magnetic core against the attack of chemical 
+
+140-
+S
+S
+NC7.M50
+120-
+Spincl:01-086-510
+100 -
+80
+S
+ntensi
+F09三
+S
+S
+S
+40-
+S
+S
+S
+20 -
+0:
+-
+20
+40
+60
+80
+208 
+ 
+species in the aqueous solution. Here, PEG was utilized to coat the optimized NPs. The FT-IR spectra 
+of the pure NCZM50, the PEG-coated CZM50 NPs and the PEG are shown in Fig. 3. For the pure 
+NPs, at around 3300 cm-1, a strong wide band exists which is attributed to the O-H stretching vibrations 
+of water molecules which are assigned to –OH group of CA absorbed by NCZM50 NPs (a structural 
+bond). The stretching vibration of C-H corresponds to the peak at ~2925 cm- 1 [38, 39]. The absorption 
+band at 1690-1760 cm-1 is due to the vibration of asymmetric carboxyl group (-COOH) [28, 40]. 
+Hence, it is suggested that CA binds to the NPs surface through carboxylate groups of citrate ions 
+[28]. Furthermore, Fe-O stretching band as the characteristic peak of magnetite NPs was located at 
+around 520 cm−1 which is attributed to the Fe-O stretching vibration bond in tetrahedral sites and the 
+absorption band in the 437 cm-1 corresponds to a Fe-O vibrating bond in octahedral sites of ferrite 
+phase [41]. Hydroxyl groups (-OH) of PEG are linked to the carboxyl group (-COOH) of citric acid 
+(CA) for coating of Zn0.3Mn0.5Fe2.2O4 NPs. As it can be seen in Fig. 3, the highest peak for PEG curve 
+showed a very small shift in PEG-coated sample. The peak at 1105 cm-1 for pure PEG were shifted to 
+lower frequencies which is a proof of C-O-C and C-O-H groups bonding with Zn0.3Mn0.5Fe2.2O4 NPs. 
+The absorption band at 2884 cm-1 can also be due to the H-C bonds stretching vibrations of the 
+polymeric chain. The peaks corresponding to the bonds, C-H and C-O-C are the strong evidence to 
+show that the synthesized magnetite NPs surface has been coated with PEG [38, 40].  
+ 
+Fig. 3. The FT-IR spectra of the pure NCZM50 and PEG-coated CZM50 NPs along with the PEG coating and 
+citric acid. 
+
+citricacid
+2.4
+Zn0.3Mn0.5Fe2.204
+PEG-Zn0.3Mn0.5Fe2.204
+PEG
+2.2
+2.0-
+1.8
+-COOH
+.6
+1.4
+C-H
+HO
+1.2
+1.0
+0.8
+0.6-
+C-O-C groups
+C-H groups
+0.4
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Wave number (cm-')9 
+ 
+ 
+The presence of PEG layer on the NPs surface was also characterized by TGA which is presented in 
+Fig. 4. The first stage of weight loss at a temperature about 32-35 °C can be related to the removal of 
+water molecules (hydroxyl ions) that are physically absorbed to the surface of the NPs. This weight 
+loss in the uncoated sample is 2.45% and in the coated sample is equal to 2.15%. The comparison of 
+the first weight loss in the two samples shows that the total water loss of the NPs is more than coated 
+NPs which is due to the total absence of water from the magnetic material structure [42]. The second 
+step, starting at about 50-300 °C, results from the loss of organic groups that were conjugated to the 
+surface of the particles. PEG desorption and subsequent evaporation were the causes of this weight 
+loss. When 7.5 mg of PEG 6000 were used, the weight loss for particles was almost 24%, indicating 
+76% iron oxide in the polymer-coated NPs. Weight losses less than 15–20% can imply that the 
+coverage of particle surface by the polymer is not complete [40]. 
+ 
+Fig. 4. The TGA result of the NCZM50 and CZM50 samples. 
+3.2. Microstructural analysis 
+Fig. 5 shows TEM micrograph and particle size distribution curve of the coated and uncoated samples. 
+By using ImageJ software to measure the diameter of at least 100 NPs, the average particle size and 
+the standard deviation was determined. As it can be seen, the synthesized NPs exhibit a rather uniform 
+size distribution, shape, and morphology. The mean particle size of the coated NPs is a bit greater than 
+that of the uncoated ones. It can be seen that NPs have become more dispersed after applying the 
+
+100
+-
+95
+Weight Percent (%)
+90
+NCZM50
+85
+CZM50
+ 08
+75
+-
+-
+1
+50
+100
+150
+200
+250
+300
+0
+Temperature (°C)10 
+ 
+coating in an aqueous medium. The average size of NPs obtained from the results of the TEM images 
+before and after coating was 6.9±1.54 nm and 9.25±1.6 nm, respectively, which indicates that the 
+polymer coating is applied on the surface of NPs at a low thickness [39]. 
+  
+Fig. 5. (a, c) TEM images and (b, d) particle size distribution histogram of the (a, b) uncoated NCZM50 and 
+(c, d) coated CZM50 NPs. 
+3.3. Stability and colloidal properties  
+The colloidal stability of magnetic fluids of NCZM50 sample was investigated using a zeta potential 
+measurement at pH=7 and various time points. The result indicated that the NCZM50 NPs sample had 
+a mean zeta potential of -48.86± 0.70 mV and a mean hydrodynamic size of 104 nm. According to 
+the results, the strong negative charge of the NPs (caused by the presence of citrate ions on their 
+surface) and the steric and electrostatic forces ensure their long-term stability in aqueous media [43]. 
+c 
+d 
+
+(a)
+70
+60
+50
+Frequency
+40
+30
+20
+10
+0
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+100nm
+15
+16
+size (nm)
+(d)
+100
+80-
+60
+40-
+20-
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+size (nm)
+75nm11 
+ 
+Poly-dispersity index (PDI) of NCZM50 sample was found to be 0.306. PDI is a parameter for 
+determining the particle size distribution of different NPs, which is obtained from photon correlation 
+spectroscopic analysis. It is a dimensionless number calculated from the autocorrelation function and 
+ranges from a value of 0.01 up to 0.7 for mono-dispersed and greater than 0.7 for poly-dispersed 
+particles [44]. In general, the particle size between 10 and 100 nm have the longest circulation time; 
+by contrast, it has been reported that particles of more than 200 nm tend to be immediately destroyed 
+by one of the MPS organs [43] and tend to be eliminated by the RES [9, 45], those with diameters <10 
+nm are removed mainly by renal filtration, and particles larger than 400 nm (minimum diameter of 
+capillaries) will be filtered by the lung [46].  
+  
+3.4. Magnetic properties 
+The particle size and magnetization saturation values of different NPs are presented in Table 2. The 
+room temperature M–H curve for NCZM50 and CZM50 samples is shown in Fig. 6. No hysteresis 
+loop can be seen and the value of magnetization sharply increases with the external magnetic field 
+strength. The M–H curve has an S-shape at the low field region, and the high field side of the curve is 
+almost linear with the external field [47]. Saturation magnetization for the NCZM50 and CZM50 NPs 
+is 55 emu/g and 38 emu/g respectively. The difference in particle size and the softening of the 
+magnetization caused by the presence of PEG can both be used to explain this mismatch [38]. The 
+magnetization curve of the CZM50 sample also revealed a negligible remnant magnetization at zero 
+field, reflecting the super-paramagnetic behavior of the ferro-fluid. Since magnetic powder has a 
+diameter much below the 20 nm cut-off expected for magnetite to show super-paramagnetic behavior, 
+the lack of hysteresis at ambient temperature is consistent with this theory [48].  
+ 
+Table 2. The size and Ms values of different NPs 
+Code 
+Chemistry 
+Size (nm) 
+Ms (emu/gr) 
+NCZM0 
+Zn0.3Fe2.7O4 
+14.5±2.7 
+47 
+NCZM25 
+Zn0.3Mn0.25Fe2.45O4 
+23.6±2.3 
+47 
+NCZM50 
+Zn0.3Mn0. 5Fe2.2O4 
+6.9±1.5 
+55 
+NCZM75 
+Zn0.3Mn0. 75Fe1.95O4 
+11.3 ±2.3 
+41 
+NCZM100 
+Zn0.3Mn1Fe1.7O4 
+6.7±2.4 
+37 
+CZM50 
+PEG coated- Zn0.3Mn0.5Fe2.2O4 
+9.3±1.6 
+38  
+ 
+
+12 
+ 
+ 
+Fig. 6. The M vs H curves of the synthesized NCZM50 and CZM50 NPs. 
+ 
+3.5. MRI analysis 
+MRI examination of the body can be performed with several coil types, depending on the design of 
+the MRI unit and the information required. Figs. 7(a-b) show T1– and T2-weighted MR images of 
+Fe3O4 and Zn-Mn ferrite solutions recorded on a 1.5-T MRI scanner at room temperature at different 
+concentrations (0.1, 0.15 and 0.2 mg/ml). As it can be seen, both T1 and T2-weighted MR images show 
+a strong dependence of signal intensity on manganese concentrations and among the Fe3O4 control 
+sample, Zn-based and Mn-Zn-based super-paramagnetic NPs, Mn-Zn ferrites represent a better MRI 
+contrast [49]. This is due to the fact that Mn2+ with five unpaired electrons, after Gd3+, is the most 
+powerful cation used as a MRI contrast agent [50]. Due to their greater paramagnetism and five 
+unpaired electrons, divalent manganese ions (Mn2+) have been shown to be a successful method of 
+increasing the r1 of ultra-small iron oxide NPs. A peculiar mixed spinel structure, a greater saturation 
+magnetization (Ms), and a high r2 of manganese doped iron oxide NPs result from the doped Mn2+ 
+with a higher magnetic moment (B=5.92) being able to fill both the tetrahedral (Td) and octahedral 
+(Oh) sites in the crystal lattice. The doped Mn2+ and ultra-small iron oxide NPs also exhibit synergetic 
+
+60 -
+CZM50
+NCZM50
+40-
+Magnetization (emu/g)
+20
+0
+40
+Magnetization (emu/g)
+20
+20
+0
+-40 -
+20
+-40
+09-
+-400
+-200
+0
+200
+400
+Applied field (Oe)
+-15000
+-10000
+-5000
+0
+5000
+10000
+15000
+Applied field (Oe)13 
+ 
+enhancement, which will further enhance both r1 and r2 of Mn-iron oxide NPs, according to the 
+embedding logic. The Mn-iron oxide NPs may therefore make superior candidates for dual-contrast 
+CA [20]. Indeed, it has recently been discovered that decreasing iron oxide NPs below 10 nm improves 
+their effectiveness as T1 contrast agents, suggesting that this approach could be employed to create 
+dual contrast agents. The utility of these NPs as T1 contrast agents is unfortunately limited by the low 
+r2/r1 values caused by the large decrease in r2 that occurred along with the increase in r1. To get over 
+this restriction, alloy-based NPs which has a high Ms are a suitable candidate to achieve NPs with 
+high MRI sensitivity [51]. The addition of Mn2+ and Zn2+ divalent cation ions to the spinel ferrite 
+structure causes the mass magnetization of the material to rise, which enhances the magnetic 
+characteristics. Therefore, the higher contrast in Zn0.3Mn0.5Fe2.2O4 NPs with higher saturation 
+magnetization can be justified [52]. As it is presented in Fig. 7, NCZM50 sample  with core diameters 
+about 6.7±1.54 nm and saturation magnetization about 55 emu/g is capable of producing dual positive 
+and negative contrast in images [26, 53]. However, the length of the polymer chain, which relates to 
+coating thickness, has a substantial impact on relaxivity as well. According to computer simulations, 
+the physical exclusion of protons from the super-paramagnetic iron oxide magnetic field and the 
+protons' residence period within the coating zone compete to decide the influence of coating thickness 
+on relaxivity.  
+As it can be seen in the Fig. 8, the surface coatings also affect the relaxivity of NPs. Laconte et al. 
+reported that the increased coating thickness would dramatically decrease the r2 and r1 relaxivity of 
+mono-crystalline magnetic NPs. Therefore it is important to note that both the chemistry of coating 
+and its thickness affect the value of r2 and r1 in which as the coating thickness increases, the ratio r2/r1 
+decreases. This is due to the inner hydrophobic layer excluding water, while the outer hydrophilic 
+PEG layer allows water to diffuse within the coating zone [53] . 
+ 
+ 
+ 
+
+14 
+ 
+ 
+Fig. 7. (a) T1-weighted and (b) T2-weighted MR images of the uncoated Fe3O4 and Mn-Zn ferrite NPs at 
+different concentrations indicated by different numbers: (1) Fe3O4 control sample, (2) NCZM0, (3) NCZM25, 
+(4) NCZM50, (5) NCZM75, and (6) NCZM100. 
+ 
+ 
+ 
+ 
+ 
+Fig. 8. (a) T1-weighted and (b) T2-weighted MR images of un-coated and uncoated samples indicated by 
+different numbers: (1) Fe3O4 control sample, (2) NCZM50, and (3) CZM50. 
+ 
+3.6. Cell viability 
+Cytotoxicity evaluations of the uncoated and coated NPs were investigated by evaluating their 
+cytotoxicity using MCF-7 cell line. The results of Alamar blue cytotoxicity assay are presented in Fig. 
+9. According to the results, a similar trend is observed in the activity of cells affected by different 
+concentrations of NPs after 24 h compared with the control group. In general, coated and uncoated 
+particles did not negatively change the cell growth process, and did not result significant reduction in 
+cell viability. In fact, a better growth was observed in the presence of coated NPs. 
+ 
+
+(a)
+(b)(a)
+(b)
+215 
+ 
+ 
+Fig. 9. The cytotoxicity assays performed on MCF-7 cells in the presence of coated and uncoated NPs after 
+24 h. 
+4. Conclusions  
+Mono-dispersed Zn0.3Mn0.5Fe2.2O4 NPs with an average size of about 6.9±1.5 nm were successfully 
+synthesized by a facile, one step citric acid-assisted hydrothermal method. The NPs were stabilized 
+with a layer of hydrophilic PEG and exhibited long-term colloidal stability in aqueous media at pH=7. 
+The magnetic properties of the uncoated and coated Zn-Mn ferrite NPs were measured as 55 and 38 
+emu/g, respectively, showing super-paramagnetic behavior at room temperature. More significantly, 
+the synthesized NPs displayed unexpectedly high T1 and T2 imaging contrast due to Zn2+ and Mn2+ 
+doping and PEG-6000 coating. The present zinc manganese iron oxide NPs coated by PEG 
+(ZnMnIONPs@PEG) are supposed to be a suitable candidate for application as T1/T2 dual contrast 
+agent, as shown by in-vitro MR imaging. Interestingly, applying low thickness of PEG layer on the 
+surface of the Zn0.3Mn0.5Fe2.2O4 NPs had no significant effect on the MR imaging. 
+ 
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+

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+Efficient Design of Helical Higher-Order Topological Insulators  
+in 3D Elastic Medium 
+Jiachen Luo1, Zongliang Du1,2*, Hui Chen3, Xianggui Ding1, Chang Liu1,2,  
+Weisheng Zhang1,2, Xu Guo1,2* 
+1State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering 
+Mechanics, Dalian University of Technology, Dalian, 116023, China 
+2Ningbo Institute of Dalian University of Technology, Ningbo, 315016, China 
+3Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics,  
+Ningbo University, Ningbo 315211, China 
+E-mail: zldu@dlut.edu.cn (ZD); guoxu@dlut.edu.cn (XG) 
+Abstract 
+Topological materials (TMs) are well-known for their topological protected properties. 
+Phononic system has the advantage of direct observation and engineering of topological 
+phenomena on the macroscopic scale. For the inverse design of 3D TMs in continuum 
+medium, however, it would be extremely difficult to classify the topological properties, 
+tackle the computational complexity, and search solutions in an infinite parameter space. 
+This work proposed a systematic design framework for the 3D mechanical higher-order 
+topological insulators (HOTIs) by combining the symmetry indicators (SI) method and the 
+moving morphable components (MMC) method. The 3D unit cells are described by the 
+MMC method with only tens of design variables. By evaluating the inherent singularity 
+properties in the 3D mechanical system, the classic formulas of topological invariants are 
+modified accordingly for elastic waves. Then a mathematical formulation is proposed for 
+designing the helical multipole topological insulators (MTIs) featured corner states and 
+helical energy fluxes, by constraining the corresponding topological invariants and 
+maximizing the width of band gap. Mechanical helical HOTIs with different symmetries 
+are obtained by this method and verified by full wave simulations. This design paradigm 
+can be further extended to design 3D TMs among different symmetry classes and space 
+groups, and different physical systems.  
+Keywords: Topological materials, Mechanical higher-order topological insulators, 
+Topology optimization, Symmetry indicators 
+ 
+
+1. Introduction 
+Metamaterials are well-known for its novel modulation of photons, phonons, and matter 
+waves in various applications. Enriching with the topological characteristics, it gives out a 
+new innovative material—topological materials (TMs), which are robust to various 
+defects1–3. Recently, the photonic and phononic TMs have attracted a great research interest 
+in engineering topological phenomena on the macroscopic scale2–12. Related topologically 
+protected states revealed some prospective applications2,3,12–23.  
+For example, the quantum spin/valley Hall topological insulators guide the energy flux in 
+a spin-locked transmission, which alternatively switches the one-way tunnel for the 
+propagated waves with immunity to defects6,8,9,24–28. That is an ideal way to improve the 
+effectivity of applications in opto-mechanics, current semiconductor and integrated circuit 
+industry13–22. For the higher-order topological materials (HOTIs), it is characterized by an 
+intensively localized topological phase within the lower dimensional domain such as the 
+edges and corners3,7,29–31. As a pioneering example of the HOTIs, the multipole topological 
+insulators (MTIs) can also provide a multipole moment enhanced topological phases, 
+where the bulk dipole is vanished3,30,31. Together with the pseudo-spin phenomenon, a 
+helical multipole-induced topological phase is inherited in the helical MTI32,33. Those 
+topological phases in the HOTIs are robust to various defects in manufacture, and show a 
+promising prospective in optical/acoustic subwavelength imaging, microelectronics, laser 
+aspects3,19,23,31,34–36.  
+Although the theoretical tight binding models have been developed, how to efficiently 
+design 3D unit cells with the demanded topological behaviors is still a crucial challenge. 
+Some typical options include tracing the featured degenerated states near the Dirac points, 
+restricting a special band structure from the band folded mechanism, keeping an obvious 
+Berry curvature (a quantity to topology), or realizing the maximal pseudo-spin energy 
+fluxes in the crossing waveguide37–44. To calculate the topological invariants, however, it 
+is very expensive to integrate the Berry curvature or its related terms in the whole Brillouin 
+zone. This issue would be more pronounced for the 3D continuous TMs, which generally 
+cover an infinite design parameter space and are more computationally expensive for 
+analysis and design optimization.  
+Luckily, the theoretical breakthrough in topological quantum chemistry gives new insight 
+into this bottleneck, from the fruitful meeting between chemistry and physics (in the real 
+and momentum space)45–48. The fundamental tool is calculating the real space orbits for 
+every band, with the aids of the elementary band representations (EBRs) or the symmetry 
+indicators (SIs). It gives out the topology in a simple linear function of the symmetry 
+
+characters at some listed point29,30,46,48. Successful applications of the SI method include 
+the classification of TMs among the whole 230 space and 1651 magnetic groups, and the 
+discovery of thousands of TMs with many uncovered for the first time47,49,50. Most recently, 
+the catalogue of topological phononic materials becomes an attractive focus4,5,11. This 
+inspires us to efficiently identify the topological properties of the 3D mechanical unit cells 
+using the SI method. It is worth to note that, in the mechanical system, rigid body motions 
+corresponding to zero energy states yield the singularity at some high symmetry point. As 
+a result, the classic formulas derived in the quantum mechanics system need to be modified 
+at first.  
+Furthermore, how to describe the 3D unit cells is an essential factor for choosing design 
+optimization method39,41,43,44,51. This is because the topological invariants are discrete 
+variables, which cannot in general effectively handled by the gradient-based algorithms. 
+The Moving Morphable Component (MMC) method could describe 3D unit cells using 
+only a few explicit geometry parameters, and this makes it suitable to guarantee a 
+computationally tractable solution process of the inverse design formulation52,53. In general, 
+we summarize three characteristics of a desired optimization framework of the TMs as: (i) 
+effective to identify the topological characters for arbitrary unit cells; (ii) suitable to 
+topological materials in different classes and different physical systems; (iii) efficient to 
+execute the solution procedure.  
+In this study, we proposed a unified optimization framework for the 3D continuum TMs 
+by combining the SI method and the MMC method. In this design framework, the helical 
+MTIs with the helical edge and helical corner states can be effectively obtained by 
+simultaneously constraining the modified fractional corner charge and pseudo-spin 
+invariants. The proposed method thoughtfully modified the topological invariants for 3D 
+elastic HOTIs according to the singularity points with zero energy, and successfully 
+obtained optimized 3D helical MTIs in different symmetries, where the in-gap corner states 
+are derived from the quadrupole moment. The numerical simulations of the transmission 
+spectra and crossing waveguide applications validated the intensive corner energy and the 
+spin-locked energy flux.  
+The rest of the paper is organized as follows: in Section 2, the governing equations for 
+elastic waves are introduced. And then based on the description method of 3D elastic unit 
+cells using the MMC method and the specialized formulas of topological invariants in 
+Sections 3 and 4, an efficient deign paradigm is proposed for the mechanical helical MTIs. 
+Optimized designs with different symmetries are presented in Section 6 together with the 
+applications in a novel crossing elastic waveguide. Finally, some concluding remarks are 
+discussed in Section 7. 
+
+2. The governing equations for elastic waves 
+In 3D elastic mechanics, the harmonic wave formulation is expressed as54 
+(������������ + ������������)∇(∇ ⋅ ������������) + ������������∇2������������ = −������������2������������������������
+(1) 
+where ������������ is the angular frequency, ������������, ������������, and ������������ are the Lame’s parameters and mass density, 
+and ������������ = (������������, ������������, ������������)⊤ denotes the displacement field.  
+Since the considered phononic crystal is periodic, the displacement field satisfies the 
+translation condition as ������������(������������ + ������������) = ������������(������������) with ������������ denoting the primitive lattice vector. 
+According to the Bloch theorem, the harmonic elastic wave propagation can be determined 
+by the following discretized equations 
+������������������������ = −������������2������������������������
+������������(������������0 + ������������)|BC = ������������(������������0)|BC ������������i������������⋅������������
+(2) 
+Here, the matrices ������������  and �������������  refer to the stiffness and mass matrixes, and ������������  is the 
+eigenvector.  
+In order to handle the periodic constraints in the above eigenvalue problem, the standard 
+Lagrange multiplier method is adopted55. By defining a Lagrange multiplier ������������, Eq. (2) can 
+be reformulated as 
+������������� + ������������2������������
+������������f
+������������
+������������ � �������������
+������������� = ������������
+(3) 
+in which the constraint matrices ������������ and ������������f are used to homogenize the eigenvalue problem.  
+Now, let us decompose the eigenvector with the solution ������������c as ������������ = ������������null������������c + ������������0, where 
+the matrix ������������null and vector ������������0 belong to the null space of ������������. An allowable value is ������������0 =
+������������. After left multiplying Eq. (3) by ������������nullf
+⊤
+ (������������nullf is the null space of ������������f
+⊤), we have the final 
+governing equation of the 3D elastic wave as 
+������������c������������c = −������������2������������c������������c
+(4) 
+where the eliminated stiffness matrix is ������������c = ������������nullf
+⊤
+������������������������null, and the eliminated mass matrix 
+is ������������c = ������������nullf
+⊤
+������������������������null. Because Eq. (3) requires ������������f ������������ = ������������, it is needless to solve for it since 
+������������ is useless, and a practical choice is setting ������������f = ������������⊤, then ������������nullf = ������������null.  
+
+3. Description of the 3D unit cells via the MMC method 
+Structural topology optimization has been successfully applied to inverse design various 
+topological metamaterials11,38–44,56. For designing the 3D elastic topological insulators, we 
+adopt the Moving Morphable Component (MMC) method40,42,52,53,56, which has the 
+advantages of the explicit geometry description and improved computational efficiency.  
+The building block in the MMC method is a set of morphable components, described by 
+some geometry parameters, such as the center coordinate, length, width, and thickness. As 
+a result, through updating those geometry parameters, every component can move, morph, 
+merge or disappear to form the optimized structure, as shown in Fig. 1. In this way, the 
+optimal parameter space will be deeply shrunk, and the solution efficiency will be 
+significantly improved.  
+In our work, each 3D component (the inclusion phase) is explicitly characterized by the 
+ellipsoid with a design variable vector ������������������������ = (������������0������������
+⊤ , ������������������������
+⊤, ������������������������
+⊤)⊤, i.e., the center coordinate 
+������������0 = (������������0, ������������0, ������������0), the length vector of semi-axes ������������ = (������������1, ������������2, ������������3), and the Euler rotation 
+angles ������������ = (������������, ������������, ������������), as shown in Fig. 1(a). In this manner, each MMC can be explicitly 
+determined by only 9 design variables. Furthermore, in a unit cell, the inclusion phase is 
+identified by the topology description function ������������������������(������������, ������������������������) for each component expressed 
+with its covered region ������������ as 
+��������������������������(������������, ������������������������) = ‖������������′‖2
+2 − 1 = �
+> 0
+if  ������������ ∈ Ω������������
+= 0
+if  ������������ ∈ ������������Ω������������
+< 0
+else
+(5) 
+In Eq. (5), the local coordinates are determined by the global coordinates ������������ and the rotation 
+matrix ������������(������������) as 
+������������������������
+′ = 1
+������������������������
+R������������������������(������������)������������������������� − ������������0�������������
+(6) 
+According to the symmetry requirement of the unit cells, only the MMCs in a reduced 
+design domain need to be optimized and they can be transformed to the rest part. 
+Furthermore, all the inclusions represented by MMCs in the design domain can be 
+smoothed by the K-S aggregation technique 57 or the Boolean operation (adopted by this 
+work). 
+
+ 
+ 
+(a) 
+(b) 
+Fig. 1. An illustration of a 3D unit cell described by the MMC method. (a) The geometric 
+description of the 3D components and (b) some representative configurations in the 
+optimization. 
+4. The SI induced topological invariants of the helical MTIs 
+The helical multipole topological insulators (MTIs)32,33, as a compound topological 
+material, should simultaneously hold the characters (or the topological invariants) from the 
+multipole moment and the pseudo-spin. In general, the calculation of topological invariants 
+is computationally expensive for the continuum unit cells. Nevertheless, recent work in 
+topological quantum chemistry reveals a rapid approach to identify topological invariants 
+through its symmetry indicators (SIs)4,5,46,49,50,58. Next, we will introduce the SI method 
+into the calculation of topological invariants, and then give out the method to design helical 
+MTIs.   
+Based on the SI method, for a spinless ������������������������=3,6-symmetric mechanic system with the time 
+reversal symmetry (TRS), we identify the eigenvalue ������������
+(������������) of the ������������̂������������ rotational operator as 
+������������
+(������������) = ������������i2������������(������������−1)/������������  = �������������(Π)�������������̂�������������������������(Π)�, ������������ ∈ [1, ������������]
+(7) 
+where ������������(Π) denotes the ������������-component of the displacement at the high-symmetry point Π. 
+The symbol  #������������
+(������������) counts the number of ������������
+(������������) below the target band gap. Compared to the 
+reference point à = (0,0), we define the SI at Рas �������������
+(������������)� = #������������
+(������������) − #Γ������������
+(������������). At the high-
+symmetry points, it satisfies ������������̂������������������������ = ������������ + ������������ with ������������ denoting the reciprocal lattice vector. 
+For the ������������3-symmetric hexagonal unit cells, the high-symmetry points include à and K in 
+the ������������3 symmetry, while for the ������������6-symmetric hexagonal unit cells, they include à in the ������������6 
+symmetry, K in the ������������3 symmetry, and M in the ������������2 symmetry, respectively. In this manner, 
+
+AZ
+Z'RY
+X
+L3
+L2Original
+Morphing
+Moving
+Mergingthe topological classification is determined completely by the corresponding SIs, such as 
+the fractional corner charge and the pseudo-spin invariants in the following contents.  
+4.1 The fractional corner charge invariants 
+For the MTIs, the fractional corner charge ������������(������������) is an effective topological invariant to 
+determine the topological corner states29–31. For the 3D mechanical topological system with 
+the TRS and ������������3 symmetry, we propose the following formulas 
+������������(3) = �1
+3 �#K������������≠1
+(3) − 1
+2 #Γ(3)� mod 1� × ��#Γ(3) + 1�mod 2�
+������������(6) = �1
+4 �#M1
+(2)� + 1
+6 �#K1
+(3)�� mod 2
+(8) 
+Here, the red terms in the function of ������������(3) are introduced to avoid the confused distinction 
+of the unpaired degenerate states6,9,10,25. As an alternative strategy, #Γ(3) counts the two-
+order degeneracy at the Γ point, and #Γ(3)/2 identically equals the #Γ2
+(3) or #Γ3
+(3). The red 
+modulo term in ������������(3) guarantees the degenerate states to be in pairs (i.e., #Γ(3) is an even 
+number). For a visualization, the Fig. 2 shows that our modification successful avoids the 
+confused distinction of degenerate states when ������������ = 4. For more details, refer to Appendix 
+A. 
+ 
+ 
+ 
+(a) 
+(b) 
+(c) 
+Fig. 2. The script of the modification procedure. (a) The singularity in mechanics and 
+degenerate states in a hexagonal unit cell. (b) and (c) The vector transformation of 
+displacement component (������������, ������������) and ������������. Here, only the degenerate states are calculated 
+under the ������������̂3 operator, and denoted as 1/2 = (������������ + ������������∗)/2 with its eigenvalues ������������ and its 
+conjugation ������������∗; the other states are calculated under the ������������̂2 operator. 
+4.2 The pseudo-spin invariants 
+For the photonic/phononic quantum spin/valley Hall effects, the protected chiral energy 
+flux can be well identified from the pseudo-spin vortex phenomenon and can be well 
+
+Degenerate
+-m=5
+1/2
+1/2
+States
+m=4
+-1
+m=3
+-1
++1
+个个
+00
+Singularity
+ky个
+(u, v)
+(-xo,yo)
+(xo, yo) x
+(-u, v)w
+(xo, Yo.
+(-xo,-yo)
+x
+wquantified by the Chen-spin or ������������2 invariants6,9,24,25. In our spinless 3D mechanical system 
+without spatial inversion symmetry, an alternative approach is adopted through tracing the 
+(broken) Dirac cone and band inversion9,10,25,38,40,42–44.  
+Practically, for the 3D ������������3 and ������������6 symmetric unit cells, the pseudo-spin invariants are 
+modified as10,29 
+������������(3) = sgn�#K2
+(3) − #K3
+(3)�
+������������(6) = sgn�#Γp
+(6) − #Γd
+(6) − 2�
+(9) 
+Here, terms #Γp
+(6) and #Γd
+(6) count the orbits p and d under the ������������̂6 operator for the Γ point, 
+respectively. Notably, the subtracted red term is introduced to correct the #Γp
+(6) due to the 
+singularity in the 3D elastic wave or the transverse electromagnetic wave (the singularity 
+has the same eigenvalue as the orbit p)5,59. For the 3D elastic wave, the singularity relates 
+to three translational motions, their displacement and vector transformation are shown in 
+Fig. 2. This modification is based on counting all occupied bands below the target band 
+gap, including the first three bands crossed through the singularity. Furthermore, the 
+counting idea keeps target bands isolated from other bands, as the SI method requires. For 
+more details, refer to Appendix B.  
+5. An efficient design paradigm of 3D mechanical helical MTIs 
+With the above topological invariants presented in Eqs. (8) and (9), we can now design the 
+helical MTIs using explicit topology optimization method. The corresponding optimization 
+formulation and solution process are introduced as follows. 
+5.1 Mathematical formulation 
+Combining the MMC-based description method and the modified formulas of topological 
+invariants in elastic medium, optimized 3D helical MTIs can be obtained by solving the 
+following mathematical formulation:  
+find
+������������ = (������������1
+⊤, … , ������������������������
+⊤, ������������)⊤
+max
+min(������������ref − max������������������������������������
+������������ , min������������������������������������
+������������+1 − ������������ref)
+s. t.
+������������c������������c = −������������2������������c������������c
+�������������(������������), ������������(������������)� = �������������ref
+(������������), ������������ref
+(������������)�
+������������min ≤ ������������ ≤ ������������max
+(10) 
+
+In the design variable vector, ������������������������ describes the ������������th component in the slab with a thickness 
+������������ (in the ������������-axis), as illustrated in Fig. 1. By denoting the eigenfrequency of the ������������th band 
+as ������������������������, the gap width between the ������������th and (������������ + 1)th bands is maximized with a target 
+mid-frequency ������������ref. The third equation in Eq. (10) is the governing equation for the 3D 
+elastic waves. Since the fractional corner charge and the pseudo-spin invariants 
+simultaneously contribute to the existence of helical corner states, the target topology 
+invariants �������������ref
+(������������), ������������ref
+(������������)� is introduced as a constraint. The last inequality persists the lower 
+and upper bounds of the design variable vector. 
+In principle, by updating the governing equation and target topological invariants, the 
+mathematical formulation in Eq. (10) can be applied for designing TMs among different 
+symmetry classes, and different physical systems. In this work, we focused on the inverse 
+design of 3D helical MTIs in elastic medium with ������������3 and ������������6 symmetries.  
+ 
+Fig. 3. The scheme of optimization for the helical MTIs. 
+5.2 Solution process 
+Since the topological invariants are quantized, gradient-based optimization algorithms 
+would be ineffective for solving Eq. (10). Thanks to the advantage of a fewer number of 
+design variables in the MMC method, the genetic algorithm (GA) is adopted here and the 
+settings are presented in Appendix C. To be specific, the flowchart for the rational design 
+of helical MTIs is shown in Fig. 3, and its solution process is summarized as follows: 
+• 
+STEP 1: Initialization of the MMC method and the GA solver.  
+The gap label ������������, the mid-frequency ������������ref, and the nonzero topological invariants 
+�������������ref
+(������������), ������������ref
+(������������)� are initialized first through a trial process, starting from ������������ = 3;  
+• 
+STEP 2: Optimal design of the first MTI.  
+
+Init.
+STEP1
+Band Order
+200 Random
+FEA
+Count Cases
+If No Case
+(Q(n), z(n))
+Q(n) ± 0&z(n) ±0?
+m=3
+Unit Cells
+(COMSOL)
+N
+Y
+m=m+1
+ STEP2
+1st MTI
+GA
+Generate
+Set Valid Para.
+'ref
+FEA
+Calculate
+Unit Cell
+Conv.?
+fref, m
+(COMSOL)
+Q(n), z(n)
+MTI Partner
+Y
+MMC
+N
+STEP3
+0,
+ref
+EndWith the parameters determined in STEP 1, solve the mathematical programming 
+Eq. (10) to obtain the first optimized MTI with the predefined invariant 
+�������������ref
+(������������), ������������ref
+(������������)� and mid-frequency ������������ref;  
+• 
+STEP 3: Optimal design of the MTI partner (if necessary).  
+With the desired topological invariants setting as �0, −������������ref
+(������������)�  and the other 
+parameters the same as STEP 2, solve Eq. (10) to obtain the optimized MTI partner 
+with the inverted pseudo-spin effect.  
+ 
+ 
+(a) 
+(b) 
+Fig. 4. The statistical charts (b) of different states at the Γ point (the partitions of p and d 
+would be decomposed into the boxed partitions without the modification in Eq. (8)) and 
+(c) of different TMs. Hint: s.p. —singularity point. 
+To illustrate the effectiveness of the proposed design framework, the statistical charts of 
+the states at the Γ point (6000 samples) and of different TMs (8000 samples) are illustrated 
+in Fig. 4(a) and 4(b), respectively. It can be found that, using Eq. (8), the states p and d are 
+successfully identified, and they take about 21.6% and 26.2% of the whole set as shown in 
+Fig. 4(a). Without the modification in Eq. (8), however, such states would be decomposed 
+to Γ2
+(3) state (22.1%), Γ3
+(3) (22.1%), and an unpaired set of state (3.5%). This unpaired set 
+would further make troubles for the calculation of the fractional corner charge invariant. In 
+Fig. 4(b), 6.6% of 8000 samples are four typical TMs (quantum valley/spin Hall 
+topological insulators (QVTIs/QSTIs), MTIs and helical MTIs), while the desired helical 
+MTIs only account for 4.0%. This validates the necessity of developing inverse design 
+paradigm for the helical MTIs.  
+
+s.p.
+31.1%
+d
+11.1%
+21.6%
+9.5%
+0.6%
+26.2%
+3
+Other
+22.1%
+Unpaired
+3.5%
+22.1%
+p
+**
+（3）
+S
+pOther
+93.4%
+0.7%
+QVTI
+2.4%
+4.0%
+Helical MTI
+QSTI
+0.2%
+MTI6. Applications of the MMC-based design framework for 3D helical MTIs 
+in elastic medium    
+In the present work, the helical MTIs are periodic in the in-plane direction and made of the 
+basic medium EP and scattering medium Fe (materials parameters and more setup details 
+are referred to Appendix C).  
+6.1 Optimal design of ������������3-symmetric mechanical helical MTIs 
+Under the optimization framework, the optimized ������������3-symmetric helical MTIs are obtained 
+in Fig. 5(a). And there is a normalized bulk band gap at 0.741-1.069 between the 6th and 
+7th bands (colored in grey in Fig. 5(a)). The symmetry behaviors of the high-symmetry 
+points are shown in the Fig. 5(b). There are three broken degenerate states (from the Dirac 
+cone) at the K point below the target bandgap, while only the third one is unpaired, and 
+implies the possibility of a pseudo-spin vortex. The phase field of this unpaired state is also 
+inserted in Fig. 5(a). The corner charge and the pseudo-spin invariants are (2/3,1).  
+In order to realize the band inversion, the corresponding MTI partner can be easily 
+constructed by applying the spatial reversal operation, or in other words, its invariants are 
+set as (0, −1). An opposite pair of ������������(3) invariants would produce a helical topological state 
+from the bulk-boundary correspondence. Moreover, a pair of zero and nonzero fractional 
+corner charges reveal the appearance of corner states18, as shown in Fig. 6(a) around the 
+normalized frequencies of 0.894 and 0.966. The latter localized corner mode is displayed 
+in the inserted diagram. 
+ 
+ 
+ 
+(a) 
+ 
+(b) 
+Fig. 5. The optimized ������������3-symmetric mechanical helical MTIs. (a) The band structure 
+inserted with the unit cell and the phase field of the unpaired state. (b) The symmetry-
+
+1.2
+(2A/2πC)
+1
+0.8
+0.6
+Freq
+0.4
+0.2
+0
+T
+M
+K
+Singularity(2)
+Band
+-(3)
+b
+b
+1
+-1
+3
+wt
+2
+-1
++1
+3
++1
++1
+m
+4
++1
++1
+5
++1
++1
+6
++1
++1
+mbehavior-table, in which the degenerate states are tagged as ������������ = ������������i2������������/3 for the K point, 
+while for the à point they are tagged as ������������. 
+ 
+ 
+(a) 
+(b) 
+ 
+ 
+(c) 
+(d) 
+Fig. 6. Simulation results of the optimized ������������3-symmetric helical MTIs. (a) The eigenvalue 
+spectrum (points are colored according to the corner energy intensity) and the energy field 
+of a corner state. (b) The transmission spectra from the probes in bulk, edge, and corner 
+area (colored in legend). (c) Energy fields tagged in (b) at the normalized frequencies of 
+0.793, 0.879, 0.966, and 1.121. (d) The energy flux and their zoom-in views of helical edge 
+states at the normalized frequency of 0.862. 
+Besides, the full-wave transmission is presented in Fig. 6(b), where energy is captured from 
+different regions around the outer bulk, the interface edges, and the interface corners. A 
+spin-down (clockwise) helical source is excited near the supercell’s center, shown as the 
+star in Fig. 6(c). The transmission reveals some edge energy peaks around the normalized 
+frequencies of 0.793 and 1.001, and some intensively localized corner states around the 
+normalized frequencies of 0.879 and 0.966. For a clear visualization, the corresponding 
+
+1.02
+0.98
+(S2A/2 TC)
+0.94
+0.9
+Freq (
+0.86
+0.82
+Index(dB)
+3
+Transmission (
+2
+-60
+Corner
+-120
+Edge
+Bulk
+0.7
+0.8
+0.9
+1
+1.1
+Freg (2 A/2πc)1
+2
+3Spin-Down
+Spin-Upbulk, edge, and corner energy fields are displayed in Fig. 6(c). In contrast to the edge gap 
+around the normalized frequency range of 0.872-1.001 (colored in light-green), those in-
+gap corner states are derived from the quadrupole moment.   
+For the verification of the helical behavior, a biased helical source off the center is excited 
+additionally, as shown in Fig. 6(d). The inserted arrow diagrams displayed the energy flux 
+near their corners and edges. We found that these two supercells had significant opposite 
+responses under different exciting helical sources (spin-up or spin-down). All their corners 
+held a clear energy vortex (clockwise or anticlockwise). Their edge energy fluxes are 
+locked by their exciting sources and only could flow forward or backward. 
+6.2 Optimal design of ������������6-symmetric mechanical helical MTIs 
+For the optimized ������������6-symmetric MTI pairs, as illustrated by the band structures shown in 
+Fig. 7(a), band gaps are observed in the normalized frequency ranges of 1.344-1.489 (up) 
+and 1.332-1.450 (below), respectively. Below the gap, there are four degenerate states 
+found at the Γ points for both cases, but only the last two states formed an unpaired double 
+Dirac cone, which features the pseudo-spin vortex. The phase fields of these unpaired states 
+are inserted in Fig. 7(a), from which the band inversion is clearly displayed. The symmetry 
+behaviors in Fig. 7(a) show that the corner charges and the modified pseudo-spin invariants 
+are (1/2,1)and (0, −1), respectively. Specifically, the pair of opposite ������������(6) invariants lock 
+the energy flux by the pseudo-spin phenomenon. In contrast, the pair of zero and nonzero 
+corner charges predict the topological corner states (according to the vanished bulk 
+polarization in ������������6-symmetric unit cells, these nonzero corner charges are only derived from 
+the quadrupole moment30). By combining these two topological characters, the topological 
+corner state will also have pseudo-spin behaviors and present as a helical corner state. For 
+a verification of this helical corner state, the eigenvalue spectrum of the supercell’s 
+simulation is shown in Fig. 7(b), and its energy density distribution, at the normalized 
+frequency of 1.426, is highly localized at corners. 
+6.3 Applications of the optimized helical MTIs in a crossing waveguide 
+As an application of the helical MTIs, a crossing waveguide (a single layer) composed of 
+the two optimized ������������3-symmetric helical TMIs in Subsection 6.1 (colored blue/yellow for 
+the original/inversed TMIs mentioned above) is developed in Fig. 8(a). Since the additional 
+pseudo-spin freedom locks the energy flux in the waveguide, two opposite transmissions 
+would be discovered when we sequentially excited the Port 1 and Port 2. By gradually 
+modulating the exciting frequency, the energy will spread through the center wall and 
+induce the output corner states.  
+
+ 
+ 
+(a) 
+(b) 
+Fig. 7. Simulation results of the optimized ������������6-symmetric helical MTIs. (a) The band 
+structures and the symmetry-behavior-tables of the optimized MTI pairs. The inserted 
+diagrams include the optimized unit cells and the ������������-directional displacement fields of the 
+unpaired states. In those tables, the degenerate states for the K point are tagged as ������������ =
+������������i2������������/3, while for the à point they are tagged as ������������. (b) The eigenvalue spectrum and the 
+inserted energy field of the corner state (points are colored according to the corner energy 
+intensity). 
+The simulations in the normalized frequency range of 0.7-1.1 are processed to test the 
+performance of the waveguide, as shown in Fig. 8(b). It is clear that when Port 1 is excited 
+at the normalized frequency of 0.776, the energy only transmits to Port 2 and Port 3, yet it 
+only transmits to Port 1 and Port 4 from Port 2. This phenomenon reveals the locked helical 
+energy flux as expected. At the normalized frequency of 0.897, the corner states in the 
+lower half of the waveguide are excited in both cases. Here, these states stay in the band 
+gap of the edge states (i.e., 0.872-1.001, refer to Appendix D for more details), and their 
+energy only localizes at corners, and no edge states exist.  
+To test the working range of the one-way transmission in this waveguide, we distinguished 
+the energy from the different ports (Port 3 or Port 4), as shown in Figs. 8(c) and 8(d). Here 
+the light-green area and yellow-solid points refer to the band gap of the edge states and the 
+states in Fig. 8(b). In this much wider frequency range of 0.749-0.861, the average 
+difference between both ports is higher than 10dB. When we reverse the exciting port, the 
+output port, which has a higher transmission, is also turned, as shown in Fig. 8(d). In this 
+frequency range, the first two edge bands, as illustrated in Appendix D, will be excited. 
+
+1.6
+Band
+a
+(2A/2πc)
+1
+-1
++1
+m
+1.2
+2
+-1
+-1
+wt
+1
+3
++1
++1
++1
++1
+0.8
+Freq (
+4
++1
++1
++1
+at
+0.6
+5
++1
+1
+m
+0.4
+6
+-1
+-1
++1
+0.2
+7
+0
+M
+K
+Singularity
+1.6
+r(2) r(3) m(2) k(3)
+Band
+a
+1.4
+b
+9
+b
+Freq (2A/2πc)
+1
+w)
+1
+2
+-1
++1
+wt
+1
+3
++1
++1
++1
++1
+0.8
+4
++1
+wt
+0.6
+5
++1
++1
++1
+m
+0.4
+6
++1
+-1
++1
+0.2
+7
++1
++1
+0
+M
+K
+Singularity1.47
+(S2A/2 TC)
+1.44
+1.41
+Freq (
+1.38
+1.35
+IndexHence, these one-way transmission results from the helical edge states. Moreover, the 
+corner states tagged with the number 2 and 4 are in the gap of the edge state but display an 
+apparent energy concentration from the exciting source. 
+ 
+ 
+(a) 
+(b) 
+ 
+ 
+(c) 
+(d) 
+Fig. 8. The crossing waveguide made of the optimized ������������3-symmetric helical MTIs. (a) The 
+sketches of the waveguide and its energy fluxes in different exciting cases (the exciting 
+line sources are tagged as stars). (b) The energy fields at the normalized frequencies of 
+0.776 and 0.897. The measured transmission of Ports 3 and 4 (c) from the exciting Port 1 
+or (d) from the exciting Port 2. Here the band gap of the edge states (light-green region) 
+and the typical states (yellow-solid points) are colored. 
+7. Concluding remarks 
+In this work, we proposed an optimization framework for the inverse design of multi-
+functional topological materials in the 3D continuous medium. By carefully manifesting 
+the degenerate states and singularity points in the elastic waves, the 3D helical multipole 
+
+Port1
+Port 2
+Port 4
+Port 3
+C
+Q2
+32
+0
+(dB)
+-50
+Transmission (
+-100
+-150
+-200
+Port 3
+-250
+Port 4
+0.7
+0.8
+0.9
+1
+1.1
+Freq (2 A/2πc)3
+Transmission (dB)
+-50
+-100
+-150
+-200
+Port 3
+-250
+-Port 4
+0.7
+0.8
+0.9
+1
+1.1
+Freq (2 A/2πc)topological insulators are well-classified by the fractional corner charge and the pseudo-
+spin invariants. With the explicit topology optimization and the symmetry indicator 
+methods, the proposed design paradigm has the advantages of (1) rapid classification of 
+the 3D topological materials and (2) efficient optimization of the 3D continuum unit cells 
+in a smaller explicit parameter space. This framework shows outstanding suitability to the 
+3D topological system and can also be generalized to other symmetry classes and space 
+groups. Besides, building up a topological materials library in continuous medium would 
+be an exciting topic for further research. 
+Methods 
+The solid mechanic simulation is performed in the commercial software COMSOL 
+MULTIPHYSICS. The default open surfaces are set as free boundaries. The Bloch theorem 
+is numerical expressed by the Floquet periodic boundaries. In common, the energy in solid 
+mechanics is consistent in distribution as the amplitude of total displacement ‖(������������, ������������, ������������)‖2
+2.  
+Acknowledgements 
+The financial supports from the National Natural Science Foundation (11821202, 
+11732004, 12002073, 12002077, 12272075, 11922204), the National Key Research and 
+Development Plan (2020YFB1709401), Dalian Talent Innovation Program (2020RQ099), 
+the Fundamental Research Funds for the Central Universities (DUT20RC(3)020, 
+DUT21RC(3)076), and 111 Project (B14013) are gratefully acknowledged. 
+Author contributions 
+X. G. and Z. D. conceived the idea and initiated the project. J. L. and Z. D. established the 
+theory. J. L and X. D. performed the numerical calculations and simulations. All the other 
+authors contributed to the discussions of the results and the manuscript preparation.  
+Declaration of competing interest 
+There are no conflicts to declare. 
+Data availability 
+Data will be made available on request. 
+ 
+
+Appendix 
+Appendix A: Modification of the fractional corner charge invariant 
+According to the results in literature30, the fractional corner charge invariant of the ������������3-
+symmetric hexagonal unit cells is 
+������������������������
+′(3) = 1
+3 �K������������≠1
+(3) � mod 1
+(A. 1) 
+where subscript ������������ equals 2 or 3 depending on the symmetry of the constructed supercell. 
+Due to the TRS and ������������3 symmetry, some two-order degenerate states are protected at the à 
+point, such as states from the linear combination of the Γ2
+(3) and Γ3
+(3), and they are 
+computationally expensive to identify clearly, especially for the unpaired degenerate 
+states6,9,10,25,28. Instead, we termed the invariant with the number of the two-order 
+degenerate states #Γ(3). To be specific, the topological character of the ������������3-symmetric 
+hexagonal unit cell is given by  
+������������‾(3) = �#Γ(3), #K2
+(3), #K3
+(3)�
+(A. 2) 
+Considering Eq. (A.2), the modified fractional corner charge invariants and the symmetry 
+behaviors are listed in Table A.1 for some possible cases.  
+Table A.1. The symmetry behaviors of the ������������3-symmetric unit cells with TRS  
+(for the fractional corner charge invariants) 
+������������������������=2
+(3)  
+������������������������=3
+(3)  
+#K2
+(3) 
+#K3
+(3) 
+#Γ(3) 
+#Γ2
+(3) 
+#Γ2
+(3) 
+1/3 
+0 
+1 
+0 
+0 
+0 
+0 
+0 
+1/3 
+0 
+1 
+0 
+0 
+0 
+0 
+2/3 
+1 
+0 
+2 
+1 
+1 
+0 
+0 
+1 
+1 
+2 
+1 
+1 
+0 
+0 
+1 
+0 
+1 
+1 
+0 
+0 
+0 
+0 
+1 
+1 
+0 
+1 
+0 
+0 
+1 
+0 
+1 
+0 
+1 
+0 
+0 
+0 
+1 
+1 
+1 
+0 
+In Table A.1, the red colored invariants ������������������������
+(3)  are modified from Eq. (A.2). This 
+modification is derived from the fact that the degenerate states #Γ(3) always appear in a 
+
+pair; or not, it is gapless 6,9,10,25,28. For the unpaired case, it is ambiguous to be tagged as 
+Γ2
+(3) or Γ3
+(3), hence the invariants ������������������������
+(3) in the last four cases cannot be solely identified by 
+the ������������‾(3) in Eq. (A.2). Therefore, when #Γ(3) is odd, the corresponding fractional corner 
+charge should be modified into zero with gapless band reality. When #Γ(3) is even, number 
+#Γ(3) can be equivalently divided as: #Γ2
+(3) = #Γ3
+(3) = #Γ(3)/2. 
+Thus, the modified fractional corner charge invariant gives 
+������������(3) = �1
+3 �#K������������≠1
+(3) − 1
+2 #Γ(3)� mod 1� × ��#Γ(3) + 1�mod 2�
+(A. 3) 
+where the red module term aims to avoid the unpaired two-order degenerate states. 
+Appendix B: Modification of the pseudo-spin invariant 
+For the ������������3 or ������������6-symmetric unit cells with the TRS, an alternative approach to get the 
+pseudo-spin invariants is to trace the broken (double) Dirac cones at the K or Γ point 
+9,10,25,38,40,42–44. Thus, their topological characters are given by 
+������������‾(3) = �#K2
+(3), #K3
+(3)�
+������������‾(6) = �#Γ1
+(2), #Γ2
+(2), #Γ(3)�
+(B. 1) 
+Here, #Γ(3) counts the two-order degenerate states of the Γ point in the ������������3 operator. The 
+modified pseudo-spin invariants and their symmetry behaviors are listed in Table B.1 and 
+Table B.2 for some possible cases.  
+Different as scaling a function5,59, the operation of a symmetry operator ������������� on a vector 
+function ������������(������������) transforms as �������������������������(������������) = �������������������������(�������������−1������������), where ������������� is the rotational operator in �������������. 
+For the present symmetry groups (������������3 or ������������6) in our paper, all group elements behave as a 
+rotation around ������������ -axis, and the transformation can be simplified as �������������������������(������������) =
+�������������������������T(�������������−1������������) + �������������������������L(�������������−1������������) , 
+where ������������ = ������������T + ������������L = (������������, ������������, 0)⊤ + (0,0, ������������)⊤  is 
+the 
+displacement vector in mechanics. This decomposed equation implies ������������T and ������������L hold the 
+same symmetry, except for the singular cases with displacement component ������������T = ������������ or 
+������������L = ������������.   
+ 
+ 
+ 
+
+Table B.1. The symmetry behaviors of the ������������3-symmetric unit cells with TRS 
+(for the pseudo-spin invariants) 
+������������(3) 
+#K2
+(3) 
+#K3
+(3) 
+1 
+1 
+0 
+1 
+2 
+0 
+-1 
+0 
+1 
+0 
+1 
+1 
+Table B.2. The symmetry behaviors of the ������������6-symmetric unit cells with TRS 
+(for the pseudo-spin invariants) 
+������������(6) 
+#Γ1
+(2) 
+#Γ2
+(2) 
+#Γ(3) 
+Orbits 
+-1 
+2 
+0 
+2 
+2d 
+1 
+0 
+2 
+2 
+2p 
+0 
+2 
+2 
+4 
+2p + 2d 
+0 
+1 
+0 
+0 
+1s 
+0 
+0 
+1 
+0 
+1f 
+0 
+1 
+2 
+2 
+2 s.p. 
+In the original SI theory29,30, the occupied bands counted in the SI method should be 
+isolated from others, and an alternative approach is to count all bands below the target band 
+gap. For the photonic and phononic systems, however, the first two or three bands always 
+converge to plane waves when |������������| → 0, where transverse modes produce two singularities 
+with ������������L = ������������5,59. In Table B.2, the red-colored data reveal the symmetry behaviors of the 
+first three bands that always cross through the singularities around the zero energy. Hence, 
+we defined the modified pseudo-spin invariants to overcount those singularities as 
+������������(3) = sgn�#K2
+(3) − #K3
+(3)�
+������������(6) = sgn�#Γp
+(6) − #Γd
+(6) − 2�
+(B. 2) 
+where the red term is the modification from the singularities. For the case of photonics, the 
+Eq. (B.2) should be further modified as the work5. And the terms #Γp
+(6) and #Γd
+(6) count 
+the p and d states at the Γ point.  
+
+Table C.1. Some typical optimized unit cells for the ������������3 and the ������������6-symmetric TMs.  
+(Here, the red data refers to the examples presented in our paper) 
+������������������������ 
+������������{������������} 
+������������{������������} 
+������������{deg} 
+������������{������������} �������������(������������), ������������(������������)� ������������max
+(������������)
+∼ ������������min
+(������������+1) 
+������������3 (0.7217,0.95,0.0275) 
+(0.3382,0.5073,0.1691) 
+(108,18,126) 
+0.55 
+(2/3, 1) 
+0.741~1.069 
+ 
+(0.4041,1.00,0.0900) 
+(0.3082,0.3082,0.2568) 
+(144,72,90) 
+0.45 
+(2/3, -1) 
+0.772~1.000 
+������������6 (0.9238,0.90,0.3250) 
+(0.3682,0.2455,0.3068) 
+(90,108,0) 
+0.65 
+(1/2, 1) 
+1.344~1.489 
+ 
+(0.8776,0.90,0.1750) 
+(0.2155,0.3232,0.2693) 
+(90, 144, 72) 
+0.50 
+(0, -1) 
+1.332~1.450 
+ 
+(0.9584,0.80,0.1800) 
+(0.1766,0.4121,0.2355) 
+(126,126,72) 
+0.60 
+(1/2, 1) 
+1.254~1.319 
+ 
+(0.8603,0.95,0.1575) 
+(0.2055,0.3596,0.3082) 
+(108,144,90) 
+0.45 
+(0, -1) 
+1.249~1.344 
+ 
+ 
+Fig. C.1. Some optimized unit cells with the ������������3 and ������������6 symmetries. The order refers to 
+the row number of Table C.1. 
+Appendix C: The setup of the optimization and the GA solver 
+For the parameters of material and optimization solver in our paper, the setup gives: the 
+basic medium is epoxy (EP) with the elastic modulus ������������0 = 4.35GPa, the Poisson’s ratio 
+������������0 = 0.37, and the mass density ������������0 = 1180kg ⋅ m−3. The scattering medium is steel (Fe) 
+with the elastic modulus ������������ = 200GPa, the Poisson’s ratio ������������ = 0.2, and the mass density 
+������������ = 7800kg ⋅ m−3. The genetic algorithm (GA) solver is set as: the population size of 100, 
+the crossover fraction of 0.9, the migration fraction of 0.3, the elite size of 5, the 
+objectivation tolerance of 1e-5, the stall generation limit of 15. The lattice constant is ������������ =
+|������������| = 1m. 
+
+3
+5
+2Table C.1 and Fig. C.1 list some typical optimized unit cells. The row number in Table 
+C.1 is consistent with the order of unit cells in Fig. C.1. For the examples in the main text, 
+we set their optimization procedure as 
+• 
+For the ������������3-symmetric helical MTIs, the broken Dirac cone appears at the K point. 
+The first unit cell is optimized with setting ������������ = 6, nonzero topological invariants 
+(2/3,1) and no specific ������������ref, which will auto-update as the mid-frequency of the 
+target gap.  
+• 
+For the ������������6-symmetric helical MTIs, the broken Dirac cone appears at the à point. 
+The first unit cell is optimized with setting ������������ = 7, �������������ref
+(������������), ������������ref
+(������������)� = (1/2,1), and 
+������������ref = 1.4. Then the MTI partner is optimized with setting ������������ = 7, �������������ref
+(������������), ������������ref
+(������������)� =
+(0, −1), and ������������ref = 1.4.  
+Appendix D: The supercell’s setup for the edge and the corner states 
+The setups for two example supercells are detailed as 
+• 
+For the ������������3-symmetric unit cells in the main text, the script of the truncated supercell 
+is shown in Fig. D.1, where it provides an approach to adjust the frequency of the 
+edge states. This truncation does not break the crystalline symmetry, and the 
+topological edge states would not vanish. In Fig. D.1(a) and D.1(b), the truncation 
+is set as ������������ = 1/4 × 2������������/√3 , and the edge gap is between 0.872-1.001. In Fig. D.1(c) 
+and (d), the eigenvalue spectrum and the crossing waveguide are simulated with the 
+truncation ������������ = 1 × 2������������/√3. The light-blue areas in Fig. D.1 refer to the supercell’s 
+structures in the main text. 
+• 
+For the ������������6-symmetric unit cell, the inner interface between the unit cell pairs can 
+be alternatively truncated as Fig. D.2. In Fig. D.2 (a) and D.2(b), the gap of the 
+edge states is found between 1.398-1.425 with the truncation ������������ = 1/2 × ������������. The 
+supercell’s script for the eigenvalue spectrum is displayed in Fig. D.2(c) with the 
+truncation ������������ = 1 × 2������������/√3 . In Fig. D.2(c), except for the inner hexagonal interface, 
+six base medium cylinders with a diameter of 0.4������������  are added to adjust the 
+supercell’s corners. The light-blue areas in Fig. D.2 refer to the supercell’s 
+structures in the main text. 
+
+ 
+ 
+(a) 
+(b) 
+ 
+ 
+(c) 
+(d) 
+Fig. D.1. The supercell’s scripts of the ������������3-symmetric unit cell. (a) The band of the edge 
+state (light-grey band belongs to the lower interface counterpart), and (b) the script of the 
+ribbon-shaped supercell in (a). The supercell’s script (c) for the eigenvalue spectrum and 
+(d) for the simulation of the crossing waveguide.  
+ 
+ 
+ 
+(a) 
+(b) 
+(c) 
+Fig. D.2. The supercell’s scripts for the ������������6-symmetric unit cell. (a) The band of the edge 
+states, (b) the script of the ribbon-shaped supercell in (a), and (c) the supercell’s script for 
+the simulation of the crossing waveguide.  
+
+Edgestates
+1.1
+(2A/2πc)
+0.9
+Freq
+0.8
+0.7
+X12A
+8A
+T8A
+7AEdgestates
+1.5
+Freq (2A/2πc)
+1.45
+1.4
+1.35
+X12A
+0.4A
+8AReference 
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+  
+

7tAyT4oBgHgl3EQfpvj9/content/tmp_files/2301.00533v1.pdf.txt

@@ -0,0 +1,2582 @@
+Analytical comparison between X(3) and X(5) models of the Bohr
+Hamiltonian
+K.R. Ajulo1; K.J. Oyewumi2
+1,2University of Ilorin, Ilorin, Nigeria.
+Abstract
+Via the inverse square potential, the solutions of the X(3) model which is a γ-rigid form of
+the X(5) critical point symmetry have been achieved. The paper presents X(3), through the
+variational technique, as another “window” through which the “pictures” of X(5) −→ SU(3)
+symmetry region can be seen. The analytical solutions of the X(3) are compared with the
+solutions of the X(5) model. Some new and unique equations connecting the two models in
+the: critical order, energy bands, spectra ratios, RL/2, and B(E2) transitional probabilities are
+presented. These equations should hold in other potentials with one-parameter such as Kratzer
+potential, Davidson potential etc. The spectra ratios and the B(E2) transitional probabilities
+are optimized via the optimization procedure. The experimental data of some selected isotopes
+are placed accordingly for the theoretical predictions. The deviations from the experiments are
+found to be quite small.
+Keywords:
+Bohr Hamiltonian, X(3), X(5), variation technique, optimization procedure, β-
+variable, γ-rigid.
+1
+Introduction
+X(3) which has been presented in [1-4] is said to be an exactly separable γ-rigid form of the X(5)
+critical point symmetry [5]. The X(3) model is defined by the collective coordinate β and two
+Euler angles since the γ is assumed to be zero unlike the case of X(5), where γ is varied around
+γ0 = 0 value in the harmonic oscillator potential [5]. This implies that, only three variables: β and
+θi are involved in the X(3) model. An exact separation of the β variable from the Euler angles is
+quite easily achievable. In the Bohr Hamiltonian model [6-9], X(5) critical point symmetry is one
+of the two critical point symmetries: it is a phase transition of the first order shape, which were
+originally proposed in the works of Iachello [10], while E(5) is the phase transition of the second
+order shape [10]. In the present work, the nuclei are taken to be γ-rigid, with the axially symmetric
+prolate shape obtained at γ0 = 0. The work presents the usefulness of a one-sided bound inverse
+square potential with one parameter. The one-parameter inverse square potential chosen is of the
+form
+V (β) =
+�
+�
+�
+�
+�
+β0
+β2 , if 0 ≤ β ≤ β0,
+∞, if β > β0,
+(1)
+where β0 is a variation parameter that changes the signatures of the nuclei, as it changes. It
+is expected that the solutions should shift forward as the β0 shifts forward and solutions should
+shift backward as the β0 shifts backward. A typical inverse square potential is bound on the left
+and unbound on the right, and it has a minimum at some positive values of β0 that forces the
+particles to infinity as β0 → 0. As a result, the particle’s energy states is one-sided, with energies
+escaping through the unbound side. The work is structured as follows: Section 2. presents the
+methodology and the solutions of X(3) model via inverse square potential. These solutions are:
+the wave functions, the normalization constants and the energy eigenvalues. The B(E2) transition
+rates are presented in Section 3. The analytical results, the numerical results, their applications in
+certain isotopes are presented and discussed in Section 4. The work is concluded and summarized
+in the Section 5.
+1E-Mail: 19-68eo001@students.unilorin.edu.ng
+2E-Mail: kjoyewumi66@unilorin.edu.ng
+1
+arXiv:2301.00533v1  [nucl-th]  2 Jan 2023
+
+2
+Methodology of X(3) model with the inverse square potential
+In the X(3) model, the Bohr Hamiltonian operator is written as [1,2]
+ˆH = − ℏ2
+2B
+�
+1
+β2
+∂
+∂β β2 ∂
+∂β +
+1
+3β2
+�
+1
+sin θ
+∂
+∂θ sin θ ∂
+∂θ +
+1
+sin2 θ
+∂2
+∂φ2
+�
+− V (β)
+�
+,
+(2)
+where the term,
+1
+sin θ
+∂
+∂θ sin θ ∂
+∂θ +
+1
+sin2 θ
+∂2
+∂φ2 ,
+(3)
+inside the bracket, represents the angular part of the Laplacian [1,2] . B, β and V (β) are respec-
+tively the mass parameter, the collective coordinate and the β-dependent potential. The wave
+equation of the Eq.(2) is
+ˆHΨ(β, θ, φ) = EΨ(β, θ, φ).
+(4)
+By the usual method of separation of variable employed in some quantum texts,
+Ψ(β, θ, φ) = χ(β)YL,M(θ, φ),
+(5)
+where YL,M(θ, φ) is the spherical harmonics and χ(β) is the radial part of Eq.(4). The separated
+angular part obtained reads [1,2]
+−
+�
+1
+sin θ
+∂
+∂θ sin θ ∂
+∂θ +
+1
+sin2 θ
+∂2
+∂φ2
+�
+YL,M(θ, φ) = L(L + 1)YL,M(θ, φ),
+(6)
+where L is the angular momentum quantum number.
+The simplified form of the radial part
+equation [1,2] ,
+� 1
+β2
+d
+dβ β2 d
+dβ − L(L + 1)
+3β2
++ 2B
+ℏ2 [E − V (β)]
+�
+χ(β) = 0
+(7)
+reads
+d2
+dβ2 χ(β) + 2
+β
+d
+dβ χ(β) − L(L + 1)
+3β2
+χ(β) − [v(β) − ϵ]χ(β) = 0,
+(8)
+where ϵ = 2B
+ℏ2 E and v(β) = 2B
+ℏ2 V (β) are the reduced energy and reduced potential respectively
+[5].
+2.1
+Determination of the wave functions
+By substituting Eq.(1) for v(β) in Eq.(8) and solving the simplified equation using MAPLE soft-
+ware, the eigenfunctions obtained read
+χs,ν,L,(β) = β−1/2 �C1,LJν(√ϵβ) + C2,LYν(√ϵβ)
+� ,
+(9)
+where C1,L and C2,L are the normalization constants associated with the Bessel functions of the first
+kind, Jν, and second kind, Yν, respectively. In the domain of Eq.(1), the critical order associated
+with the X(3) model in Eq.(9) is
+νX(3) =
+�
+L
+3 (L + 1) + β0 + 1
+4.
+(10)
+If a boundary condition χs,ν,L(β0) = 0 is considered, then C2,L,nYν(√ϵβ) vanishes and the wave
+functions become
+χs,ν,L(β) = β−1/2 �C1,LJν(√ϵβ)
+� .
+(11)
+2
+
+2.2
+Determination of the energy eigenvalues and the spectral ratio
+The procedure for finding the eigenvalues is written in ref. [11]. If the first condition of the listed
+procedure is considered, then the acceptable expression for the energy eigenvalues is written as:
+Es,L,nβ = ℏ2
+2B k2
+s,ν,nβ,
+k2
+s,ν,nβ = ϵs,ν,nβ,
+k2
+s,ν,nβ = xnβ,s,ν
+β0
+,
+(12)
+where s = nβ +1, xs,ν,nβ is the s-th zeros of the Bessel function of order ν. The energy eigenvalues
+of the β-part in ℏω = 1 unit reads:
+ϵs,L,nβ = 2nβ + 1 + νX(3) = 2nβ + 1 +
+�
+L
+3 (L + 1) + β0 + 1
+4 :
+nβ = 0, 1, 2, ...
+(13)
+For the X(3) model, the ground state energy levels are defined with s = 1, the quasi-β1 levels are
+defined with s = 2 and the quasi-β2 levels are defined with s = 3. L = 0, 2, 4, 6... There exist no
+γ-bands in the X(3) model because, γ0 = 0.
+Eq.(13) is the similar to the energy eigenvalues obtained in the β-part of X(5) model [12], the
+difference is observed in their critical orders where
+νX(5) =
+�
+L
+3 (L + 1) + β0 + 9
+4.
+(14)
+Since s = nβ + 1, ϵs,L,nβ can be reduced to ϵs,L, then the spectra ratios can be written as
+RL/2 = ϵs,L − ϵ1,0
+ϵ1,2 − ϵ1,0
+.
+(15)
+2.3
+Determination of the normalization constants and the complete wave func-
+tions
+The normalization condition for the Hamiltonian operator in Eq.(2) is written as [1,2]
+� β0
+0
+β2 | χs,ν,L,nβ(β) |2 dβ = 1,
+(16)
+such that
+| χs,ν,L,nβ(β) |2→ 0
+for
+β → 0;
+| χs,ν,L,nβ(β) |2 β2 → 0
+for
+β → ∞.
+(17)
+If these conditions are satisfied, then
+� β0
+0
+β2 | χs,ν,L,nβ(β) |2 dβ < β0.
+Using the identity [13-14]
+Jν(√ϵβ)Jν(√ϵβ) =
+∞
+�
+nβ=0
+�1
+2
+√ϵβ
+�2ν+2nβ
+(2ν + nβ + 1)nβ
+nβ![Γ(ν + nβ + 1)]2
+(18)
+in Eq.(16), the simplified normalization constants read
+C1,L,nβ =
+�
+����
+�
+nβ=0,1,2,3...
+(η)nβ
+�ks,ν,nβ
+2
+�ξ−2
+β(ξ)
+0
+nβ!
+ξ
+�
+Γ
+�ξ
+2
+��2
+�
+����
+−1/2
+,
+(19)
+where
+ξ = 2ν + 2nβ + 2,
+η = 2ν + nβ + 1
+and
+(η)nβ = η(η + 1)(η + 2)...(η + nβ − 1),
+(20)
+3
+
+with (η)0 = 1. Hence, Eq.(11) becomes
+χs,ν,L,nβ(β) =
+�
+����
+�
+nβ=0,1,2,3...
+(η)nβ
+�ks,ν,nβ
+2
+�ξ−2
+β(ξ)
+0
+nβ!
+ξ
+�
+Γ
+�ξ
+2
+��2
+�
+����
+−1/2
+β−1/2Jν(√ϵβ).
+(21)
+3
+B(E2) transition rates
+The electric quadrupole operator is written as [1,2]
+T E2
+µ
+= tβ
+�
+D(2)
+µ,0(θi) cos γ + 1
+√
+2
+�
+D(2)
+µ,2(θi) + D(2)
+µ,−2(θi)
+�
+sin γ
+�
+,
+(22)
+where D(θi) are the Wigner functions of the Euler angle and t is known as a scale factor. For
+γ0 = 0,
+T E2
+µ
+= tβ
+�4π
+5 Y2µ(θ, φ).
+(23)
+The B(E2) [1,2,5,15] is written as
+B(E2; sL −→ s′L′) =
+1
+2sL + 1|
+�
+s′L′||T E2||sL
+�
+|2,
+(24)
+= 2s′L′ + 1
+2sL + 1 B(E2; s′L′ −→ sL).
+(25)
+Eq.(24) or Eq.(25) has been solved in ref. [1] as:
+B(E2; sL −→ s′L′) = t2 �
+CL′0
+L0,20
+�2 I2
+sL;s′L′,
+(26)
+where the coefficients, CL′0
+L0,20 are the Clebsch-Gordan coefficients, and
+IsL;s′L′ =
+� β0
+0
+βχs,ν,L,nβ(β)χs′,ν′,L′,n′
+β(β)β2dβ,
+(27)
+are the integrals over β.
+4
+Numerical results, analytical results, applications and discus-
+sion
+Some important solutions for the collective model of Eq.(2) are the energy levels, the spectra ratios
+and the B(E2) transitions. Their theoretical predictions are important when energy spectra are
+assigned to the states for which experimental data are not available. The numerical calculations,
+the analytical comparisons and how the search for the experimental realizations of the model was
+achieved are discussed accordingly in this section.
+Both the X(3) and the X(5) have their critical orders, ν(L, β0), from their Bessel functions which
+describes their energy spectra. Firstly, in the comparison of the Eq.(10) and Eq.(14), it can be
+deduced from the numerical computation of ν, shown in Table 1., that
+νX(3)(β0 = c + 2) = νX(5)(β0 = c) :
+c = 0, 1, 2, ...
+(28)
+In both cases, it increases with increase in the angular momentum, L, and with increase in the
+variation parameter, β0. These effects of L and β0 in ν are also seen in the energy values of Eq.(13).
+4
+
+Table 1: The comparison in the critical order, ν, of the X(5) [12], with the ν of Eq.(10).
+ν(L)
+L
+β0 = 0
+β0 = 2
+β0 = 4
+β0 = 6
+β0 = 102
+β0 = 100
+X(3)
+X(5)
+X(3)
+X(5)
+X(3)
+X(5)
+X(3)
+X(5)
+X(3)
+X(5)
+0
+0.500
+1.500
+1.500
+2.062
+2.062
+2.500
+2.500
+2.062
+10.112
+10.112
+2
+1.500
+2.062
+2.062
+2.500
+2.500
+2.872
+2.872
+2.500
+10.210
+10.210
+4
+2.630
+2.986
+2.986
+3.304
+3.304
+3.594
+3.594
+3.304
+10.436
+10.436
+6
+3.775
+4.031
+4.031
+4.272
+4.272
+4.500
+4.500
+4.272
+10.782
+10.782
+8
+4.924
+5.123
+5.123
+5.315
+5.315
+5.500
+5.500
+5.315
+11.236
+11.236
+10
+6.076
+6.238
+6.238
+6.397
+6.397
+6.551
+6.551
+6.397
+11.786
+11.786
+β0 = 1
+β0 = 3
+β0 = 5
+β0 = 7
+β0 = 101
+β0 = 103
+0
+1.118
+1.803
+1.803
+2.291
+2.291
+2.693
+2.693
+3.041
+10.062
+10.259
+2
+1.803
+2.291
+2.291
+2.693
+2.693
+3.041
+3.041
+3.354
+10.161
+10.356
+4
+2.814
+3.149
+3.149
+3.452
+3.452
+3.731
+3.731
+3.990
+10.388
+10.579
+6
+3.905
+4.153
+4.153
+4.387
+4.387
+4.610
+4.610
+4.823
+10.735
+10.920
+8
+5.025
+5.220
+5.220
+5.408
+5.408
+5.590
+5.590
+5.766
+11.191
+11.369
+10
+6.158
+6.318
+6.318
+6.474
+6.474
+6.627
+6.627
+6.776
+11.747
+11.913
+Figure 1: (a) Comparison in the energy levels of the X(3) and X(5) models [15] at β0 = 2 from the gsb up
+to the quasi-β2 band. (b): the variation of the critical order, ν, of the X(5) as a function of β0, is compared
+with ν of the X(3) at constant angular momenta, L = 0, 2 and L = 4.
+5
+
+16
+5
+βo= 2
+14
+4.5
+12
+X(3)
+B2
+4 
+L=4
+3.5
+L=4
+L=0
+rgy
+10
+X(3)
+X(5)
+L=2
+0X(5)
+3
+8
+β2
+X(3)
+X(5)
+X(5)
+-β1
+V 2.5
+L=0
+6
+X(5)
+β1
+gsb
+2
+4
+X(5)
+X(3)
+gsb
+1.5
+2
+X(3)
+0
+0.5
+0
+2
+4
+6
+8
+10
+12
+14
+0
+(a)
+7
+(b)
+0
+2
+4
+8
+10Table 2: Ground state energies, the energies of the quasi-β1 and the quasi-β2 denoted by nβ =
+0, s = 1; nβ = 1, s = 2; and nβ = 2, s = 3 respectively for the X(3) and X(5) symmetry [12] in
+ℏω = 1 unit.
+β0 = 2
+β0 = 3
+β0 = 4
+β0 = 15
+L
+nβ = 0;
+s = 1
+X(3)
+X(5)
+X(3)
+X(5)
+X(3)
+X(5)
+X(3)
+X(5)
+0
+2.500
+2.031
+2.803
+2.146
+3.062
+2.250
+4.905
+3.077
+2
+3.062
+2.250
+3.291
+2.346
+3.500
+2.436
+5.153
+3.194
+4
+3.986
+2.652
+4.149
+2.726
+4.304
+2.797
+5.682
+3.445
+6
+5.031
+3.136
+5.153
+3.194
+5.272
+3.250
+6.408
+3.795
+8
+6.123
+3.658
+6.220
+3.704
+6.315
+3.750
+7.265
+4.212
+10
+7.238
+4.198
+7.318
+4.237
+7.397
+4.276
+8.205
+4.671
+12
+8.365
+4.750
+8.433
+4.783
+8.500
+4.816
+9.201
+5.161
+14
+9.500
+5.308
+9.559
+5.337
+9.617
+5.366
+10.233
+5.670
+nβ = 1;
+s = 2
+0
+4.500
+4.031
+4.803
+4.146
+5.062
+4.250
+6.905
+5.077
+2
+5.062
+4.250
+5.291
+4.346
+5.500
+4.436
+7.153
+5.194
+4
+5.986
+4.652
+6.149
+4.726
+6.304
+4.797
+7.682
+5.445
+6
+7.031
+5.136
+7.153
+5.194
+7.272
+5.250
+8.408
+5.795
+8
+8.123
+5.658
+8.220
+5.704
+8.315
+5.750
+9.265
+6.211
+10
+9.238
+6.198
+9.318
+6.237
+9.397
+6.276
+10.205
+6.671
+12
+10.365
+6.750
+10.433
+6.783
+10.500
+6.816
+11.201
+7.161
+14
+11.500
+7.308
+11.559
+7.337
+11.617
+7.366
+12.233
+7.670
+nβ = 2;
+s = 3
+0
+6.500
+6.031
+6.803
+6.146
+7.062
+6.250
+8.905
+7.077
+2
+7.062
+6.250
+7.291
+6.346
+7.500
+6.436
+9.153
+7.194
+4
+7.986
+6.652
+8.149
+6.726
+8.304
+6.797
+9.682
+7.445
+6
+9.031
+7.136
+9.153
+7.194
+9.272
+7.250
+10.408
+7.795
+8
+10.123
+7.658
+10.220
+7.704
+10.315
+7.750
+11.265
+8.211
+10
+11.238
+8.198
+11.318
+8.237
+11.397
+8.276
+12.205
+8.671
+12
+12.365
+8.750
+12.433
+8.783
+12.500
+8.816
+13.201
+9.161
+14
+13.500
+9.308
+13.559
+9.337
+13.617
+9.366
+14.233
+9.670
+6
+
+Figure 2: (a) The plots showing the values of β0 at which energies are minimum. (b) The rate of energy
+with respect to β0, showing non stationary property of β0.
+Table 3: Comparison of the ground state spectra ratios, defined in Eq.(15), of the inverse square
+potential in the X(3) model at different values of the β0, compared with the X(5) [12]. It can be
+seen that X(3)(β0 = ∞) ≈ X(5)(β0 = ∞).
+Ls,nβ
+β0 = 0
+β0 = 0
+β0 = 2
+β0 = 2
+β0 = 4
+β0 = 4
+β0 = ∞
+β0 = ∞
+X(3)
+X(5)
+X(3)
+X(5)
+X(3)
+X(5)
+X(3)
+X(5)
+gsb
+01,0
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+21,0
+1.000
+1.000
+1.000
+1.000
+1.000
+1.000
+1.000
+1.000
+41,0
+2.130
+2.646
+2.646
+2.834
+2.834
+2.938
+3.296
+3.296
+61,0
+3.275
+4.507
+4.507
+5.042
+5.042
+5.372
+6.806
+6.808
+81,0
+4.424
+6.453
+6.453
+7.421
+7.421
+8.508
+11.413
+11.423
+101,0
+5.576
+8.438
+8.438
+9.887
+9.887
+11.881
+16.991
+17.013
+121,0
+6.728
+10.445
+10.445
+12.404
+12.404
+15.686
+23.409
+23.450
+141,0
+7.882
+12.465
+12.465
+14.951
+14.951
+19.740
+30.544
+30.611
+β0 = 1
+β0 = 1
+β0 = 3
+β0 = 3
+β0 = 5
+β0 = 5
+β0 = 15
+β0 = 15
+01,0
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+21,0
+1.000
+1.000
+1.000
+1.000
+1.000
+1.000
+1.000
+1.000
+41,0
+2.476
+2.756
+2.756
+2.893
+2.893
+2.946
+3.128
+3.148
+61,0
+4.070
+4.812
+4.812
+5.224
+5.224
+5.529
+6.058
+6.136
+81,0
+5.706
+6.995
+6.995
+7.767
+7.767
+8.638
+9.508
+9.690
+101,0
+7.360
+9.243
+9.243
+10.424
+10.424
+11.915
+13.297
+13.620
+121,0
+9.024
+11.525
+11.525
+13.145
+13.145
+15.854
+17.307
+17.800
+141,0
+10.694
+13.829
+13.829
+15.907
+15.907
+19.899
+21.468
+22.152
+7
+
+L=2 
+L=4 :
+6
+L=2 
+L=4 
+6
+Po
+10
+15
+0
+5
+10
+15
+-0.005-
+-1.2 -
+-0.010-
+-1.4卡
+8v
+8v-1.6
+-0.015
+log
+SPo
+-1.8 -
+-0.020-
+-2 -
+-0.025
+(a)
+[b]
+-2.2-Table 4: RL/2 ratios, defined in Eq.(15), of the quasi-β1 and quasi-β2 bands of the inverse square
+potential in the X(3) model at different values of the β0.
+Ls,nβ
+β0 = 0
+β0 = 1
+β0 = 2
+β0 = 3
+β0 = 4
+β0 = 15
+β0 = ∞
+quasi-β1
+02,1
+2.000
+2.921
+3.562
+4.094
+4.562
+8.058
+20.124
+22,1
+3.000
+3.921
+4.562
+5.094
+5.562
+9.058
+21.124
+42,1
+4.130
+5.397
+6.208
+6.850
+7.395
+11.187
+23.420
+62,1
+5.275
+6.991
+8.069
+8.906
+9.603
+14.115
+26.929
+82,1
+6.424
+8.626
+10.014
+11.090
+11.982
+17.567
+31.537
+102,1
+7.576
+10.281
+11.999
+13.337
+14.449
+21.356
+37.116
+122,1
+8.728
+11.945
+14.007
+15.619
+16.965
+25.366
+43.534
+142,1
+9.882
+13.615
+16.027
+17.923
+19.513
+29.526
+50.668
+quasi-β2
+03,2
+4.000
+5.842
+7.123
+8.188
+9.123
+16.117
+40.249
+23,2
+5.000
+6.842
+8.123
+9.188
+10.123
+17.117
+41.249
+43,2
+6.130
+8.318
+9.769
+10.944
+11.957
+19.245
+43.545
+63,2
+7.275
+9.912
+11.630
+13.000
+14.165
+22.174
+47.054
+83,2
+8.424
+11.547
+13.576
+15.184
+16.544
+25.625
+51.662
+103,2
+9.576
+13.201
+15.561
+17.431
+19.010
+29.414
+57.240
+123,2
+10.728
+14.866
+17.568
+19.713
+21.527
+33.424
+63.658
+143,2
+11.882
+16.536
+19.589
+22.018
+24.074
+37.584
+70.792
+Figure 3: The ϵs,L
+ϵ1,2
+of the X(3). (b): the ϵs,L
+ϵ1,2
+of the X(5) both at constant angular momenta,
+L = 0...10 are plotted against the variation parameter, β0.
+8
+
+1.8
+X(3)
+1.20-
+x(5)
+1.6
+L=10.
+1.15-
+1.4-
+L=8
+es,L1.10
+8
+1,2 1.2
+E1,2
+=4
+1.05-
+1.0 -
+L=2
+O
+0.8-
+1.00
+=2
+:0
+6
+10
+2
+3
+6
+o
+9
+(a)
+10
+Po
+(b)
+PoFigure 4: (a) The comparison in the R4/2 of the X(3) and X(5) for β0 = ∞ and for different values of
+the β0,max labelled as X(3)−var and X(5)−var respectively, peculiar to each angular momentum. (b): the
+comparison in the R0/2 of the X(3) and X(5) for β0 = ∞ and for different values of the β0,max labelled as
+X(3)−var and X(5)−var respectively, peculiar to each angular momentum.
+Figure 5: (a) and (b) The visual plots of the potentials correspond to R4/2 and R0/2 respectively.
+The values of β0 used correspond to X(3)-var and X(5)-var in the gsb and quasi-β1 bands.
+9
+
+70
+35
++X(3)-var
+60
+30
+X(5)-var
+x(3)-var
+50
+25
+(gsb)
++x(5)-var
+X(3)- βo = 00
+20
+40
+X(3)-βo = 00
+R
+R
+15
+→X(5)-Po = 00
+30
+10
+20
+5
+10
+0
+0
+0
+4
+8
+10
+12
+14
+16
+0
+2
+4
+6
+8
+10
+12
+14
+2
+6
+16
+7
+(b)
+(a)
+7X (3)
+x(5)
+X (3) 
+ X (5)
+F00S
+gsb
+1-005
+β1
+250-
+00
+200
+300-
+V(β) 150 
+V(β)
+200
+100-
+100-
+50-
+(a)
+0.2
+03
+(q)
+0.1
+to
+0.5
+0.1
+0.2
+to
+0.5
+β
+βFigure 6: (a) and (b) present the RL/2 ratios for the ground state and the quasi-β1 bands of the
+X(3) model of inverse square potential respectively, at different values of β0 compared with X(3)-
+IW and 162Dy. (c): the RL/2 ratios for the quasi-β2 bands of the X(3) model of inverse square
+potential at different values of β0 compared with X(3)-IW [1]. It appears that the gsb solutions of
+X(3) at β0 = ∞ lie on the experimental data of 162Dy, which is a typical SU(3) candidate. The
+available data on the first exited state lie very close to one another.
+Figure 7: (a) and (b) present the RL/2 ratios for the ground state and the quasi-β1 bands of the
+X(3) and the X(5) models of inverse square potentials respectively, obtained at different values of
+β0,max, labeled X(3)-var and X(5)-var, are compared with the 172−180Os chain.
+10
+
+55
+35
+50
+O βo= 0
+×βo = 2
+O-βo= 0
+30
+X-βo= 2
+45
+→βo = 3
+→βo = 15
+→βo= 3
+βo= 15
+162 DY
+40
+-X(3)-IW
+10-βg=8
+25
+_162 Dy
+35
+X(3)-IW
+(qs)
+30
+20
+RL/2
+25
+RL/2
+15
+20
+15
+10
+10
+5
+5
+(b) 
+0
+(a) 0
+0
+2
+4
+6
+8
+10
+12
+14
+0
+2
+4
+8
+10
+12
+14
+6
+75
+70
+O-βo= 0
+*-βo= 2
+65
+βo= 3
+60
+-βo= 15
+55
+βo= 8
+-X(3)-IW
+45
+40
+z/7
+35
+R
+30
+25
+20
+15
+10
+5
+(c)
+0
+0
+2
+4
+6
+10
+12
+14
+825
+30
+72
+1760s
+172
+1760s
+25
+20
+180
+Os
+1780s
+b
+Os
+20
+β
+←x(3)-var
+O-X(5)-var
+L/2
+-x(3)-var
+-x(5)-var
+10
+R
+R
+10
+5
+(a)
+(b) 0
+0
+6
+8
+10
+12
+14
+0
+2
+6
+8
+10
+LFigure 8: The Neutron-β0 distribution is employed to show the relative positions of 104−108Ru,
+120−126Xe, 184−188Pt and 172−180Os along their common chain.
+Figure 9: The B(E2) transition rates of the X(3) normalized to the B(E2 : 21,0 → 01,0) = 100
+units within: (a) the ground state bands at β0 = 0, 1, 2, ∞ and B(E2)-var compared with the
+X(3)-IW [1], X(5) experimental data [34] and 158Gd [35], which is a typical SU(3) candidate.
+(b): the β1 state bands at β0 = 0, 1, 2, ∞ and B(E2)-var compared with the X(3)-IW [1] and
+158Gd. (c): the β2 state bands at β0 = 0, 1, 2, ∞ and B(E2)-var compared with the X(3)-IW [1].
+[Note:-IW denotes infinite well potential.]
+11
+
+120
+188Pt
+110
+186 Pt
+184Pt
+100
+180Os
+178 0s
+1760s
+90
+N
+126
+80
+Xe
+124 Xe
+122 Xe
+70
+108Ru
+120
+Xe
+60
+104Ru
+106Ru
+50
+0
+2
+4
+6
+8
+10
+12
+14
+βo1200
+1200
+-βo= 0
+→βo = 1
+-βo= 0
+→βo= 1
+→βo = 2
+×βo=8
+1000
+1000
+→βo = 2
+￥β=8
+-* B(E2)-var
+X(3)-IW
+米一
+B(E2)-var
+- X(3)-IW
+800
+800
+ X(5)-Exp
+600
+600
+400
+2
+2
+400
+E
+E
+200
+B
+200
+(b)
+(a)
+0
+2
+4
+6
+8
+10
+12
+0
+0
+2
+4
+6
+8
+10
+12
+1400
+βo= 0
+←βo= 1
+1200
+→ β = 2
+￥β= 8
+1000
+* B(E2)-var
+-0-. X(3)-IW
+800
+600
+2
+400
+200
+B
+(c)
+0
+0
+2
+4
+6
+8
+10
+12Table 5: The RL/2 ratios, defined in Eq.(15), for the X(3) version of inverse square potential,
+labelled X(3)-var, calculated at different values of β0,max, are compared with the X(3)-IW [1].
+[Note: IW denotes infinite well potential].
+Ls,nβ
+β0,max
+X(3)-var
+X(3)-IW
+gsb
+01,0
+β0
+0.000
+0.000
+21,0
+β0
+1.000
+1.000
+41,0
+0.844
+2.440
+2.440
+61,0
+1.576
+4.244
+4.230
+81,0
+2.033
+6.383
+6.350
+101,0
+2.143
+8.666
+8.780
+121,0
+2.695
+11.421
+11.520
+141,0
+3.643
+14.573
+14.570
+quasi-β1
+02,1
+0.815
+2.703
+2.870
+22,1
+2.101
+4.619
+4.830
+42,1
+3.729
+7.255
+7.370
+62,1
+5.213
+10.327
+10.290
+82,1
+6.098
+13.493
+13.570
+102,1
+6.855
+16.908
+17.180
+122,1
+8.106
+21.009
+21.140
+quasi-β2
+03,2
+2.524
+7.701
+7.650
+23,2
+4.497
+10.553
+10.560
+43,2
+6.523
+14.088
+14.190
+63,2
+8.567
+18.172
+18.220
+83,2
+10.438
+22.613
+22.620
+103,2
+11.932
+25.999
+-
+123,2
+13.011
+28.928
+-
+12
+
+Table 6: The spectra ratios for the X(3) version of inverse square potential are compared with
+the experimental data. The values of the β0 and the quality factor, σ, used during the fittings are
+recorded.
+Ls,nβ
+102Mo
+102Mo
+104Ru
+104Ru
+106Ru
+106Ru
+108Ru
+108Ru
+120Xe
+120Xe
+122Xe
+122Xe
+Exp
+Theor
+Exp
+Theor
+Exp
+Theor
+Exp
+Theor
+Exp
+Theor
+Exp
+gsb
+41,0
+2.510
+2.566
+2.480
+2.468
+2.660
+2.662
+2.750
+2.623
+2.470
+2.522
+2.500
+2.571
+61,0
+4.480
+4.483
+4.350
+4.299
+4.800
+4.805
+5.120
+5.102
+4.330
+4.263
+4.430
+4.500
+81,0
+6.810
+6.703
+6.480
+6.488
+7.310
+7.199
+8.020
+7.999
+6.510
+6.361
+6.690
+6.759
+101,0
+9.410
+8.989
+8.690
+8.702
+10.020
+9.920
+11.310
+11.495
+8.900
+8.497
+9.180
+9.216
+121,0
+-
+11.498
+-
+11.878
+-
+12.334
+-
+12.879
+-
+10.362
+-
+11.985
+141,0
+-
+13.302
+-
+14.522
+-
+15.073
+-
+15.591
+-
+12.640
+-
+14.759
+β1
+02,1
+2.350
+3.009
+-
+2.569
+3.670
+3.678
+-
+4.462
+2.820
+3.000
+3.470
+3.492
+22,1
+3.860
+4.301
+4.230
+4.233
+-
+5.127
+-
+5.902
+3.950
+4.203
+4.510
+4.608
+42,1
+-
+6.289
+5.810
+5.921
+-
+7.443
+-
+8.285
+5.310
+5.899
+-
+6.792
+62,1
+-
+8.691
+-
+8.009
+-
+10.001
+-
+11.099
+-
+7.958
+-
+9.172
+82,1
+-
+11.319
+-
+11.101
+-
+12.519
+-
+14.532
+-
+10.739
+-
+11.829
+β2
+03,2
+-
+7.437
+-
+6.589
+-
+8.388
+-
+10.099
+-
+6.664
+-
+7.722
+23,2
+-
+9.000
+-
+8.202
+-
+10.200
+-
+11.811
+-
+8.007
+-
+9.402
+43,2
+-
+11.009
+-
+10.341
+-
+12.049
+-
+12.972
+-
+10.229
+-
+11.538
+β0
+1.464
+0.963
+-
+2.121
+1.830
+1.221
+1.494
+σ
+0.569
+0.310
+-
+0.422
+0.276
+0.481
+0.295
+124Xe
+124Xe
+126Xe
+126Xe
+148Nd
+148Nd
+184Pt
+184Pt
+186Pt
+186Pt
+188Pt
+188Pt
+Exp
+Theor
+Exp
+Theor
+Exp
+Theor
+Exp
+Theor
+Exp
+Theor
+Exp
+Theor
+gsb
+41,0
+2.480
+2.498
+2.420
+2.463
+2.490
+2.500
+2.670
+2.721
+2.560
+2.567
+2.530
+2.890
+61,0
+4.370
+4.418
+4.210
+4.291
+4.240
+4.187
+4.900
+5.001
+4.580
+4.399
+4.460
+4.503
+81,0
+6.580
+6.596
+6.270
+6.325
+6.150
+6.007
+7.550
+7.679
+7.010
+6.768
+6.710
+6.666
+101,0
+8.960
+8.726
+8.640
+8.382
+8.190
+7.986
+10.470
+10.585
+9.700
+8.863
+9.180
+8.969
+121,0
+-
+11.998
+-
+10.479
+-
+9.769
+-
+12.807
+-
+11.999
+-
+11.378
+141,0
+-
+15.079
+-
+12.681
+-
+11.246
+-
+15.367
+-
+14.875
+-
+14.242
+β1
+02,1
+3.580
+3.402
+3.380
+3.201
+3.040
+3.082
+3.020
+3.546
+2.460
+2.798
+3.010
+3.209
+22,1
+4.600
+4.498
+4.320
+4.241
+3.880
+4.006
+5.180
+5.452
+4.170
+4.381
+4.200
+4.397
+42,1
+5.690
+6.289
+5.250
+5.862
+5.320
+5.589
+7.570
+7.943
+6.380
+6.599
+-
+6.583
+62,1
+-
+8.564
+-
+7.828
+7.120
+7.411
+11.040
+11.157
+8.360
+8.581
+-
+8.603
+82,1
+-
+11.900
+-
+10.062
+-
+9.752
+-
+14.601
+-
+12.007
+-
+12.100
+β2
+03,2
+-
+7.334
+-
+6.759
+-
+6.249
+-
+10.122
+-
+7.389
+-
+7.603
+23,2
+-
+8.888
+-
+8.009
+-
+7.442
+-
+11.900
+-
+8.942
+-
+9.162
+43,2
+-
+10.022
+-
+9.287
+-
+9.051
+-
+12.998
+-
+10.208
+-
+10.441
+β0
+1.101
+-
+0.949
+1.111
+2.639
+1.469
+4.950
+σ
+0.279
+-
+0.299
+0.347
+0.475
+0.600
+0.729
+13
+
+Table 7: The B(E2) transition rates of the X(3) model at β0 = 0, 1, 2, ∞ and its values obtained
+at β0,max peculiar to each angular momentum, normalized to the B(E2; 21,0 → 01,0) = 100 units
+are compared with the X(3)-IW model [1] and with the experimental data of X(5) [34]. [Note:
+-IW denotes infinite well potential.]
+L(i)
+s,nβ
+L(f)
+s,nβ
+β0 = 0
+β0 = 1
+β0 = 2
+β0 = ∞
+β(i)
+0,max → β(f)
+0,max
+B(E2) − var
+X(3)-IW
+176Os-Exp
+21,0
+01,0
+100.000
+100.000
+100.000
+100.000
+β0 → β0
+100.000
+100.00
+100.00
+41,0
+21,0
+237.513
+190.935
+178.005
+143.992
+0.844 → β0
+189.495
+189.90
+193.00
+61,0
+41,0
+380.702
+286.006
+270.996
+167.292
+1.576 → 0.844
+250.995
+248.90
+267.00
+81,0
+61,0
+523.695
+384.599
+369.090
+185.328
+2.033 → 1.576
+293.038
+291.40
+297.00
+101,0
+81,0
+667.003
+486.036
+469.991
+202.099
+2.143 → 2.033
+324.599
+323.80
+352.50
+121,0
+101,0
+810.954
+587.658
+559.744
+229.986
+2.695 → 2.143
+350.710
+349.50
+-
+141,0
+121,0
+954.746
+690.364
+671.484
+253.007
+3.643 → 2.695
+371.992
+370.70
+-
+22,1
+02,1
+166.813
+160.292
+152.428
+69.929
+2.011 → 0.815
+78.922
+80.60
+-
+42,1
+22,1
+320.619
+260.000
+243.186
+123.888
+3.729 → 2.101
+139.471
+140.10
+-
+62,1
+42,1
+470.001
+355.240
+343.376
+156.031
+5.213 → 3.729
+181.730
+182.40
+-
+82,1
+62,1
+617.075
+456.792
+439.962
+189.542
+6.098 → 5.213
+213.899
+215.50
+-
+102,1
+82,1
+763.927
+557.317
+542.129
+215.763
+6.855 → 6.098
+242.026
+242.40
+-
+122,1
+102,1
+899.983
+656.999
+639.677
+233.499
+8.106 → 6.855
+268.543
+265.10
+-
+142,1
+122,1
+1009.079
+759.642
+736.660
+251.643
+9.441 → 8.106
+281.320
+-
+-
+23,2
+03,2
+233.504
+221.942
+209.888
+56.684
+4.497 → 2.524
+72.090
+73.50
+-
+43,2
+23,2
+401.982
+327.461
+311.072
+82.831
+6.523 → 4.497
+118.990
+120.50
+-
+63,2
+43,2
+559.801
+422.880
+408.564
+116.085
+8.567 → 6.523
+154.892
+154.20
+-
+83,2
+63,2
+708.989
+523.436
+512.997
+139.859
+10.438 → 8.567
+183.019
+181.20
+-
+103,2
+83,2
+858.095
+624.555
+609.096
+166.646
+11.932 → 10.438
+202.222
+-
+-
+123,2
+103,2
+1003.933
+727.909
+715.990
+182.910
+13.011 → 11.932
+218.753
+-
+-
+143,2
+123,2
+1151.239
+832.003
+819.115
+202.421
+14.629 → 13.011
+229.986
+-
+-
+14
+
+Secondly, the exact relationship between the νX(3) and the νX(5) stated in Eq.(28) does not reflect
+in the exact comparison of their energy levels. That is, it can be inferred from the results that
+ϵX(3)(β0 = c + 2) ̸= ϵX(5)(β0 = c),
+(29)
+because the total energy of the X(5) contains the γ-part solutions. However, the relation
+ϵgs,L = 2 + ϵβ1,L = 4 + ϵβ2,L,
+(30)
+holds in all the levels for both X(3) and the β-part of X(5): this third remark is shown in the
+Table 2.
+Another significant remark is such that, the values of ν, for the case of X(5) at L = 2, correspond
+to those of X(3), at L = 0. This is shown in Table 1. and the visual comparison is shown with
+the lines in Figure 1(b). Analytically, the behaviour of the energies of the X(5) and the X(3)
+at constant value of variation parameter, β0, is shown in the Figure 1(a). The critical orders,
+ν(L, β0), of the X(5) and that of the X(3), which define their energy levels, are plotted against the
+variation parameter, β0, at constant angular momenta and shown in the Figure 1(b): it is shown,
+with the numerical values of ν, in the Table 1., that
+νX(5)(L = 0) = νX(3)(L = 2)
+∀
+β0.
+(31)
+The derivatives of ν with respect to the β0 are shown in Figures 2(a) and 2(b). The first and the
+second derivatives are carried out in order to show the stationary properties of β0 and the values
+of β0 at which the energy is minimum.
+The variation of the ratio ϵs,L
+ϵ1,2
+with respect to the variation parameter, β0, for both X(3) and
+X(5) are respectively shown in the Figures 3(a) and 3(b). For all values of β0, its values increase
+at L = 0, are constant at L = 2, that is ϵs,L
+ϵ1,2
+=1 and decrease at L > 2 .
+The ground state bands (gsb) are defined with s = 1;
+nβ = 0. The quasi-β1 bands and the
+quasi-β2 bands are defined by s = 2;
+nβ = 1 and s = 3;
+nβ = 2 respectively. The γ bands do
+not exist for X(3) model because, γ0 = 0. The increase in the angular momentum, L, at constant
+value of β0, increases the energies, in all energy levels. Also, at constant values of the angular
+momentum, the increase in the β0 increases the energy levels. The Table 2. shows the numerical
+solutions of Eq.(13) obtained for the ground states and the β-bands at β0 = 2, 3, 4 and at β0 = 15.
+The Figure 4(a) shows the comparison, in the R4/2, of the X(3) with X(5) at β0 = ∞ and at
+β0,max unique to each angular momentum, labelled as X(3)−var and X(5)−var respectively. The
+comparison in the R0/2 of the X(3) with X(5), at β0 = ∞ and at different values of the β0,max
+peculiar to each angular momentum, labelled as X(3)−var and X(5)−var respectively is shown in
+Figure 4(b).
+The ‘nature’ of critical point symmetry transitions for different isotopes, constrained to one-
+parameter potentials, can be investigated using a variational technique. This technique was used
+in ref. [11] to retrieve the U(5) and O(6) ground state bands from the E(5) within the domain
+of the one-parameter inverse square potential. The technique has also been used in ref. [12] and
+employed in ref. [16] to construct ‘image’ of the X(5) critical symmetry and to construct the Z(5)
+critical symmetry respectively. The forward variation of the ‘control parameter’, β0, causes the
+nuclei transition from X(5) to SU(3) (i.e. X(5) −→ SU(3) transition symmetry). The nature of
+the critical symmetry or the nuclear shape phase region under investigation predicts the directions
+of the variation: whether forward variation or backward variation, and also depends on the poten-
+tial’s boundary conditions. The rate of change of RL/2(β0) is maximized for each L by using this
+approach. As shown in Table 3., each angular momentum is considered and treated separately in
+terms of the variation parameter, β0, as the critical values of RL/2 are distinct. Each value of β0
+implies a distinct potential with which the energy is maximized. The method is comparable to
+the “normal” variational principle used in some quantum books, in which trial wave functions are
+chosen and energy is minimized.
+15
+
+The comparisons of the ground state spectra ratios, defined in Eq.(15) with the X(5) model [11],
+at different values of β0 corresponds to the potentials are displayed in Figures 5(a) and 5(b) and
+also shown in Table 3: the visual comparison is shown in Figures 6(a), 6(b) and 6(c). It can
+be observed that the solutions of X(3)(β0 = ∞) ≈ X(5)(β0 = ∞). The forward variation of β0
+shifts the solutions to X(3). The solutions leave X(3) and approach SU(3) as β0 tends to ∞.
+The available experimental data of 162Dy [17], which is a typical SU(3) candidate are placed for
+comparison in Figure 6(a) and Figure 6(b). This is another remark that isotopes which have X(3)
+signatures must lie between U(5) −→ SU(3) symmetrical plane.
+The two other important relations that can be deduced from the comparison are:
+RL/2(gsb) = 2 + RL/2(β1) = 4 + RL/2(β2),
+(32)
+at β0 = 0 as shown numerically in the Table 3. and Table 4. This is an observable effect or a
+signature from Eq.(30) while the effect of Eq.(28) is observed in the spectral ratios of X(3) and
+X(5) such that
+RX(3)
+L/2 (β0 = c + 2) = RX(5)
+L/2 (β0 = c) :
+c = 0, 1, 2, ...
+(33)
+In order to obtained the exact solutions of RL/2 ratios rather than vary β0, the technique of
+optimizing β0 employed in refs. [11,12,15,16] and others has been used to obtained the solutions
+of RL/2 at certain values of the β0 peculiar to the angular momenta. These special values of β0 are
+labelled β0,max and they produce exact solutions labelled X(3)-var, shown in Table 5. The values
+obtained at different values of β0,max, are compared with X(3)-IW solutions. For all β0,max, 00,0
+and 20,0 levels yield 0.000 and 1.000 respectively. β0,max increases with increase in the angular
+momentum and its values are obtained at the points where the increases in β0 become steep.
+d
+dβ0
+RL/2|β0=max is achieved via a numerical procedure as
+d2
+dβ2
+0
+RL/2 vanished. The RL/2 ratios
+for the ground state and the quasi-β1 bands of the X(3) and the X(5) models of inverse square
+potentials obtained at different values of β0,max, labeled X(3)-var and X(5)-var, are compared with
+the experimental data of 172,176,178,180Os [18-21] chain, as shown in Figures 7(a) and 7(b). The
+ground state solutions of the X(3) for L = 0 up to L = 10 are in good agreement with 172Os while
+those of X(5) are seen lying closer to 176Os than 178Os and 180Os: the generalized comparison is
+moderate in the first excited state. This suggests that 172Os is a good candidate for X(3) model
+while 176Os shows a signature of X(5) model.
+The RL/2 theoretical predictions of the X(3) model are compared with the experimental data of
+some selected isotopes: 102Mo [22], 104−108Ru [23-25], 120−126Xe [26-29], 148Nd [30] and 184−188Pt
+[31-33] as shown in Table 6. Each energy level is normalized to the particular 20,0 state. The energy
+obtained in Eq.(13) is fitted with the experimental energy of each of the isotopes considered. The
+equivalent values of the β0 for the isotopes are recorded. The quality factor, σ, used is obtained
+from
+σ =
+��m
+i [(Rs,L)Exp
+i
+− (Rs,L)Theor
+i
+]2
+m − 1
+,
+(34)
+where m is the number of available experimental states, (Rs,L)Exp
+i
+and (Rs,L)Theor
+i
+represent the
+experimental and the theoretical spectral ratios of the ith levels normalized to the ground state
+with L = 2, s = 1 and nβ = 0 respectively.
+Against the neutron numbers, N, of the chains of the isotopes: 104−108Ru, 120−126Xe, 184−188Pt,
+172−180Os, considered for the comparison, the neutron-β0 distribution, showing the relative posi-
+tions of the isotopes, is shown in the Figure 8.
+The comparison in the ground state, the quasi-β1 bands and the quasi-β2 bands of the B(E2)
+transition probabilities at β0 = 0, 1, 2 and β0 = ∞, normalized to the B(E2 : 21,0 → 01,0) = 100
+units with the X(3)-IW [1] and experimental data on X(5) [34] are presented in the Table 7. The
+values of β0,max peculiar to each angular momentum, obtained from the optimization of β0, in
+16
+
+Table 5., are employed to compute the optimized B(E2) transition probabilities, labelled B(E2)-
+var. The visuals of these comparisons are shown in the Figures 9(a), 9(b) and 9(c). In order to
+show the nature of the solutions along the X(5) −→ SU(3) symmetry region, the experimental
+data on the 158Gd [35], which is a typical SU(3) candidate, are placed for comparison in Figures
+9(a) and 9(b): the solutions at β0 → ∞ are seen lying close to the 158Gd [35]. The values of the
+B(E2) transition probabilities decrease as the variation parameter, β0, increases: they increase as
+the angular momentum increases. The forward variation, as the β0 increases, pushes the solutions
+to X(5) and the solutions tend to the SU(3) as β0 tends to ∞.
+5
+Conclusion
+The X(3) solutions of the Bohr Hamiltonian are obtained by solving the radial function of the
+Hamiltonian with an inverse square potential with the aid of MAPLE software. Analytically, an
+expression for the energy levels is determined from the zeros of the Bessel functions. Through the
+use of the variational approach and the optimization procedure, the spectra ratios and the B(E2)
+transition probabilities are computed. The analytical solutions of the X(3) model are compared
+with the X(5) model of the inverse square potentials. It is worth noting that, X(3) model is
+another “window” through which X(5) and SU(3) “pictures” can be seen: X(3) lies between U(5)
+and SU(3), hence, X(5) lies between X(3) and SU(3). It has been shown via variational procedure,
+that the solutions shift to X(5) from X(3) and approach SU(3) as the variation parameter shifts
+forward.
+The theoretical predictions on RL/2 and B(E2) with the experimental data for some selected
+isotopes are found to be proficient in the gsb and moderate in other levels. This is shown as the
+theoretical deviations from the experiments are quite small.
+The same manner in which the Davidson potential is employed in ref. [4], the employment of the
+one parameter-dependent inverse square potential in the form of Eq.(1), its properties, is efficient
+in the variational procedure. Eq.(1) is also a good choice of potential which can be employed for
+the description of the nuclei transition at the critical points. For the comparison of X(3) and
+X(5) models of Bohr Hamiltonian, with the same formalism employed in this work, it is expected
+that Equations (28), (29), (30), (31), (32) and (33) should hold in any one-parameter-dependent
+potential domain such as Kratzer potential, Davidson potential and others.
+Data availability statement
+All the sources of data included in this article for comparison purpose, are cited and referenced
+accordingly, in the article.
+Funding Information
+No funding of any form is received for the course of this work.
+References
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+19
+

8NE2T4oBgHgl3EQflQeg/content/tmp_files/2301.03987v1.pdf.txt

@@ -0,0 +1,1376 @@
+API Entity and Relation Joint Extraction from Text via
+Dynamic Prompt-tuned Language Model
+QING HUANG, Jiangxi Normal University, School of Computer Information Engineering, China
+YANBANG SUN∗, Jiangxi Normal University, School of Computer Information Engineering, China
+ZHENCHANG XING, CSIRO’s Data61 & Australian National University, College of Engineering and
+Computer Science, Australia
+MIN YU†, Jiangxi Normal University, School of Computer Information Engineering, China
+XIWEI XU, CSIRO’s Data61, Australia
+QINGHUA LU, CSIRO’s Data61, Australia
+Extraction of Application Programming Interfaces (APIs) and their semantic relations from unstructured
+text (e.g., Stack Overflow) is a fundamental work for software engineering tasks (e.g., API recommendation).
+However, existing approaches are rule-based and sequence-labeling based. They must manually enumerate the
+rules or label data for a wide range of sentence patterns, which involves a significant amount of labor overhead
+and is exacerbated by morphological and common-word ambiguity. In contrast to matching or labeling API
+entities and relations, this paper formulates heterogeneous API extraction and API relation extraction task as
+a sequence-to-sequence generation task, and proposes AERJE, an API entity-relation joint extraction model
+based on the large pre-trained language model. After training on a small number of ambiguous but correctly
+labeled data, AERJE builds a multi-task architecture that extracts API entities and relations from unstructured
+text using dynamic prompts. We systematically evaluate AERJE on a set of long and ambiguous sentences
+from Stack Overflow. The experimental results show that AERJE achieves high accuracy and discrimination
+ability in API entity-relation joint extraction, even with zero or few-shot fine-tuning.
+Additional Key Words and Phrases: API Entity, API Relation, Joint Extraction, Dynamic Prompt
+ACM Reference Format:
+Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu. 2023. API Entity and
+Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model. 1, 1 (January 2023), 20 pages.
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+1
+INTRODUCTION
+Application Programming Interfaces (APIs) are important software engineering artifacts that can
+be frequently found in a wide range of natural language texts, from official API references and
+tutorials to informal online forums. Meanwhile, API relations are also embedded in these texts.
+For example, the text “To manipulate data you actually need executeUpdate() rather than execute-
+Query()” in the Stack Overflow (SO) post 1 describes the Function-Replace relation [1] between
+executeUpdate() and executeQuery(). This API relation reveals that we should replace executeQuery()
+with executeUpdate() to solve the question in the post, i.e., “why cannot issue data manipulation
+statements with executeQuery()”. API entity and relation extraction from unstructured texts is
+∗Y. Sun and Q. Huang are co-first authors.
+†M. Yu is the corresponding author.
+1https://stackoverflow.com/questions/1905607
+Authors’ addresses: Qing Huang, Jiangxi Normal University, School of Computer Information Engineering, Nanchang,
+Jiangxi, China, qh@whu.edu.cn; Yanbang Sun, Jiangxi Normal University, School of Computer Information Engineering,
+Nanchang, Jiangxi, China, ybsun@jxnu.edu.cn; Zhenchang Xing, CSIRO’s Data61 & Australian National University, College
+of Engineering and Computer Science, Canberra, Australia, zhenchang.xing@data61.csiro.au; Min Yu, Jiangxi Normal
+University, School of Computer Information Engineering, Nanchang, Jiangxi, China, myu@jxnu.edu.cn; Xiwei Xu, CSIRO’s
+Data61, Sydney, Australia, xiwei.xu@data61.csiro.au; Qinghua Lu, CSIRO’s Data61, Sydney, Australia, qinghua.lu@data61.
+csiro.au.
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+arXiv:2301.03987v1  [cs.SE]  10 Jan 2023
+
+2
+Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu
+Table 1. Three types of ambiguities for API entities and relations.
+PostID
+Sentence
+#47871272
+You need to override remove() in your iterator.
+#14200489
+This code is invalid since l.remove() is called during iteration over l.
+#60017952
+You may be calling iterator.remove more than once.
+#34682267
+By default, printWriter calls flush in println, whereas it doesn’t do this in print.
+#703396
+If the idea is to ::::
+print integer stored as doubles...
+#322715
+linkedlist and arraylist are two different implementations of the list interface.
+#33405095
+nextline() will read the entire line, but next() will only read the next word.
+#355089
+StringBuffer is synchronized, StringBuilder is not.
+Note: API mention is tagged with an underline; common word is tagged with a wavy line.
+fundamental for efficiently accessing and applying API knowledge to various software engineering
+tasks. Once extracted, these entities and relations can be organized into structured knowledge
+(particularly in the form of knowledge graphs) to support a variety of software engineering tasks
+such as API linking [2, 3], API recommendation [4, 5], and API comparison [6].
+There are currently two main types of approaches for extracting API entities from unstructured
+text. The first is a rule-based approach such as language-convention based regular expressions [7, 8],
+island parsing [9, 10] and heuristic rule matching [1, 6, 11]. Because it is impossible to manually
+enumerate the rules that adapt to all sentence patterns, it suffers from rule design overhead. The
+second is a sequence labeling based approach such as CRF [2, 12] and Bi-LSTM-CRF [13]. Because
+it is impossible to manually label entities for a large amount of sentences, it suffers from data
+labeling overhead. Compared with API entity extraction, relation extraction from software text is
+rather primitive, which relies on either API syntax (e.g., a class declares a method) [6], special-tag
+annotated relations (e.g., “see also” keyword and hyperlink-based method) [14], or some ad-hoc
+relation phrases (e.g., “differ in” and “be similar to”) [1]. Same as rule-based entity extraction, these
+relation extraction methods suffer from rule-design overhead. We refer to rule design overhead
+and data labeling overhead as labor overhead in this work.
+This labor overhead is exacerbated by three types of ambiguities, which necessitate the manual
+design of more rules or the labeling of more data to distinguish ambiguous sentences. Morphological
+ambiguity, which includes abbreviations, synonyms, and misspellings, is one type of ambiguity [12].
+It is common in informal discussions, because people rarely write full API names that exactly match
+the API names in the library [15]. Three sentences in the first row of Table 1, for example, shows
+three morphological variations of API java.util.iterator.remove(), that either omit some prefixes
+and special symbols (e.g., package names, class names, and “()”), or are preceded by user-defined
+variable names. The second type is common-word ambiguity between common words and API
+mentions [12], which occurs because people frequently write API method names without proper
+punctuation, parentheses, and uppercase letters. For example, as shown in the second row of Table 1,
+even though the word print appears in two sentences, print in the first sentence refers to the API
+java.io.printwriter.print(), whereas print in the second one is only a verb. The final type is expression
+ambiguity of API semantic relations, which is caused by changes in sentence patterns. In general,
+the same API relation can be expressed in multiple sentence patterns. For example, as shown in the
+third raw of Table 1, three sentence patterns are used in the three sentences, all of which express
+the Behavioral-Difference relation between API entities. They are “API1 and API2 are different”,
+“API1 does one thing, but API2 does the other thing”, and “API1 is (adjective), API2 is not”.
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model
+3
+To alleviate the labor overhead, we devise a novel idea of extracting API entities and relations
+using a large pre-trained language model (LLM). LLM stores a large amount of prior knowledge
+and can serve as a neural knowledge base of real-world entities and relations [16]. In addition, LLM
+can provide better model initialization [17] and strong learning capbility. By fine-tuning a LLM
+with a small set of domain-specific training data, we can prompote the LLM to identify as many
+API entities and relations as possible. In order to make LLM be more discriminative, the training
+data should contain sufficient morphological and common-word ambiguity, and the API entities
+and relations should be labeled correctly. To reduce manual labeling, we devise morphology and
+verb-based data augmentation strategies to generate more ambiguous data but correctly labeled
+sentences for the LLM fine-tuning.
+Existing work [18] separates API entity extraction and relation extraction as two tasks, leaving
+relation extraction heavily reliant on entity extraction results, which leads to error propagation [19].
+Instead, we consider API entity extraction and relation extraction as correlated tasks and adopt
+a unified model for joint entity and relation extraction, inspired by the recent work on universal
+information extraction (UIE [20]). However, the LLM (i.e., T5 [21]) in UIE often fails with complex
+sentences, particularly long and ambiguous sentences containing API entities and various relations,
+because it only uses one static prompt to recognize all types of API relations. To tackle this issue, we
+design a dynamic prompt generator, inspired by the input-dependent prompt tuning method [22],
+that generates dynamic prompts for a small number of potentially relevant relations at inference
+time based on the actual input sentences, rather than relying on the same static prompt for all
+inputs. When confronted with complex sentences, the more relation types to recognize, the more
+noise it suffers from, the more difficult it is for LLM to understand to what extent a complex
+sentence contains certain relations. As our dynamic prompt reduces the number of relation types
+to recognize and mitigate noise interference, it improves the extraction accuracy of API relations.
+In this paper, we propose a API Entity-Relation Joint Extraction framework, called AERJE. It
+consists of a dynamic prompt generator and a joint entity-relation extractor. The kernel of the
+prompt generator is a BERT-based text classifier that is used to classify the input text. Each class
+represents an API relation and the prompt generator generates dynamic prompts based on the
+top-N possible API relations. The generated dynamic prompt and the input sentence are fed into
+the joint entity-relation extractor to extract the API entities and relations contained in the text. In
+our current implementation, the joint entity-relation extractor is Transformer-base LLM (T5). The
+prompt generator and the entity-relation extractor are fine-tuned in an end-to-end manner.
+No model, to the best of our knowledge, can simultaneously extract both API entities and
+relations. AERJE, on the other hand, achieves an F1 score of 96.51% for API entity extraction, which
+is approximately 6% higher than the state-of-the-art API entity recognition model ARCLIN [13]
+and 7% higher than APIReal [2], and an F1 score of 81.20% for API relation extraction. Then, we
+evaluate the impact of intrinsic factors (two data augmentation strategies and the number of API
+relations in the dynamic prompts) on performance. Our experiments find that data augmentation
+helps to improve AERJE’s discriminative capability for API entities and relations, and the dynamic
+prompts with four API relations can significantly improve AERJE’s extraction accuracy. Finally,
+we assess AERJE’s generalization and ability to extract API entities and relations in low-resource
+scenarios (i.e., less than 0.8% fine-tuning data) and find that, even under low resource conditions,
+our AERJE still has strong extraction ability, outperforming APIReal [2] and ARCLIN [13].
+The main contributions of this paper are as follows:
+• Conceptually, we are the first to formulate heterogeneous API extraction and API relation
+extraction tasks as a uniform sequence-to-sequence generation task, and propose AERJE, an
+API entity-relation joint extraction framework based on pre-trained LLMs.
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+4
+Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu
+In fact, collections.sort() has
+already been migrated to
+list.sort().
+You better using getint()
+instead of get().
+You can use
+double.parsedouble() to
+convert a string to a double.
+Input Sentences
+Bert
+Dynamic
+prompt
+ 
+ 
+    (
+        ( API: getint()
+            ( function replace: get() )
+        )
+        ( API: get() )
+    )
+    (
+        ( API: collection.sort() )
+        ( API: list.sort() )
+    )
+[spot] API [asso] function replace [asso] efficiency comparison [asso] type conversion 
+[text] You better using getint() instead of get().
+[spot] API [asso] function similarity [asso] logic constraint [asso] type conversion 
+[text] In fact, collections.sort() has already been migrated to list.sort().
+[spot] API [asso] logic constraint [asso] type conversion [asso] function similarity 
+[text] You can use double.parsedouble() to convert a string to a double. 
+Joint Entity-Relation Extractor
+Linear
+P(Relation)
+top-3 Relations
+a
+b
+c
+Ⅰ 
+Ⅱ
+CLS
+E
+...
+E
+E
+D
+...
+D
+D
+Latent Vector
+...
+b
+c
+a
+b
+c
+a
+Ⅲ
+Dynamic Prompt Generator
+T5
+ 
+    (
+        ( API: string
+            ( type conversion: double )
+        )
+        ( API: double )
+        ( API: double.parsedouble() )
+    )
+Structured Extraction Language
+......
+......
+......
+Fig. 1. Overall Framework of AERJE. The Dynamic prompt’s bold font represents the semantic relation to be
+extracted from the input sentence. II.b lacks a bold font because I.b contains no semantic relation.
+• We devise two data augmentation strategies in order to obtain more ambiguous but correctly
+labeled sentences. Learning such sentences enables AERJE to be more discriminative for API
+entities and relations.
+• Unlike the single task model, we build a multi-task architecture that encodes the structures
+of entity and relation extraction into a unified structure language for extracting API entities
+and relations simultaneously.
+• Instead of using a single static prompt with all types of API relations for all sentences, we
+design a dynamic prompt based on relation classification, which reduces the number of
+relation types to recognize, eliminates noise interference, and lowers the difficulty of relation
+extraction.
+• We systematically evaluate the AERJE’s intrinsic factors, performance, generalization, and
+few-shot learning capabilities. It is the first approach to extract API entities and relations
+simultaneously, and it achieves superior performance than independent API extraction and
+API relation extraction. Our data package can be found here2, the code will be released after
+the paper is accepted.
+2
+APPROACH
+We formulate heterogeneous API extraction and API relation extraction tasks as a uniform sequence-
+to-sequence generation task, and propose a novel model AERJE to accomplish it. As shown in
+Fig. 1, AERJE consists of a dynamic prompt generator and a joint entity-relation extractor. The
+dynamic prompt generator generates dynamic prompts based on the input texts (one at a time).
+The input text is then appended to the prompt to form a whole input that is fed into the joint
+entity-relation extractor, which generates a structured extraction language sequence with API
+entities and relations.
+2.1
+Dynamic Prompt Generator
+This section describes how to build a prompt that unifies heterogeneous API extraction and API
+relation extraction tasks, followed by a discussion of how to design dynamic prompt to improve
+AERJE’s API relation extraction performance.
+2.1.1
+Prompt Construction for Multi-tasking. In order to extract both API entities and API relations
+from an input text, the prompt consists of API entity type, API relation type, and the input text,
+which are labeled by [spot], [asso] and [text], respectively. For example, “[spot] API [asso] function
+2https://anonymous.4open.science/r/AERJE-6DBF/README.md
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model
+5
+replace [asso] efficiency comparison [text] You better using getint() instead of get()” represents
+an API entity type “API”, two relation types “function replace” and “efficiency comparison”, and
+an input text “You better using getint() instead of get()”. In this work, we consider a generic API
+entity type “API” and seven relation types defined in [1], including “function similarity”, “behavior
+difference”, “logic constraint”, “type conversion”, “function collaboration”, “efficiency comparison”,
+“function replace”. Note that more fine-grained API entity types can be used, such as “class”,
+“method”, “field” [23], but we leave it as our future work.
+2.1.2
+Dynamic Prompt Generation. As stated in Section 1, the more relation types there are, the
+harder it is for T5 to determine which types of relation the API entities in the input text belong to,
+especially when the sentence is long and ambiguous. If we adopt the static prompt that includes
+all seven relation types, the relation extraction performance of the model will decrease (cf. RQ3).
+As a result, we design a dynamic prompt generation method to make the content of the prompt
+more accurate and instructive for the complex input text. The dynamic prompts, as shown in II.a of
+Fig.1, contain only the top-N relations and provide better guidance to the subsequent T5-supported
+joint entity-relation extractor. Here, the prompt generator is implemented as a text classifier which
+predicts the API relations present in the input text. We use a BERT-based classifier because the
+pre-training task (i.e., Next Sentence Prediction) of BERT [24] is consistent with our task, both
+of which are classification tasks. Given a sentence containing API entities (see I.a of Fig. 1), the
+BERT-based classifier outputs the probability that the sentence belongs to each semantic relation;
+the top-3 relations are then chosen as candidate relations. Finally, entity type, candidate relations,
+and input sentence are connected by labels (i.e., [spot], [asso], [text]) to generate the dynamic
+prompt (see II.a of Fig. 1).
+Note that the BERT-based classifier in our current implementation aims to narrow the scope and
+provide candidate relations, and it cannot replace the API relations extractor. When the candidate
+relations classified by the classifier do not fit these entities in the sentence, the extractor does not
+force a relation to be selected from the incorrect candidate relations, but instead assumes that no
+relation exists between these entities. For example, given a sentence with no relations between API
+entities (see I.b of Fig.1), the dynamic prompt generator generates a dynamic prompt (see II.b of
+Fig.1). Based on such a dynamic prompt, the subsequent extractor will not extract relations from
+the sentence as none of the candidate relation types is applicable to the input sentence.
+To summarize, too many candidate relations may reduce the extractor’s ability to recognize
+them, while too few candidate relations may cause the extractor to miss the correct relations. As a
+result, we should investigate the appropriate number of candidate relations (cf. RQ3).
+2.2
+API Joint Entity-Relation Extractor
+We adopt a structured extraction language (SEL) [20] to encode the structures of entity extraction
+and relation extraction into a unified representation, so that heterogeneous API extraction and
+API relation extraction tasks can be modeled uniformly within a sequence-to-sequence generation
+framework. The first sequence refers to the dynamic prompt, while the second sequence refers to
+the SEL sequence.
+2.2.1
+Structured Extraction Language. SEL sequence is proposed to encode different information
+extraction structures via the hierarchical spotting-associating structure. Fig. 2.a shows its universal
+format. “Spot Name: Info Span” denotes various entity type and the object of a specific entity type;
+“Asso Name: Info Span” denotes various relation types and the associated object of a specific relation
+type. Fig. 2.b shows the concrete SEL sequence in our work. “API: getint()” represents that “getint()”
+is an API entity; “function replace: get()” represents that the relation between “getint()” and “get()”
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+6
+Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu
+    ( 
+       ( Spot Name:  Info Span 
+        （ Asso Name:  Info Span ) 
+       ) 
+      (  Spot Name: Info Span ) 
+    ) 
+    ( 
+       ( API:  getint() 
+        （ function replace:  get() ) 
+       ) 
+       ( API: get() ) 
+    ) 
+a
+b
+Fig. 2. Specialization of Universal Structured Extraction Language.
+is “function replace”. From this concrete SEL sequence, we can extract API entities and relations
+simultaneously as it unifies the structure of API entities and relations.
+2.2.2
+SEL Sequence Generation. We implement our API joint entity-relation extractor as the
+sequence-to-sequence generation framework:
+�
+𝑦1, . . . ,𝑦|𝑦|
+�
+= JE(
+�
+𝑝1, . . . , 𝑝 |𝑝 |
+�
+)
+(1)
+where JE is a Transformer-based LLM,
+�
+𝑝1, . . . , 𝑝 |𝑝 |
+� is the dynamic prompt,
+�
+𝑦1, . . . ,𝑦|𝑦|
+� is the
+linearized SEL sequence that contains the API entities and relations to be extracted. In this frame-
+work, we feed the dynamic prompt into the LLM (as shown in Fig. 1.II), and the LLM generates the
+SEL sequence (as shown in Fig. 1.III), from which we can obtain API entities and relations. The
+dynamic prompt to the JE can also be written in the format described in Section 2.1.1:
+�
+𝑝1, . . . , 𝑝 |𝑝 |
+�
+=[[ spot ], . . . [ spot ] . . . ,
+[ asso ], . . . , [ asso ] . . . ,
+[ text ],𝑥1,𝑥2, . . . ,𝑥 |𝑥 |
+�
+(2)
+where 𝑥 =
+�
+𝑥1, . . . ,𝑥 |𝑥 |
+�
+denotes the input text.
+To better illustrate the framework’s internal mechanics, an encoder-decoder-style architecture is
+introduced. Given the dynamic prompt 𝑝, JE computes the hidden representation H =
+�
+p1, . . . , p|𝑝 |
+�
+of each token:
+H = Encoder �𝑝1, . . . , 𝑝 |𝑝 |
+�
+(3)
+where Encoder(·) is a Transformer encoder. Then JE decodes the prompt into a SEL sequence in an
+auto-regressive style. At the step 𝑖 of decoding, JE generates the 𝑖-th token 𝑦𝑖 in the SEL sequence
+and the decoder state h𝑑
+𝑖 as following:
+𝑦𝑖, h𝑑
+𝑖 = Decoder
+��
+H; h𝑑
+1, . . . , h𝑑
+𝑖−1
+��
+(4)
+Decoder(·) is a Transformer decoder that predicts the conditional probability 𝑝 (𝑦𝑖 | 𝑦 <𝑖, 𝑝) of
+token 𝑦𝑖 until the end symbol <eos> is output.
+2.3
+Model Training
+This section describes data collection and augmentation, model training, which includes training a
+BERT-based classifier and fine-tuning a Transformer-based LLM.
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model
+7
+2.3.1
+Data Collection. Given that the relation types we consider are all from a knowledge graph
+of Java APIs [1], we randomly chose 5,000 Java-tagged posts from the Stack Overflow data dump 3.
+Each post is accompanied by its answers and post tags (such as “java”, “arrays”, “java.lang”). We
+choose the most voted answers from the posts to ensure the quality of the training data, but we
+exclude code snippets and all HTML tags because the focus of our study is informal text. All the
+answers are then splitted into sentences using spaCy 4, yielding 28,140 sentences. Every sentence is
+accompanied by multiple category tags from the post to which it belongs. Then, for each sentence,
+we parse it into tokens using the software-specific tokenizer [12] which preserves the integrity of
+an API mention. iterator.remove(), for example, is treated as a single token. Finally, we crawl all
+APIs in JDK 1.8 5, and use these APIs to filter out the sentences containing API entities, as inspired
+by a previous study [13], based on the following criteria:
+• Because of the large number of morphological ambiguities, a token may be an API entity if it
+partially matches any of the crawled APIs (e.g., remove() and java.util.Iterator.remove()).
+• Since API mentions usually end with “()”, the token is treated as an API entity if it contains
+“()”.
+• API mentions typically include “.” to indicate a function call (e.g., iterator.remove(), or l.remove());
+thus, if token contains “.”, we consider it to be an API entity .
+After filtering, we obtain 9,111 sentences that may contain API entities. However, this is rough
+sentence filtering. In order to do accurate sentence filtering, We invite 12 master students (all
+with more than five years Java experience) to examine the API entities and annotate the semantic
+relations between APIs in order to further verify whether these sentences contain API entities and
+the seven types of API relations we aim to extract. We train the annotators prior to annotation to
+ensure that they can recognize these API relations in the text. After training, the annotators were
+divided into six groups, with two students from each group annotating the same content. After the
+annotation, we assign two authors to deal with the annotation results’ conflicts, and the Cohen’s
+Kappa [25] coefficient is 0.859 (i.e., almost perfect agreement). As a result, we get a total of 2917
+sentences, with 2471 containing only entities and 446 containing both entities and relations.
+2.3.2
+Data Augmentation. To improve the AERJE’s ability to recognize API entities and relations
+from long and ambiguous sentences, we devise two data augmentation strategies to obtain more
+ambiguous sentences for model training.
+Morphology based Mutation. Inspired by [13], we change the form of each API entity in the
+sentence. Specifically, we replace the API entity itself with the final piece of its fully qualified name.
+For example, iterator.remove() is replaced with remove() or remove.
+Verb based Mutation. We use spaCy to locate the verbs on which each API entity relies, and
+then replace those verbs with synonyms, as Liu et al. [26] do to obtain similar question titles. As
+shown in the seventh sentence of Table 1, we replace “read” with “load”. However, because spaCy
+may not obtain the correct API entity, we must identify the dependency between the API entity’s
+subtoken and the verb to ensure the mutation quality. For example, there is a dependency between
+“nextline” and “read”, so we can reliably mutate “read” with synonyms.
+Our data augmentation strategy does not include sentence pattern mutation [26], which uses
+different sentence patterns to present the same API relation between the same API entities. Unlike
+the morphology-based and verb-based mutation, this mutation is not reliable in software text
+which demands stricter semantics than general text. The sentence pattern mutation could result in
+sentence structure reconstruction, which would likely change the sentence semantics, contaminate
+3Retrieved June 6, 2022 from https://archive.org/download/stackexchange/
+4https://spacy.io
+5https://docs.oracle.com/javase/8/docs/api
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+8
+Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu
+the training data, and compromise AERJE training. For example, the original sentence “StringBuffer
+is synchronized, StringBuilder is not” may be mutated into “StringBuffer and StringBuilder differ in
+synchronized”. The original sentence indicates that StringBuffer is synchronized and StringBuilder
+is asynchronous, but the mutated sentence does not specify who is synchronous or asynchronous.
+We obtain 2,334 sentences as the initial training set and 583 sentences as the initial test set in an
+8:2 ratio from the 2,917 sentences collected. The number of sentences after applying the two data
+augmentation strategies to the initial training and test sets is 10,678 and 2,686, referred to as the
+final training set and the final test set, respectively. This final training set is used to fine-tune the
+LLM-based extractor, and the final test set is used to test the fine-tuned extractor. Here, we split
+the sentences into training and testing sets and then mutated them. This ensures that the sentence
+before and after the mutation is in the same set, preventing the leaking of training data into the test
+set (e.g., one sentence in the training set and its mutation in the test set). Furthermore, we obtain
+1,639 sentences with both entities and relations as the classifier training set from the final training
+set. Similarly, we obtain 387 sentences with both entities and relations as the classifier test set from
+the final test set.
+2.3.3
+BERT-based Relation Classifier Training. We choose BERT [24] as a relation classifier because
+its pre-training task (i.e., Next Sentence Prediction) is consistent with our task, both of which are
+classification tasks. However, the implementation of relation classifier is not limited to BERT, we
+can also use TextCNN [27] and FastText [28]. In our current implementation, we use the BERT-base
+classifier to classify each input sentence into N relation types. Based on the N relation types,
+dynamic prompt generator generates the corresponding dynamic prompt.
+A mask language model (BERT) [24] and a linear layer comprise the classifier. Due to the seven
+API relation types, the linear layer’s output dimension is set to 7. We obtain the latent vector from
+the CLS token when we enter the sentence into BERT. The latent vector obtained from the CLS token
+characterizes the sentence features better than other positions, resulting in better classification
+performance. The latent vector is then fed into the linear layer, which produces a vector with seven
+dimensions, each corresponding to a relation type. Finally, the classifier is trained on the classifier
+training set. In back propagation, we use the cross-loss entropy to calculate the classifier’s loss and
+adjust the BERT and linear layer parameters. The loss function is formulated as follows, where
+𝑧 = [𝑧0, . . . ,𝑧𝐶−1] represents the linear layer’s output result, and C represents the sentence’s label.
+Loss(𝑧,𝑐) = −𝑧[𝑐] + log
+�𝐶−1
+∑︁
+𝑗=0
+exp(𝑧[𝑗])
+�
+(5)
+2.3.4
+LLM-based Extractor Fine-tuning. We use the pre-trained T5-v1.1-large model [21] as the
+LLM in our current implementation because T5’s training objective aligns perfectly with our
+formulation of the API entity and relation extraction task as a sequence to sequence generation
+task. Furthermore, studies [29, 30] confirm that T5 is capable of capturing rich text information and
+demonstrate its effectiveness in a variety of downstream NLP tasks. Our approach is not limited to
+T5, but can use any Transformer-based LLM.
+In order to fine-tune T5, we convert each labeled sentence in the final training set into a SEL
+sequence (y), then feed it into the dynamic prompt generator to obtain its dynamic prompt (p), and
+finally construct the labeled corpus: De = {(𝑝,𝑦)}. On the labeled corpus, we fine-tune T5 for 50
+epoch with batch size 10 using the Adam optimizer with a learning rate of 1e-4, linear scheduling
+with a warming up proportion of 6%, and the teacher-forcing cross-entropy loss:
+LFT =
+∑︁
+(𝑝,𝑦) ∈De
+− log 𝑃 (𝑦 | 𝑝;𝜃𝑒,𝜃𝑑)
+(6)
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model
+9
+where 𝜃𝑒 and 𝜃𝑑 are the parameter of encoder and decoder, respectively.
+3
+EXPERIMENTAL SETUP
+This section starts with five questions about AERJE’s performance, followed by a description of the
+experimental setup, which includes the dataset, baseline, and evaluation metrics.
+3.1
+Rearch question
+• RQ1: Effectiveness of Data Augmentation
+• RQ2: Optimal Num. of Relation Types for Dynamic Prompt
+• RQ3: Joint Extraction Performance of AERJE
+• RQ4: Generalization Ability of AERJE
+• RQ5: AERJE’s Performance in Low-Resource Scenario
+3.2
+Dataset
+As described in section 2.3.2, there are three groups of data sets. The first group refers to the
+sentences collected initially, some of which contain only entities and others contain both entities
+and relations.
+• The initial training set consists 2,334 sentences, of which 362 contain both entities and
+relations.
+• The initial test set consists 583 sentences, of which 84 contain both entities and relations.
+The second group refers to the sentences after applying the two data augmentation strategies,
+some of which contain only entities and others contain both entities and relations.
+• The final training set with a total of 10,678 sentences, 1639 of which contain both entities
+and relations.
+• The final test set with a total of 2,686 sentences,387 of which contain both entities and
+relations.
+The third group refers to the sentences containing both entities and relations in the final training
+and testing sets.
+• The classifier training set with a total of 1,639 sentences.
+• The classifier test set with a total of 387 sentences.
+3.3
+Baselines
+Our AERJE is capable of API entity-relation joint extraction. However, to the best of our knowledge,
+no previous work has focused on extracting both API entities and relations from unstructured texts
+at the same time. As a result, we can only compare AERJE with the existing work in the respective
+fields of API entity extraction and API relation extraction.
+For API entity extraction, there are rule-based methods (such as regular Expressions [7, 8]),
+heuristic rule matching methods [1, 6, 11], and sequence-labeling based methods (such as AR-
+CLIN [13] using BI-LSTM as encoder and CRF as decoder, APIReal [2] using only CRF). Since the
+performance of the first two classes of methods is not as good as that of the last class of methods [13],
+we choose ARCLIN, APIReal as baselines. We obtain them source code from Github 6 7, and label
+the API entities and non-entities in the sentences with the “BIO” tag (i.e., “B”: the beginning of
+API entity segment, “I”: the inside of API entity segment, “O”: non-entity). Then we create pairs
+of original sentences and labeled sentences to train these two baselines. Finally, we the trained
+models on the final test set, from which we obtain API entities based on the “BIO” tag.
+6https://github.com/YintongHuo/ARCLIN
+7https://github.com/baolingfeng/APIExing
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+10
+Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu
+For API relation extraction, there are only rule matching methods that rely on API syntax [6],
+special-tag annotated relations [14], or some ad-hoc relation phrases [1]. It is very difficult to
+re-implement these methods due to the rule-design overhead. Furthermore, it is impractical to apply
+these methods as we assume plain texts without any special annotations. Instead, we implement a
+variant 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 (𝐷𝑃𝐺 means the dynamic prompt generator), which uses a static prompt
+with all 7 relation types to evaluate the performance of the full-version of AERJE.
+In addition, we also implement two other variants of AERJE as our baseline. One is 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒,
+which separate API entity and relation extraction into two independent tasks. For API entity
+extraction, the prompt contains only “[spot] API”. For API relation extraction, the prompt contains
+only “[asso] relation type”. 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 still uses dynamic prompt in relation extraction. After
+relation extraction, we merge the extracted entities and relations as the final results of 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒.
+e.g., the extracted entities getint(), get() and relation function replace are merged as (API: getint()
+(function replace: get())). We compare 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 with AERJE to understand the effectiveness of
+joint entity-relation extraction. Meanwhile, the entity extraction results of 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 is equivalent
+to fine-tuning pre-trained model for entity extraction, and its final results is equivalent to fine-
+tuning pre-trained model for relation extraction. Therefore, 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 also reflects the capability
+of fine-tuning pre-trained model for entity and relation extraction separately. Another variant is
+𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒, which replace T5-v.1.1-large in AERJE with a smaller model backbone, i.e., T5-v1.1-base.
+We use it to explore the impact of large pre-trained language models on AERJE performance.
+All variants use the same hyper-parameters as AERJE and remain constant across experimental
+scenarios. Note that SEL used in AERJE has been demonstrated to be effective in the extraction
+task [20]. As such, we do not to verify the effectiveness of SEL in AERJE.
+3.4
+Evaluation Metrics
+We use Precision, Recall, and F1 score as metrics to evaluate the performance of AERJE and baseline
+models on our test set. Precision means what percentage of API entities and relations extracted
+are correct, recall means what percentage of the real API entities and relations are extracted, and
+F1 score is the harmonic mean of precision and recall. It is important to note that the relation
+is only correct if the relation type and corresponding entities are both correct. In context of our
+work, we are not concerned with the top-N relation classification accuracy. As long as the top-N
+includes relevant relation types, the extractor does not care about the order of these relation types.
+Furthermore, a sentence may have 2 or more relations, which renders the top-1 accuracy irrelevant.
+Finally, as the extractor has the capability of ruling out irrelevant relation types in the prompt, it is
+also not necessary to evaluate the classification precision and recall at N.
+4
+EXPERIMENTAL RESULTS
+This section delves into five research questions to evaluate and discuss the AERJE’s performance.
+4.1
+RQ1: Effectiveness of Data Augmentation
+4.1.1
+Motivation. To reduce manual labeling effort and improve model training, we devise two data
+augmentation strategies. We want to investigate if ambiguous but correctly annotated sentences
+obtained through two data augmentation strategies could improve AERJE’s discriminative capability
+for extracting API entities and relations, in order to demonstrate the effectiveness of two data
+augmentation strategies.
+4.1.2
+Methodology. We set up two scenarios 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 and 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 (𝐷𝐴 means the data
+augmentation). 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 is trained on the initial training set, while 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 is trained on
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model
+11
+Table 2. Impact of data augmentation strategy on AERJE
+Strategy
+Entity
+Relation
+P
+R
+F1
+P
+R
+F1
+𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴
+97.57
+95.48
+96.51
+86.54
+76.48
+81.20
+𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴
+95.11
+92.19
+93.63
+77.71
+75.66
+76.67
+the final training set. Both 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 and 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴are tested on the same final test set. This
+setting allows us to compare the effectiveness of data augmentation.
+4.1.3
+Result. Table 2 shows the experimental results. In terms of API entity extraction, 𝐴𝐸𝑅𝐽���𝑤𝐷𝐴
+has precision, recall, and F1-scores of 97.57%, 95.48%, and 96.51%, while 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 has precision,
+recall, and F1-scores of 95.11%, 92.19%, and 93.63%. 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴’s precision, recall, and F1-score are
+all higher than 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴’s, with 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴’s recall and F1-score being about 3% higher.
+In terms of API relation extraction,𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 has precision, recall, and F1-score of 86.54%, 76.48%,
+and 81.20%, while 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 has precision, recall, and F1-score of 77.71%, 75.66%, and 76.67%.
+𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴’s precision, recall, and F1-score are all higher than 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴’s. The precision of
+𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 is 8.83% higher than that of 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴, and the F1-score of 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 is 4.53% higher
+than that of 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴. This demonstrates that fine-tuning 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 using a large number
+of ambiguous sentences with API relations benefits 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 to distinguish between relations
+and non-relations, as well as between correct and incorrect relations. In contrast, 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 has
+not been fine-tuned on ambiguous sentences and thus does not perform as well as 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴. For
+example, an ambiguous sentence “you want to read up on processbuilder to launch the exe file
+and then waitfor() to wait until the process is complete”. 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 correctly extracts two API
+entities, ProcBuilder and waitfor(), as well as the “logic constraint” relation between them, from the
+sentence. In contrast, 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 only extracts one API waitfor() from the sentence. This shows
+AERJE’s capability to extract API entities and relations from ambiguous sentences can be improved
+by fine-tuning with the augmentated data.
+AERJE’s discriminative capability for API entities and relations can be improved by fine-tuning
+it with ambiguous but correctly labeled sentences obtained through the data augmentation
+strategies.
+4.2
+RQ2: Optimal Num. of Relation Types for Dynamic Prompt
+4.2.1
+Motivation. As described in section 2.1.2, given an input sentence, the dynamic prompt
+generator employs the BERT-based classifier to predict a set of candidate relation types, which are
+then included in the dynamic prompt to guide the subsequent joint entity-relation extractor. In this
+RQ, we would like to investigate how many candidate relation types (i.e., top-N classifier results)
+can provide the most effective guidance to the extractor.
+4.2.2
+Methodology. We exhaust all cases of N values (from 1 to 6) in the dynamic prompt generator,
+then fine-tune AERJE on the same final training set and test it on the same final test set to select the
+most appropriate N value based on experimental results. We do not test N=7 because it is essentially
+the static prompt with all seven relation types (i.e., 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 studied in RQ3).
+4.2.3
+Result. As shown in Table 3, changing the N value has small effect on entity extraction
+because N represents the number of relation types in the dynamic prompt which does not directly
+affect entity extraction. At N=3, AERJE achieves the marginally best F1-score 96.51% for API entity
+extraction.
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+12
+Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu
+Table 3. Model results for different values of N
+top-N
+Entity
+Relation
+P
+R
+F1
+P
+R
+F1
+1
+96.88
+94.23
+95.54
+75.92
+71.92
+73.87
+2
+97.04
+95.18
+96.10
+77.75
+73.80
+75.72
+3
+97.57
+95.48
+96.51
+86.54
+76.48
+81.20
+4
+97.84
+94.39
+96.08
+83.51
+73.22
+78.03
+5
+96.72
+94.39
+95.54
+77.90
+73.30
+75.53
+6
+96.44
+94.75
+95.59
+75.35
+72.61
+73.95
+For relation extraction, changing the N value has larger effect on both precision and recall. As
+N increases, both precision and recall improve until N=3. When N=3, the precision, recall and
+F1-score of AERJE reaches the highest 86.54%, 76.48% and 81.20%, respectively. This means that
+the correct API relation type is most likely covered in the top-3 candidate relations predicted
+by the classifier. When N is less than 3, however, the F1-score of AERJE in relation extraction
+decreases because the top-N candidate relations may miss the correct relation type. Here is an
+example: “A TreeMap has the same limitation (as does a HashMap, which also breaks when the
+hashcode of its elements changes after insertion)”. When N=2, classifier predicts two relations
+between TreeMap and Hashmap, including “behavior difference” and “logic constraint” , but ignores
+the “function similarity” relation. This ignored relation is at the third relation predicted by the
+classifier. However, when N is greater than 3, the F1-score of AERJE in relation extraction decreases
+because the dynamic prompt may contain some incorrect relation types, which may mislead the
+extractor. This misleading effect has bigger impact on precision than on recall.
+The optimal number of relation types for dynamic prompt should be set to 3. This not only
+ensures that the majority of the correct relation types appear in the dynamic prompts, but it
+also prevents the dynamic prompts from containing too many noise relation types which may
+make the model sacrifice precision for recall.
+4.3
+RQ3: Joint Extraction Performance of AERJE
+4.3.1
+Motivation. We would like to evaluate AERJE’s performance in API entity and relation joint
+extraction, compared with the state-of-the-art methods for API entity extraction and API relation
+extraction. Note that only our AERJE can achieve joint API entity and relation extraction.
+4.3.2
+Methodology. AERJE is compared to APIReal and ARCLIN for API entity extraction, and
+three variant models (i.e., 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺, 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒, and 𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒) for both API entity and
+relation extraction. Note that the entity and relation extraction results by 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 represents
+the capability of fine-tuning the pre-trained model for the two tasks separately. All models are
+trained and tested on the same final training and test sets. Details on configuration can be found in
+Section 3.3.
+4.3.3
+Result. Table 4 shows the evaluation result of AERJE and five baselines on final test sets. We
+see that AERJE’s F1-score is higher 7.5% than APIReal’s F1-score and 5.7% than ARCLIN’s F1-score
+on API entity extraction. Compared with the three variant models, AERJE’s F1-score for API entity
+extraction is only slightly lower (0.18%) than the best performer (i.e., 96.69% by 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒), but
+AERJE’s F1-score for relation extraction is 6.83% higher than that of the second best performer
+(74.37% by 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒).
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model
+13
+Table 4. Comparison of Overall Performance
+Model
+Entity
+Relation
+P
+R
+F1
+P
+R
+F1
+APIReal
+89.13
+88.90
+89.01
+-
+-
+-
+ARCLIN
+94.76
+87.17
+90.81
+-
+-
+-
+AERJE
+97.57
+95.48
+96.51
+86.54
+76.48
+81.20
+𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒
+98.03
+95.38
+96.69
+82.83
+67.47
+74.37
+𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺
+97.52
+95.78
+96.64
+75.38
+70.62
+72.92
+𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒
+96.39
+95.15
+95.77
+75.97
+70.28
+73.01
+For APIReal and ARCLIN performance on API entity extraction, both AERJE and it variant
+models outperform them largely. This superior performance is due to the backbone large pre-
+trained language models (T5) in AERJE. During the pre-training, T5 learns linguistic and semantic
+knowledge in text and has powerful abilities in word and sentence representations. Through fine-
+tuning, the semantic knowledge packed in the T5 can be transferred to the downstream tasks and
+benefit API entity extraction.
+The amount of knowledge in the T5 also affects the AERJE’s performance on API entity and
+relation extraction. Compared with AERJE, the F1-score of 𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒 is reduced by 0.74% and 8.19%
+in API entity extraction and API relation extraction, respectively. The decrease of 𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒’s F1
+score on API entity extraction is very small compared with the decrease on API relation extraction.
+It is because the number of sentences containing API entities in the final training set is 6 times
+more than the number of sentences containing both API entity and relation (i.e., 10,678 vs 1,639).
+Sufficient fine-tuning data for API entity extraction allows the basic T5 model to achieve the
+equivalent performance on API entity extraction as the large T5. In contrast, the relation extraction
+is more complex than the entity extraction, and the amount of fine-tuning data is smaller. In such
+case, the basic T5 cannot compete with the large T5.
+For 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 and AERJE, they achieve almost the same entity extraction performance. How-
+ever, in terms of API relation extraction, AERJE’s F1-score (81.20%), precision (86.54%) and recall
+(76.48%) are much higher than 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒’s F1-score (74.37%), precision (82.83%) and recall (67.47%),
+respectively. This suggests that fine-tuning pre-trained model for API entity extraction individually
+or jointly with API relation extraction does not affect the quality of API entity extraction. But
+joint entity and relation extraction is much more effective for the relation extraction task than
+fine-tuning the model just for the relation extraction.
+For 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 and AERJE, they also achieve almost the same entity extraction performance.
+This is due to the fact that dynamic prompt only affects the relation type, not the entity type. In
+terms of API relation extraction, AERJE’s precision (86.54%), recall (76.48%) and F1-score (81.20%) are
+higher than 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺’s precision (75.38%), recall (70.62%) and F1-score (72.92%), respectively.
+This is because 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 uses the same static prompt that includes all seven relation types
+for all input sentences. The more types of relations there are in the prompt, the more noise the
+prompt is, and the more difficult it is for AERJE to identify and extract the correct relations in the
+input sentence. In contrast, AERJE’s use of dynamic prompt reduces the number of relation types
+to recognize, improving its ability to extract API relations.
+Standing on the shoulder of large pre-trained language model (T5), AERJE outperforms traditional
+sequence labeling models for API entity extraction. Dynamic prompt has no impact on API
+entity extraction, but can largely boost the performance of API relation extraction. Fine-tuning
+the pre-trained model jointly is much more effective than fine-tuning the model just for one
+task, which makes joint entity-relation extraction more accurate on both tasks than separate
+entity and relation extraction.
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+14
+Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu
+4.4
+RQ4: Generalization Ability of AERJE
+4.4.1
+Motivation. Each API comes with its own API package, which often have different forms.
+Furthermore, as APIs from different packages support diverse functionalities, the texts in which
+they appear may be different in content and linguistic properties. It is impossible for AERJE to see
+all API packages during fine-tuning. In this RQ, we want to investigate if AERJE can recognize
+APIs and their relations from the API packages that it does not see during fine-tuning.
+4.4.2
+Methodology. In order to collect as much data from different packages as possible, we
+combine the final training set and the final test set into a new data set with a total of 13,364
+sentences. Every sentence, as stated in Section 2.3.1, is accompanied by multiple post tags, some of
+which show the relationship between the sentence and the API package. For example, the tag “io” is
+associated with the package name “java.io”. Therefore, we filter out sentences with package names
+by matching each tag of a sentence to any JDK 1.8 package name. Here is a partial match, which
+means it matches a portion of the package name, for example, “swing” can match “javax.swing”.
+And then we pool the package names that appear with the sentences and select the three package
+names that appear the most frequently (i.e., javax.swing, java.io, and java.util). Finally, we gather
+1651 sentences whose tags match these three package names.
+To ensure the correctness of the sentences obtained through approximate match, we invite six
+students (who have previously participated in annotation) and divide them into three groups to
+annotate sentences from three different packages. Two students in each group annotate the same
+sentences. They independently determine whether the API entities in each sentence are from the
+specific package (i.e., java.io, java.util, javax.swing). Here is an example “you can use lines() method
+in BufferedRead” for java.io package. The sentence is annotated as True, since the API entities
+line() and BufferedRead only correspond to java.io. Instead, if any API entity in the sentence do
+not belong to specific package, the sentence is annotated as False. Then we assign an author to
+handle conflicts between the group members. Finally, we obtain 999 sentences that strictly matched
+these packages names. Cohen’s Kappa [25] coefficient is 0.795 (i.e., substantial agreement). The
+data details for each package are as follows:
+• The java.io dataset has 235 sentences, 51 of which contain both entities and relations. 12 of
+the 51 sentences are non-augmented sentences.
+• The javax.swing dataset has 435 sentences, 76 of which contain both entities and relations.
+14 of the 76 sentences are non-augmented sentences.
+• The java.util dataset has 329 sentences, 68 of which contain both entities and relations. 18 of
+the 68 sentences are non-augmented sentences.
+Our AERJE and baseline models are all trained on one of the three datasets and tested on the
+two others. As AERJE outperforms its variants. We don’t consider these variants here.
+4.4.3
+Result. Table 5 shows the results that reflect each model’s generalization ability. For API
+entity extraction, AERJE’s F1-score achieves 95.05%, when trained on the java.util dataset, far
+exceeding APIReal’s F1-score (61.27%) and ARCLIN’s F1-score (58.50%). We attribute this to the
+underlying LLM on which AERJE is built. As Qiu et al. [17] show, LLM provides better model
+initialization, which usually leads to better generalization performance on the target tasks. Similar
+observations can be made for training the models on the java.io dataset and the javax.swing dataset.
+Generally, ARCLIN and APIReal may perform well on either precision or recall, but not both and
+thus poor F1-score. In contrast, AERJE is very stable with much better precision and recall and
+with only small fluctuations in F1-scores across the experiments.
+For API relation extraction, AERJE’s F1-score is 40.98%, 79.99% and 68.48% when trained on
+the java.io, java.util and javax.swing datasets, respectively. In the across-package training-testing
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model
+15
+Table 5. Comparison of Generalization Ability
+Model
+java.io
+java.util
+javax.swing
+Entity
+Relation
+Entity
+Relation
+Entity
+Relation
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+APIReal
+85.02
+36.97
+51.53
+-
+-
+-
+98.11
+44.54
+61.27
+-
+-
+-
+98.99
+25.62
+40.70
+-
+-
+-
+ARCLIN
+95.93
+70.84
+81.50
+-
+-
+-
+98.55
+41.60
+58.50
+-
+-
+-
+98.64
+56.57
+71.90
+-
+-
+-
+AERJE
+92.00
+89.35
+90.66
+45.87
+37.03
+40.98
+95.17
+94.93
+95.05
+78.68
+81.35
+79.99
+93.89
+89.91
+91.86
+96.92
+52.94
+68.48
+Table 6. Experimental results in a low-resource scenario
+Model
+1-Shot
+5-Shot
+10-Shot
+Entity
+Relation
+Entity
+Relation
+Entity
+Relation
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+APIReal
+80.30
+15.92
+26.57
+-
+-
+-
+86.17
+60.67
+71.20
+-
+-
+-
+83.94
+68.59
+75.49
+-
+-
+-
+ARCLIN
+55.07
+62.38
+58.50
+-
+-
+-
+74.64
+72.91
+73.76
+-
+-
+-
+83.58
+75.52
+79.34
+-
+-
+-
+AERJE
+72.76
+85.06
+78.43
+9.34
+44.49
+15.44
+79.09
+91.62
+84.90
+31.68
+65.96
+42.80
+82.47
+93.74
+87.74
+35.00
+72.94
+47.30
+setting, the performance of AERJE degrades, compared with the non-across-package setting (see
+Table 4). However, when trained on the java.util dataset, AERJE’s F1-score (79.99%) is only about
+1% less than non-across-package setting (81.20%). This suggests that AERJE is capable of dealing
+with the data drift across different packages. In addition, different across-package training-testing
+settings also bring different results. When using java.util for training AERJE, its F1-score is about
+39% higher than the F1-score of AERJE trained on java.io. First, due to java.io having fewer sentences
+with relations than java.util (51 vs 68). Second, java.io data has fewer non-augmented sentences
+with relations than java.util (12 vs 18), which makes java.io data less diverse than java.util.
+Our AERJE has a strong generalization ability in face of the data drift across different API
+packages. This ability comes from the generalization ability of the underlying LLM.
+4.5
+RQ5: AERJE’s Performance in Low-Resource Scenario
+4.5.1
+Motivation. Labor overhead means that the data available for training is limited. In this RQ,
+we want to investigate how well AERJE perform when trained with the extremely small amount of
+training data.
+4.5.2
+Methodology. We conduct a K-shot experiment, where K can be 1, 5, or 10. To begin the
+K-shot experiment, we randomly select K sentences from the final training set for each relation
+type. Then we choose K sentences at random from the final training set that contain only entities
+but no relations. This yields a training set containing 8*k sentences. Finally, we train our AERJE
+and baseline models on this training set and test them on the final test set. Note that, to avoid the
+influence of random sampling, we repeat each K-shot experiment ten times with different samples.
+4.5.3
+Result. For API entity extraction, Table 6 shows the performance of each model in three
+low-resource scenarios (i.e., 1-shot, 5-shot, and 10-shot) where AERJE significantly outperforms
+APIReal and ARCLIN. Especially, in the 1-shot scenario, AERJE’s F1-score is 78.43%, which is
+significantly higher than APIReal’s (26.57%) and ARCLIN’s (58.50%). Compared to APIReal and
+ARCLIN, the LLM-based AERJE has a large amount of prior knowledge from the LLM pre-training.
+As the fine-tuning shot increases, the accuracy of AERJE improves fast, especially on F1-score,
+reaching the F1-score 84.90% at 5-shot and 87.74% at 10-shot.
+For API relation extraction, in the 1-shot scenario, AERJE does not perform well, but it still
+magically achieves the recall 44.49%. However, with only 4 more examples (at 5-shot), the F1-score
+of AERJE significantly increases from below 16% at 1-shot to about 43% at 5-shot. This suggests
+that the underlying LLM can quickly adapt to the API relation extraction task that it does not see
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+16
+Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu
+during pre-training with only a few examples. In the few-shot setting, we see that precision is
+much more difficult to improve than recall. It could be due to the ambiguities of relations, i.e. the
+same type of relation can be expressed in very different forms (as shown in table 1), while different
+types of relations may be expressed in the similar forms (e.g., “API1 be ADJ to API2��� represents
+function similarity or function opposite relation). With only a few examples of each type of relation,
+it makes learning to distinguish between them more difficult. Furthermore, AERJE’s F1-score for
+entity extraction is 78.43% at one-shot, while its F1-score for relation extraction is only 15.44%. The
+primary cause for this is that the training set of one-shot contains almost all API entity ambiguity
+types but only a few API relation ambiguity types. As a result, relation extraction is more difficult
+than entity extraction.
+AERJE can quickly adapt the underlying LLM to the API entity and relation extraction tasks with
+only a small number of fine-tuning data. Prior knowledge in LLM enables this quick adaptation.
+Relation extraction is much harder than entity extraction in the few-shot setting.
+5
+DISCUSSION
+The major threat to internal validity is the manual labeling of training and testing datasets. Incorrect
+human labels could harm modeling training and testing. To mitigate this threat, we invited two
+students to annotate the same content and assigned an author to resolve disagreements in the
+labeling results. However, even humans can’t always tell if a token references an API, especially
+when it comes to common nouns that reference basic computing concepts, such as policy and time,
+which can be either basic noun concepts or APIs (java.security.policy class, java.time package). We
+take a conservative strategy, i.e., common nouns as API entities, unless both annotators agree.
+The threat to external validity is three-fold. The first external threat is that we only collect data
+on Stack Overflow. Although our model performed well on the SO data set, we intend further to
+validate its generalization performance in the other data sources (e.g., Java Tutorial8, SitePoint9,
+and Reddit10). The second external threat is that AERJE has only been tested on Java packages. We
+chose Java because previous work [1, 6, 11] has demonstrated how difficult it is to extract these
+API entities and relations from it. In the future, we plan to expand AERJE to other programming
+languages (such as Python and C#). The third external threat stems from two AERJE components:
+the BERT-based classifier and the T5-based extractor. There are numerous alternative models for
+both components of the model. TextCNN [27] and FastText [28] can be used to build the classifier.
+It is possible to use BART [31] and GPT-3 [32] to implement the extractor. In the future, we will
+compare two AERJE components with alternative models to determine the best performing model.
+6
+RELATED WORK
+API entity and relation extraction is a fundamental work in software engineering. It is useful
+in the construction of knowledge graphs; extracted structured API knowledge can help with
+many software engineering tasks such as API linking [2, 3, 8, 33], API misuse detection [11], API
+recommendation [4, 5], and API comparison [6]. This section describes the methods for extracting
+API entities and relations from unstructured text.
+Bacchelli [7, 34] and Dagenais [3] detect class and method mentions in developer emails, docu-
+mentation and forum posts using regular expressions of distinct orthographic features. Ren [11],
+Huang [1], and Liu [6] extract entities from SO posts using the HTML <code> tag. Bacchelli et
+al. [10] extract coarse-grained structured code fragments from natural language text with island
+8http://www.java2s.com
+9https://www.sitepoint.com/
+10https://www.reddit.com
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model
+17
+parsing. Huang [1], and Liu [6] extract semantic relations between entities based on syntactic
+patterns. However, their API entity and relation extraction method from natural language text
+relies on unique orthographic features of APIs, and suffer from the rule design overhead.
+To mitigate the overhead of rule design, researchers extract API entities using machine learning
+methods. Ye et al. [2] propose APIReal, which uses CRF to identify API entities. They label the
+API entities and non-entities in the sentence with the “BIO” tag and form the pair of the labeled
+sequence and the sequence. They then train CRF on these pairs, and use the trained CRF to label
+the input text, from which they obtain API entities with the “BI” or “B” tag. Huo et al. [13], on
+the other hand, propose ARCLIN, which uses BI-LSTM as encoder and CRF as decoder to identify
+API entities, rather than just CRF. However, these methods suffer from data labeling overhead
+because preparing a large number of high-quality training data for these sequence labeling models
+is unrealistic.
+To solve the two overhead issues mentioned above, researchers use LLM to extract entities. Li
+et al. [35] use BERT and Yan et al. [36] use XLNet [37] to extract entities in the natural language
+domain. These models, however, are limited to a single natural language processing task, i.e., the
+entity extraction only. In order to realize joint extraction of multiple tasks, researchers propose LLM-
+based unified architectural models, such as UIE [20] and OpenUE [38]. In particular, UIE proposes
+SEL to encode different information extraction structures via the hierarchical spotting-associating
+structure. Motivated by this, we consider adapting UIE to the joint API entity-relation extraction.
+However, UIE is not good at dealing with complex sentences, particularly long and ambiguous
+sentences containing API entities and various relations, because UIE has only one static prompt to
+identify all types of API relations. As a result, when confronted with ambiguous sentences, the
+more relation types to recognize, the more noise interference, and the lower the UIE recognition
+rate. In contrast, we propose LLM-based AERJE, which extracts API entities and relations from
+unstructured complex sentences at the same time. Different from UIE, our dynamic prompt design
+could generate a small number of potentially relevant relations for input text to eliminate noise
+interference and lessens the difficulty of API relation extraction.
+7
+CONCLUSION AND FUTURE WORK
+In this paper, we are the first to formulate heterogeneous API extraction and API relation extraction
+task as a sequence-to-sequence task, and proposes AERJE to extract API entities and relations
+from unstructured text simultaneously using pre-trained LLM and dynamic prompt learning. The
+systematic evaluation of AERJE is conducted on a set of long and ambiguous sentences from Stack
+Overflow. The experimental results show that AERJE’s ability to extract API entities and relations
+can be activated with a small amount of data, allowing it to accurately identify API entities and
+relations from complex text that the model has never seen during fine-tuning. In the future, we
+will carry out the plans mentioned in the discussion and apply AERJE to any software engineering
+task supported by API entity and relation extraction, such as API linking, API search, and API
+recommendation.
+ACKNOWLEDGMENTS
+The work is partly supported by the National Nature Science Foundation of China under Grant
+(Nos.62262031, 61902162), the Nature Science Foundation of Jiangxi Province (20202BAB202015),
+the Central Guided Local Science and Technology Development Special Project (20222ZDH04090),
+the Graduate Innovative Special Fund Projects of Jiangxi Province (YC2021-S308, YC2022-S258).
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+18
+Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu
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+, Vol. 1, No. 1, Article . Publication date: January 2023.
+
+20
+Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu
+QING HUANG received the M.S degree in computer application and
+technology from Nanchang University, in 2009, and the PH.D. degree in
+computer software and theory from Wuhan University, in 2018. He is
+currently an Assistant Professor with the School of Computer and Informa-
+tion Engineering, Jiangxi Normal University, China. His research interests
+include information security, software engineering and knowledge graph.
+Yanbang Sun is a second-year master student at the School of Computer
+and Information Engineering, Jiangxi Normal University, China. His re-
+search interests include software engineering and knowledge graph.
+Zhenchang Xing is a Senior Research Scientist with Data61, CSIRO,
+Eveleigh, NSW, Australia. In addition, he is an Associate Professor in
+the Research School of Computer Science, Australian National University.
+Previously, he was an Assistant Professor in the School of Computer Sci-
+ence and Engineering, Nanyang Technological University, Singapore, from
+2012-2016. His main research areas are software engineering, applied data
+analytics, and human-computer interaction.
+MIN YU is a Professor in Communication, Electronic Engineering, and
+Computer Science at Jiangxi Normal University, was a visiting scholar at
+the University of California, Irvine, the USA, and interested in Distributed
+computing, Wireless Sensor Network, and Indoor Positioning.
+Xiwei Xu is a Senior Research Scientist with Architecture& Analytics
+Platforms Team, Data61, CSIRO. She is also a Conjoint Lecturer with UNSW.
+She started working on blockchain since 2015. Her main research interest
+is software architecture. She also does research in the areas of service
+computing, business process, and cloud computing and dependability.
+Qinghua Lu is a Senior Research Scientist with Data61, CSIRO, Eveleigh,
+NSW, Australia. She has published more than 100 academic papers in
+international journals and conferences. Her research interests include the
+software architecture, blockchain, software engineering for AI, and AI
+ethics.
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+

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+The q-neighbor Ising model on multiplex networks with partial overlap of nodes
+A. Krawiecki and T. Gradowski
+Faculty of Physics, Warsaw University of Technology,
+Koszykowa 75, PL-00-662 Warsaw, Poland
+The q-neighbor Ising model for the opinion formation on multiplex networks with two layers in
+the form of random graphs (duplex networks), the partial overlap of nodes, and LOCAL&AND spin
+update rule was investigated by means of the pair approximation and approximate Master equations
+as well as Monte Carlo simulations. Both analytic and numerical results show that for different fixed
+sizes of the q-neighborhood and finite mean degrees of nodes within the layers the model exhibits
+qualitatively similar critical behavior as the analogous model on multiplex networks with layers in
+the form of complete graphs. However, as the mean degree of nodes is decreased the discontinuous
+ferromagnetic transition, the tricritical point separating it from the continuous transition and the
+possible coexistence of the paramagnetic and ferromagnetic phases at zero temperature occur for
+smaller relative sizes of the overlap. Predictions of the simple homogeneous pair approximation
+concerning the critical behavior of the model under study show good qualitative agreement with
+numerical results; predictions based on the approximate Master equations are usually quantitatively
+more accurate, but yet not exact. Two versions of the heterogeneous pair approximation are also
+derived for the model under study, which, surprisingly, yield predictions only marginally different
+or even identical to those of the simple homogeneous pair approximation. In general, predictions of
+all approximations show better agreement with the results of Monte Carlo simulations in the case
+of continuous than discontinuous ferromagnetic transition.
+I.
+INTRODUCTION
+Investigation of the opinion formation process by means of nonequilibrium models has become a firmly established
+research field in statistical physics in the last decades [1]. Many results in this area were obtained using models with
+agents’ opinions represented by spins with discrete (in most cases two) states obeying stochastic dynamics described
+by various rates at which agents change (e.g., flip) their opinions, e.g., the majority-vote model [2–7], the noisy voter
+model [8–10], different versions of the noisy nonlinear and q-voter model [11–20] and the q-neighbor Ising model [21–
+24]. In particular, much effort was devoted to determining conditions under which the above-mentioned models exhibit
+phase transition from a disordered paramagnetic (PM) state in which each opinion appears with the same probability
+to an ordered ferromagnetic (FM) state with one dominant opinion as the parameter controlling the level of stochastic
+noise in the model is varied, measuring the agents’ uncertainty in decision making. In this context the presence of
+the first-order FM transition, or even transition to a frozen FM phase is of prime importance, with abrupt occurrence
+of a dominant opinion as well as possible hysteresis and bistability of the PM and FM phases [4, 5, 12–21, 24].
+Following the growing interest in the dynamical processes on complex networks [25] agents in the models for the
+opinion formation are often located in the nodes and interact via edges of complex networks reflecting a complicated
+structure of social interactions [3–7, 9, 13–16, 18–20, 23, 24]. In this case analytic predictions concerning the critical
+behavior of the models based on the mean-field approximation (MFA) need not exhibit quantitative agreement with
+results of Monte Carlo (MC) simulations, hence, more accurate approaches based on the pair approximation (PA)
+[26–30] and approximate Master equations (AMEs) [28–30] were applied to describe theoretically the observed phase
+transitions [10, 14–16, 18–20, 24].
+Recently much attention has been devoted to combining complex networks in order to create even more complicated
+and heterogeneous structures known in general as ”networks of networks” [31]. An important class of such structures
+is formed by multiplex networks (MNs) which consist of a fixed set of nodes connected by various sets of edges called
+layers [31–33]. In the simplest case, the layers are independently generated random networks with a full overlap of
+nodes, i.e., with each node belonging to all layers, which means it has at least one attached edge from each layer.
+In turn, in MNs with partial overlap of nodes, there are nodes belonging only to some rather than all layers. In
+particular, in the case of MNs with two layers (duplex networks) and partial overlap of nodes, the nodes are divided
+into a class of nodes belonging to both layers and forming the overlap, and two other classes, each consisting of nodes
+belonging only to one of the two layers [34–36] (the node overlap should not be confused with the link overlap [37–39]
+which is negligible in the case of independently generated layers). FM phase transition in equilibrium models on MNs
+was studied, e.g., in the Ising model [40, 41] and a related Ashkin-Teller model [42]. Analogously, FM transition in
+nonequilibrium models for the opinion formation on MNs was studied, e.g., in the majority vote model [44, 45], the
+q-voter model [46–48] and the q-neighbor Ising model [49]. As expected, the critical properties of the nonequilibrium
+models, in particular the extension or confinement of the range of parameters for which the first-order transition occurs,
+strongly depend on the way in which the multiplexity affects the spin-flip rate. In this respect, very interesting seems
+arXiv:2301.03107v1  [cond-mat.stat-mech]  8 Jan 2023
+
+2
+the q-neighbor Ising model with LOCAL&AND spin update rule [50], which so far has been studied by MC simulations
+and in the MF approximation on duplex networks with full and partial overlap of nodes and with layers in the form of
+fully connected graphs [49]. In this model, the flip probability per unit time for the spins in nodes belonging to only
+one layer (i.e., outside the overlap) is given by a Metropolis-like rate, but with a local field produced only by a subset
+of q randomly chosen neighboring spins (q-neighborhood), and for the spins in nodes belonging to both layers (i.e.,
+within the overlap) it is given by a product of two above-mentioned rates evaluated separately for each layer. With
+the increase of the relative size of the overlap, and depending on the size of the q-neighborhood, suppression of the
+first-order transition, appearance of a tricritical point separating first- and second-order FM transition, and possible
+coexistence of the PM and FM phases even in zero temperature were observed in the model [49].
+In this paper, the q-neighbor Ising model on MNs with partial overlap of nodes, with layers in the form of complex
+networks and with the LOCAL&AND spin update rule is studied by means of MC simulations and theoretically in
+the framework of the PA and AMEs. It should be noted that the q-neighbor Ising model is used here as a convenient
+example since the results can be readily compared with the above-mentioned ones for the limiting case of the model
+on MNs with layers in the form of complete graphs [49], and the PA and AMEs used here can be easily generalized
+to other models for the opinion formation with similar structure of interactions. In order to make large systems of
+AMEs numerically tractable in this paper only the case of duplex networks with layers in the form of homogeneous
+random networks is considered; nevertheless, such MNs exhibit certain overlap-induced inhomogeneity since the nodes
+within and outside the overlap form distinct classes characterized by different degrees within the individual layers
+(both non-zero or one zero and one non-zero). Thus also the flip rates for the spins located in nodes belonging to
+distinct classes are different; a related q-voter model with quenched disorder, with agents divided into subpopulations
+according to different rates of the opinion change, has been recently considered [15].
+The aim of this paper is first to provide a general formulation of the PA and AMEs, which take into account to
+a different extent the above-mentioned inhomogeneity of nodes, for models on MNs with partial overlap of nodes.
+For this purpose, first, the homogeneous PA for models on MNs with a full overlap of nodes [47] is extended to the
+case with partial overlap. For nodes belonging to different classes this simplest form of the PA takes into account the
+inhomogeneity of the average directions of spins (opinions) but neglects possible inhomogeneity of the distributions
+of directions of neighboring spins within each layer. For the q-neighbor Ising model predictions of this approximation
+concerning the FM phase transition show surprisingly good agreement with results of MC simulations for a wide
+range of the size of the q-neighborhood, the mean degrees of nodes within layers and the size of the overlap. Then,
+the most advanced approximation based on the AMEs for models on MNs with the full overlap of nodes [45] and
+weighted networks [51] is extended to the case of models on MNs with partial overlap of nodes. Finally, two kinds
+of heterogeneous PA, the fully heterogeneous PA [15] and the AMEs-based heterogeneous PA [28–30] are applied
+to models on MNs with partial overlap of nodes. Both versions of the PA take into account, to a different extent,
+the above-mentioned inhomogeneity of distributions of directions of neighboring spins within each layer and are in
+general intermediate with respect to the accuracy of predictions between the homogeneous PA and the AMEs. For
+the q-neighbor Ising model under study, it turns out that their predictions are only marginally different or even
+identical with these of the homogeneous PA. On the other hand, predictions based on the AMEs show slightly better
+quantitative agreement with the results of MC simulations, in particular for smaller mean degrees of nodes within
+layers. In general, predictions of all approximations concerning the first-order FM transition (e.g., location and width
+of the hysteresis loop) are quantitatively worse than those concerning the second-order transition (e.g., location of
+the critical point). Besides, the aim of this paper is also to study in detail the phase diagram for the q-neighbor Ising
+model on MNs with partial overlap of nodes and with layers with a finite mean degree of nodes. It is shown that the
+critical behavior of this model resembles qualitatively that of the analogous model on MNs with layers in the form
+of fully connected graphs [49]. However, as the mean degree of nodes is decreased, the first-order FM transition, the
+tricritical point separating it from the second-order transition, and the possible coexistence of the PM and FM phases
+occur for smaller relative sizes of the overlap, while the range of the occurrence of the second-order FM transition is
+broadened correspondingly.
+II.
+THE MODEL
+A.
+Multiplex networks with partial overlap of nodes
+MNs consist of a fixed set of nodes connected by several sets of edges; the set of nodes with each set of edges
+forms a network which is called a layer of a MN [32, 33]. Henceforth, the nodes are indexed by i, i = 1, 2, . . . N,
+and the subsequent layers are denoted as G(L), L = A, B, . . . Lmax. In the case of MNs with a full overlap of nodes
+each node belongs to all layers, i.e., each node has at least one edge from each layer attached to it. In general, MNs
+with partial overlap of nodes are defined as MNs in which nodes may belong to (i.e., may have attached edges from)
+
+3
+some rather than all layers, given that each node belongs to at least one layer. Henceforth, the number of nodes
+belonging to the layer G(L) is denoted as N (L). In this paper, it is assumed that the sets of edges for the subsequent
+layers G(L) are generated independently and form complex random networks with N (L) nodes. As a result, multiple
+connections between nodes are not allowed within the same layer, but the same nodes belonging to several layers can
+be accidentally connected by multiple edges belonging to different layers. A simple example of the MN with partial
+overlap of nodes is that with only two layers G(A), G(B), called a duplex network, and with n nodes belonging to
+both layers which form the overlap (0 ≤ n ≤ N); then, N = N (A) + N (B) − n. Furthermore, if both layers contain
+the same number of nodes N (A) = N (B) = ˜N it is possible to introduce a single parameter r = n/ ˜N, also called the
+overlap. Then, the nodes are divided into three subsets: ˜N − n = N(1 − r)/(2 − r) nodes belonging only to the layer
+G(A), N(1 − r)/(2 − r) nodes belonging only to the layer G(B) and n = Nr/(2 − r) nodes belonging both to G(A) and
+G(B).
+The numbers of edges attached to the node i (degrees) within the individual layers G(L) are denoted as k(L)
+i
+; if the
+node i does not belong to the layer G(L) then k(L)
+i
+= 0. In the case of MNs with independently generated layers the
+degrees of nodes belonging to the individual layers G(L), i.e., these with k(L)
+i
+> 0, are drawn from probability distri-
+butions P
+�
+k(L)�
+which characterize the layers as complex networks. For a given node i a vector of its degrees within
+the individual layers ki =
+�
+k(A)
+i
+, k(B)
+i
+, . . . k(Lmax)
+i
+�
+, with possible zero components in the case of MNs with partial
+overlap of nodes, is called a multidegree of the node. The multidegree distribution P(k) = P
+�
+k(A)
+i
+, k(B)
+i
+, . . . k(Lmax)
+i
+�
+characterizes the MN as a complex ”network of networks”; in the case of MNs with the full overlap of nodes and
+independently generated layers, it is obviously P(k) = �Lmax
+L=A P(k(L)). In the formulas below, averages are evaluated
+over the multidegree distribution, e.g., ⟨k(L)⟩ = N −1 �N
+i=1 k(L)
+i
+= �
+k P(k)k(L) is the mean degree of nodes within
+the layer G(L) (note that the average is over all N nodes rather than N (L) nodes belonging to the layer G(L)). As a
+simple example, in this paper the q-neighbor Ising model is considered on a duplex network with partial overlap of
+nodes and with the two independently generated layers in the form of random regular graphs (RRGs) with K edges
+attached to each node belonging to the layer, the same numbers of nodes N (A) = N (B) = ˜N and the overlap r, for
+which the multidegree distribution is
+P (k) = P
+�
+k(A), k(B)�
+= 1 − r
+2 − rδk(A),Kδk(B),0 +
+r
+2 − rδk(A),Kδk(B),K + 1 − r
+2 − rδk(A),0δk(B),K,
+(1)
+and ⟨k(A)⟩ = ⟨k(B)⟩ = ˜NK/N = K/(2 − r).
+B.
+The q-neighbor Ising model on multiplex networks with partial overlap of nodes
+The q-neighbor Ising model [21–24, 49] is a nonequilibrium variant of the Ising model used to investigate the process
+of opinion formation. In this paper the above-mentioned model is considered on MNs with partial overlap of nodes
+and layers in the form of complex networks; the MF version of this model, on MNs with layers in the form of fully
+connected graphs, was studied in Ref. [49]. The main interest is in the FM transition which can occur in the q-neighbor
+Ising model with decreasing effective temperature T, which measures the level of internal noise (uncertainty in agents’
+decision making).
+In order to introduce the model under study, it is convenient to start with the q-neighbor Ising model on (monoplex)
+networks which can be regular, complex, or fully connected graphs [21–24, 49]. In this model agents with two possible
+opinions on a given subject are represented by two-state spins σi = ±1, i = 1, 2, . . . N placed in the nodes and
+interacting via edges of the network. It is assumed that these interactions prefer identical orientations of spins in the
+connected nodes, which is reflected in the spin-flip rate. Thus, interactions between spins with opposite directions in
+general increase the probability that one of the spins flips, i.e., the corresponding agent changes opinion, and edges
+representing these interactions are called active links. The dynamics of the q-neighbor Ising model on networks is a
+modification of that of the kinetic Ising model with the Metropolis spin-flip rate in which, at each time step, each
+spin interacts only with its q randomly chosen neighbors. MC simulations of the model are performed using random
+asynchronous updating of spins, with each MC simulation step (MCSS) corresponding to updating all N spins. Nodes
+are picked randomly and for each picked node q its neighbors are chosen randomly and without repetitions, which
+form the q-neighborhood of the picked node. Then, the spin in the picked node is flipped with probability given by a
+Metropolis-like formula,
+E (l; T, q) = min {1, exp[−2(q − 2l)/T]} ,
+(2)
+where l is the number of nodes belonging to the q-neighborhood occupied by spins with a direction opposite to that
+of the spin in the picked node, i.e., the number of active links attached to the picked node leading to nodes within
+
+4
+the chosen q-neighborhood (notation in Eq. (2) emphasizes that T, q, are parameters of the model). As a result, the
+flip rate for a picked spin given that it is placed in a node with degree k which has in total i active links attached
+(0 ≤ i ≤ k) is
+f (i; T|k) =
+1
+�k
+q
+�
+q
+�
+l=0
+�i
+l
+��k − i
+q − l
+�
+E (l; T, q) =
+1
+�k
+i
+�
+q
+�
+l=0
+�k − q
+i − l
+��q
+l
+�
+E (l; T, q) .
+(3)
+The q-neighbor Ising model on complete graphs for q = 3 exhibits second-order FM transition, while for q ≥ 4
+first-order FM transition occurs with a clearly visible hysteresis loop. Width of the hysteresis loop in general increases
+with q, though for q > 4 there are oscillations superimposed on this trend such that loops for the consecutive odd
+values of q are narrower than for the neighboring even values of q [21]. The same is true for the model on networks
+with finite mean degree ⟨k⟩ provided that q ≪ ⟨k⟩. However, as q is increased and becomes comparable with ⟨k⟩ the
+hysteresis loop becomes narrower and eventually disappears, and the FM transition becomes second-order [24].
+In the q-neighbor Ising model on MNs with full or partial overlap of nodes, interactions take place within individual
+layers with respective, independently chosen q-neighborhoods. Then, spins flip according to a probabilistic rule which
+combines the effect of the above-mentioned interactions. In this paper the LOCAL&AND spin update rule is used [50]
+according to which the spin in the picked node flips if interaction with every q-neighborhood from every layer suggests
+flip; consequently, the probability of the spin-flip is given by a product of the Metropolis-like factors (2) corresponding
+to all layers containing the picked node. The LOCAL&AND rule is assumed in this paper since it usually leads to
+richer phase diagrams than other methods of including the multiplex character of the network of interactions in the
+spin-flip rate [46–49]. Eventually, in numerical simulations of the q-neighbor Ising model on MNs with partial overlap
+of nodes and the LOCAL&AND spin update rule, each MCSS is performed as follows.
+(i.) A node i, 1 ≤ i ≤ N, with multidegree ki is picked randomly.
+(ii.) From each layer G(L) containing the picked node a set of its q neighbors (q-neighborhood) is chosen randomly
+and without repetitions; it is assumed that 0 < q ≤ k(L)
+i
+. Sets from different layers are chosen independently,
+thus the same node can by chance belong to two or more q-neighborhoods if it is a neighbor of the picked node
+within two or more layers.
+(iii.) The Metropolis-like factor for the picked node is evaluated separately for each layer G(L),
+E
+�
+l(L); T, q
+�
+= min
+�
+1, exp[−2(q − 2l(L))/T]
+�
+(4)
+where l(L) is the number of nodes in the q-neighborhood in the layer G(L) occupied by spins with direction
+opposite to that of the spin in the picked node; note that if a node does not belong to G(L) then q = l(L) = 0
+and E(T, 0, 0) = 1.
+(iv.) Due to the LOCAL&AND spin update rule, the spin σi in the picked node flips with probability
+E (l; T, q) =
+Lmax
+�
+L=A
+E
+�
+l(L); T, q
+�
+,
+(5)
+where l =
+�
+l(A), l(B), . . . l(Lmax)�
+; and obviously l(L) = 0 if the picked node does not belong to the layer G(L)
+(i.e., l is a vector of numbers of active links from the individual layers attached to the picked node which lead
+to nodes within the respective q-neighborhoods).
+(v.) Steps (i.)-(iv.) are repeated until all N spins are updated without repetition.
+Hence, the flip rate for a spin placed in a node with multidegree k =
+�
+k(A), k(B), . . . k(Lmax)�
+and with the numbers of
+attached active links within the individual layers i(L), 0 ≤ i(L) ≤ k(L), given by the corresponding components of the
+vector i =
+�
+i(A), i(B), . . . i(Lmax)�
+assumes a multiplicative form,
+f (i; T |k) =
+Lmax
+�
+L=A
+f
+�
+i(L); T|k(L)�
+(6)
+(note that if a node does not belong to the layer G(L) there is k(L) = i(L) = 0 and f (0; T|0) ≡ 1).
+The q-neighbor Ising model on a duplex network with layers in the form of complete graphs and partial overlap
+of nodes, and with the LOCAL&AND spin update rule exhibits FM phase transition already for q ≥ 1 [49]. This
+
+5
+transition is in general second-order, with some exceptions. For q = 2 the transition is first-order for 1/2 < r < 1,
+with a clearly visible hysteresis loop, and for rc < r ≤ 1/2, where rc = 2(3
+√
+2 − 4) = 0.4853 . . ., the coexistence of
+the FM and PM phases is observed as the temperature is decreased below a critical value down to T = 0; for r < rc
+there is no phase transition and the PM phase remains the only stable phase down to T = 0. For q ≥ 4 the transition
+for small r is first-order and for larger r is second-order. The first- and second-order transitions are separated by a
+tricritical point at r = rT CP (q) which for q = 4 occurs at a particularly high value of r, and for q > 4 is an increasing
+function of q, but again with oscillations between the consecutive odd and even values of q superimposed on this
+trend. Remarkably, for r = 1 the FM transition is always second-order for any q, i.e., full overlap of nodes suppresses
+discontinuous transition. In this paper, it is investigated how the phase diagram of the model changes if the layers of
+the MN are complex networks with a finite mean degree of nodes.
+III.
+THEORY
+A.
+Pair approximation
+In the case of spin models on networks, the effect of the network topology (e.g, of the degree distribution or the mean
+degree of nodes) on the observed phase transitions often can be more accurately described in the framework of the
+PA than by the usual MFA [26–30]. In particular, this was demonstrated for the q-neighbor Ising model on complex
+networks [24] and a sort of stochastic q-voter model on MNs with a full overlap of nodes [47]. In both above-mentioned
+studies the networks, or the layers of the MNs, were homogeneous complex networks (e.g., RRGs), thus the simplest
+homogeneous PA was enough to reproduce quantitatively results of MC simulations in a wide range of the parameters
+of the models. As mentioned in Sec. I & II MNs with partial overlap of nodes retain some multiplexity-induced
+inhomogeneity even if the layers are homogeneous complex networks. Nevertheless, in this section the homogeneous
+PA derived in Ref. [47] for a wide class of models with various spin update rules on MNs with the full overlap of nodes
+is presented in a more general form which makes it applicable to models on MNs with partial overlap of nodes, in
+order to find, inter alia, to what extent it can be used to explain critical behavior of systems with multiplicity-induced
+inhomogeneity.
+The advantage of the PA consists in that it takes into account dynamical correlations between pairs of interacting
+agents (spins). In the framework of the homogeneous PA, macroscopic quantities characterizing a model with two-
+state spins on MNs are concentrations ck of spins directed up located in nodes with multidegree k (with possible zero
+components in the case of MNs with partial overlap of nodes) as well as concentrations b(L) of active links within
+separate layers G(L). The homogeneous character of the PA allows for the simplification that the latter concentrations
+are averaged over all nodes belonging to a given layer and do not depend on the multidegrees of the connected nodes.
+Consequently, it is assumed that conditional probabilities θ(L)
+j
+, j ∈ {↑, ↓}, that an active link within the layer G(L) is
+attached to a node given that it is occupied by spin with direction j are also independent of the multidegree of the
+node. These probabilities can be evaluated as ratios of the number of attachments of active links to nodes with spins
+with direction j, independently of their multidegrees, within the layer G(L), which is N⟨k(L)⟩b(L)/2, and the number
+of attachments of all links within GL to such nodes, which is �
+k NP (k) k(L)ck,j, where ck,↑ = ck, ck,↓ = 1 − ck, thus
+θ(L)
+↑
+=
+b(L)
+2 �
+k P (k) k(L)ck,↑/⟨k(L)⟩ =
+b(L)
+2 �
+k P (k) k(L)ck/⟨k(L)⟩,
+(7)
+θ(L)
+↓
+=
+b(L)
+2 �
+k P (k) k(L)ck,↓/���k(L)⟩ =
+b(L)
+2
+�
+1 − �
+k P (k) k(L)ck/⟨k(L)⟩
+�
+(8)
+The core approximation made in the PA for models on MNs is that the numbers of active links i(L) attached
+to a node with degrees k(L) within individual layers G(L) (0 ≤ i(L) ≤ k(L)) occupied by spin with direction j obey
+independent binomial distributions with parameters θ(L)
+j
+given by Eq. (7,8). Then, the rates at which the concentration
+ck increases or decreases are given by averages of the spin-flip rate, Eq. (6), over the appropriate joint distributions
+of the number of active links within all layers which have a multiplicative form
+P(j, i|k) =
+Lmax
+�
+L=A
+Bk(L),i(L)
+�
+θ(L)
+j
+�
+,
+(9)
+where Bk,i(θ) =
+�k
+i
+�
+θi(1 − θ)k−i denotes the binomial factor and, formally, B0,0(θ) ≡ 1. Hence, the equation for the
+time dependence of ck can be written as a rate equation,
+
+6
+∂ck
+∂t =
+�
+j∈{↑,↓}
+(−1)δj,↑ck,j
+�
+i
+Lmax
+�
+L=A
+Bk(L),i(L)
+�
+θ(L)
+j
+�
+f (i; T |k) ,
+(10)
+where �
+i ≡ �k(A)
+i(A)=0 . . . �k(Lmax)
+i(Lmax)=0.
+In order to obtain an equation for the time dependence of the concentrations of active links b(L) one should observe
+that each flip of a spin (irrespective of its direction) in a picked node with multidegree k with the numbers of active
+links attached given by the components of the vector i results in the change of the numbers of active links within
+the individual layers G(L) by k(L) − 2i(L), since then i(L) previously active links become inactive and k(L) − i(L)
+previously inactive links become active. The corresponding changes in the concentrations of active links b(L) are thus
+�
+k(L) − 2i(L)�
+/(N⟨k(L)⟩/2). As in Eq. (10), such changes connected with the flip of a spin with direction j occur at a
+rate given by the average of the spin-flip rate, Eq. (6), over the appropriate joint distributions of the number of active
+links attached to the picked node, Eq. (9). Due to the homogeneous character of the PA, in order to obtain time
+dependence of b(L) further averaging over all nodes occupied by spins with direction j should be performed, which
+is equivalent to averaging over the probability distribution P(k)ck,j that a node with multidegree k is occupied by
+a spin with direction j. Eventually, taking into account that nodes are picked and spins are updated within time
+intervals 1/N, for a given layer G(L′) it is obtained that
+∂b(L′)
+∂t
+=
+2
+⟨k(L′)⟩
+�
+j∈{↑,↓}
+�
+k
+P (k) ck,j
+�
+i
+Lmax
+�
+L=A
+Bk(L),i(L)
+�
+θ(L)
+j
+�
+f (i; T |k)
+�
+k(L′) − 2i(L′)�
+,
+(11)
+where L′ = A, B . . . Lmax.
+In particular, let us consider the q-neighbor Ising model on a MN with two layers in the form of RRGs and partial
+overlap of nodes, with the multidegree distribution given by Eq. (1). Then, the nodes are divided into three classes,
+these belonging only to the layer G(A) with multidegree k = (K, 0), only to the layer G(B) with k = (0, K) and to
+the overlapping part of G(A) and G(B), with k = (K, K). The macroscopic quantities to be used in the homogeneous
+PA are thus concentrations of spins directed up in the nodes belonging to the subsequent classes c(K,0), c(0,K), c(K,K)
+and concentrations of active links in the two layers b(A), b(B). Since both layers are identical, with N (A) = N (B) = ˜N,
+stable solutions of the system of equations (10), (11) are limited to the subspace with c(0,K) = c(K,0), b(A) = b(B) ≡ b;
+moreover, according to Eq. (7,8) there is θ(A)
+j
+= θ(B)
+j
+≡ θj. Using Eq. (1), (3), (6), performing summations in Eq.
+(10), (11) as in Ref. [24] and introducing functions R(θ; T, q) and S(θ; T, K, q) to shorten notation,
+R(θ; T, q) =
+q
+�
+l=0
+Bq,l (θ) E(l; T, q),
+(12)
+S(θ; T, K, q) =
+q
+�
+l=0
+Bq,l (θ) [(K − q)θ + l]E(l; T, q),
+(13)
+the following system of three equations for the time dependence of the macroscopic quantities in the homogeneous
+PA is obtained,
+dc(K,0)
+dt
+=
+�
+1 − c(K,0)
+�
+R (θ↓; T, q) − c(K,0)R (θ↑; T, q)
+(14)
+dc(K,K)
+dt
+=
+�
+1 − c(K,K)
+�
+[R (θ↓; T, q)]2 − c(K,K) [R (θ↑; T, q)]2
+(15)
+db
+dt = 2
+K (1 − r)
+��
+1 − c(K,0)
+�
+[KR (θ↓; T, q) − 2S (θ↓; T, K, q)] + c(K,0) [KR (θ↑; T, q) − 2S (θ↑; T, K, q)]
+�
++ 2
+K r
+��
+1 − c(K,K)
+�
+[KR (θ↓; T, q) − 2S (θ↓; T, K, q)] R (θ↓; T, q)
++ c(K,K) [KR (θ↑; T, q) − 2S (θ↑; T, K, q)] R (θ↑; T, q)
+�
+,
+(16)
+where
+θ↑ =
+b
+2
+�
+(1 − r)c(K,0) + rc(K,K)
+�,
+(17)
+θ↓ =
+b
+2
+�
+1 − (1 − r)c(K,0) − rc(K,K)
+�.
+(18)
+
+7
+Other macroscopic quantities of interest are the concentration of spins directed up in each layer, i.e., the fraction of ˜N
+nodes occupied by such spins, which is ˜c = (1 − r)c(K,0) + rc(K,K), the concentration of spins directed up in the whole
+MN, i.e., the fraction of N nodes occupied by such spins, which is c = 2(1−r)
+2−r c(K,0) +
+r
+2−rc(K,K), and the resulting
+magnetization of the MN m = 2c − 1. Note that in the limiting case of layers in the form of fully connected graphs
+there is b = ˜N 2˜c(1 − ˜c)/[ ˜N( ˜N − 1)/2] ≈ 2˜c(1 − ˜c) and θ↓ = ˜c, θ↑ = 1 − ˜c; after inserting this into Eq. (14) and (15)
+equations for the concentrations c(K,0), c(K,K) in the MF approximation are reproduced [49], as expected.
+Natural extension of the homogeneous PA consists in taking into account heterogeneity of the concentrations of
+the (possibly active) links connecting classes of nodes with different multidegrees, so that, instead of the average
+concentration b(L) of active links within the layer G(L), e.g., concentrations of classes of active links connecting spins
+in nodes with multidegrees k, k′ within the layer G(L) become separate macroscopic quantities characterizing the
+model.
+This leads to the most advanced and accurate version of the PA called fully heterogeneous PA [15, 27];
+corresponding equations for the macroscopic quantities for spin models on MNs with partial overlap of nodes, in
+particular for the q-neighbor Ising model under study, are given in Appendix A. In the latter case solutions of these
+equations show that in the stationary state concentrations of active links (strictly speaking, of their ends called bonds)
+belonging to different classes indeed show noticeable heterogeneity; nevertheless, this does not lead to the values of
+magnetization noticeably different from these predicted by the homogeneous PA. Thus, magnetization curves and
+phase diagrams for the model under study obtained from the fully heterogeneous PA are practically indistinguishable
+from those obtained from the homogeneous PA and do not show better agreement with the results of MC simulations.
+B.
+Approximate Master equations
+A more accurate approximation for the study of spin models on MNs with partial overlap of nodes is based on
+approximate Master equations (AMEs) for the densities of spins directed up ck,m and down sk,m which are located
+in nodes with multidegree k and have m(L) neighboring spins directed up within the consecutive layers G(L), which
+is denoted as m =
+�
+m(A), m(B) . . . m(Lmax)�
+. In the thermodynamic limit and for mutually uncorrelated layers in the
+form of random networks with finite mean degrees
+�
+k(L)�
+possibility that a pair of nodes is connected simultaneously
+by edges within different layers can be neglected. Thus, in the AMEs it is assumed that in a single simulation step
+for a given node the allowed changes of the number of neighboring spins directed up are m → m ± e(L), where e(L)
+is a unit vector with Lmax components and only L-th component equal to one, while simultaneous changes of many
+components of m, e.g., m → m ± e(L) ± e(L′), L ̸= L′, etc., cannot occur. Under the above-mentioned assumptions,
+the AMEs in a general form are [45, 51]
+dsk,m
+dt
+= −Fk,msk,m + Rk,mck,m
++
+Lmax
+�
+L=A
+�
+−β(L)
+s
+�
+k(L) − m(L)�
+sk,m + β(L)
+s
+�
+k(L) − m(L) + 1
+�
+sk,m−e(L)
+�
++
+Lmax
+�
+L=A
+�
+−γ(L)
+s
+m(L)sk,m + γ(L)
+s
+�
+m(L) + 1
+�
+sk,m+e(L)
+�
+,
+(19)
+dck,m
+dt
+= −Rk,mck,m + Fk,msk,m
++
+Lmax
+�
+L=A
+�
+−β(L)
+i
+�
+k(L) − m(L)�
+ck,m + β(L)
+i
+�
+k(L) − m(L) + 1
+�
+ck,m−e(L)
+�
++
+Lmax
+�
+L=A
+�
+−γ(L)
+i
+m(L)ck,m + γ(L)
+i
+�
+m(L) + 1
+�
+ck,m+e(L)
+�
+.
+(20)
+In Eq. (19), (20) the first two terms account for the effect of a flip of a spin in a node with multidegree k and the
+remaining terms account for the average effect of the flips of spins in the neighboring nodes, irrespective of their
+multidegrees. In terms of Sec. II B the flip rate for a spin directed down occupying a node with multidegree k with
+m neighboring spins directed up is Fk,m = f (m; T |k) and that for a spin directed up Rk,m = f (k − m; T |k).
+The remaining average rates can be estimated by evaluating the ratios (at a given time step) of the average number
+of edges connecting spins with a given direction such that one of these spins flips to the average total numbers
+of these edges [28, 29]; in the case of models on MNs this should be done separately for each layer [45, 51].
+Thus β(L)
+s
+=
+� �
+m
+�
+k(L) − m(L)�
+Fk,msk,m
+�
+/
+� �
+m
+�
+k(L) − m(L)�
+sk,m
+�
+, γ(L)
+s
+=
+� �
+m
+�
+k(L) − m(L)�
+Rk,mck,m
+�
+/
+
+8
+� �
+m
+�
+k(L) − m(L)�
+ck,m
+�
+,
+β(L)
+i
+=
+� �
+m m(L)Fk,msk,m
+�
+/
+� �
+m m(L)sk,m
+�
+,
+γ(L)
+i
+=
+� �
+m m(L)Rk,mck,m
+�
+/
+� �
+m m(L)ck,m
+�
+, where L = A, B, . . . Lmax, �
+m ≡ �k(A)
+m(A)=0
+�k(B)
+m(B)=0 . . . �k(Lmax)
+m(Lmax)=0 and ⟨. . .⟩ denotes average
+over the multidegree distribution P (k), as usually. Natural initial conditions for the system of equations (19), (20)
+are sk,m(0) = (1−c(0)) �Lmax
+L=A Bk(L),m(L)(c(0)), ck,m(0) = c(0) �Lmax
+L=A Bk(L),m(L)(c(0)), where 0 < c(0) < 1 is arbitrary.
+In particular, in the case of the q-neighbor Ising model on a MN with two layers in the form of RRGs and partial over-
+lap of nodes, with the multidegree distribution P(k) given by Eq. (1), there are three classes of nodes with k = (0, K),
+k = (K, 0) and k = (K, K). The corresponding spin flip rates are F(K,0),(m(A);0) = f
+�
+m(A); T |K
+�
+, F(0,K),(0;m(B)) =
+f
+�
+m(B); T |K
+�
+, F(K,K),(m(A),m(B)) = f
+�
+m(A); T |K
+�
+f
+�
+m(B); T |K
+�
+and R(K,0),(m(A),0) = f
+�
+K − m(A); T |K
+�
+,
+R(0,K),(0,m(B)) = f
+�
+K − m(B); T |K
+�
+, R(K,K),(m(A),m(B)) = f
+�
+K − m(A); T |K
+�
+f
+�
+K − m(B); T |K
+�
+, with f (m; T |K )
+given by Eq. (3). Hence, the system (19), (20) consists of 2(K +1)2+4(K +1) equations and can be solved numerically
+for moderate K. The quantities of interest, e.g., the concentration c of spins directed up in the MN and the magnetiza-
+tion m = 2c−1 can be evaluated as in Sec. III A using c(K,0) = �K
+m(A)=0 c(K,0),(m(A),0), c(0,K) = �K
+m(B)=0 c(0,0),(0,m(B)),
+c(K,K) = �K
+m(A)=0
+�K
+m(B)=0 c(K,K),(m(A),m(B)).
+The AMEs are a starting point for a more elaborate approximation representing another formulation of the het-
+erogeneous PA [28–30, 45, 51] which takes into account the possible heterogeneity due to different multidegrees k
+of nodes of both the concentrations ck of spins directed up and of the conditional probabilities that a link attached
+to a node is active or, equivalently, leads to a spin with a given (say, up) direction. A general formulation of such
+AMEs-based heterogeneous PA for spin (two-state) models on (monoplex) networks by Gleeson [28, 29] was extended
+to the case of weighted networks [51] and, partly, MNs [45]. It is believed that due to the approximations made the
+AMEs-based heterogeneous PA is in general more accurate than the homogeneous PA and less accurate than the
+fully heterogeneous PA mentioned in Sec. III A. In this paper the AMEs-based heterogeneous PA is applied to spin
+models on MNs with partial overlap of nodes, in particular to the q-neighbor Ising model under study; equations
+for the macroscopic quantities are given in Appendix B. Surprisingly, it turns out that in the stationary state the
+above-mentioned conditional probabilities that a node has a link leading to a spin directed up do not depend on
+whether the node belongs or not to the overlap. Hence, predictions of the AMEs-based heterogeneous PA concerning
+the FM transition in the model under study are identical to those of the homogeneous PA from Sec. III A, so they
+are not further discussed.
+IV.
+RESULTS
+The main results concerning the FM transition in the q-neighbor Ising model on MNs with partial overlap of nodes
+and with layers in the form of complete graphs have been summarized in Sec. II B. These results were obtained in the
+MF approximation and confirmed by MC simulations [49]. In this section first predictions of the homogeneous PA
+of Sec. III A concerning the FM transition in the q-neighbor Ising model on MNs with partial overlap of nodes and
+with layers in the form of RRGs are presented and compared with results of MC simulations. In this case, noticeable
+discrepancies occur between theoretical and numerical results, in particular concerning the first-order FM transition.
+As pointed out in Sec. III, the more advanced fully and AMEs-based heterogeneous PA yield results practically
+indistinguishable or even identical to the homogeneous PA, thus their predictions are only briefly mentioned in the
+Appendix. Finally, it is verified in which cases and to what extent theoretical predictions are improved by using the
+AMEs of Sec. III B.
+In the framework of the homogeneous PA of Sec. III A stationary values of the magnetization m vs. T, corresponding
+to different thermodynamic phases, are given by stable fixed points of the system of equations (14-16) with ˙c(K,0) =
+˙c(K,K) = ˙b = 0; for certain ranges of parameters r, q, K many stable fixed points can coexist for given T, and their
+basins of attraction are then separated by stable manifolds of unstable fixed points. The homogeneous PA predicts
+various critical behavior of the model under study as the temperature T is varied, depending on r, q, K which are
+fixed: first- and second-order FM phase transition, the coexistence of the PM and FM phases for T → 0 and absence
+of the FM transition. At high temperatures, the only stable fixed point is that with m = 0 corresponding to the PM
+phase. In the case of the second-order FM transition, this fixed point loses stability as the temperature is decreased
+below the critical value Tc, and simultaneously a pair of symmetric stable fixed points with m > 0 and m < 0 occurs
+via a supercritical pitchfork bifurcation, corresponding to the two symmetric FM phases. In the case of the first-order
+transition two symmetric pairs of stable and unstable fixed points with m > 0 and m < 0 occur simultaneously
+via two saddle-node bifurcations as the temperature is decreased below the upper critical value T (2)
+c
+, and the two
+above-mentioned stable fixed points correspond to the two symmetric FM phases. As the temperature is further
+decreased both the FM and PM fixed points remain stable (coexist) until the PM point loses stability via a subcritical
+
+9
+(c)
+(d)
+(b)
+(a)
+FIG. 1.
+The curves show magnetization m vs. temperature T obtained from the homogeneous PA for different K (green
+solid lines, both stable and unstable fixed points of the system of equations (14-16) are shown) and from the MFA of Ref.
+[49] (black solid lines), for q = 2, K = 200, 100, 50, 20, 10, 4 (from left) and (a) r = 0.49, (b) r = 0.5, as well as for q = 4,
+K = 500, 200, 100, 50, 20, 10 and (c) r = 0.05, (d) r = 0.15.
+pitchfork bifurcation at the lower critical temperature T (1)
+c
+(T (1)
+c
+< T (2)
+c
+) by colliding simultaneously with the two
+above-mentioned unstable fixed points; coexistence of the PM and FM phases for T (1)
+c
+< T < T (2)
+c
+leads to the
+occurrence of the hysteresis loop in the magnetization curves m(T). Eventually, for T < T (1)
+c
+the only stable fixed
+points remain these corresponding to the two symmetric FM phases. In the case of the coexistence of the FM and
+PM phases for T → 0 a pair of symmetric stable FM fixed points occurs at T = T (2)
+c
+as in the case of the first-order
+transition, but these FM points, as well as the PM fixed point, remain stable (coexist) as T → 0. Finally, it can also
+happen that fixed points corresponding to the FM phase do not exist for any T > 0, thus the FM transition is absent
+and the only stable phase for T → 0 is the PM one.
+Exemplary curves m(T) predicted by the homogeneous PA for the model under study with different K and selected
+values of r are shown in Fig. 1 for the most interesting cases q = 2 and q = 4; in the former case, the MFA (valid
+for the model on MNs with layers in the form of fully connected graphs with K → ∞) predicts occurrence of all
+above-mentioned kinds of the critical behavior for different ranges of r, while in the latter one it predicts occurrence
+of the first-order transition for a particularly wide range of small r. The curves m(T) for q = 2 are drawn in Fig. 1(a)
+for r = 0.49, and in Fig. 1(b) for r = 0.5, i.e., for the values of r within or at the border of the interval rc < r < 0.5
+where the MFA predicts coexistence of the FM and PM phases for T → 0. In contrast, for the model on MNs with
+layers in the form of RRGs the homogeneous PA for r = 0.49 (Fig. 1(a)) predicts second- or first-order FM transition
+for small and moderate K, respectively; the critical temperature(s) decrease and the width of the hysteresis loop
+increases with K. Only for large K coexistence of the FM and PM phases for T → 0 is predicted by the PA, and the
+curves m(T) approach those resulting from the MFA, as expected. For r = 0.5 (Fig. 1(b)) only second- or first-order
+FM transitions for finite K are predicted by the PA, with the lower critical temperature for the first-order transition
+
+10
+(a)
+(b)
+FIG. 2. Critical behavior predicted by the homogeneous PA for the model with q = 2 (left and middle panels) and q = 4 (right
+panel) and different r, K; filled circles — continuous FM transition, open circles — discontinuous FM transition, filled squares
+— coexistence of the FM and PM phases for T → 0, crosses — absence of the transition.
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+FIG. 3. Results of MC simulations, predictions of the PA and AMEs for the model with q = 2, K = 20 and (a) r = 0.45, (b)
+r = 0.46, (c) r = 0.47, (d) r = 0.50, (e) r = 0.60, (f) r = 0.70; blue dots — results of MC simulations with FM initial conditions
+and increasing temperature, red dots — results of MC simulations with PM initial conditions and decreasing temperature, black
+dots — predictions of the AMEs for both FM (c(0) = 1) and PM (c(0) = 0.5) initial conditions and increasing or decreasing
+temperature, respectively, green solid lines — predictions of the PA as in Fig. 1.
+
+60Q0-0000-011
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+FIG. 4. As in Fig. 3 but for q = 4. (a) K = 20, r = 0.05, (b) K = 20, r = 0.10, (c) K = 20, r = 0.15, (d) K = 10, r = 0.10,
+(e) K = 50, r = 0.10, (f) k = 10, r = 0.05.
+T (1)
+c
+> 0. The curves m(T) for q = 4 are drawn in Fig. 1(c) for r = 0.05, and in Fig. 1(d) for r = 0.15, i.e., for the
+values of r where the MFA predicts first-order FM transition with a wide and narrow hysteresis loop, respectively.
+For the model on MNs with layers in the form of RRGs the homogeneous PA for small r = 0.05 (Fig. 1(c)) similarly
+predicts the first-order FM transition for moderate and large K, while for larger r = 0.15 (Fig. 1(d)) it predicts
+the second-order FM transition already for moderate K and the first-order FM transition only for large K. Again,
+the critical temperature(s) decrease, and the width of the hysteresis loop increases with K, and the curves m(T)
+eventually approach these resulting from the MFA.
+It may be inferred from Fig. 1 that the homogeneous PA predicts for the q-neighbor Ising model on MNs with
+partial overlap of nodes and layers in the form of RRGs with finite K the same critical behavior as the MFA for
+the model on analogous MNs with layers in the form of complete graphs, only for different ranges of the overlap r.
+This conclusion is supported by Fig. 2 where the critical behavior predicted by the PA is summarized for the former
+model with fixed q = 2 and q = 4 and different K, r. For all K and r = 1 (full overlap of nodes), both PA and MFA
+predict continuous FM transition with decreasing T, i.e., the first-order transition observed in the model on monoplex
+networks is suppressed. However, for both q = 2, 4 and finite K the PA predicts the second-order FM transition also
+for a range of r below r = 1 which is broadened with decreasing K. As a consequence, for q = 2 (Fig. 2, left and
+middle panels) the PA predicts that the range of the occurrence of the first-order FM transition is shifted toward
+smaller values of r. Similarly, for a narrow range of still smaller values of the overlap the PA predicts the coexistence
+of the FM and PM phases for T → 0, but for small K this kind of critical behavior is expected at r significantly
+below the interval rc < r < 0.5 obtained from the MFA. Finally, it is predicted that the range of small r for which
+the FM transition is absent for decreasing K is narrowed. Eventually, for very small K = 4 comparable with q only
+continuous FM transition is expected for any r, and all other kinds of critical behavior are suppressed. For q = 4 (Fig.
+2, right panel) the range of small r for which the PA predicts the first-order FM transition is substantially diminished
+with decreasing K.
+In order to verify predictions of the homogeneous PA, MC simulations of the q-neighbor Ising model with q = 2, 4
+on large MNs with various parameters r, K were performed and the magnetization curves m(T) were obtained for
+random PM initial conditions σi = ±1, i = 1, 2, . . . N and decreasing temperature as well as for FM initial conditions
+σi = +1, i = 1, 2 . . . N and increasing temperature. Comparison with MC simulations shows that the homogeneous
+PA qualitatively captures modification of the critical behavior of the model under study due to finite values of the
+
+00000000-012
+mean degree of the layers K, but its quantitative predictions, though much improved in comparison with those from
+the MFA valid for large K, are not exact (Fig. 3, 4). For fixed q, K the PA approximately predicts the ranges
+of the overlap r where different kinds of critical behavior should occur. However, as a rule, these predictions are
+overestimated and in MC simulations the particular kinds of critical behavior appear for smaller values of r than
+estimated from the PA. For example, for q = 2 the ranges of appearance of the coexistence of the FM and PM phases
+for T → 0 (Fig. 3(a,b)) and of the second-order FM transition (Fig. 3(e,f)) in MC simulations are, respectively, shifted
+and extended toward smaller values of r than expected from the PA. Consequently, in the case of the first-order FM
+transition for q = 2 (Fig. 3(c,d)) and q = 4 (Fig. 4(a,b,e,f)) the lower and upper critical temperatures T (1)
+c
+, T (2)
+c
+are
+underestimated and the width of the hysteresis loop is overestimated by the PA in comparison with these obtained
+from MC simulations; it is interesting to note that discrepancies between the theoretical and numerical values of T (2)
+c
+are usually smaller than those for T (1)
+c
+. Similarly, in the case of the second-order FM transition for q = 2 (Fig. 3(f))
+and q = 4 (Fig. 4(c,d)) the critical temperature Tc is underestimated by the PA in comparison with that obtained
+from MC simulations. In general, the curves m(T) evaluated from the PA show better agreement with those obtained
+from MC simulations in the case of the second-order than the first-order FM transition.
+In order to investigate the critical behavior of the model under study by means of appropriate AMEs as defined
+in Sec. III B, Eq. (19,20) were solved numerically with various initial conditions and the curves m(T) were obtained
+using long-time asymptotic values of the concentrations of spins directed up c(K,0),(m(A),0), etc., to evaluate stationary
+values of the magnetization. As expected, predictions of the AMEs usually show comparable or better agreement
+with the results of MC simulations than those of the homogeneous PA. This is particularly visible in the case of the
+second-order FM transition in the model under study with small K and q = 2 (Fig. 3(f)) and q = 4 (Fig. 4(d)),
+where the theoretical and numerical curves m(T) coincide very well and the critical temperature Tc is predicted
+correctly. However, the ranges of the overlap predicted by the AMEs for which different kinds of critical behavior
+occur are still shifted toward slightly higher values of r than obtained from MC simulations (see Fig. 3(a), where the
+AMEs predict the absence of the FM transition rather than coexistence of the FM and PM phases observed in MC
+simulations, and Fig. 3(e), where the AMEs predict the first-order FM transition with a narrow hysteresis loop rather
+than the second-order transition). In the case of the coexistence of the FM and PM phases for T ��� 0 for q = 2 (Fig.
+3(b)) and the first-order FM transition for q = 2 (Fig. 3(c,d)) and q = 4 (Fig. 4(a,b,e,f)) predictions of the AMEs
+concerning the upper critical temperature T (2)
+c
+are usually better than those of the homogeneous PA, but the lower
+critical temperature T (1)
+c
+is again usually underestimated and the width of the hysteresis loop is overestimated. In
+general, some improvement of theoretical predictions by the AMEs in comparison with the homogeneous PA can be
+seen for small and moderate K; for large K the curves m(T) obtained from the AMEs and PA coincide (Fig. 4(e)).
+V.
+DISCUSSION AND CONCLUSIONS
+In this paper the q-neighbor Ising model on MNs with partial overlap of nodes and layers in the form of random
+networks was investigated; as an example, the model on MNs with two layers in the form of RRGs was studied in
+detail. Both theoretical considerations based on the homogeneous PA and AMEs as well as MC simulations show
+that for given q ≥ 1 and finite mean degree of nodes K, and for varying overlap r and temperature T the model
+exhibits qualitatively similar critical behavior as the q-neighbor Ising model on MNs with partial overlap of nodes
+and layers in the form of complete graphs. In particular, for any q and full overlap of nodes r = 1 the first-order
+FM transition is suppressed and only the second-order transition appears with decreasing T. Besides, for decreasing
+K continuous rather than discontinuous FM transition is observed for an increasing range of large (for q = 2) and
+large and moderate (for q > 2) values of r below r = 1. As a consequence, for decreasing K the ranges of r for
+which the model exhibits the first-order FM transition (for q ≥ 2) and the coexistence of the FM and PM phases for
+T → 0 (for q = 2) are shifted toward smaller values. It should be mentioned that in the q-neighbor Ising model on
+(monoplex) networks the first-order FM transition is also suppressed for small K comparable with q [24]; in contrast,
+in the model on MNs this suppression is due to the overlap of nodes and occurs for any K. The q-neighbor Ising
+model was used here as an example, and related models for the opinion formation on MNs with partial overlap of
+nodes can be studied using similar numerical and analytic methods; however, the expected qualitative changes of the
+observed critical behavior with r will be probably less spectacular, since, e.g., in the case of the q-voter model even
+for r = 1 the first-order FM transition is not suppressed [47].
+For the model under study with large K predictions of the simple homogeneous PA and more advanced system of
+AMEs converge to these of the MFA and agree quantitatively with the results of MC simulations. For finite K the
+predicted curves m(T) and critical temperature(s) differ quantitatively from the numerically obtained ones; usually,
+the particular kinds of critical behavior are predicted to occur for smaller values of the overlap than observed in MC
+simulations. In general, predictions of both PA and AMEs show better agreement with the results of MC simulations
+
+13
+in the case of the continuous than discontinuous FM transition. Predictions based on the AMEs are comparable to
+or better than those of the PA; in particular, the critical temperature for the second-order FM transition and the
+upper critical temperature for the first-order transition are more accurately predicted. Nevertheless, both PA and
+AMEs qualitatively correctly capture changes of the critical behavior of the model with varying parameters K, r
+characterizing the underlying MN.
+Two versions of the heterogeneous PA were derived for the model under study, the more accurate fully heterogeneous
+PA and the less accurate AMEs-based heterogeneous PA, which take to a different extent into account heterogeneity
+of the distribution of the active links due to inhomogeneity of the nodes. In both cases, systems of equations for
+the macroscopic quantities characterizing the model are significantly larger and more complicated than that in the
+homogeneous PA but do not lead to a noticeable improvement of theoretical predictions concerning the magnetization
+curves and phase diagrams for the model. This suggests that the simple homogeneous PA is as reliable as more
+advanced versions of the PA in the study of the critical behavior of systems with multiplexity-induced inhomogeneity.
+Only using much larger systems of AMEs in certain cases quantitatively improves agreement between theoretical
+predictions and results of MC simulations of the above-mentioned models.
+APPENDIX
+A.
+Fully heterogeneous pair approximation
+The following outline of the fully heterogeneous PA for models on MNs is an extension of that for the q-voter models
+with quenched disorder on networks, with two populations of agents differing by the spin-flip rates [15]. The fully
+heterogeneous PA uses the assumption that the probability that a spin directed up or down in a node with multidegree
+k has a given number of attached edges leading to spins directed up or down in nodes with multidegree ˜k obeys a
+binomial distribution; this assumption is valid for each layer and for any pair k, ˜k separately, and the related binomial
+distributions are assumed to be independent. The macroscopic quantities characterizing a model with two-state spins
+on a MN are concentrations ck of spins directed up in nodes with multidegree k and concentrations ek,˜k,(L)
+j,˜j
+(= e
+˜k,k,(L)
+˜j,j
+)
+of bonds (ends of edges) attached within the layer G(L) to nodes with multidegree k containing spins with direction
+j ∈ {↓, ↑} such that at the other end of the edge there is a node with multidegree ˜k containing spin with direction
+˜j (normalized to the total number of bonds N⟨k(L)⟩ within G(L)). According to the above-mentioned assumptions
+the joint probability that a node with multidegree k containing spin with direction j has i =
+�
+i(A), i(B), . . . i(Lmax)�
+active bonds (ends of active links) attached within the consecutive layers, pointing at nodes with arbitrary multidegree
+containing spins with opposite direction −j, has a multiplicative form
+P(j, i|k) =
+Lmax
+�
+L=A
+Bk(L),i(L)
+�
+αk,(L)
+j
+�
+,
+(21)
+where αk,(L)
+j
+= �
+k′ ek,k′,(L)
+j,−j
+/ �
+k′
+�
+j′∈{↓,↑} ek,k′,(L)
+j,j′
+are conditional probabilities that an active bond is attached to
+a node with multidegree k and spin with direction j (similar to θ(L)
+↑
+, θ(L)
+↓
+given by Eq. (7, 8)). In order to evaluate
+the change in the concentration ek,˜k,(L)
+j,˜j
+due to, e.g., flipping the spin with direction j in a node with multidegree
+k, it is necessary to know the numbers of bonds y, z attached to this node within the layer G(L) pointing at nodes
+with multidegree ˜k given that these bonds are active (˜j = −j) or inactive (˜j = j), respectively. These numbers
+obey binomial distributions Bi(L),y
+�
+βk,˜k,(L)
+j,−j
+�
+, Bk(L)−i(L),z
+�
+γk,˜k,(L)
+j,j
+�
+, respectively, where the conditional probabilities
+are βk,˜k,(L)
+j,−j
+= ek,˜k,(L)
+j,−j
+/ �
+k′ ek,k′,(L)
+j,−j
+, γk,˜k,(L)
+j,j
+= ek,˜k,(L)
+j,j
+/ �
+k′ ek,k′,(L)
+j,j
+. Then the rate equations for the macroscopic
+concentrations of spins directed up are
+∂ck
+∂t =
+�
+j∈{↑,↓}
+(−1)δj,↑ck,j
+�
+i
+Lmax
+�
+L=A
+Bk(L),i(L)
+�
+αk,(L)
+j
+�
+f (i; T |k) ,
+(22)
+while the rate equations for the concentrations of active and inactive bonds within the layers contain terms such as
+d
+dtek,˜k,(L′)
+j,−j
+=
+1
+⟨k(L′)⟩P(k)ck,j
+�
+i
+Lmax
+�
+L=A
+Bk(L),i(L)
+�
+αk,(L)
+j
+� i(L′)
+�
+y=0
+Bi(L′),y
+�
+βk,˜k,(L′)
+j,−j
+�
+(−y)f (i; T |k) + . . .
+(23)
+
+14
+d
+dtek,˜k,(L′)
+j,j
+=
+1
+⟨k(L′)⟩P(k)ck,j
+�
+i
+Lmax
+�
+L=A
+Bk(L),i(L)
+�
+αk,(L)
+j
+� k(L′)−i(L′)
+�
+z=0
+Bk(L′)−i(L′),z
+�
+γk,˜k,(L′)
+j,j
+�
+(−z)f (i; T |k) + . . .
+(24)
+It can be seen that Eq. (22) resembles Eq. (10) in the homogeneous PA. Concerning the equations for the concentrations
+of bonds, e.g., Eq. (23) states that a flip of the spin with direction j in node with multidegree k, which has i(L′)
+active bonds attached within the layer G(L′), out of which y bonds point at nodes with multidegrees ˜k, decreases the
+concentration ek,˜k,(L′)
+j,−j
+by y/
+�
+N⟨k(L′)⟩
+�
+; such a flip occurs with probability P(k)ck,jf (i; T |k) within a time interval
+1/N; and the final input to the rate equation (23) is obtained by averaging the above-mentioned change over the
+probability distributions Bk(L′),i(L′)
+�
+αk,(L)
+j
+�
+for i(L′) and Bi(L′),y
+�
+βk,˜k,(L′)
+j,−j
+�
+for y; etc.
+In the case of the q-neighbor Ising model on MNs with partial overlap of nodes and with two layers in the form
+of RRGs, with the multidegree distribution P (k) given by Eq. (1), there are three classes of nodes with k = (K, 0),
+k = (0, K) and k = (K, K), and two layers G(L), L = A, B. Taking into account the symmetry of the model under
+study and general symmetry conditions for the concentrations ek,˜k,(L)
+j,˜j
+the solutions of the system of equations (22 - 24)
+can be constrained to a 12-dimensional subspace c(K,0) = c(0,K), e(K,0),(K,0),(A)
+j
+,
+j′
+= e(K,0),(K,0),(A)
+j′
+,
+j
+= e(0,K),(0,K),(B)
+j
+,
+j′
+=
+e(0,K),(0,K),(B)
+j′
+,
+j
+≡ e(K,0),(K,0)
+j
+,
+j′ , e(K,0),(K,K),(A)
+j
+,
+j′
+= e(K,K),(K,0),(A)
+j′
+,
+j
+= e(0,K),(K,K),(B)
+j
+,
+j′
+= e(K,K),(0,K),(B)
+j′
+,
+j
+≡ e(K,0),(K,K)
+j
+,
+j′ ,
+e(K,K),(K,K),(A)
+j
+,
+j′
+= e(K,K),(K,K),(A)
+j′
+,
+j
+= e(K,K),(K,K),(B)
+j
+,
+j′
+= e(K,K),(K,K),(B)
+j′
+,
+j
+≡ e(K,K),(K,K)
+j
+,
+j′
+, j, j′ ∈ {↓, ↑}. Thus, as in
+Ref. [15], there are effectively only two classes of agents located in nodes with k = (K, 0) and k = (K, K), differing
+by the spin-flip rates (6). Besides, the distributions of the number of links pointing at nodes belonging to each class
+given that these links are active or inactive are fully determined by the conditional probabilities βk,k
+j,−j, γk,k
+j,j for the
+links within each class. Taking this into account and performing summations in Eq. (22 - 24) as in Ref. [24] the
+following system of equations for the macroscopic quantities is obtained in the fully heterogeneous PA for the model
+under study,
+dc(K,0)
+dt
+=
+�
+1 − c(K,0)
+�
+R
+�
+α(K,0)
+↓ ; T, q
+�
+− c(K,0)R
+�
+α(K,0)
+↑ ; T, q
+�
+(25)
+dc(K,K)
+dt
+=
+�
+1 − c(K,K)
+� �
+R
+�
+α(K,K)
+↓
+; T, q
+��2
+− c(K,K)
+�
+R
+�
+α(K,K)
+↑
+; T, q
+��2
+(26)
+d
+dte(K,0),(K,0)
+↑
+,
+↑
+= −2(1 − r)
+K
+c(K,0)γ(K,0),(K,0)
+↑
+,
+↑
+�
+KR
+�
+α(K,0)
+↑ ; T, q
+�
+− S
+�
+α(K,0)
+↑ ; T, K, q
+��
++ 2(1 − r)
+K
+�
+1 − c(K,0)
+�
+β(K,0),(K,0)
+↓
+,
+↑ S
+�
+α(K,0)
+↓ ; T, K, q
+�
+(27)
+d
+dte(K,0),(K,0)
+↓
+,
+↓
+= 2(1 − r)
+K
+c(K,0)β(K,0),(K,0)
+↑
+,
+↓ S
+�
+α(K,0)
+↑ ; T, K, q
+�
+− 2(1 − r)
+K
+�
+1 − c(K,0)
+�
+γ(K,0),(K,0)
+↓
+,
+↓
+�
+KR
+�
+α(K,0)
+↓ ; T, q
+�
+− S
+�
+α(K,0)
+↓ ; T, K, q
+��
+(28)
+d
+dte(K,K),(K,K)
+↑
+,
+↑
+= −2r
+K c(K,K)γ(K,K),(K,K)
+↑
+,
+↑
+�
+KR
+�
+α(K,K)
+↑
+; T, q
+�
+− S
+�
+α(K,K)
+↑
+; T, K, q
+��
+R
+�
+α(K,K)
+↑
+; T, q
+�
++ 2r
+K
+�
+1 − c(K,K)
+�
+β(K,K),(K,K)
+↓
+,
+↑
+S
+�
+α(K,K)
+↓
+; T, K, q
+�
+R
+�
+α(K,K)
+↓
+; T, K, q
+�
+(29)
+d
+dte(K,K),(K,K)
+↓
+,
+↓
+= 2r
+K c(K,K)β(K,K),(K,K)
+↑
+,
+↓
+S
+�
+α(K,K)
+↑
+; T, K, q
+�
+R
+�
+α(K,K)
+↑
+; T, K, q
+�
+− 2r
+K
+�
+1 − c(K,K)
+�
+γ(K,K),(K,K)
+↓
+,
+↓
+�
+KR
+�
+α(K,K)
+↓
+; T, q
+�
+− S
+�
+α(K,K)
+↓
+; T, K, q
+��
+R
+�
+α(K,K)
+↓
+; T, q
+�
+(30)
+d
+dte(K,0),(K,0)
+↑
+,
+↓
+= 1 − r
+K
+c(0,0)
+�
+−β(K,0),(K,0)
+↑
+,
+↓ S
+�
+α(K,0)
+↑ ; T, K, q
+�
++ γ(K,0),(K,0)
+↑
+,
+↑
+�
+KR
+�
+α(K,0)
+↑ ; T, q
+�
+− S
+�
+α(K,0)
+↑ ; T, K, q
+���
++ 1 − r
+K
+�
+1 − c(K,0)
+� �
+γ(K,0),(K,0)
+↓
+,
+↓
+�
+KR
+�
+α(K,0)
+↓ ; T, q
+�
+− S
+�
+α(K,0)
+↓ ; T, K, q
+��
+− β(K,0),(K,0)
+↓
+,
+↑ S
+�
+α(K,0)
+↓ ; T, K, q
+��
+(31)
+d
+dte(K,K),(K,K)
+↑
+,
+↓
+= r
+K c(K,K)
+�
+−β(K,K),(K,K)
+↑
+,
+↓
+S
+�
+α(K,K)
+↑
+; T, K, q
+�
+
+15
++ γ(K,K),(K,K)
+↑
+,
+↑
+�
+KR
+�
+α(K,K)
+↑
+; T, q
+�
+− S
+�
+α(K,K)
+↑
+; T, K, q
+���
+R
+�
+α(K,K)
+↑
+; T, q
+�
++ r
+K
+�
+1 − c(K,K)
+� �
+γ(K,K),(K,K)
+↓
+,
+↓
+�
+KR
+�
+α(K,K)
+↓
+; T, q
+�
+− S
+�
+α(K,K)
+↓
+; T, K, q
+��
+− β(K,K),(K,K)
+↓
+,
+↑
+S
+�
+α(K,K)
+↓
+; T, K, q
+��
+R
+�
+α(K,K)
+↓
+; T, q
+�
+(32)
+d
+dte(K,0),(K,K)
+↑
+,
+↑
+= −1 − r
+K
+c(K,0)
+�
+1 − γ(K,0),(K,0)
+↑
+,
+↑
+� �
+KR
+�
+α(K,0)
+↑ ; T, q
+�
+− S
+�
+α(K,0)
+↑ ; T, K, q
+��
++ 1 − r
+K
+�
+1 − c(K,0)
+� �
+1 − β(K,0),(K,0)
+↓
+,
+↑
+�
+S
+�
+α(K,0)
+↓ ; T, K, q
+�
+− r
+K c(K,K)
+�
+1 − γ(K,K),(K,K)
+↑
+,
+↑
+� �
+KR
+�
+α(K,K)
+↑
+; T, q
+�
+− S
+�
+α(K,K)
+↑
+; T, K, q
+��
+R
+�
+α(K,K)
+↑
+; T, q
+�
++ r
+K
+�
+1 − c(K,K)
+� �
+1 − β(K,K),(K,K)
+↓
+,
+↑
+�
+S
+�
+α(K,K)
+↓
+; T, K, q
+�
+R
+�
+α(K,K)
+↓
+; T, q
+�
+(33)
+d
+dte(K,0),(K,K)
+↓
+,
+↓
+= 1 − r
+K
+c(K,0)
+�
+1 − β(K,0),(K,0)
+↑
+,
+↓
+�
+S
+�
+α(K,0)
+↑ ; T, K, q
+�
+− 1 − r
+K
+�
+1 − c(K,0)
+� �
+1 − γ(K,0),(K,0)
+↓
+,
+↓
+� �
+KR
+�
+α(K,0)
+↓ ; T, q
+�
+− S
+�
+α(K,0)
+↓ ; T, K, q
+��
++ r
+K c(K,K)
+�
+1 − β(K,K),(K,K)
+↑
+,
+↓
+�
+S
+�
+α(K,K)
+↑
+; T, K, q
+�
+R
+�
+α(K,K)
+↑
+; T, q
+�
+− r
+K
+�
+1 − c(K,K)
+� �
+1 − γ(K,K),(K,K)
+↓
+,
+↓
+� �
+KR
+�
+α(K,K)
+↓
+; T, q
+�
+− S
+�
+α(K,K)
+↓
+; T, K, q
+��
+R
+�
+α(K,K)
+↓
+; T, q
+�
+(34)
+d
+dte(K,0),(K,K)
+↑
+,
+↓
+= −1 − r
+K
+c(K,0)
+�
+1 − β(K,0),(K,0)
+↑
+,
+↓
+�
+S
+�
+α(K,0)
+↑ ; T, K, q
+�
++ 1 − r
+K
+�
+1 − c(K,0)
+� �
+1 − γ(K,0),(K,0)
+↓
+,
+↓
+� �
+KR
+�
+α(K,0)
+↓ ; T, q
+�
+− S
+�
+α(K,0)
+↓ ; T, K, q
+��
++ r
+K c(K,K)
+�
+1 − γ(K,K),(K,K)
+↑
+,
+↑
+� �
+KR
+�
+α(K,K)
+↑
+; T, q
+�
+− S
+�
+α(K,K)
+↑
+; T, K, q
+��
+R
+�
+α(K,K)
+↑
+; T, q
+�
+− r
+K
+�
+1 − c(K,K)
+� �
+1 − β(K,K),(K,K)
+↓
+,
+↑
+�
+S
+�
+α(K,K)
+↓
+; T, K, q
+�
+R
+�
+α(K,K)
+↓
+; T, q
+�
+(35)
+d
+dte(K,0),(K,K)
+↓
+,
+↑
+= 1 − r
+K
+c(K,0)
+�
+1 − γ(K,0),(K,0)
+↑
+,
+↑
+� �
+KR
+�
+α(K,0)
+↑ ; T, q
+�
+− S
+�
+α(K,0)
+↑ ; T, K, q
+��
+− 1 − r
+K
+�
+1 − c(K,0)
+� �
+1 − β(K,0),(K,0)
+↓
+,
+↑
+�
+S
+�
+α(K,0)
+↓ ; T, K, q
+�
+− r
+K c(K,K)
+�
+1 − β(K,K),(K,K)
+↑
+,
+↓
+�
+S
+�
+α(K,K)
+↑
+; T, K, q
+�
+R
+�
+α(K,K)
+��
+; T, q
+�
++ r
+K
+�
+1 − c(K,K)
+� �
+1 − γ(K,K),(K,K)
+↓
+,
+↓
+� �
+KR
+�
+α(K,K)
+↓
+; T, q
+�
+− S
+�
+α(K,K)
+↓
+; T, K, q
+��
+R
+�
+α(K,K)
+↓
+; T, q
+�
+,
+(36)
+where the significant conditional probabilities are
+α(K,0)
+↓
+=
+e(K,0),(K,0)
+↓
+,
+↑
++ e(K,0),(K,K)
+↓
+,
+↑
+e(K,0),(K,0)
+↓
+,
+↑
++ e(K,0),(K,K)
+↓
+,
+↑
++ e(K,0),(K,0)
+↓
+,
+↓
++ e(K,0),(K,K)
+↓
+,
+↓
+α(K,0)
+↑
+=
+e(K,0),(K,0)
+↑
+,
+↓
++ e(K,0),(K,K)
+↑
+,
+↓
+e(K,0),(K,0)
+↑
+,
+↓
++ e(K,0),(K,K)
+↑
+,
+↓
++ e(K,0),(K,0)
+↑
+,
+↑
++ e(K,0),(K,K)
+↑
+,
+↑
+α(K,K)
+↓
+=
+e(K,K),(K,K)
+↓
+,
+↑
++ e(K,K),(K,0)
+↓
+,
+↑
+e(K,K),(K,K)
+↓
+,
+↑
++ e(K,K),(K,0)
+↓
+,
+↑
++ e(K,K),(K,K)
+↓
+,
+↓
++ e(K,K),(K,0)
+↓
+,
+↓
+α(K,K)
+↑
+=
+e(K,K),(K,K)
+↑
+,
+↓
++ e(K,K),(K,0)
+↑
+,
+↓
+e(K,K),(K,K)
+↑
+,
+↓
++ e(K,K),(K,0)
+↑
+,
+↓
++ e(K,K),(K,K)
+↑
+,
+↑
++ e(K,K),(K,0)
+↑
+,
+↑
+,
+(37)
+β(K,0),(K,0)
+↓
+,
+↑ =
+e(K,0),(K,0)
+↓
+,
+↑
+e(K,0),(K,0)
+↓
+,
+↑
++ e(K,0),(K,K)
+↓
+,
+↑
+, β(K,0),(K,0)
+↑
+,
+↓ =
+e(K,0),(K,0)
+↑
+,
+↓
+e(K,0),(K,0)
+↑
+,
+↓
++ e(K,0),(K,K)
+↑
+,
+↓
+
+16
+2.1
+2.2
+2.3
+2.4
+2.5
+2.6
+2.7
+T
+−1.0
+−0.5
+0.0
+0.5
+1.0
+m
+FIG. 5. The curves show magnetization m vs. temperature T obtained from the homogeneous PA (solid lines) and from the
+heterogeneous PA (symbols) for q = 4, K = 20 and r = 0.1, 0.15, 0.2 (from left to right).
+β(K,K),(K,K)
+↓
+,
+↑
+=
+e(K,K),(K,K)
+↓
+,
+↑
+e(K,K),(K,K)
+↓
+,
+↑
++ e(K,K),(K,0)
+↓
+,
+↑
+, β(K,K),(K,K)
+↑
+,
+↓
+=
+e(K,K),(K,K)
+↑
+,
+↓
+e(K,K),(K,K)
+↑
+,
+↓
++ e(K,K),(K,0)
+↑
+,
+↓
+,
+(38)
+γ(K,0),(K,0)
+↓
+,
+↓ =
+e(K,0),(K,0)
+↓
+,
+↓
+e(K,0),(K,0)
+↓
+,
+↓
++ e(K,0),(K,K)
+↓
+,
+↓
+, γ(K,0),(K,0)
+↑
+,
+↑ =
+e(K,0),(K,0)
+↑
+,
+↑
+e(K,0),(K,0)
+↑
+,
+↑
++ e(K,0),(K,K)
+↑
+,
+↑
+γ(K,K),(K,K)
+↓
+,
+↓
+=
+e(K,K),(K,K)
+↓
+,
+↓
+e(K,K),(K,K)
+↓
+,
+↓
++ e(K,K),(K,0)
+↓
+,
+↓
+, γ(K,K),(K,K)
+↑
+,
+↑
+=
+e(K,K),(K,K)
+↑
+,
+↑
+e(K,K),(K,K)
+↑
+,
+↑
++ e(K,K),(K,0)
+↑
+,
+↑
+.
+(39)
+Concentration ˜c of spins directed up within each layer and concentration c of spins directed up in the MN are defined
+in the same way as in Sec. III A. Natural initial conditions for the system of equations (25 - 36) are c(K,0)(0) =
+c(K,K)(0) = ρ0, e(K,0),(K,0)
+↑
+,
+↑ (0) = (1 − r)2ρ2
+0, e(K,0),(K,0)
+↑
+,
+↓ (0) = (1 − r)2ρ0(1 − ρ0), e(K,0),(K,0)
+↓
+,
+↓ (0) = (1 − r)2(1 − ρ0)2,
+e(K,K),(K,K)
+↑
+,
+↑
+(0) = r2ρ2
+0, e(K,K),(K,K)
+↑
+,
+↓
+(0) = r2ρ0(1 − ρ0), e(K,K),(K,K)
+↓
+,
+↓
+(0) = r2(1 − ρ0)2, e(K,0),(K,K)
+↑
+,
+↑
+(0) = (1 − r)rρ2
+0,
+e(K,0),(K,K)
+↓
+,
+↓
+(0) = (1 − r)r(1 − ρ0)2, e(K,0),(K,K)
+↑
+,
+↓
+(0) = (1 − r)2ρ0(1 − ρ0) = e(K,0),(K,0)
+↓
+,
+↑ (0) = (1 − r)rρ0(1 − ρ0), where
+ρ0 can be chosen arbitrarily.
+As mentioned in Sec. III A the magnetization curves obtained from the fully heterogeneous PA are practically
+indistinguishable from those obtained from the homogeneous PA. This is illustrated by examples in Fig. 5.
+B.
+AMEs-based heterogeneous pair approximation
+The AMEs-based heterogeneous PA again uses the assumption that the probability that a spin directed up or down
+in a node with multidegree k has a given number of neighboring spins directed up obeys a binomial distribution; for
+models on MNs this assumption is made for each layer separately, and the related binomial distributions are assumed
+to be independent. Hence, in contrast with the homogeneous PA, in the AMEs-based heterogeneous PA it is taken
+into account that for a node with multidegree k occupied by a spin with downward or upward direction the respective
+probabilities ϑ(L)
+k , η(L)
+k
+that a randomly chosen neighboring node within the layer G(L) is occupied by a spin directed
+upward can depend on k. However, in contrast with the fully heterogeneous PA developed in Appendix A, all active
+or inactive edges attached to a given node within a given layer are treated in the same way and obey common binomial
+distributions [28, 29]. As mentioned in Sec. III B, these two assumptions should make the AMEs-based heterogeneous
+PA more accurate than the homogeneous PA and less accurate than the fully heterogeneous PA. Eventually, in the
+AMEs-based heterogeneous PA the time-dependent macroscopic quantities are the density ck of spins directed up in
+nodes with multidegree k as well as the above-mentioned probabilities ϑ(L)
+k , η(L)
+k .
+In terms of the densities ck,m and sk,m used in the AMEs, Eq. (19), (20) the above-mentioned macroscopic
+quantities can be expressed as ck = �
+m ck,m = 1 − �
+m sk,m, ϑ(L)
+k
+= �
+m m(L)sk,m/
+�
+k(L) (1 − ck)
+�
+, η(L)
+k
+=
+
+17
+�
+m m(L)ck,m/
+�
+k(L)ck
+�
+.
+Then, the core approximation for the AMEs-based heterogeneous PA can be made, ac-
+cording to which sk,m ≈ (1 − ck) �Lmax
+L=A Bk(L),m(L)
+�
+ϑ(L)
+k
+�
+, ck,m ≈ ck
+�Lmax
+L=A Bk(L),m(L)
+�
+η(L)
+k
+�
+. The latter approxi-
+mation should be made in Eq. (19), (20) as well as in the definitions of the average rates β(L)
+s
+, . . . , γ(L)
+i
+, so that,
+e.g., β(L′)
+s
+≈ ¯β(L′)
+s
+=
+�
+(1 − ck) �
+m
+�
+k(L′) − m(L′)�
+Fk,m
+�Lmax
+L=A Bk(L),m(L)
+�
+ϑ(L)
+k
+� �
+/
+�
+(1 − ck) k(L′) �
+1 − ϑ(L′)
+k
+� �
+, etc.
+Differentiating the definitions of ck, ϑ(L)
+k , η(L)
+k
+with respect to time and using Eq. (19), (20) with the above-mentioned
+approximations yields the following system of equations for the time dependence of the macroscopic quantities in the
+heterogeneous PA,
+dck
+dt = −ck
+�
+m
+Rk,m
+Lmax
+�
+L=A
+Bk(L),m(L)
+�
+η(L)
+k
+�
++ (1 − ck)
+�
+m
+Fk,m
+Lmax
+�
+L=A
+Bk(L),m(L)
+�
+ϑ(L)
+k
+�
+,
+(40)
+dϑ(L′)
+k
+dt
+=
+�
+m
+�
+ϑ(L′)
+k
+− m(L′)
+k(L′)
+� �
+Fk,m
+Lmax
+�
+L=A
+Bk(L),m(L)
+�
+ϑ(L)
+k
+�
+−
+ck
+1 − ck
+Rk,m
+Lmax
+�
+L=A
+Bk(L),m(L)
+�
+η(L)
+k
+��
++¯β(L′)
+s
+�
+1 − ϑ(L′)
+k
+�
+− ¯γ(L′)
+s
+ϑ(L′)
+k
+,
+(41)
+dη(L′)
+k
+dt
+=
+�
+m
+�
+η(L′)
+k
+− m(L′)
+k(L′)
+� �
+Rk,m
+Lmax
+�
+L=A
+Bk(L),m(L)
+�
+η(L)
+k
+�
+− 1 − ck
+ck
+Fk,m
+Lmax
+�
+L=A
+Bk(L),m(L)
+�
+ϑ(L)
+k
+��
++¯β(L′)
+i
+�
+1 − η(L′)
+k
+�
+− ¯γ(L′)
+i
+η(L′)
+k
+,
+(42)
+where L′ = A, B . . . Lmax. The above equations are very similar to those obtained in the AMEs-based heterogeneous
+PA for the spin models on (monoplex) networks [28, 29]; in particular, terms containing β(L)
+s
+, . . . , γ(L)
+i
+with L ̸= L′
+do not occur in Eq. (41), (42) for ϑ(L′)
+k
+, η(L′)
+k
+since the respective terms from Eq. (19), (20) sum up to zero in the
+derivation. It should be mentioned that the AMEs can also be a starting point to obtain the homogeneous PA from
+Sec. III A by assuming that the probability that a spin directed down has within the layer G(L) a neighboring spin
+directed up does not depend on k and can be expressed as the average θ(L)
+↓
+= ⟨�
+m m(L)sk,m⟩/⟨k(L) (1 − ck)⟩ [28, 29].
+In the case of the q-neighbor Ising model on MNs with partial overlap of nodes and with layers in the form of
+RRGs, with the multidegree distribution P (k) given by Eq. (1), there are three classes of nodes with k = (K, 0),
+k = (0, K) and k = (K, K), and two layers G(L), L = A, B, thus the system of equations (40-42) is 11-dimensional.
+Due to the symmetry of the model solutions of these equations should be constrained to a subspace c(0,K) = c(K,0),
+ϑ(A)
+(K,0) = ϑ(B)
+(0,K) ≡ ϑ(0,K), η(A)
+(K,0) = η(B)
+(0,K) ≡ η(0,K), ϑ(A)
+(K,K) = ϑ(B)
+(K,K) ≡ ϑ(K,K), η(A)
+(K,K) = η(B)
+(K,K) ≡ η(K,K) which
+reduces the number of equations to six. Performing summations in Eq. (40-42) as in Ref. [24] the following system of
+equations for the macroscopic quantities is obtained in the AMEs-based heterogeneous PA for the model under study,
+dc(K,0)
+dt
+= −c(K,0)R
+�
+1 − η(K,0); T, q
+�
++
+�
+1 − c(K,0)
+�
+R
+�
+ϑ(K,0); T, q
+�
+,
+(43)
+dϑ(K,0)
+dt
+= ϑ(K,0)
+�
+R
+�
+ϑ(K,0); T, q
+�
+−
+c(K,0)
+1 − c(K,0)
+R
+�
+1 − η(K,0); T, q
+��
+− 1
+K
+�
+S
+�
+ϑ(K,0); T, K, q
+�
+−
+c(K,0)
+1 − c(K,0)
+�
+KR
+�
+1 − η(K,0); T, q
+�
+− S
+�
+1 − η(K,0); T, K, q
+���
++ ¯βs
+�
+1 − ϑ(K,0)
+�
+− ¯γsϑ(K,0),
+(44)
+dη(K,0)
+dt
+= η(K,0)
+�
+R
+�
+1 − η(K,0); T, q
+�
+− 1 − c(K,0)
+c(K,0)
+R
+�
+ϑ(K,0); T, q
+��
+− 1
+K
+��
+KR
+�
+1 − η(K,0); T, q
+�
+− S
+�
+1 − η(K,0); T, K, q
+��
+− 1 − c(K,0)
+c(K,0)
+S
+�
+ϑ(K,0); T, K, q
+��
++ ¯βi
+�
+1 − η(K,0)
+�
+− ¯γiη(K,0),
+(45)
+dc(K,K)
+dt
+= −c(K,K)
+�
+R
+�
+1 − η(K,K); T, q
+��2 +
+�
+1 − c(K,K)
+� �
+R
+�
+ϑ(K,K); T, q
+��2 ,
+(46)
+dϑ(K,K)
+dt
+= ϑ(K,K)
+��
+R
+�
+ϑ(K,K); T, q
+��2 −
+c(K,K)
+1 − c(K,K)
+�
+R
+�
+1 − η(K,K); T, q
+��2
+�
+− 1
+K
+�
+S
+�
+ϑ(K,K); T, K, q
+�
+R
+�
+ϑ(K,K); T, q
+�
+
+18
+−
+c(K,K)
+1 − c(K,K)
+�
+KR
+�
+1 − η(K,0); T, q
+�
+− S
+�
+1 − η(K,0); T, K, q
+��
+R
+�
+1 − η(K,K); T, q
+��
++ ¯βs
+�
+1 − ϑ(K,K)
+�
+− ¯γsϑ(K,K),
+(47)
+dη(K,K)
+dt
+= η(K,K)
+��
+R
+�
+1 − η(K,K); T, q
+��2 − 1 − c(K,K)
+c(K,K)
+�
+R
+�
+ϑ(K,K); T, q
+��2
+�
+− 1
+K
+��
+KR
+�
+1 − η(K,0); T, q
+�
+− S
+�
+1 − η(K,0); T, K, q
+��
+R
+�
+1 − η(K,K); T, q
+�
+−1 − c(K,K)
+c(K,K)
+S
+�
+ϑ(K,K); T, K, q
+�
+R
+�
+ϑ(K,K); T, q
+��
++ ¯βi
+�
+1 − η(K,K)
+�
+− ¯γiη(K,K),
+(48)
+where the average rates are
+¯βs =
+�1 − r
+2 − r
+�
+1 − c(K,0)
+�
+K
+�
+1 − ϑ(K,0)
+�
++
+r
+2 − r
+�
+1 − c(K,K)
+�
+K
+�
+1 − ϑ(K,K)
+��−1
+×
+�1 − r
+2 − r
+�
+1 − c(K,0)
+� �
+KR
+�
+ϑ(K,0); T, q
+�
+− S
+�
+ϑ(K,0); T, K, q
+��
++
+r
+2 − r
+�
+1 − c(K,K)
+� �
+KR
+�
+ϑ(K,K); T, q
+�
+− S
+�
+ϑ(K,K); T, K, , q
+��
+R
+�
+ϑ(K,K); T, q
+��
+,
+(49)
+¯γs =
+�1 − r
+2 − rc(K,0)K
+�
+1 − η(K,0)
+�
++
+r
+2 − rc(K,K)K
+�
+1 − η(K,K)
+��−1
+×
+�1 − r
+2 − rc(K,0)S
+�
+1 − η(K,0); T, K, q
+�
++
+r
+2 − rc(K,K)S
+�
+1 − η(K,K); T, K, q
+�
+R
+�
+1 − η(K,K); T, q
+��
+,
+(50)
+¯βi =
+�1 − r
+2 − r
+�
+1 − c(K,0)
+�
+Kϑ(K,0) +
+r
+2 − r
+�
+1 − c(K,K)
+�
+Kϑ(K,K)
+�−1
+×
+�1 − r
+2 − r
+�
+1 − c(K,0)
+�
+S
+�
+ϑ(K,0); T, K, q
+�
++
+r
+2 − r
+�
+1 − c(K,K)
+�
+S
+�
+ϑ(K,K); T, K, q
+�
+R
+�
+ϑ(K,K); T, q
+��
+,
+(51)
+¯γi =
+�1 − r
+2 − rc(K,0)Kη(K,0) +
+r
+2 − rc(K,K)Kη(K,K)
+�−1
+×
+�1 − r
+2 − rc(K,0)
+�
+KR
+�
+1 − η(K,0); T, q
+�
+− S
+�
+1 − η(K,0); T, K, q
+��
++
+r
+2 − rc(K,K)
+�
+KR
+�
+1 − η(K,K); T, q
+�
+− S
+�
+1 − η(K,K); T, K, q
+��
+R
+�
+1 − η(K,K); T, q
+��
+.
+(52)
+Concentration ˜c of spins directed up within each layer and concentration c of spins directed up in the MN are defined in
+the same way as in Sec. III A. Natural initial conditions for the system of equations (43-48) are ϑ(K,0)(0) = η(K,0)(0) =
+ϑ(K,K)(0) = η(K,K)(0) = ˜c(0), while c(K,0)(0), c(K,K)(0) can be chosen arbitrarily.
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+

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+Solar Physics
+DOI: 10.1007/•••••-•••-•••-••••-•
+Reconstruction of the Sunspot Number Source
+Database and the 1947 Zurich Discontinuity
+Fr´ed´eric Clette1 · Laure Lef`evre1 ·
+Sabrina Bechet1 · Renzo Ramelli2 ·
+Marco Cagnotti3
+© Springer ••••
+Abstract The recalibration of the sunspot number series, the primary long-
+term record of the solar cycle, requires the recovery of the entire collection of
+raw sunspot counts collected by the Zurich Observatory for the production of
+this index between 1849 and 1980.
+Here, we report about the major progresses accomplished recently in the con-
+struction of this global digital sunspot number database, and we derive global
+statistics of all the individual observers and professional observatories who pro-
+vided sunspot data over more than 130 years.
+First, we can announce the full recovery of long-lost source-data tables covering
+the last 34 years between 1945 and 1979, and we describe the unique information
+available in those tables. We then also retrace the evolution of the core observing
+team in Zurich and of the auxiliary stations. In 1947, we find a major disruption
+in the composition of both the Zurich team and the international network of
+auxiliary stations.
+This sharp transition is unique in the history of the Zurich Observatory and
+coincides with the main scale-jump found in the original Zurich sunspot number
+series, the so-called “Waldmeier” jump. This adds key historical evidence ex-
+plaining why methodological changes introduced progressively in the early 20th
+century could play a role precisely at that time. We conclude on the remaining
+steps needed to fully complete this new sunspot data resource.
+Keywords: Sunspots, statistics; Solar Cycle, observations
+� F. Clette
+frederic.clette@oma.be
+1
+Royal Observatory of Belgium, 3 Avenue Circulaire, 1180 Brussels, Belgium
+2
+Istituto Ricerche Solari Locarno (IRSOL), Universit`a della Svizzera italiana, Via
+Patocchi 57, 6600 Locarno, Switzerland
+3
+Specola Solare Ticinese, Via ai Monti 146, 6605 Locarno, Switzerland
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 1
+arXiv:2301.02429v1  [astro-ph.SR]  6 Jan 2023
+
+Clette et al.
+1. Introduction
+Our knowledge of the long-term evolution of the solar cycle is largely based on
+the historical observations of sunspots since the newly invented telescope was
+aimed at the Sun for the first time in 1610. Two main indices were built from
+those sunspot observations. The sunspot number (hereafter SN) was initiated by
+Rudolf Wolf in 1850 (Wolf, 1856; Friedli, 2016). This daily index combines the
+total group count and the total spot count and its goes back to 1700. Much more
+recently, Hoyt and Schatten (1998a,b) introduced the sunspot group number
+(hereafter GN), which only uses the total group count, but was constructed back
+to the very first telescopic observations in 1610. Both indices are abundantly
+used by most studies of the long-term evolution of solar activity and Sun-Earth
+relations, as constraints for validating physical models of the solar dynamo, and
+for calibrating various parameters relevant to space weather and space climate
+(geomagnetic and ionospheric indices, cosmogenic radionucleides).
+However, significant disagreements between the sunspot number and group
+number series over their common time interval indicated that either series or
+both suffered from inhomogeneities. This prompted various efforts to identify
+flaws and biases in both series, which led to the release of the first revised
+versions of the group number (Svalgaard and Schatten, 2016, “backbone” GN)
+and of the sunspot number (Clette et al., 2014; Clette and Lef`evre, 2016, SN
+Version 2.0). Regarding the GN, further corrections and improvements have been
+proposed over recent years, but we will not develop this ongoing work here (see
+e.g. Chatzistergos et al., 2017; Willamo, Usoskin and Kovaltsov, 2017; Svalgaard
+and Schatten, 2016; Svalgaard, 2020; Usoskin, Kovaltsov and Kiviaho, 2021).
+However, a key element that supported this revision effort was the expansion
+and correction of the GN database collecting all original observed group counts
+(Vaquero et al., 2016). This work, which started from the original database
+assembled over many years by Hoyt and Schatten (1998a,b), is still continuing
+now, and already allowed new improved reconstructions of the GN directly from
+the base source data. As highlighted by Mu˜noz-Jaramillo and Vaquero (2019),
+the recovery of all existing historical observations is crucial for future progresses
+in such reconstructions of past solar activity.
+By contrast, the current revised SN series was reconstructed from source data
+only for the recent decades, since 1981, when the production of the SN moved
+from the Zurich Observatory to the Royal Observatory of Belgium, where it is
+still maintained today (Clette et al., 2007, 2016). Indeed, the data processing was
+then computerized, and all collected data from the worldwide network of con-
+tributing stations are preserved in digital form (more than 500,000 observations
+from 285 stations). On the other hand, for the entire Zurich period before 1981,
+the corrected SN series was obtained by deriving and applying correction factors
+to the original Zurich SN series, as provided by Wolf and his successors (Clette et
+al., 2014; Clette and Lef`evre, 2016). This approach already allowed to correct the
+main flaws present in the original SN series and affecting long segments of this
+series, in particular a sharp 18% upward jump in 1947 (see Clette and Lef`evre,
+2016, for the details), but it faces limitations for finer corrections.
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 2
+
+Sunspot Number Database and the 1947 Zurich Discontinuity
+This more indirect and limited approach was imposed by two main constraints
+that are specific to the history of the sunspot number. While the GN was directly
+built from the whole set of available observations, the Zurich SN was mostly
+based on the sunspot counts from the Zurich Observatory, which acted as pilot
+station. The data from auxiliary stations were mostly used to fill in the daily
+gaps due, e.g., to bad weather in Zurich, and they thus only played a secondary
+role in the production of the early part of the SN (Clette et al., 2014; Dudok de
+Wit, Lef`evre and Clette, 2016; Friedli, 2016, 2020). As a consequence, the sources
+of inhomogeneity are predominantly associated with a single reference station,
+and are thus very different from the GN, which requires other diagnostics.
+However, the other major restriction was the absence of a global digital
+database of the source data collected by Wolf and his successors. As we will
+describe later in this article, only part of those data were published, and none
+of those data were converted into digital form. The inaccessibility of the Zurich
+source data prevents researchers from getting access to a huge amount of detailed
+information and to essential metadata. The recovery of this vast collection can
+feed full statistical analyses by current state-of-the-art methods and lead to an
+improved index, independent of all assumptions and practices adopted over the
+years by Wolf and his successors at the Observatory of Zurich.
+This is what motivated a collective effort to recover and digitize all those
+original source data. Major progresses have been accomplished over the past
+few years. In this article, we report on those major advances. In Section 2, we
+first present the global digitization of the published data, available in printed
+form, and complemented by deeper archives of hand-written logbooks. Based
+on the resulting global chronology of all contributing observers assembled in
+Section 3, we summarize the temporal evolution of the sources on which the SN
+was founded. In Section 4, we then present the recent recovery of the long-lost
+Waldmeier archives, and we describe the contents of those new tables. Based
+on the now-continuous historical timeline, we show the occurrence of a double
+discontinuity in the composition of the Zurich team of observers (Section 5) and
+the network of auxiliary stations (Section 6). In Section 7, we finish by concluding
+on the overall Zurich history emerging from this early exploration of the new SN
+database, and on the prospects and upcoming tasks.
+2. Complete Digitization of Published Tables (1849-1944)
+2.1. The Zurich Printed Data: Full Survey of the Mitteilungen
+The Zurich sunspot number produced by Wolf and his successors is based on
+three types of data:
+•
+the raw counts from the Zurich staff: essentially, the director and the assis-
+tants in Zurich, and also in the course of the 20th century, other assistants
+stationed in the Arosa and Locarno observatories in southern Switzerland.
+•
+the counts sent to the Observatory of Zurich by external auxiliary observers,
+either individual solar observers or professional observatories.
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 3
+
+Clette et al.
+•
+the historical observations collected by Wolf over the course of his entire
+career, which extend the first two sets of data before 1849 and back to
+1610. Most of those numbers were recounted by Wolf himself from original
+documents (Friedli, 2020).
+Most of this material was published on a yearly basis in the bulletins of the Zurich
+Observatory, the Astronomische Mitteilungen der Eidgen¨ossischen Sternwarte
+Z¨urich (hereafter Mitteilungen). This is a fundamental resource for any future
+recomputation of the SN series. As noted in the introduction, a large part of
+those data were never directly used for the production of the sunspot number,
+as on most days, the SN was simply the raw Wolf number from the Zurich
+Observatory.
+In each issue of the Mitteilungen, the source data are listed in a series of
+numbered rubrics at the end of the issue. The rubric series starts in 1857 (Volume
+3, page 126) and ends in 1930 (Volume 122, page 41), at the 1727th entry,
+forming all together a very comprehensive census of all data collected by the
+Zurich Observatory. Systematic observations by the Zurich observers (with the
+director and his assistants listed separately from 1870 onward) and by auxiliary
+observers are presented in yearly tables (Figure 1) with, for each observed day,
+the number of groups g and number of spots s, in the standard format g.s.
+The table is preceded by a brief description of the observer, mainly his/her
+name, the general location (city), and in most cases, the kind of telescope used
+for the observations (aperture, focal length and magnification). Symbols are
+sometimes added in the table to mark changes on a daily basis. The symbol
+may identify a specific observer when there are several observers working in the
+same observatory. In other cases, it marks a change of location or instrument. A
+prominent example involves Wolf himself, who observed either with the standard
+83 mm Fraunhofer refractor mounted permanently at the Zurich Observatory or
+with smaller portable refractors (Friedli, 2016, 2020). This auxiliary information
+can thus prove essential for the proper exploitation of the raw data.
+Sometimes, when Wolf includes a new observer who already collected spot
+counts over many years, a long multi-year table is published with all those past
+observations. Key examples are the tables for Staudacher (Vol. 4, 1857), Schwabe
+(Vol. 10 , 1859), Flaugergues (Vol. 13 , 1861), Carrington (Vol. 35, 1873) or
+Pastorff (Vol. 36, 1875). Finally, next to the tables, many rubrics mention small
+isolated data sets, or even unique spot counts found in old documents during
+searches that Wolf did in libraries all over Europe. These are mostly single
+sunspot sightings that are embedded in textual descriptions, e.g. spots noticed
+at the occasion of solar eclipses. Although they may be individually important,
+all together, they form only a tiny fraction of Mitteilungen data (< 1%), and
+they are less exploitable because they cannot be calibrated.
+2.2. The Digitization: First Milestone
+So far, this large collection of data was completely inaccessible in digital form,
+by contrast with the GN database, which includes all raw group counts collected
+by Hoyt and Schatten (1998a,b) and was recently expanded by Vaquero et al.
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 4
+
+Sunspot Number Database and the 1947 Zurich Discontinuity
+Figure 1. Facsimile of a typical yearly table, as published in the Mitteilungen (first page
+going up to early June). This table lists all daily observations from A.Wolfer for the year 1890.
+Each column gives the date followed by the total number of groups and total number of spots,
+separated by a dot. A star symbol is added for some days, and marks the days when the
+observations were made occasionally with a different telescope (On these days, Wolfer used a
+small portable “Parisian” telescope with a 40 mm aperture).
+(2016). Although there is a rather wide overlap between the GN and SN data and
+many observers are common to both data sets, the GN database unfortunately
+contains only the number of groups. Therefore, the number of spots can only
+be found in the Zurich data, as it was required to compute the SN. This thus
+motivates the construction of a complete SN database, equivalent to the existing
+GN database.
+As a first major step, in 2018, a full encoding of the Mitteilungen data tables
+was undertaken at the World Data Center Sunspot Index and Long-term Solar
+Observations (SILSO), with the help of students for the bulk encoding work.
+By the end of 2019, all the data tables have been digitized, forming the first
+version of the SN database, which includes all data published between 1849,
+when R. Wolf undertook the production of the sunspot number, and 1944, when
+the last director, Max Waldmeier, decided to cease publishing raw data in print.
+This database now contains 205,000 individual daily sunspot counts. Isolated
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 5
+
+625) Alfred Wolfer, Beobachtungen der Sonnenfecken
+auf der Sternwarte in Zurich im Jahre 1890. (Fortsetzung
+zu 604.)
+1890
+1890
+1890
+1890
+1890
+1
+1|1.1
+II
+14/1.1
+III
+17/0.0
+IV
+15|1.2
+V
+10/1.11
+21
+1.1
+1610.0*
+18|0.0
+161.3
+11/2.11
+41.1
+200.0
+190.0
+170.0
+122.13
+1.1
+210.0
+210.0
+18/0.0
+140.0
+6
+2.5
+2210.0
+22/0.0
+19/0.0
+15/0.0
+18/0.0
+25|0.0
+23|1.3
+20|0.0*
+160.0
+19/1.3*
+26/0.0
+240.0
+21/0.0
+17|3.11
+20/1.3*
+270.0
+26/0.0
+22/0.0
+18|2.11
+24/0.0
+28/1
+1.1
+270.0
+230.0
+192.6
+250.0
+III
+11
+1.1
+280.0
+240.0
+203.6
+26|0.0
+2
+0.0
+290.0
+25/1.1
+22|2.5
+270.0
+3|1
+1.1
+300.0
+260.0
+23|1.1
+280.0
+411
+1.6
+31/0.0
+270.0
+24|1.1
+29
+0.0
+1.6
+IV
+10.0
+28/1.3
+25/0.0
+30|1.2
+7
+1.5
+20.0
+291.7
+261.10
+31/1.6
+8]
+1.10
+40.0*
+30/1.11
+270.0
+II
+11.3
+9|1
+1.10
+50.0*
+V
+1|1.1
+290.0
+2|0.0*
+10/1.16
+60.0
+2|0.0
+300.0
+30.0
+11/1.11
+70.0
+30.0
+310.0
+40.0
+12|1.6
+9/0.0*
+4/0.0
+VI
+1/0.0
+50.0
+13|1.3
+10/0.0
+5|0.0
+20.0
+1010.0
+14/1
+1.3
+122.10
+60.0
+30.0
+110.0
+15/1
+1.1
+.13/2.8
+70.0
+一
+4|0.0
+12|0.0
+16/0.0
+141.1
+9|1.2
+一
+5|1.5
+NB. Die mit * bezeichneten Beobachtungen sind mit einem
+kleinern Fernrohr gemacht, welchem etwa der Factor 1,5 zukommt.
+April1891,
+**Clette et al.
+numbers mentioned in textual rubrics are not yet included, but we plan to add
+them later on, for the sake of historical completeness.
+Next to the daily separate counts of spots and groups, the database includes
+metadata derived from annotations in the printed tables. When daily symbols
+indicated regular changes of observers or instruments and when each subset
+included a large number of days, we split the data included in common tables,
+and attached the subsets to distinct observers. So, an observer may appear in
+different incarnations, corresponding to different instruments and/or locations,
+which thus require a different calibration and should not be mixed.
+Currently, this first major input to the SN database is subjected to a thorough
+quality control, fixing typos, date inconsistencies and occasional ambiguities in
+observer names. Meanwhile, we looked for other data sources that can help
+recovering information that proved to be missing in the Mitteilungen. One of
+the gaps happens in the early part of the SN database.
+2.3. Wolf’s Sourcebook and Wolfer’s Global Register
+Indeed, before 1870, the information about the core observations made by Wolf
+and his assistants is incomplete. A single “master” yearly table contains all the
+counts used to produce the sunspot number. It thus consists mainly of the counts
+made by Wolf, which are thus largely complete. On the other hand, data from
+other observers, assistants or external observers, are only inserted on days when
+the primary observer could not observe. As a consequence, between 1864, when
+the first assistants were recruited, and 1869, only a small fraction of the data
+from the Zurich assistants appear in the Mitteilungen, as Wolf’s own data fill a
+majority of days.
+Moreover, before 1864, Wolf’s main auxiliary observer was Samuel Heinrich
+Schwabe. However, although a significant fraction of the daily counts were from
+Schwabe, Wolf did not mark them in the published tables before 1859, as he
+first considered Schwabe’s numbers fully equivalent to his own. This now makes
+it impossible to distinguish Wolf’s primary counts from rescaled numbers from
+Schwabe during the first 10 years of the Wolf series. This important information
+about the primary Zurich observers is thus largely incomplete between 1849 and
+1870.
+Fortunately, two additional sources that provide full tables of the base counts
+were preserved, and are archived at the ETH Zurich University Archives of
+the Eidgen¨ossische Technische Hochschule (ETH). One of them is the so-called
+Wolf’s sourcebook (Wolf, 1878, catalogue entry Hs368:46). Those handwritten
+yearly tables gather all daily numbers forming the sunspot number series from
+1610 to 1877 (Figure 2). In fact, these are the master tables assembled by Wolf
+(Friedli, 2016). Those tables provide two kinds of unique information. Firstly,
+right from the start of Wolf’s yearly census in 1849, they include symbols identi-
+fying the source observer for each daily number. This additional information will
+thus allow to remove the ambiguity in the early Mitteilungen tables. Moreover,
+each yearly table indicates the personal k coefficient that was actually used
+by Wolf, a precious information that can be crossed with the few occasional
+mentions by Wolf of changes in his k calculations.
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 6
+
+Sunspot Number Database and the 1947 Zurich Discontinuity
+Figure 2. Facsimile of the table for 1860 in Wolf’s hand-written sourcebook, which covers the
+period 1610 to 1877 (ETH catalogue entry Hs368:46). The layout is similar to the equivalent
+yearly table published in the Mitteilungen, but it contains very important additional infor-
+mation. Symbols indicate for each day, from which observer the daily sunspot number was
+obtained. One can see that most of the observations were from Wolf, as primary observer. For
+each auxiliary observer, including here Schwabe and Carrington, a list at bottom left mentions
+the personal k coefficient that was used to rescale the raw numbers, to match the scale of
+Wolf’s own numbers.
+Moreover, as the copying and typesetting process for the publication in the
+Mitteilungen most probably led to errors and typos, the original sourcebook
+provides the ground truth and will allow fixing those occasional mistakes in the
+master database. Thanks to the efforts of the Wolf Gesellschaft (Friedli, 2016),
+Wolf’s sourcebook was digitized from 1849 to 1877, when the collection ends.
+While the tables can now be consulted online at URL http://www.wolfinstitute.
+ch/data-tables.html, this extended information must still be merged with the pri-
+mary Mitteilungen database. This work is now in preparation. Finally, the yearly
+tables in the sourcebook actually go back to the very first sunspot observations
+in the early 17th century. Although this part is less substantial, those data tables
+for years before 1849 must still be digitized.
+However, like in the Mitteilungen, the sourcebook does not contain the full set
+of raw observations collected by Wolf from the auxiliary observers and from his
+assistants, between 1849 and 1870, in particular, the observations from Schwabe.
+However, a larger set of handwritten tables also exists at the ETH Zurich
+University archives (Wolfer, 1909, catalogue entry Hs1050:227). This series is a
+standardized compilation of all data and metadata published in the Mitteilungen,
+up to 1908 (Figure 3). This huge register was first produced by Wolf, and after
+Wolf’s death in 1893, it was continued by A. Wolfer and his assistants until 1909,
+as a base for a global verification of the sunspot number series. In this collection,
+there is a separate table by observer and by year. Therefore, the full data set is
+included, even data that were never used for the calculation of the daily Zurich
+sunspot number. In particular, there are also many data series from before 1700,
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 7
+
+147
+8
+6.32
+92
+.21
+121
+11 114
+10.5
+90
+7
+81
+3.19
+cm
+.43
+10.3
+12
+70.9
+34
+52
+132
+.34
+104
+10
+9
+74s
+96
+G
+52
+35
+94
+97
+.32
+037
+3
+.29
+2.8
+^.
+.32
+.22
+115
+10
+23
+121
+116
+.2 
+.7
+:8
+14
+73
+h
+5 .192
+761
+131
+64
+44
+4
+3.5
+154
+128
+8.36
+11
+5.22
+74
+57
+2.8.
+718
+.2.
+4.
+.9
+36
+C
+16
+3.15
+4 5
+.24
+85
+.1c
+4.7.
+103
+710
+34
+.31
+mm
+w
+53
+11
+99
+.25.
+.21
+14
+112
+7
+5.12
+双
+3 0
+50
+.19 
+4.14
+6x
+.15
+127
+.61
+114
+4.15
+好
+15
+6
+.9:
+.13
+30
+72
+15
+118
+6.33
+94
+.26
+848
+60
+4
+.13
+39
+3 7
+55
+9
+8t:
+10 4
+w
+5.1
+89529858848
+51
+91
+7
+6.13
+2 4
+104
+8.23
+103
+.4
+5.18
+6.13
+.34
+94
+1
+4.14
+6.22
+85
+.16
+:6
+4 . 7
+83
+19
+25
+4.22
+5.10
+.43
+113
+2.2
+6.1元
+64
+97
+85
+19
+6.41
+104
+89
+93
+93
+5.18
+91
+6.23.
+6
+.11
+3h
+58
+7.30
+1α1
+6.24
+84
+6.5
+99
+101
+5.25
+5.2
+.14
+w
+.135
+94
+8.31
+to,
+136
+13
+3k
+65
+6.18
+45
+133
+11.34
+14 4
+73
+6
+.13
+双
+.9
+6.8
+s1
+6o
+54
+128
+10.65
+9.17
+160
+.33
+93
+.19
+.18
+58
+8.25
+708
+49
+50
+9.13
+160
+G
+6
+84
+.21
+712
+46
+.39
+12,
+7.28
+6
+g.11
+3.30
+49
+1.61
+.16
+1u 4
+93
+27
+77.
+111
+36
+w
+10 1
+10 .19,
+132
+164
+10. 43
+143
+10.64
+11 .160
+.33
+123
+2 1
+154
+78
+29
+11. 83
+8.16
+.
+25
+44
+109
+23
+125
+114
+94
+10.47
+11.72
+-24
+114
+3 18
+6c
+19
+86
+2
+8.31
+992
+10.46
+2.11
+33
+4.20
+3
+.17
+59
+33
+3.32.
+10 2.
+5.16
+4.8
+5
+6.37
+97
+7.44
+114
+10.31
+4.
+5.19
+103
+5.13
+13
+3 .
+tei
+6
+K
+6.13
+10 9
+w
+44
+M.
+116.7
+100,3
+92,2
+107,1
+108,6
+M.
+9011
+97·9
+95,6
+M.
+M.
+81,5
+88.0
+98.9 
+71.4
+Bemerkungen :
+Bemerkungen:
+T,00
+= 1,50
+2
+Jehwae
+= 1,25
+1859
+=
+(amington
+=
+7, 03
+webs
+ht.si.
+2h
+fmeatmth!
+0,47
+4o Vergl.m*tw,k,3
+=
+Jhon
+=
+7,11
+46Clette et al.
+Figure 3. Facsimile of one page from the register of hand-written tables compiled by R. Wolf
+and continued by A. Wolfer, and covering the entire period 1610 - 1908 (ETH catalogue entry
+Hs1050:227). This page shows the yearly table for Flaugergues in 1796. The layout is similar
+to the yearly tables in Wolf’s sourcebook, but here, all daily observations are listed for each
+observer. On the right, literal citations and detailed indications are often included to clarify
+the interpretation of the tabulated numbers. This series of tables thus gives a complete and
+well-standardized view of all data collected by Wolf and Wolfer, including data that were not
+used to produce the daily sunspot number, and also data and metadata that were not published
+in the Mitteilungen.
+which were never used by Wolf, as he decided to compute the sunspot number
+only from 1700 onwards.
+Still, the tables in this complete register may prove invaluable for crossing
+this information collected long ago by Wolf with other recovered observations
+of the same observers. They also indicate which data were known by Wolf and
+his collaborators at the epoch when they produced the Zurich numbers. The
+scanning of this large set of tables is now planned at the ETH Library in Zurich.
+When this step will be completed, the encoding into a database will require
+substantial additional work.
+3. Chronology of the Data
+Although series of data and metadata still needs to be added, the database is now
+largely complete between 1849 and 1944, and we have now already a complete
+chronology of all the observers who provided data to the Zurich Observatory
+between 1849 and 1980, i.e. during the entire Zurich era. This allows us to derive
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 8
+
+1196
+Jlru yuyur I 100
+1796
+区
+X
+11] rap 4 110
+1.8
+o.0
+1.s0s
+0.0
+00
+0.0
+2
+3. 4
+6.0
+2 ·
+00
+0.0
+3
+0.0
+0. 0
+0.0
+0.0
+n wlit luhi
+4
+2.4
+0·0
+0.0
+2.8
+0.0
+13( Jlg imum un
+0 .0
+0.0
+2.3
+0.0
+2.9
+0.0
+0.0
+0.0
+lauhuy sd de ylun ln grrus du
+6
+2.5
+H
+7
+0.0
+2.3
+0.0
+2. ~
+0.0
+hin
+1.1
+0.0
+2. ~
+0.0
+9
+0.0
+1.1
+0.0
+1.1
+0-0
+1.~
+S g' hu nuir 'ni rlri wn
+(0
+1.2
+0.0
+1.1
+0.0
+1.(
+(
+2·3
+1.7
+le g muud ri lesrli;
+2.3
+6.0
+2.3
+0.0
+1.1
+0.0
+13
+1.7
+0.0
+1.1
+0-0
+1.s
+1-9
+14
+0.0
+1·3
+0.0
+1. G
+1.f
+0.0
+1·3
+0.0
+duervrd ln us, tuhis
+16
+1-3
+0.0
+1.5
+0.0
+0.0
+1、-
+1.6
+13
+0.0
+0.0
+0.0
+0.0
+2-
+C
+2.14
+1. 2
+1.-.
+Tvmts
+8/
+o.d
+1.9
+1.1
+- 3
+mliimms dyuiy ci muir
+19
+1.2
+1·2
+0.6
+0.0
+20
+Cro
+2 -(2
+1.2
+1.1
+0.0
+21.
+00
+nrrt l rcle
+0.0
+2
+- 2
+23
+0.0
+1.1
+0.0
+1.2
+Iy) Nn umus d tuulur liyinss
+-
+24
+0.0
+1. 1
+0.0
+1-1
+25
+ 2
+1. 1
+1.1
+0.0
+0.0
+26
+0.0
+1-1
+0.0
+0.d
+0.0
+29
+1.4
+0.0
+rry rid rut rnh chs yurli.
+1.4
+1.1
+24
+0..
+21
+2-2
+6.0
+0-0
+0.0
+Cmui le misie d dis y.
+30
+4-u
+1.8
+0.0
+0.0
+0.0
+0.0
+2.2
+1.2
+0.0Sunspot Number Database and the 1947 Zurich Discontinuity
+Figure 4. Evolution of the number of stations for each year contained in the data tables
+published in the Mitteilungen of the Zurich Observatory (gray curve). After 1919, when the
+Zurich Observatory ceased to publish all the data, the total number of contributing stations is
+plotted in blue, based on the annual list of stations. Between 1919 and 1944, the data from a
+subset of observers were still included, but after 1945, none of the source data were published.
+The two vertical shaded bands mark the two world wars, which both definitely left an imprint
+on the Zurich sunspot data set.
+some global statistics of the observers and the time interval over which they were
+active, which provides very interesting new insights in the construction of the
+Zurich SN.
+Figure 4 shows the number of stations for each year. This illustrates the
+evolution of the input data, as published in the Mitteilungen. The number of
+stations steadily increased from 1865 to 1896, when it reaches about 20 sta-
+tions and then drops slightly, but remaining above 15. This corresponds to the
+continuous recruiting of new additional external observers by Wolf and later by
+Wolfer. This evolution is completely disrupted in 1919. At the end of World
+War I (WWI), Wolfer adds many new observers. The number of stations passes
+the 40 mark, doubling the size of what becomes a true international network.
+However, probably for financial reasons, Wolfer then decides not to publish all
+data anymore (Friedli, 2020). Only the numbers from the Zurich observers and 7
+to 9 primary external observers are still published each year. Although some of
+those privileged external observers had been important long-term contributors
+by 1919, the selection criteria are unclear and were not explained by Wolfer.
+But another drop of the number of tabulated data happens in 1926, when
+William Otto Brunner succeeds Wolfer as director of the Zurich Observatory.
+Brunner then decides to publish only the data from the Zurich team (Brunner,
+1927). None of the data from the network are published after that year. The
+only exception is Karl Rapp, a private observer, who observed in Locarno,
+Switzerland, from 1940 to 1957. Rapp was actually trained in the same way
+as assistants at the main observatory in Zurich, and was thus treated as an
+internal observer over his whole observing career. Although Brunner states in
+1927 that the external data from auxiliary stations are archived and can be
+consulted on request (Brunner, 1927, page 188) (Friedli, 2020, Section 3.3),
+searches undertaken over past years failed to recover those archives. So far, only
+the data for 1944 were found in a single unpublished manuscript, referenced
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 9
+
+WWI
+WW II
+60
+count
+40
+Station
+20
+Unpublished
+Published
+1860
+1880
+1900
+1920
+1940
+1960
+1980
+Time (years)Clette et al.
+Hs1050:14 in the ETH Zurich University archives, which contains all calculation
+sheets for that single year (Friedli, 2020).
+Then in 1945, when Max Waldmeier becomes the new director, the publication
+of source data ceases completely, as can be seen in Figure 4. By then, the volume
+of data collected in Zurich had further increased, with almost 60 contributing
+stations (blue curve in Figure 8), making their publication bulky and costly.
+The Mitteilungen then switch to a different format. The thick yearly volumes
+become a series of shorter thematic issues, with articles about diverse research
+topics developed by Waldmeier. The sunspot number gets a more limited space,
+compared to the earlier volumes published by Wolfer and Brunner, which were
+almost entirely dedicated to sunspots. Again, during this last period of the Zurich
+history, all the original data were saved like before in archives at the observatory
+in Zurich.
+However, since the closing of the Zurich Observatory in 1980, those archives
+somehow went lost. This created a major 35-year data gap in the raw data
+collection on which the Zurich sunspot number is based. This wide gap falls at
+a critical moment, as one of the main scale jumps identified in the Zurich series
+falls in 1947, thus precisely within this time interval (Clette et al., 2014; Clette
+and Lef`evre, 2016). The raw input data are thus essential to reconstruct the
+methodological changes that took place in Zurich at that epoch and may have
+caused this inhomogeneity. Moreover, this gap creates a critical missing link
+between the early Zurich epoch, up to Brunner, and the modern international
+sunspot number produced in Brussels since 1981, for which all data are preserved
+in a computer-accessible digital database.
+4. The Original Waldmeier Source Tables (1945-1980)
+4.1. A Serendipitous and Complete Recovery
+Fortunately, in late 2018 and early 2019, a serendipitous finding by the staff of the
+Specola Solare Ticinese Observatory in Locarno (https://www.specola.ch/e/),
+followed by subsequent searches, allowed to recover the entire Waldmeier data
+archive (1945 – 1979), which was in fact dispersed over three locations: the Specola
+Observatory (26 years, 1945 – 1970), the Royal Observatory of Belgium in Brus-
+sels (4 years, 1971 – 1974), and in the deep storage of the ETH Zurich University
+archives in Zurich (5 years, 1975 – 1979). This dispersion seems to be due to
+the rather tumultuous closure of the Zurich Observatory (for an evocation of
+that transition, see Stenflo, 2016). Except for copies of the years 1975 – 1979
+on microfiches at the ETH archives, the fragmented original collection was also
+stored without inclusion in any inventory or catalogue.
+This recovery is a breakthrough, and given the amount of data collected over
+those 35 years, it will keep researchers busy for many years. Indeed, we estimate
+that those tables contain about 350,000 individual daily numbers, thus more
+than in all published tables from 1849 to 1944. In a first step, all the elements of
+this archive were brought together again at the ETH Zurich University archives.
+They are now fully cataloged (Waldmeier, 1980), and the ETH archives have
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 10
+
+Sunspot Number Database and the 1947 Zurich Discontinuity
+Figure 5. Facsimile of a typical handwritten yearly table from the complete 1945 – 1980
+collection of source tables that was recovered in 2018 – 2019. This table lists the data from
+H. M¨uller, one of the assistants observing at the Zurich Observatory with the standard 8-cm
+Fraunhofer refractor, for the year 1960 (ETH catalogue entry Hs1304.8:16.3; DOI: 10.7891/e–
+manuscripta-87290). For each day, the table gives the number of groups, the total number
+of spots, the calculated personal k value relative to the primary observer (Waldmeier), and
+a sky quality index. For each column, monthly sums and the mean k coefficient are given at
+the bottom. The yearly totals and the mean k coefficient for the whole year are appended
+at the lower right. Here, k equals 0.52 and thus differs by more than 15% from Waldmeier’s
+target value of 0.6, revealing a significant dispersion of the Wolf numbers from the assistants,
+although they were expected to be closely aligned on the primary observer.
+completed the digitization of the whole collection in 2020. The scans of all tables
+are now accessible online on the digital platform for manuscript material from
+Swiss libraries and archives at https://www.e-manuscripta.ch/ (ETH catalogue
+entry Hs 1304.8). Now, in order to make all the data computer-readable, all
+those tables need to be encoded. This work has now just started at the Royal
+Observatory of Belgium.
+4.2. The Waldmeier Yearly Tables: a Key to the Zurich Method
+The Waldmeier archive consists in yearly handwritten tables, one per observer,
+and each one on a separate sheet. Over the period 1945 – 1980, there was an
+average of 50 stations each year. All tables adopt the same standard format,
+with one column per month. Figure 5 illustrates the typical layout of one sheet,
+here with the table for H. M¨uller, one of the Zurich observers, for the year 1960.
+External auxiliary stations are presented with exactly the same layout. Each
+table lists all daily observations provided by the observer. The number of spots
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 11
+
+Hs
+1304.8:16.3
+Methode: 
+Beobachter:
+Jahr...
+V
+VI
+VII
+IX
+X
+IX
+IV
+VIII
+IIX
+II
+III
+I
+1960
+g, f
+g, f
+g, f
+g, f
+k
+k
+g, f
+k
+f 
+k
+g, f
+J‘8
+g, f
+k
+k
+k
+g,
+f
+k
+k
+k
+k
+ots
+C.82#0.S2
+8.1024
+11.2532130.46
+2.93
+0.46
+13.2082-30-516623
+￥.182
+1
+8.14
+1-21230.935
+0.55
+3.193
+0.69
+8.1
+0.53
+3.21420.49
+0.50
+15.4420
+M.MiUn...
+2
+12.82+30.531
+15.0.51
+0.50
+3.M
+3.340.46
+3.242-0.50
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+n-190230.51
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+0.51
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+L.622-3o.n.143
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+10.MLL0.53
+.1950.53
+1.81,
+01.1621420.49
+ Sonnendurchmesser:
+26
+8.83元
+10.81
+6.4030.52
+16.543-4
+8hoFstrw
+8.14
+0.58
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+M.2080.41.1590.44
+27
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+10.1930.481.81
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+88
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+4:1
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+6.89
+Nr.:
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+21
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+25
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+0.52
+0.54
+0.52
+0.53
+0.54
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+0-18
+0.54J
+0.53
+0.52
+0.51
+M
+45.20
+1960
+N
+11462%3
+k= 0.52 (0.515)Clette et al.
+and groups are given separately, exactly like in the tables published earlier in
+the Mitteilungen.
+This essential piece of information, which was so far entirely lost, will allow
+to determine for each day exactly how the observers were separating sunspot
+groups, on the one hand, and counting sunspots on the other hand. In partic-
+ular, it will help clarifying and quantifying the use of weighted sunspot counts,
+an alternate counting method adopted by the Zurich observers, in particular
+by Waldmeier himself. This alternate counting rule, in which large spots with
+extended penumbra are counted as more than 1, is suspected to be the cause of
+the 18% upward jump that affected the original SN series in 1947 (Clette et al.,
+2014; Clette and Lef`evre, 2016; Svalgaard, Cagnotti and Cortesi, 2017). Indeed,
+recent double counts, using the regular Wolf formula or weighted counts, were
+made at the Specola Observatory during several years, between 2003 and 2015,
+and led exactly to the same inflation of the sunspot number as the one found in
+the Zurich series after 1947 (Clette et al., 2014; Svalgaard, Cagnotti and Cortesi,
+2017). The recovered tables are thus providing the same kind of evidence, but
+over 35 years, including the epoch when the jump occurred.
+The tables also include the monthly and yearly mean k personal coefficients
+computed by the Zurich Observatory, a very important piece of metadata to
+understand how Zurich was treating the source observations. In particular, k
+coefficients are given for all Zurich assistants, and also the associated observers
+of the Specola station in Locarno. As all internal observers were assumed to align
+themselves on the primary observer (Waldmeier during that period), without
+applying any rescaling by a personal k coefficient, those internal yearly k values
+can bring invaluable insights on how and to what extent assistants managed to
+actually align themselves on the primary reference in their daily raw observa-
+tions. As this internal practice was introduced by Wolf, as soon as 1870, when he
+started to combine his own counts with those of his first assistants, this can thus
+help in the understanding of the Zurich number production well before 1945.
+In this collection, the most important tables are the yearly tables for the
+primary observer, Max Waldmeier (Figure 6). They provide unique information
+about three key aspects of the resulting Zurich sunspot numbers. Firstly, those
+tables were the master tables from which the daily sunspot number was derived
+for each day of the year. Therefore, they include raw counts and the resulting
+Wolf number for each day of the year. They thus provide a complete day-by-day
+census of how each daily SN was derived.
+Secondly, most of the days contain the personal counts by Waldmeier, who
+had the role of base reference. Therefore, this is the yearly table of raw group
+and sunspot counts by the primary observer, which allows tracking changes in
+Waldmeier’s own daily observations. For instance, Waldmeier was sometimes on
+mission at the coronagraph of the astronomical station in Arosa, then observing
+from high altitude with an alternate telescope. The counts for those days may de-
+viate from the base reference scale defined by the standard Fraunhofer refractor
+used on the front terrace of the observatory in downtown Zurich. Fortunately,
+Waldmeier marked the days when he observed from Arosa, which will allow
+analyzing the consequences of this site alternation.
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 12
+
+Sunspot Number Database and the 1947 Zurich Discontinuity
+Figure 6. Facsimile of the primary table for Max Waldmeier in 1957, extracted from the
+complete 1945 – 1980 collection of source tables (ETH catalogue entry Hs1304.8:13.2; DOI:
+10.7891/e-manuscripta-87246). Such tables are particularly important, as Waldmeier was
+the pilot observer of the Zurich sunspot number over that 35-year interval. They include
+various annotations that allow retracing day-by-day, how Waldmeier himself was observing,
+and which alternate number was used on days when he could not observe. They thus contain
+essential information about the Zurich data processing that cannot be found in any other
+Zurich document.
+Thirdly, the days in which Waldmeier could not observe are filled with num-
+bers from local assistants or from the stations in Arosa or Locarno (Karl Rapp
+until 1 April 1957 and the Specola Observatory starting on 1 October 1957).
+As can be seen in Figure 6, those days are also marked in the tables with a
+symbol identifying which alternate observer was used. Finally, as those tables
+record the provisional values issued immediately at the end of each month, on the
+remaining missing days when none of the local stations had managed to observe
+the Sun, the numbers were simply interpolated between adjacent days, and those
+dates are marked as “interpolated”. These are the few days which were later
+replaced by definitive values calculated using k-normalized Wolf numbers from
+the auxiliary stations, according to a standard method, of which the principle
+can be reconstructed from a few reference documents (Friedli, 2020).
+Those master tables thus provide almost all the keys that were badly miss-
+ing to reconstruct the method and practices implemented in Zurich, and most
+probably, to retrace persisting changes or local inconsistencies in the Zurich
+processing.
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 13
+
+Methode: 
+Beobachter:
+Jahr: ...
+T
+II
+III
+IV
+V
+VI
+VII
+VIII
+IX
+X
+XI
+XII
+Tmnisiris he kdaf'vzah lm.
+R
+g, f
+R
+R
+g, f
+R
+g, f
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+g,f
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+R
+g, f
+R
+g, f
+R
+g, f
+R
+R
+1954
+g, f
+A
+g, f
+doc
+244/20.242
+150
+105
+14.145,
+153
+140
+118
+13.134),
+[M,202 ]
+187
+15.2M
+216
+[9.160,]
+[12.114,
+dr.
+is.
+(180)
+13.6oz
+11.1642
+164
+13.130
+(121)
+14.2002
+204
+2
+[12.152,]
+20.200
+240
+3-4
+13,244,
+206
+ Bemerkungen : ...mumliri.
+[13.209,]
+9.1382
+131
+3
+12.85,
+123
+12.226,
+20.215,
+Ghz
+4５
+10.1
+121
+3.130,
+106
+18.268,
+33
+(210)
+10.362
+10.182
+19.144
+12.882
+12.110,
+h8't
+13.230
+230
+200
+18.250
+.Objektivoffnung:
+[20.20b.]
+６7
+16.83
+10.80
+14.183
+18.248
+daz
+17.189
+1313,
+9.136,
+da
+ Sonnenfleckenbeobachtungen 
+1128,
+8.160,
+8
+15.176,
+150
+12.122
+10.145
+250
+13130
+18
+1M.130
+9
+.
+10.132
+163
+8.190,
+162
+6.120,
+12.81,
+214
+13.149
+158
+1313
+195
+1
+15.1602
+9.160,
+9.230,
+[2.144
+14.98
+10
+150
+152471
+Its 1304.8:13.2
+11
+13112
+145
+10.88
+113
+2
+10.235,
+1.124
+140
+110
+12.130
+12845678022230
+(16)
+(160)
+9702
+17.263,
+260
+20.16b
+13.1
+11.87,
+NA
+160
+忆忆忆
+22.1
+14
+[9.70,]
+140
+13.204]
+10.165,
+M.1152
+1
+169
+44.192]
+10
+1b,239,
+yr
+139
+11.134
+181
+1948
+M448
+121
+46.287
+100
+.Vergr....
+126
+(120
+13.116
+8h
+15.1602
+186
+[18.184]
+150
+14.102
+.
+150
+8.102
+10
+13,296
+6.90
+19.283
+10
+149
+16.315
+18.184
+15.175
+13,288
+183
+15.204*
+12.138
+6yx.
+13.155
+dn
+18.197
+16.1463
+dn
+4.195,
+(126)
+12.122
+104
+15.249
+20.235
+11.282,
+22
+35
+12.232
+20.3754
+150
+.. mdue. hidgnelhitit... Nr.:
+170
+138
+127
+13.136,
+15.333
+10
+200
+24.355
+17.160
+864
+1.134
+146/10.113
+128
+17.202
+21.307
+(Otl)
+hhzhl
+42.224,1
+3
+10
+ Sonnendurchmesser:
+12.10%
+20.277
+11.1463
+(1%0)
+14,256
+238
+171
+19.259,
+19.1203
+14.683
+125
+12b
+12.1234
+14+)
+12.196h
+20.234,
+186
+29
+1215,
+117
+16.135
+154
+[12.184]
+142
+50
+23.228,
+30
+9.513
+88
+15136
+155
+14.143,]
+142
+M2,244,1
+213
+171
+21340
+33011.1bog
+162
+24.217,
+11.43,
+92
+12.1221
+M45
+15.40,]
+152
+22,290
+306
+21.215,
+31
+4320
+5255
+$108
+6164
+6023
+5041
+330
+4867
+8149
+62.18
+7251
+32.69
+N
+28
+31
+30
+[] +8
+31
+31
+31
+[13]+4+2*
+30
+[2] +3
+30
+30
+31
+B] +16++*
+10
+[28]
+31
+[4] +13
+1a7+12
+3
+[10]+ 1
+34
+152.3
+145.2
+164.8
+105.6
+194.3
+12.6
+M
+157.0
+944.3
+116.8
+262.9
+233.9
+1954
+69318
+355
+180
+29,6.02 184) -182
+ponoredus fora mMd
+30.b[12.244,]-200Clette et al.
+Figure 7. Timelines of the active observing periods of all Zurich observers. In red (top group),
+the primary observers and in orange (bottom group), the assistants. In purple, the observers
+of the auxiliary station in Locarno, who were considered as members of the Zurich core group.
+The vertical shaded band marks World War II and the vertical dashed line indicates the time
+when the 1947 scale jump occurs in the original SN series. The bottom plot gives the number
+of active Zurich observers for each year.
+5. A Major Disruption: Zurich Observers
+Although the above data still need to be digitized, we now have the full list
+of observers who contributed year-by-year to the Zurich sunspot number up to
+1980. By assembling the timelines of each individual observer, we could map
+how their observing period overlaps with other observers. Figure 7 retraces the
+observing periods for all Zurich primary observers and all the assistants, between
+1850 and 1960.
+In this figure, Schwabe is included among the assistants (orange group) al-
+though he was an external observer. Indeed, before Wolf could recruit his first
+assistants in the newly founded Zurich Observatory in 1865, he used Schwabe’s
+numbers as primary alternate source for filling the gaps in his own observations,
+and even initially considered Schwabe’s numbers as fully equivalent to his own
+(personal k = 1) before 1859. We also included K. Rapp in the associated Lo-
+carno station (purple group) although he contributed before the establishment
+of the Specola Observatory by Waldmeier in 1957, starting in 1940. Indeed,
+both Brunner and Waldmeier always included Rapps’s data together with the
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 14
+
+Wolf ST
+WolfPR
+Wolfer
+Brunner,wlo.
+Waldmeier
+Schwabe
+Fretz
+Weilenmann
+Mever
+Billwiller
+Fauquez
+Hoffler
+Broger
+Observers
+Biske
+Buser Arosa
+Brunner Ass
+Muller
+Beck
+Muller,E
+Wile
+Lemans
+Scheidlegger
+Frick
+Hermes
+Riesen
+Zelenka
+Durst
+Pfister
+Rapp
+Keller
+Schmidt
+ilszak
+Cbrtesi
+Pittini
+1840
+1860
+1880
+1900
+1920
+1940
+1960
+1980
+Time (years)Sunspot Number Database and the 1947 Zurich Discontinuity
+Zurich data in the Mitteilungen, even when the data of all the other external
+stations were not published anymore. Rapp was also trained to follow the Zurich
+observing methods, and can thus be considered as an internal member of the
+Zurich group of stations. Finally, although Waldmeier, the last primary observer
+(red group), started observing as an assistant in 1936, his participation was
+partly interrupted, as explained below.
+For the period before 1944, the resulting chronology reveals a few interesting
+facts. In particular, one of Wolfer’s assistants, Max Broger, had a very long
+observing career (40 years, 1896 to 1935). He actually observed over more years
+than several primary observers. As he observed in parallel with Wolfer and then
+with Brunner, his observations can provide an essential link to check the Wolfer-
+Brunner homogeneity.
+This touches the fundamental issue of the weighted sunspot counts used by
+the Zurich Observatory, as mentioned in the previous section. Indeed, Clette
+et al. (2014), Clette and Lef`evre (2016), and Svalgaard, Cagnotti and Cortesi
+(2017) conclude that this alternate counting method is the most likely cause of
+the 1947 scale jump in the original SN series. However, the timing and sharpness
+of the jump seem to be contradicted by the fact that this weighting practice was
+introduced progressively well before 1947, in the early 20th century by Wolfer
+(Cortesi et al., 2016; Svalgaard, Cagnotti and Cortesi, 2017). Although Wolfer
+himself never used it for his own counts (Svalgaard, Cagnotti and Cortesi, 2017),
+this practice was implemented to help assistants aligning their raw counts on the
+reference of Wolfer, the primary observer. This could be verified by taking the
+counts on occasional days when only a single big spot was visible on the Sun.
+Then, when one of Wolfer’s assistants, W.O. Brunner, took over as director and
+as primary observer in 1926, he continued to use weighted counts, but this time as
+primary observer. Although this marks the moment when the break with Wolf’s
+original methodology occurred, Brunner managed to maintain the stability of his
+counts, as found by Svalgaard, Cagnotti and Cortesi (2017). When Waldmeier
+took his succession in 1945, after being assistant for a few years, he thus just
+continued an established practice. So, apparently, this chronology does not match
+at all the abrupt occurrence of a jump in 1947, two years after Waldmeier became
+the new reference observer, a status that he kept for the next 35 years without
+any other noticeable transition.
+Now, by retracing the composition of the network of collaborating observers,
+we found evidence of a major transition that occurred between 1945 and 1947.
+The change was twofold. Firstly, at the Zurich Observatory, although Waldmeier
+became director in 1945, the former director, W.O. Brunner, actually continued
+observing during one year until December 1945 (see Figure 7). Moreover, his
+primary assistant, W. Brunner-Hagger, who was part of the team since 1928,
+continued until August 1946. This actually marks the moment when the link
+with the former Zurich core team is broken. As shown in Figure 7, in 1945,
+Waldmeier starts to recruit new assistants. However, the first one, Beck, worked
+in parallel with Brunner only during a few months, when solar activity was
+rather low, and he left the observatory already in 1949. Then follows a succession
+of other assistants who also leave after only a few years. This means that the
+overlap between the old and new team was extremely limited and that for several
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 15
+
+Clette et al.
+years the Zurich team was very unstable, contrary to the Brunner team that had
+remained unchanged for nearly 20 years.
+So, the internal stability of the Zurich system during the 1945 Brunner-
+Waldmeier transition rested only on Waldmeier himself. This is unprecedented
+in the entire Zurich history. Indeed, the stability of the Wolf-Wolfer transition
+benefited from a 17-year period, during which Wolf and Wolfer observed jointly.
+Although the Wolfer-Brunner joint period was shorter (3 years, 1926-1928),
+another assistant, Broger brought a solid reference to bridge the Wolfer-Brunner
+transition, as he had worked jointly with Wolfer for 30 years, since 1896, and
+then continued for 10 years together with Brunner, until 1935.
+Finally, although Waldmeier started collaborating with the Zurich Obser-
+vatory in 1936, he did not contribute during three years, from 1939 to 1941
+because of the onset of World War II. Moreover, as he was strongly involved in
+coronagraph observations, he worked for a large part of his time at the Arosa
+station, rather than as an ordinary assistant observing side by side with Brunner
+in Zurich. We also note that the last years before 1946 fell in a minimum of
+the solar cycle, when the low sunspot activity makes mutual comparisons less
+accurate. Therefore, all those circumstances reduced the effective overlap period
+between Brunner and Waldmeier.
+6. A Major Disruption: Auxiliary Stations
+In parallel with the Zurich internal transition, another major and unprecedented
+disruption also occurred just after 1945, but now for the Zurich auxiliary sta-
+tions. Although those external data were not at the core of published sunspot
+numbers, they definitely provided a wide ensemble of independent data series
+against which the Zurich numbers were continuously compared. Moreover, all
+external stations derived their counts using Wolf’s original definition, without
+any weighting. Therefore, the auxiliary data were not affected by the introduc-
+tion of Zurich’s internal weighting practice, and in that sense, they provided
+the only base against which the Zurich team could infer that their weighted
+numbers remained coherent with the unwheighted Wolf numbers that formed
+the original SN series until Wolfer’s retirement in 1926 (Clette et al., 2014;
+Svalgaard, Cagnotti and Cortesi, 2017). This continuous bench-marking could
+only work if at any given time, there was a large number of active auxiliary
+stations which had already contributed data during many past years, preferably
+over one or more full solar cycles.
+Figure 8 shows the timelines of all auxiliary stations that contributed obser-
+vations to the Zurich Observatory since the mid-19th, over a duration longer
+than 11 years, i.e. a full solar cycle. This subset of long-duration stations is
+indeed the most important for the long-term calibration and stability of the
+series. We distinguished the professional observatories from the individual ama-
+teur observers, which reveals a deep evolution in the composition of the Zurich
+observing network. While a large majority of stations were individual observers
+before World War II (WWII), professional observatories dominate the network
+after WWII.
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 16
+
+Sunspot Number Database and the 1947 Zurich Discontinuity
+Figure 8. Timelines of the active observing periods of all external stations that sent data
+to Zurich until the observatory was closed in 1980. The stations are ordered according to the
+starting date of their series. The top series (dark blue) gathers the professional observatories
+and the bottom group (light blue) gathers the individual amateur observers. The vertical
+shaded band marks World War II and the vertical dashed line indicates the 1947 scale jump
+in the original Zurich series. The bottom plot gives the total number of active stations per
+year. The light-blue section indicates unpublished data that have not been recovered yet in
+the Zurich archives.
+However, a much more drastic change is also caused by WWII. In Figure 8,
+we see that, starting in 1938, long-time contributing stations cease to send data,
+one after the other. When WWII ended, none of those stations, which gave an
+external benchmark for the earlier Zurich SN, had survived. During the war,
+given the steep drop of contributing stations, Brunner and Waldmeier called to
+the rescue a large number of local Swiss amateur astronomers, but this local
+network was quickly changing, as most observers contributed only for one year
+or at best a few years (therefore, they do not appear in Figure 8). None of those
+observers were long-term observers in the preceding Zurich network established
+by Wolfer and Brunner.
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 17
+
+Observers
+Obs/year
+40
+20
+0
+1860
+1880
+1900
+1920
+1940
+1960
+1980
+Time (years)Clette et al.
+Then, just after the war, Waldmeier quickly undertakes the construction of
+a new international network. The number of stations grows steeply and reaches
+about 50 stations (see Figure 4), a number that will remain rather stable until
+1980. As noted before, this new network includes many professional observa-
+tories, which since then, have delivered observations over very long durations.
+In fact, some of them are still contributing nowadays to the worldwide SILSO
+network, and thus provide an invaluable long-term reference spanning up to 75
+years, since 1945. However, none of those new stations were part of the pre-
+1940 long-term network. Therefore, the context in which the sunspot number
+was produced after 1945 was largely disconnected from the context surrounding
+this production before 1940. This further weakened the thin internal continuity
+within the Zurich Observatory.
+In order to give a more quantitative measure of this second disruption, we
+summed the number of past observed years already accumulated by all stations
+that were active on a given year. Figure 9 (top plot) shows the temporal evolution
+of this total number, which gives a measure of the total amount of past informa-
+tion that the Zurich Observatory had at its disposal for past comparisons and the
+verification of their stability relative to independent observers. As expected, the
+evolution is characterized by a steady increase in the total amount of available
+data. The only interruption in this trend is the steep drop during WWII, when
+the count suddenly drops back to the values of the early 20th century. After
+WWII, there is a recovery, but it takes about 15 years before the amount of
+past reference data comes back to the value just before WWII. Afterwards, the
+amount of past data from active stations continues to grow and finally stabilizes
+in the 1970’s.
+If we divide this total number of past observed years by the number of active
+stations, we obtain the mean past duration over which stations active at a given
+time have been observing before that time (Figure 9; bottom plot). This mean
+duration quantifies the past memory built into the SN system. Between 1860 and
+1890, this mean duration increases. This marks the progressive recruiting of the
+first auxiliary observers by Wolf. Then, the mean duration largely stabilizes until
+1926, i.e. the Wolfer-Brunner transition. The only feature is a temporary peak
+associated with WWI, which thus left only a minor imprint in this evolution.
+Thanks to the many new observers recruited after WWI, and who continue
+observing until WWII, the mean duration grows to almost 15 years in 1938.
+Then, WWII again produces a steep drop, by a factor of two. In 1945 and the
+decade that follows, the mean memory range falls back to about 7 years, a level
+that was not encountered since 1880, i.e. the epoch when Wolf was still recruiting
+his first associated observers. After this dramatic shortening of the past memory,
+there was a steady recovery. However, it is only around 1965, 20 years after
+WWII, that the pre-WWII mean memory range is recovered. It continues to
+rise until 1980, when the Zurich Observatory was closed. This continuous trend
+largely rests on the long-term contribution from the professional observatories
+that entered the network just after WWII. Figure 9 thus illustrates that the
+years immediately following WWII were abruptly affected by a major loss of
+past references, and that this loss had no equivalent in the history of the Zurich
+SN number.
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 18
+
+Sunspot Number Database and the 1947 Zurich Discontinuity
+Figure 9. Evolution of the amount of past data available for each year at Zurich. The upper
+plot gives the total number of preceding observed years by all the stations active on a given
+year. After an almost continuous increase, a sharp drop occurs just after WWII. The lower
+plot shows the mean number of preceding observed years per station, for all stations active on
+a given year. The rise after 1925 indicates the growing participation of stations with very long
+duration, but a drop to 19th century levels marks the late 1940’s and early 1950’s.
+Although the above indicators are indirect contextual elements, the fact that
+this unique double discontinuity in the history of the Zurich sunspot number
+production coincides with the jump revealed by the SN series itself is a very
+strong indication that the sharp SN scale jump was a consequence of this abrupt
+and radical change in the base data input. Until 1946, the potential biasing effect,
+which was present since the weighted counting method had been introduced,
+had been kept under control thanks to the double stabilizing effect of long-term
+internal and external observers who did not change their counting practices.
+This stabilizing continuity was clearly broken between 1946 and 1947, which
+suddenly opened the way for the biasing effect inherent to the weighted counts, as
+evidenced by the 1947 upward jump. This new contextual evidence thus explains
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 19
+
+800
+Amount of past data
+600
+400
+200
+0
+1860
+1880
+1900
+1920
+1940
+1960
+1980
+Time (years)
+25
+duration (years)
+20
+15
+past
+10
+Mean
+5
+0
+1860
+1880
+1900
+1920
+1940
+1960
+1980
+Time (years)Clette et al.
+simultaneously the delayed effect of the weighting practice and the abruptness
+of the jump.
+7. Conclusion
+Over just a few years, we thus achieved major progress in the construction of
+the SN database. Now, about two thirds of the existing source data are recorded
+in digital form. We can now also report on the recovery of a major missing part
+of this collection, the yearly source tables of the Waldmeier era from 1945 to
+1980. This fills the main gap in the SN database and provides the missing link
+between the contemporary index and the rest of this long series before 1945
+and back to 1700. While significant work is still needed to digitize those newly
+recovered documents, the global panorama that the SN database now offers
+made it possible to establish the complete chronology of contributing stations
+and observers. We found that the two world wars had deep consequences on
+the production of the SN by the Zurich Observatory. WWI brought a major
+expansion of the network of auxiliary observers, but without disrupting the
+internal practices and organization of the Zurich sunspot observers.
+On the other hand, after WWII, we find a double disruption in the Zurich
+system. A complete renewal of the Zurich observing team occurred between 1946
+and 1947, with almost no overlap between the old team, which had remained
+mostly unchanged for more than 20 years, and the new team progressively built
+by Waldmeier between 1946 and 1950. Moreover, after the loss of most of the
+external observers active over the decades preceding WWII, between 1938 and
+1945, an entirely new worldwide network is established after the war with entirely
+different stations. The narrow correspondence of this drastic and unprecedented
+structural change with the 18% SN scale-jump diagnosed in the SN series pro-
+vides strong historical evidence that a sharp jump in the SN exactly at that
+moment is a real and logical consequence. Although the suspected cause, i.e. the
+introduction of the size-based weighting of the spot counts, was introduced much
+earlier in the practice of Zurich assistants, our now-complete timeline explains
+why it only led to actual consequences when this sharp and unprecedented
+discontinuity in the Zurich system took place.
+All together, those recovered tables open the way to future major steps in the
+end-to-end calibration of the sunspot number series. Full statistical diagnostics of
+the actual stability of each separate Zurich observer, which was simply postulated
+since the epoch of Wolf, will allow disentangling in detail the causes of anomalies
+found in the heritage series. Much more importantly, those data open the way for
+a full recalculation of the sunspot number, starting again from the full set of raw
+input data. This recalculation will use new advanced computer-based processing
+methods, which exploit the entire set of data instead of mostly using the numbers
+of the single primary observer, as was the case in the original Zurich series. This
+should improve further the stability and accuracy of the sunspot number in the
+interval 1945-1980, where so far, SN Version 2 consisted only in a correction
+factor applied to the original Zurich SN series. This would also finally bridge the
+gap separating the current international sunspot number from the early epoch
+before 1945.
+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 20
+
+Sunspot Number Database and the 1947 Zurich Discontinuity
+However, a partial gap still remains. Although all observations made in Zurich
+from Wolf in 1849 to Waldmeier in 1980 now finally form a complete and unin-
+terrupted thread, we still miss the unpublished archives from the Brunner era.
+Therefore, efforts are still continuing to try recovering the last missing data from
+the network of the auxiliary stations between 1919 and 1944. Hopefully, this will
+finally bring the last touch to this digital database that will feed sunspot science
+and long-term solar-cycle studies for many years.
+Acknowledgments
+This work and the team of the World Data Center SILSO (http://www.
+sidc.be/silso/), which produces the international sunspot number and maintains the sunspot
+database used in this study, are supported by Belgian Solar-Terrestrial Center of Excellence
+(STCE, http://www.stce.be) funded by the Belgian Science Policy Office (BelSPo). This work
+was also supported by the International Space Science Institute (ISSI, Bern, Switzerland) via
+the International Team 417 “Recalibration of the Sunspot Number Series”, chaired by M.
+Owens and F. Clette (https://www.issibern.ch/teams/sunspotnoser/). Specola Solare Ticinese
+acknowledges the financial support provided by Canton Ticino through the Swisslos fund
+and by the Federal Office of Meteorology and Climatology MeteoSwiss, in the framework of
+GCOS. We would like to thank Thomas Friedli for digitizing and making available the original
+sourcebook by R. Wolf via the web site of the Rudolf Wolf Society (http://www.wolfinstitute.
+ch). We also thank the ETH Library (https://library.ethz.ch/en/), and in particular Evelyn
+Boesch, of the Hochschularchiv, for the deep searches in the catalogues and archives, and for
+giving us access to original historical documents from the Zurich Observatory. We also thank
+Olivier Lemaˆıtre for developing the software and computer database, Stephen Fay and Shreya
+Bhattasharya for the quality control, and last but not least, we are also grateful to the summer-
+job students who patiently and carefully encoded all numbers tabulated in the original paper
+documents: Elfaniel Hermel, Esther-Lauren M’Bilo and Mael Panouillot.
+Disclosure of Potential Conflicts of Interest
+The authors declare that they have no conflicts of interest.
+References
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+Hs 1304.8, 0.4 Laufmeter, unpublished manuscript, Zurich.
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+SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 22
+

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+Stochastic Model of Organizational State Transitions in a Turbulent Pipe Flow
+Robert J¨ackel1,3, Bruno Magacho2,3, Bayode Owolabi2,3,
+Luca Moriconi2,3∗, David J.C. Dennis3, and Juliana B.R. Loureiro1,3
+1Programa de Engenharia Mecˆanica, Coordena¸c˜ao dos Programas de P´os-Gradua¸c˜ao em Engenharia,
+Universidade Federal do Rio de Janeiro, C.P. 68503, CEP: 21941-972, Rio de Janeiro, RJ, Brazil
+2Instituto de F´ısica, Universidade Federal do Rio de Janeiro,
+Av. Athos da Silveira Ramos 149, CEP: 21941-909, Rio de Janeiro, RJ, Brazil
+and
+3Interdisciplinary Center for Fluid Dynamics, Universidade Federal do Rio de Janeiro,
+R. Moniz Arag˜ao 360, CEP: 21941-594, Rio de Janeiro, Brazil
+Turbulent pipe flows exhibit organizational states (OSs) that are labelled by discrete azimuthal
+wavenumber modes and are reminiscent of the traveling wave solutions of low Reynolds number
+regimes. The discretized time evolution of the OSs, obtained through stereoscopic particle image
+velocimetry, is shown to be non-Markovian for data acquisition carried out at a structure-resolved
+sampling rate. In particular, properly defined time-correlation functions for the OS transitions are
+observed to decay as intriguing power laws, up to a large-eddy time horizon, beyond which they
+decorrelate at much faster rates. We are able to establish, relying upon a probabilistic descrip-
+tion of the creation and annihilation of streamwise streaks, a lower-level Markovian model for the
+OS transitions, which reproduces their time-correlated behavior with meaningful accuracy. These
+findings indicate that the OSs are distributed along the pipe as statistically correlated packets of
+quasi-streamwise vortical structures.
+Notwithstanding the large body of knowledge accumu-
+lated since the landmark experiments of Reynolds [1],
+turbulent pipes comprise flow patterns which have re-
+mained surprisingly unsuspected until recent years. They
+can be depicted as relatively organized sets of wall-
+attached low-speed streaks coupled to pairs of counter-
+rotating quasi-streamwise vortices [2–4]. These organi-
+zational states (OSs) actually characterize the turbulent
+velocity fluctuations at high Reynolds numbers and are
+topologically similar to traveling waves – a class of ex-
+act (but unstable) low-Reynolds number solutions of the
+Navier-Stokes equations [5, 6].
+As for traveling waves, the OSs can be classified by the
+number of low-speed streaks they contain. Observation
+tells us, however, that this quantity changes in an ap-
+parently random way along the turbulent pipe. For the
+sake of illustration, Fig. 1 shows a transition between
+OSs, visualized from a pair of cross-sectional snapshots
+of the flow obtained through stereoscopic particle image
+velocimetry (sPIV).
+The existence of spatial transitions among the OS
+modes suggests, within the perspective of dynamical sys-
+tems, that the turbulent pipe flow could be described as
+a chaotic attractor and its unstable periodic orbits in a
+phase space of much reduced dimensionality [7–11]. In
+connection with this circle of ideas, we are motivated to
+study the OS transitions in the framework of stochastic
+processes, focusing particular attention on their recurrent
+dynamics.
+To start, let u = u(r, θ) be, in polar coordinates, the
+fluctuating streamwise component of the velocity field
+∗Corresponding author: moriconi@if.ufrj.br
+FIG. 1: Example of a transition between organized states,
+as sampled out from our measurements, which are associated
+to two and three low-speed streaks. Blue and red colors re-
+fer, respectively, to negative and positive streamwise velocity
+fluctuations around the mean (the systematic procedure to
+ascertain a well-defined number of low-speed structures to a
+given flow snapshot is discussed in the text).
+defined over a fixed pipe’s cross-sectional plane. We may
+introduce, accordingly, the instantaneous spectral power
+density,
+I(kn) =
+����
+� 2π
+0
+dθeiknθfuu(r0, θ)
+����
+2
+,
+(1)
+where
+fuu(r0, θ) =
+� 2π
+0
+dθ′u(r0, θ′)u(r0, θ′ + θ) ,
+(2)
+kn = n ∈ Z+ is an azimuthal wavenumber, and r0 is a ref-
+erence radial distance which falls within the log-region of
+the pipe’s turbulent boundary layer. Empirical evidence
+shows that I(kn) is in general peaked at some clearly
+dominant wavenumber ¯k (to be identified to the number
+of snapshotted low-speed streaks), which can be used to
+label the probed velocity profile u(r, θ). As time evolves,
+arXiv:2301.05344v1  [physics.flu-dyn]  13 Jan 2023
+
+3
+22
+FIG. 2: Statistical results for the OS mode ¯k = 5. Left image:
+positive (red) and negative (blue) level curves of Ruu, defined
+by |Ruu(r − r0|¯k)| = 5% and 10% of (Ruu)max, with the
+reference point r0 depicted as a black dot. Right image: a
+closer look at the averaged streamwise velocity fluctuations
+(red for positive, blue for negative), conditioned on u(r0) > 0.
+The cross-sectional averaged velocity field reveals the vortical
+structures that are usually coupled with velocity streaks.
+I(kn) changes, and so does the wavenumber position of
+its dominant peak.
+Therefore, if u(r, θ) is recorded at
+equally spaced time intervals ∆, the dynamical evolu-
+tion of the pipe turbulent field can be mapped into the
+stochastic process
+S ≡ {¯k(t), ¯k(t + ∆), ¯k(t + 2∆), ... } .
+(3)
+In order to investigate the still very open statistical prop-
+erties of S, we have performed a pipe flow experiment, at
+Reynolds number Re = 24415, in the large pipe rig facil-
+ity of the Interdisciplinary Nucleus for Fluid Dynamics
+(NIDF) at the Federal University of Rio de Janeiro. The
+pipe’s diameter and length are, respectively, D = 15 cm
+and L = 12 m. By means of sPIV, with sampling rate
+of 10 Hz (i.e., ∆ = 0.1 s), we have collected 104 cross-
+sectional snapshots of the flow, each one containing the
+three components of the turbulent velocity field over a
+uniform grid of size 78 × 78. It turns out that all the ob-
+served OS modes fall into the range 0 ≤ ¯k ≤ ¯kmax = 10.
+Our experimental data has been validated with the
+help of previous benchmark pipe flow experiments [12],
+through the inspection of the performance of first and
+second order single-point statistics for the streamwise
+component of velocity field.
+We have also attained a
+further validation of the entire measured velocity field,
+from the evaluation of particularly defined streamwise
+velocity-velocity correlation functions conditioned on the
+OS modes ¯k, more precisely,
+Ruu(∆r|¯k) ≡ E[u(r0)u(r0 + ∆r)|¯k] ,
+(4)
+which has its level curves depicted in Fig. 2, for the case
+¯k = 5, in close correspondence with the results of Ref. [4].
+The first immediate question that can be raised about
+the stochastic process S is whether it is Markovian or
+|λh| 
+10−3
+10−2
+10−1
+100
+10−3
+10−2
+10−1
+100
+h = time lag between sPIV snapshots / Δ
+0
+1
+2
+3
+0
+1
+2
+3
+FIG. 3: Eigenvalues of the probability transition matrices for
+the original process (h = 1) and a decimated one (h = 2).
+The dashed lines should intercept eigenvalue pairs if S were
+a Markovian process.
+not. Of course, while it is not possible to answer this
+in full rigor, one may check if the Chapman-Kolmogorov
+(CK) equation holds for the time series (3), a necessary
+condition for S to be Markovian [13]. The CK equation
+would imply that the eigenvalues of the transition prob-
+ability matrix for OS modes separated by the time inter-
+val h∆ can be represented, in some arbitrary ordering, as
+the set of powers {λh
+1, λh
+2, ..., λh
+¯kmax}. A straightforward
+computation of the transition matrix eigenvalues for the
+cases h = 1 and h = 2 indicates, however, that S is not
+Markovian; see Fig. 3.
+We expect that the decimated process for h large
+enough is essentially Markovian, since in this situation
+the OS modes become weakly correlated. The transition
+to Markovian behavior can be alternatively addressed
+from the analysis of correlation functions which we intro-
+duce as it follows. Taking 0 ≤ m, m′ ≤ ¯kmax, let Vm(t)
+and Mm′m(t) be, respectively, the components of vector
+and matrix valued stochastic processes derived from S as
+Vm(t) =
+�
+1,
+if ¯k(t) = m
+0,
+otherwise
+(5)
+and
+Mm′m(t) =
+�
+1,
+if ¯k(t) = m and ¯k(t + ∆) = m′
+0,
+otherwise .
+(6)
+Define, now, the correlation functions
+˜F(t − t′) ≡ E[V(t) · V(t′)] − (E[V])2 ,
+(7)
+˜G(t − t′) ≡ Tr
+�
+E[MT(t)M(t′)] − E[M]TE[M]
+�
+, (8)
+and their normalized versions,
+F(t − t′) ≡
+˜F(t − t′)
+˜F(0)
+, G(t − t′) ≡
+˜G(t − t′)
+˜G(0)
+.
+(9)
+
+O3
+F(t-t')
+10−3
+10−2
+10−1
+100
+10−3
+10−2
+10−1
+100
+|t-t'|	(s)
+10−1
+100
+10−1
+100
+δt
+(a)
+G(t-t')
+10−2
+10−1
+100
+10−2
+10−1
+100
+|t-t'|	(s)
+10−1
+100
+10−1
+100
+δt
+(b)
+FIG. 4: The time-dependent correlation functions defined in
+(9) are noticed to decay as power laws for |t − t′| ≤ δt ≈ 2 s.
+The dotted lines in (a) and (b) have scaling exponent −1 for
+both F(t − t′) and G(t − t′).
+It is not difficult to see that F(t − t′) and G(t − t′)
+describe, respectively, the correlations of returning OS
+modes and transitions which are apart from each other
+by the time interval |t − t′|. They are plotted in Fig. 4
+and are noticed to have interesting power law decays
+(with the same approximate scaling exponent −1) up to
+|t − t′| ≡ δt ≈ 20∆ = 2 s, which suggests some sort of
+self-similarity across the spatial distribution of about ten
+OS modes (their mean lifetime is 0.2 s ≈ δt/10).
+For
+time separations larger than δt, the correlation func-
+tions become suddenly undersampled, meaning that they
+crossover to a faster law of decay, probably exponential,
+as it should be for the putative asymptotic Markovian
+behavior of a large-time decimated S.
+It is worth emphasizing that the non-Markovian nature
+of S does not mean at all that it cannot be modeled as
+a Markov process defined in terms of lower-level state
+variables. In this connection, it is reasonable to assume
+that there is a combinatoric degeneracy factor
+Ω(¯kmax, m) =
+�¯kmax
+m
+�
+(10)
+associated to a given OS mode ¯k = m. We simply mean
+here that the m wall-attached low-speed streaks can be
+spatially arranged for this particular mode in Ω(¯kmax, m)
+different ways, since the pipe’s cross-sectional plane is
+taken to hold at most ¯kmax low-speed streak channels.
+The phase space of the “microscopic” state variables
+for the underlying Markovian model of S is spanned,
+therefore, by all the possible sets of ¯kmax streak bits,
+X ≡ {s1, s2, ..., s¯kmax}, where
+si =
+�
+1,
+if the i-th streak channel is active
+0,
+otherwise .
+(11)
+We postulate, now, that the time evolution of the micro-
+scopic states X is produced from the independent fluctu-
+ations of streak bits, which have persistence probabilities
+that depend on the total number of active streak chan-
+nels, that is the OS label m. In this way, we define qm
+and pm to be the persistence probabilities for any given
+streak bit to keep its value 0 or 1, respectively, along
+subsequent sPIV snapshots. There are, thus, four dif-
+ferent types of streak bit flips, which appear in different
+occurrence numbers for a given OS mode transition, as
+summarized in Table I.
+Transition Type # of Streak Channels Transition Prob.
+0 → 0
+n1
+qm
+0 → 1
+n2
+1 − qm
+1 → 0
+n3
+1 − pm
+1 → 1
+n4
+pm
+TABLE I: Definition of the four possible transition types for
+the streak channel states, together with the notations for their
+occurrence numbers and individual transition probabilities.
+Above, m = n3 + n4 labels the OS mode.
+The parameters reported in Table I are related to
+the OS mode transition m → m′, where m = n3 + n4
+and m′ = n2 + n4. The transition probability between
+any specific pair of associated microstates is, as a conse-
+quence, qn1
+m (1−qm)n2(1−pm)n3pn4
+m . Taking into account,
+furthermore, the role of degeneracy factors, we may write
+the transition probability between the OS modes m and
+m′ as
+
+4
+Tm′m =
+�¯kmax
+m
+�−1 ¯kmax
+�
+n1=0
+¯kmax
+�
+n2=0
+¯kmax
+�
+n3=0
+¯kmax
+�
+n4=0
+δ(n1 + n2 + n3 + n4, ¯kmax)δ(n3 + n4, m)δ(n2 + n4, m′) ×
+×
+�¯kmax
+n1
+��¯kmax − n1
+n2
+��¯kmax − n1 − n2
+n3
+�
+qn1
+m (1 − qm)n2(1 − pm)n3pn4
+m .
+(12)
+Using, from now on, ¯kmax = 10, the Markovian model
+just introduced may not appear very phenomenologically
+attractive at first glance, since Tm′m is parametrized by
+a large number of unknown parameters (q0, q1, ..., q9 and
+p1, p2, ..., p10).
+Note, however, that there are, in prin-
+ciple, 90 independent entries in the empirical transition
+matrix (the one derived from the sPIV measurements),
+so the model is rather underdetermined (as we would ex-
+pect for a phase-space reduced description of turbulent
+fluctuations).
+Instead of attempting to provide a detailed and com-
+putationally costly model of the empirical transition ma-
+trix, we address a much simpler approach, where we focus
+on the asymptotic probability eigenvector of the modeled
+transition matrix,
+P = (P1, P2, ..., P10) ,
+(13)
+which satifies to TP = P, that is, �10
+m=0 Tm′mPm = Pm′.
+Here, Pm is the probability that the OS mode m be ob-
+served in the statistically stationary regime. In an analo-
+gous way, denoting by P∞ the empirical probability vec-
+tor, determined from the sPIV measurements, we are in-
+terested to find the set of probabilities qm and pm that
+minimize the quadratic error
+d({qm}, {pm}) ≡ ||P − P∞||2 .
+(14)
+While, as already commented, the original problem is
+underdetermined, the optimization scheme related to
+Eq. (14) is not: as a matter of fact, we would have to
+model the 9 independent probability entries of (13) by
+means of the 20 probability parameters qm and pm. To
+reduce this large overdeterminacy, we rely on a few phe-
+nomenological inputs:
+(i) We assume that we can model the observed coher-
+ence (time persistence) of low-speed streaks by a single
+mode-independent and not small probability parameter
+p, where p = p2 = p3 = ... = p10;
+(ii) P0 turns out to be negligible, so we suppress transi-
+tions from the OS mode m = 1 to m = 0, by imposing
+that p1 = 1 (other transitions to the mode m = 0 from
+modes m ̸= 1 are possible, but they are of O((1 − p)2).
+Therefore, we end up with 11 parameters (q0, q1, ..., q9
+and p) to locate the minimum value of (14). The result
+is a slightly overdetermined system, but if besides P∞,
+the correlation functions F(t − t′) and G(t − t′) turn out
+to be well reproduced with the same set of probability
+parameters, as an extra bonus, then the model can be
+taken as physically appealing. That is the heuristic setup
+that we have in mind.
+We have resorted to a straightforward Monte Carlo
+procedure to obtain the set of qm’s that minimizes (14)
+for various fixed values of p. We find, as shown in Fig. 5,
+Min{q}[d({q},p)]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+p	(=	p2	=	p3	...	=	p10)
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+FIG. 5: Minimization of the quadratic distance d({q}, p) for
+various values of p.
+Occurrence Probability (%)
+0
+5
+10
+15
+20
+0
+5
+10
+15
+20
+OS Mode (k)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+	
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+FIG. 6: The occurrence probability of OS modes obtained
+from the experiment (dots) and from the stochastic model
+(open circles: p = 0.86; crosses: p = 0.95), defined by the
+transition matrix elements (12).
+
+5
+F(t-t')
+10−3
+10−2
+10−1
+100
+10−3
+10−2
+10−1
+100
+|t-t'|	(s)
+10−1
+100
+10−1
+100
+(a)
+G(t-t')
+10−2
+10−1
+100
+10−2
+10−1
+100
+|t-t'|	(s)
+10−1
+100
+10−1
+100
+δt
+δt
+(b)
+10−3
+10−2
+10−1
+	
+2
+4
+6
+8
+	
+FIG. 7: Empirical (dots) and modeled (crosses) correlation
+functions F(t−t′) and G(t−t′). Crosses refer, in (a) and (b),
+respectively, to modeling parameters p = 0.86 and p = 0.95.
+The semi-log plot in the inset of (b) indicates the simple ex-
+ponential form of G(t − t′) at large enough |t − t′|.
+that the quadratic error quickly drops for p ≥ 0.85. The
+modeled asymptotic probabilities for the occurrence of
+OS modes are excellently compared, in Fig. 6, to the
+empirical ones for the cases p = 0.86 and p = 0.95. These
+are the values of p that lead to good accounts of F(t−t′)
+and G(t−t′), as reported in Fig. 7. The related values of
+the probabilities qm are listed in Table II. Even if a point
+of subjective concern, the uncertainty of about 10% in
+the definition of p should be taken as relatively small,
+vis a vis the model’s accuracy in predicting the decaying
+profiles of the OS correlation functions.
+p
+q0
+q1
+q2
+q3
+q4
+q5
+q6
+q7
+q8
+q9
+0.86 0.53 0.96 0.95 0.92 0.92 0.85 0.95 0.75 0.86 1.0
+0.95 0.22 0.98 0.98 0.97 0.97 0.96 0.97 0.93 0.94 0.49
+TABLE II: The list of probabilities qm’s which describe
+the persistence of inactive streak channels, for the cases
+p = 0.86 and p = 0.95.
+Also evidenced in the inset Fig. 7 is the exponential
+decay profile of the modeled G(t − t′) for time intervals
+larger than δt.
+At present, this point rests as a pre-
+diction of the modeling scenario introduced in this work,
+akin with the observed sudden undersampling of the time
+series for larger decimations. We note that the crossover
+to the faster exponential decay of correlation functions
+takes place at δt ≈ 2D/U, where U is the bulk flow ve-
+locity. Thus, the physical picture that emerges is that
+the OSs are packed as chains of low-speed streaks and
+vortical structures which are strongly correlated within
+sizes that scale with the pipe’s diameter, although they
+are merged along the entire turbulent flow.
+To summarize, we have investigated the stochastic
+properties of the non-Markovian OS mode transitions in a
+turbulent pipe flow, recovering them as a surjective map-
+ping of a lower-level Markov process. The essential idea
+that underlies the model construction is that a given OS
+mode may be associated to several spatial arrangements
+of its low-speed streaks into a fixed number of “streak
+channels” which azimuthally partition the pipe’s cross
+section.
+We find that the Markov model can account for the
+scaling behavior of specifically introduced correlation
+functions of OS mode transitions. Further work is in or-
+der, not only to enlarge the size of sPIV ensembles, but
+to address, in an analytical way, the very unexpected self-
+similar dynamics of the OS mode transitions. We point
+out that the dynamical scaling range of the recurrent OS
+transitions reflects the existence of finite-sized OS packets
+along the pipe flow, correlated at integral length scales
+(i.e., the pipe’s diameter).
+An interesting theoretical direction to pursue is related
+to the use of instanton techniques [14] to evaluate the
+transition probabilities between unstable flow configura-
+tions as are the OS modes. In the turbulence or transi-
+tional context, instantons are taken, respectively, as ex-
+treme events or flow configurations that dominate the
+probability measures in the weak coupling limit. They
+have been successfully applied to a number of fluid dy-
+namic problems, as in geophysical models, homogeneous
+turbulence and the laminar-turbulent transition in shear
+flows [15–17].
+We conclude by noting that the findings here presented
+are likely to add relevant phenomenological information
+to the discussion of fundamentally important issues in
+pipe flow turbulence, as drag control and particle-laden
+dynamics, once they are closely connected to the statis-
+tical features of near-wall coherent structures [18–23].
+Acknowledgments
+This work was partially supported by the Conselho
+Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico
+(CNPq) and by Funda¸c˜ao Coppetec/UFRJ (project num-
+ber 20459). L.M. thanks E. Marensi for enlightening dis-
+cussions about the phenomenology of traveling waves.
+
+6
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+[3] T.M. Schneider, B. Eckhardt, and J. Vollmer, Phys. Rev.
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+

K9E4T4oBgHgl3EQf8A5c/content/tmp_files/load_file.txt

@@ -0,0 +1,380 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf,len=379
+page_content='Stochastic Model of Organizational State Transitions in a Turbulent Pipe Flow  Robert J¨ackel1,3, Bruno Magacho2,3, Bayode Owolabi2,3,  Luca Moriconi2,3∗, David J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Dennis3, and Juliana B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Loureiro1,3  1Programa de Engenharia Mecˆanica, Coordena¸c˜ao dos Programas de P´os-Gradua¸c˜ao em Engenharia,  Universidade Federal do Rio de Janeiro, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 68503, CEP: 21941-972, Rio de Janeiro, RJ, Brazil  2Instituto de F´ısica, Universidade Federal do Rio de Janeiro,  Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Athos da Silveira Ramos 149, CEP: 21941-909, Rio de Janeiro, RJ, Brazil  and  3Interdisciplinary Center for Fluid Dynamics, Universidade Federal do Rio de Janeiro,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Moniz Arag˜ao 360, CEP: 21941-594, Rio de Janeiro, Brazil  Turbulent pipe flows exhibit organizational states (OSs) that are labelled by discrete azimuthal  wavenumber modes and are reminiscent of the traveling wave solutions of low Reynolds number  regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' The discretized time evolution of the OSs, obtained through stereoscopic particle image  velocimetry, is shown to be non-Markovian for data acquisition carried out at a structure-resolved  sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' In particular, properly defined time-correlation functions for the OS transitions are  observed to decay as intriguing power laws, up to a large-eddy time horizon, beyond which they  decorrelate at much faster rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' We are able to establish, relying upon a probabilistic descrip-  tion of the creation and annihilation of streamwise streaks, a lower-level Markovian model for the  OS transitions, which reproduces their time-correlated behavior with meaningful accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' These  findings indicate that the OSs are distributed along the pipe as statistically correlated packets of  quasi-streamwise vortical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Notwithstanding the large body of knowledge accumu-  lated since the landmark experiments of Reynolds [1],  turbulent pipes comprise flow patterns which have re-  mained surprisingly unsuspected until recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' They  can be depicted as relatively organized sets of wall-  attached low-speed streaks coupled to pairs of counter-  rotating quasi-streamwise vortices [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' These organi-  zational states (OSs) actually characterize the turbulent  velocity fluctuations at high Reynolds numbers and are  topologically similar to traveling waves – a class of ex-  act (but unstable) low-Reynolds number solutions of the  Navier-Stokes equations [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  As for traveling waves, the OSs can be classified by the  number of low-speed streaks they contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Observation  tells us, however, that this quantity changes in an ap-  parently random way along the turbulent pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' For the  sake of illustration, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 1 shows a transition between  OSs, visualized from a pair of cross-sectional snapshots  of the flow obtained through stereoscopic particle image  velocimetry (sPIV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  The existence of spatial transitions among the OS  modes suggests, within the perspective of dynamical sys-  tems, that the turbulent pipe flow could be described as  a chaotic attractor and its unstable periodic orbits in a  phase space of much reduced dimensionality [7–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' In  connection with this circle of ideas, we are motivated to  study the OS transitions in the framework of stochastic  processes, focusing particular attention on their recurrent  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  To start, let u = u(r, θ) be, in polar coordinates, the  fluctuating streamwise component of the velocity field  ∗Corresponding author: moriconi@if.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='ufrj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='br  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 1: Example of a transition between organized states,  as sampled out from our measurements, which are associated  to two and three low-speed streaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Blue and red colors re-  fer, respectively, to negative and positive streamwise velocity  fluctuations around the mean (the systematic procedure to  ascertain a well-defined number of low-speed structures to a  given flow snapshot is discussed in the text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  defined over a fixed pipe’s cross-sectional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' We may  introduce, accordingly, the instantaneous spectral power  density,  I(kn) =  ����  � 2π  0  dθeiknθfuu(r0, θ)  ����  2  ,  (1)  where  fuu(r0, θ) =  � 2π  0  dθ′u(r0, θ′)u(r0, θ′ + θ) ,  (2)  kn = n ∈ Z+ is an azimuthal wavenumber, and r0 is a ref-  erence radial distance which falls within the log-region of  the pipe’s turbulent boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Empirical evidence  shows that I(kn) is in general peaked at some clearly  dominant wavenumber ¯k (to be identified to the number  of snapshotted low-speed streaks), which can be used to  label the probed velocity profile u(r, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' As time evolves,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='05344v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='flu-dyn]  13 Jan 2023  3  22  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 2: Statistical results for the OS mode ¯k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Left image:  positive (red) and negative (blue) level curves of Ruu, defined  by |Ruu(r − r0|¯k)| = 5% and 10% of (Ruu)max, with the  reference point r0 depicted as a black dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Right image: a  closer look at the averaged streamwise velocity fluctuations  (red for positive, blue for negative), conditioned on u(r0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  The cross-sectional averaged velocity field reveals the vortical  structures that are usually coupled with velocity streaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  I(kn) changes, and so does the wavenumber position of  its dominant peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Therefore, if u(r, θ) is recorded at  equally spaced time intervals ∆, the dynamical evolu-  tion of the pipe turbulent field can be mapped into the  stochastic process  S ≡ {¯k(t), ¯k(t + ∆), ¯k(t + 2∆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  (3)  In order to investigate the still very open statistical prop-  erties of S, we have performed a pipe flow experiment, at  Reynolds number Re = 24415, in the large pipe rig facil-  ity of the Interdisciplinary Nucleus for Fluid Dynamics  (NIDF) at the Federal University of Rio de Janeiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' The  pipe’s diameter and length are, respectively, D = 15 cm  and L = 12 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' By means of sPIV, with sampling rate  of 10 Hz (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=', ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='1 s), we have collected 104 cross-  sectional snapshots of the flow, each one containing the  three components of the turbulent velocity field over a  uniform grid of size 78 × 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' It turns out that all the ob-  served OS modes fall into the range 0 ≤ ¯k ≤ ¯kmax = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Our experimental data has been validated with the  help of previous benchmark pipe flow experiments [12],  through the inspection of the performance of first and  second order single-point statistics for the streamwise  component of velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  We have also attained a  further validation of the entire measured velocity field,  from the evaluation of particularly defined streamwise  velocity-velocity correlation functions conditioned on the  OS modes ¯k, more precisely,  Ruu(∆r|¯k) ≡ E[u(r0)u(r0 + ∆r)|¯k] ,  (4)  which has its level curves depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 2, for the case  ¯k = 5, in close correspondence with the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  The first immediate question that can be raised about  the stochastic process S is whether it is Markovian or  |λh|  10−3  10−2  10−1  100  10−3  10−2  10−1  100  h = time lag between sPIV snapshots / Δ  0  1  2  3  0  1  2  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 3: Eigenvalues of the probability transition matrices for  the original process (h = 1) and a decimated one (h = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  The dashed lines should intercept eigenvalue pairs if S were  a Markovian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Of course, while it is not possible to answer this  in full rigor, one may check if the Chapman-Kolmogorov  (CK) equation holds for the time series (3), a necessary  condition for S to be Markovian [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' The CK equation  would imply that the eigenvalues of the transition prob-  ability matrix for OS modes separated by the time inter-  val h∆ can be represented, in some arbitrary ordering, as  the set of powers {λh  1, λh  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=', λh  ¯kmax}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' A straightforward  computation of the transition matrix eigenvalues for the  cases h = 1 and h = 2 indicates, however, that S is not  Markovian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  We expect that the decimated process for h large  enough is essentially Markovian, since in this situation  the OS modes become weakly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' The transition  to Markovian behavior can be alternatively addressed  from the analysis of correlation functions which we intro-  duce as it follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Taking 0 ≤ m, m′ ≤ ¯kmax, let Vm(t)  and Mm′m(t) be, respectively, the components of vector  and matrix valued stochastic processes derived from S as  Vm(t) =  �  1,  if ¯k(t) = m  0,  otherwise  (5)  and  Mm′m(t) =  �  1,  if ¯k(t) = m and ¯k(t + ∆) = m′  0,  otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  (6)  Define, now, the correlation functions  ˜F(t − t′) ≡ E[V(t) · V(t′)] − (E[V])2 ,  (7)  ˜G(t − t′) ≡ Tr  �  E[MT(t)M(t′)] − E[M]TE[M]  �  , (8)  and their normalized versions,  F(t − t′) ≡  ˜F(t − t′)  ˜F(0)  , G(t − t′) ≡  ˜G(t − t′)  ˜G(0)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content="  (9)  O3  F(t-t')  10−3  10−2  10−1  100  10−3  10−2  10−1  100  |t-t'| (s)  10−1  100  10−1  100  δt  (a)  G(t-t')  10−2  10−1  100  10−2  10−1  100  |t-t'| (s)  10−1  100  10−1  100  δt  (b)  FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 4: The time-dependent correlation functions defined in  (9) are noticed to decay as power laws for |t − t′| ≤ δt ≈ 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  The dotted lines in (a) and (b) have scaling exponent −1 for  both F(t − t′) and G(t − t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  It is not difficult to see that F(t − t′) and G(t − t′)  describe, respectively, the correlations of returning OS  modes and transitions which are apart from each other  by the time interval |t − t′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' They are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 4  and are noticed to have interesting power law decays  (with the same approximate scaling exponent −1) up to  |t − t′| ≡ δt ≈ 20∆ = 2 s, which suggests some sort of  self-similarity across the spatial distribution of about ten  OS modes (their mean lifetime is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='2 s ≈ δt/10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  For  time separations larger than δt, the correlation func-  tions become suddenly undersampled, meaning that they  crossover to a faster law of decay, probably exponential,  as it should be for the putative asymptotic Markovian  behavior of a large-time decimated S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  It is worth emphasizing that the non-Markovian nature  of S does not mean at all that it cannot be modeled as  a Markov process defined in terms of lower-level state  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' In this connection, it is reasonable to assume  that there is a combinatoric degeneracy factor  Ω(¯kmax, m) =  �¯kmax  m  �  (10)  associated to a given OS mode ¯k = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' We simply mean  here that the m wall-attached low-speed streaks can be  spatially arranged for this particular mode in Ω(¯kmax, m)  different ways, since the pipe’s cross-sectional plane is  taken to hold at most ¯kmax low-speed streak channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  The phase space of the “microscopic” state variables  for the underlying Markovian model of S is spanned,  therefore, by all the possible sets of ¯kmax streak bits,  X ≡ {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=', s¯kmax}, where  si =  �  1,  if the i-th streak channel is active  0,  otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  (11)  We postulate, now, that the time evolution of the micro-  scopic states X is produced from the independent fluctu-  ations of streak bits, which have persistence probabilities  that depend on the total number of active streak chan-  nels, that is the OS label m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' In this way, we define qm  and pm to be the persistence probabilities for any given  streak bit to keep its value 0 or 1, respectively, along  subsequent sPIV snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' There are, thus, four dif-  ferent types of streak bit flips, which appear in different  occurrence numbers for a given OS mode transition, as  summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Transition Type # of Streak Channels Transition Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  0 → 0  n1  qm  0 → 1  n2  1 − qm  1 → 0  n3  1 − pm  1 → 1  n4  pm  TABLE I: Definition of the four possible transition types for  the streak channel states, together with the notations for their  occurrence numbers and individual transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Above, m = n3 + n4 labels the OS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  The parameters reported in Table I are related to  the OS mode transition m → m′, where m = n3 + n4  and m′ = n2 + n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' The transition probability between  any specific pair of associated microstates is, as a conse-  quence, qn1  m (1−qm)n2(1−pm)n3pn4  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Taking into account,  furthermore, the role of degeneracy factors, we may write  the transition probability between the OS modes m and  m′ as  4  Tm′m =  �¯kmax  m  �−1 ¯kmax  �  n1=0  ¯kmax  �  n2=0  ¯kmax  �  n3=0  ¯kmax  �  n4=0  δ(n1 + n2 + n3 + n4, ¯kmax)δ(n3 + n4, m)δ(n2 + n4, m′) ×  ×  �¯kmax  n1  ��¯kmax − n1  n2  ��¯kmax − n1 − n2  n3  �  qn1  m (1 − qm)n2(1 − pm)n3pn4  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  (12)  Using, from now on, ¯kmax = 10, the Markovian model  just introduced may not appear very phenomenologically  attractive at first glance, since Tm′m is parametrized by  a large number of unknown parameters (q0, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=', q9 and  p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=', p10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Note, however, that there are, in prin-  ciple, 90 independent entries in the empirical transition  matrix (the one derived from the sPIV measurements),  so the model is rather underdetermined (as we would ex-  pect for a phase-space reduced description of turbulent  fluctuations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Instead of attempting to provide a detailed and com-  putationally costly model of the empirical transition ma-  trix, we address a much simpler approach, where we focus  on the asymptotic probability eigenvector of the modeled  transition matrix,  P = (P1, P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=', P10) ,  (13)  which satifies to TP = P, that is, �10  m=0 Tm′mPm = Pm′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Here, Pm is the probability that the OS mode m be ob-  served in the statistically stationary regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' In an analo-  gous way, denoting by P∞ the empirical probability vec-  tor, determined from the sPIV measurements, we are in-  terested to find the set of probabilities qm and pm that  minimize the quadratic error  d({qm}, {pm}) ≡ ||P − P∞||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  (14)  While, as already commented, the original problem is  underdetermined, the optimization scheme related to  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' (14) is not: as a matter of fact, we would have to  model the 9 independent probability entries of (13) by  means of the 20 probability parameters qm and pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' To  reduce this large overdeterminacy, we rely on a few phe-  nomenological inputs:  (i) We assume that we can model the observed coher-  ence (time persistence) of low-speed streaks by a single  mode-independent and not small probability parameter  p, where p = p2 = p3 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' = p10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  (ii) P0 turns out to be negligible, so we suppress transi-  tions from the OS mode m = 1 to m = 0, by imposing  that p1 = 1 (other transitions to the mode m = 0 from  modes m ̸= 1 are possible, but they are of O((1 − p)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Therefore, we end up with 11 parameters (q0, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=', q9  and p) to locate the minimum value of (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' The result  is a slightly overdetermined system, but if besides P∞,  the correlation functions F(t − t′) and G(t − t′) turn out  to be well reproduced with the same set of probability  parameters, as an extra bonus, then the model can be  taken as physically appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' That is the heuristic setup  that we have in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  We have resorted to a straightforward Monte Carlo  procedure to obtain the set of qm’s that minimizes (14)  for various fixed values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' We find, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 5,  Min{q}[d({q},p)]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='5  p (= p2 = p3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' = p10)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 5: Minimization of the quadratic distance d({q}, p) for  various values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Occurrence Probability (%)  0  5  10  15  20  0  5  10  15  20  OS Mode (k)  0  1  2  3  4  5  6  7  8  9  10  0  1  2  3  4  5  6  7  8  9  10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 6: The occurrence probability of OS modes obtained  from the experiment (dots) and from the stochastic model  (open circles: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='86;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' crosses: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='95), defined by the  transition matrix elements (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content="  5  F(t-t')  10−3  10−2  10−1  100  10−3  10−2  10−1  100  |t-t'| (s)  10−1  100  10−1  100  (a)  G(t-t')  10−2  10−1  100  10−2  10−1  100  |t-t'| (s)  10−1  100  10−1  100  δt  δt  (b)  10−3  10−2  10−1  2  4  6  8  FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 7: Empirical (dots) and modeled (crosses) correlation  functions F(t−t′) and G(t−t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Crosses refer, in (a) and (b),  respectively, to modeling parameters p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='86 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  The semi-log plot in the inset of (b) indicates the simple ex-  ponential form of G(t − t′) at large enough |t − t′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  that the quadratic error quickly drops for p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' The  modeled asymptotic probabilities for the occurrence of  OS modes are excellently compared, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 6, to the  empirical ones for the cases p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='86 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' These  are the values of p that lead to good accounts of F(t−t′)  and G(t−t′), as reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' The related values of  the probabilities qm are listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Even if a point  of subjective concern, the uncertainty of about 10% in  the definition of p should be taken as relatively small,  vis a vis the model’s accuracy in predicting the decaying  profiles of the OS correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  p  q0  q1  q2  q3  q4  q5  q6  q7  q8  q9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='49  TABLE II: The list of probabilities qm’s which describe  the persistence of inactive streak channels, for the cases  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='86 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Also evidenced in the inset Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 7 is the exponential  decay profile of the modeled G(t − t′) for time intervals  larger than δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  At present, this point rests as a pre-  diction of the modeling scenario introduced in this work,  akin with the observed sudden undersampling of the time  series for larger decimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' We note that the crossover  to the faster exponential decay of correlation functions  takes place at δt ≈ 2D/U, where U is the bulk flow ve-  locity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Thus, the physical picture that emerges is that  the OSs are packed as chains of low-speed streaks and  vortical structures which are strongly correlated within  sizes that scale with the pipe’s diameter, although they  are merged along the entire turbulent flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  To summarize, we have investigated the stochastic  properties of the non-Markovian OS mode transitions in a  turbulent pipe flow, recovering them as a surjective map-  ping of a lower-level Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' The essential idea  that underlies the model construction is that a given OS  mode may be associated to several spatial arrangements  of its low-speed streaks into a fixed number of “streak  channels” which azimuthally partition the pipe’s cross  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  We find that the Markov model can account for the  scaling behavior of specifically introduced correlation  functions of OS mode transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Further work is in or-  der, not only to enlarge the size of sPIV ensembles, but  to address, in an analytical way, the very unexpected self-  similar dynamics of the OS mode transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' We point  out that the dynamical scaling range of the recurrent OS  transitions reflects the existence of finite-sized OS packets  along the pipe flow, correlated at integral length scales  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=', the pipe’s diameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  An interesting theoretical direction to pursue is related  to the use of instanton techniques [14] to evaluate the  transition probabilities between unstable flow configura-  tions as are the OS modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' In the turbulence or transi-  tional context, instantons are taken, respectively, as ex-  treme events or flow configurations that dominate the  probability measures in the weak coupling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' They  have been successfully applied to a number of fluid dy-  namic problems, as in geophysical models, homogeneous  turbulence and the laminar-turbulent transition in shear  flows [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  We conclude by noting that the findings here presented  are likely to add relevant phenomenological information  to the discussion of fundamentally important issues in  pipe flow turbulence, as drag control and particle-laden  dynamics, once they are closely connected to the statis-  tical features of near-wall coherent structures [18–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Acknowledgments  This work was partially supported by the Conselho  Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico  (CNPq) and by Funda¸c˜ao Coppetec/UFRJ (project num-  ber 20459).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' thanks E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Marensi for enlightening dis-  cussions about the phenomenology of traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  6  [1] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Reynolds, Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
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+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
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+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
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+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
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+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
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+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  Valad˜ao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
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+page_content='  [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
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+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
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+page_content=' Moin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
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+page_content=' Chandran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
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+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Fu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Wine,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Holloway, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Chung, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Smits, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  12, 5805 (2021) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  [21] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Gallorini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Quadrio, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Gatti, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Fluids  7, 114602 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  [22] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Wang and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Richter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 861, 901 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content='  [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Brandt and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Coletti, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}
+page_content=' 54,  159 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'}

KNAyT4oBgHgl3EQff_hn/content/tmp_files/2301.00350v1.pdf.txt

@@ -0,0 +1,894 @@
+Rawat and Soni et. al. 2023 
+1 
+ 
+            Anisotropic Light-Matter Interactions in Single Crystal 
+Topological Insulator Bismuth Selenide  
+Divya Rawat, Aditya Singh, Niraj Kumar Singh and Ajay Soni* 
+School of Physical Sciences, Indian Institute of Technology Mandi, Mandi, 175005, HP India 
+*Author to whom correspondence should be addressed: ajay@iitmandi.ac.in 
+ 
+Anisotropy of light-matter interactions in materials give remarkable information about the 
+phonons and their interactions with electrons. We report the angle-resolved polarized Raman 
+spectroscopy of single-crystal of Bi2Se3 to obtain the elements of Raman tensor for understanding 
+the strength of polarization along different crystallographic orientations. Intensity variation in the 
+polar plots corresponding to 𝐸𝑔
+1 ~ 37 cm-1,  𝐴1𝑔
+1  ~71 cm-1, 𝐸𝑔
+2 ~ 130 cm-1, and 𝐴1𝑔
+2  ~ 173 cm-1 
+suggests the higher differential polarizability along cross-plane (bc-plane). The polar patterns and 
+the differences in elements of the Raman tensor provides the evidence of the fundamental electron-
+phonon and anisotropic light matter interactions in Bi2Se3.  
+ 
+Keywords: Bismuth Selenide, Anisotropic behaviour, Polarization Raman spectroscopy, Raman 
+tensor, Electron-phonon interactions 
+ 
+ 
+
+Rawat and Soni et. al. 2023 
+2 
+ 
+I. 
+INTRODUCTION 
+Light-matter interaction helps to understand the many body physics and fundamentals of 
+the electron and phonon coupling in materials.[1,2] Exploring the optical properties can provide 
+significant understanding of the (an)-isotropic interaction of light along with the electronic 
+susceptibility and permittivity (dielectric constant) of the materials. [3,4] Generally, the electric 
+field vector (𝐸⃗ ) of the incident and the scattered light are related through a complex matrix, known 
+as Raman tensor (Ʈ) associated with the polarizability (α) of materials along three crystallographic 
+orientations.[5] Recently, several layered materials such as MoS2 [6], WS2 , MoSe2 [5], PdTe2 [7] 
+have been studied using Raman spectroscopy by controlling the polarization vector of incident and 
+scattered light, to understand the dynamics of phonons along the different orientation of the crystal. 
+Layered chalcogenide materials have been known for their anisotropic carrier relaxation times, 
+which mainly arises due to their intriguing crystal structures and inherent anharmonicity.[8,9] 
+Additionally, the Raman studies on ternary chalcogenides, Bi2GeTe4, Sb2SnTe4 have shown that 
+electronic topological properties can also be coupled with phonons, which has been shown by the 
+anomalous thermal behaviour of the Raman modes associated with bonds involved heavy elements. 
+[8] Though several  chalcogenide quantum materials have been explored extensively for their exotic 
+electronic phenomena such as Shubnikov-de Haas quantum oscillations, [10] weak 
+(anti)localization [11], thermoelectricity, superconductivity, charge-density waves and topological 
+quantum insulating properties, yet the coupling of their topological electrons with phonons is less 
+explored. [12-14] Bi2Se3 is one of the layered chalcogenides which has a fascinating layered crystal 
+structure of five atoms (quintuple layers) stacked with van der Waals (vdWs) gaps and a crystal 
+unit cell is composed of three quintuple layers. [15] Primarily, the topological studies on Bi2Se3 has 
+a focus on investigating surface and bulk electronic structures using magneto-transport and angle-
+resolved photoemission spectroscopy studies, phonon dispersion, [16-19], but there are 
+imperceptible reports on the anisotropic response of the inelastic light scattering. Since the 
+topological quantum phenomena are associated with electrons, electron-phonon and electron-
+photon interactions [3,20], thus the investigation of the anisotropy of the electron-phonon-photon 
+interaction, dynamics of phonon and evaluation of Raman-tensor are very important to explore. In 
+this regard, the polarized Raman spectroscopy can provide a significant information about the light 
+sensitive responses of single crystals along various orientations by controlling the polarization of 
+both the incident and scattered photons to acquire the evidences of electron-phonon interactions 
+and anisotropic behaviour. [21] In this work, we have discussed the angle resolved polarized Raman 
+spectroscopy (APRS) to corroborate the interaction between the polarized light (𝑘𝑖) and the 
+
+Rawat and Soni et. al. 2023 
+3 
+ 
+crystallographic orientation of the single crystal Bi2Se3. The isotropic and anisotropic behaviour of 
+phonons are studied with the rotation of crystal along two different configurations in ab-plane 
+(𝑘𝑖||c-axis) and bc-plane (𝑘𝑖||a-axis), respectively. The observed anisotropic behaviour and 
+polarizability of in-plane (𝐸𝑔) and out-of-plane (𝐴1𝑔) modes are quantified from the Raman tensor’s 
+elements. Our results open the opportunities to understand the role of anisotropic light-matter and 
+electron-phonon interactions by both classical as well as quantum treatment of the Raman tensors 
+obtained from the APRS analysis. The experimental details of synthesis and characterization of the 
+single crystal are mentioned in supplemental materials.[22] 
+ 
+ 
+FIG. 1. (a) Electron microscopy image of the fractured cross section of layered Bi2Se3, (b) 
+Powder X-ray diffraction pattern of single crystal showing the typical orientation along the 
+c-axis, (inset: photograph of the grown sample), (c) Schematic of the crystal structure 
+comprises of quintuple layers stacked with a weak Van der Waals gap, (d) Normalized 
+Raman spectra and (e) Schematic of the atomic displacements of the 𝐸𝑔, and 𝐴1𝑔 modes. 
+ 
+The layered nature of the grown Bi2Se3 is shown in FESEM image (Fig. 1 (a)) and the  XRD 
+pattern in Fig. 1 (b), which confirms the orientation of the grown sample along c-axis.[23]  Rietveld 
+refinement of the XRD pattern of powdered Bi2Se3 provides the lattice parameters a =b ~ 4.13 Å, 
+c ~ 28.63 Å, and unit cell volume (V) ~ 425 Å3, (Fig. S1 of supplemental materials [22]). The 
+residual resistance ratio (RRR ~ 2.11) has been evaluated from the low temperature resistance 
+measurement (Fig. S2 of supplemental materials [22]), which shows a generate electron transport 
+in a high quality of single crystal. [22] Bi2Se3 crystallizes in a rhombohedral crystal structure with 
+
+(0)
+(o)
+(d)
+ntensity (arb.units)
+2)
+(b)
+20
+160
+200
+Intensity (arb.units)
+Ramanshift (cm*)
+(e)
+600
+(
+20 (deg)Rawat and Soni et. al. 2023 
+4 
+ 
+space group R3̅m (166), which is comprised of quintuple layers (SeI-Bi-SeII-Bi-SeI) separated by 
+weak vdW gap represented in Fig. 1(c). Here, SeI and SeII represents the different chemical 
+environment of Se atoms in the unit cell. [24,25] The primitive unit cell of Bi2Se3 has fifteen zone-
+center vibrational modes, three acoustic and twelve optical, which can be represented by: Г =
+ 2𝐸𝑔 + 2𝐴1𝑔 + 2𝐸𝑢 + 2𝐴1𝑢, where 𝐴1𝑔 and doubly degenerate 𝐸𝑔 are Raman active modes, 
+whereas 2𝐴1𝑢, 2𝐸𝑢 are the infra-red active modes.[24] The normalized room temperature Raman 
+spectra, having modes at ~ 37 cm-1 (𝐸𝑔
+1), ~ 71 cm-1 (𝐴1𝑔
+1 ), ~ 130 cm-1 (𝐸𝑔
+2), and ~ 173 cm-1 (𝐴1𝑔
+2 ), is 
+shown in Fig. 1(d) and the corresponding schematics of atomic displacements are illustrated in Fig. 
+1(e). The modes 𝐴1𝑔
+1  (𝐴1𝑔
+2 ) and 𝐸𝑔
+1 (𝐸𝑔
+2) have a different polarizability as they involve the out-of-
+plane and in-plane displacements in symmetric (anti-symmetric) stretching, respectively. Thus, 
+angle-resolved polarized spectra (APRS) is an important tool to provide the detailed information 
+on the interaction of the light along the different orientations of the crystal for estimation of 
+elements of Raman tensor.   
+ 
+FIG. 2. Schematic representation of the two configurations used for APRS studies on Bi2Se3 
+crystal, where polarized laser (ki) incidents along (a) c-axis (on ab-plane) and (b) normal to c-
+axis (bc -plane).  Here, ω and θ correspond to the angle between electric polarization vector (𝑒𝑖) 
+of incident light with a-axis (in ab-plane) and b-axis (in bc-plane), respectively.  
+ 
+
+(a)
+(inab-plane)
+(b)
+(in bc-plane)
+532nm Laser
+532nmLaser
+kill = (c-axis)
+E(e)
+Sn-
+E(e)
+D
+x(a-axis)
+y(b-axis)
+xisRawat and Soni et. al. 2023 
+5 
+ 
+Fig. 2 represents the two configurations used for the APRS measurements, where 
+crystallographic axes a, b, and c are taken as equivalent to x, y, and z axes of rotating stage. For the 
+first configuration (Fig. 2 (a)), the incident laser (ki) is parallel to the c-axis and electric polarization 
+vector (𝑒𝑖)  is making an angle ω with the a-axis (in ab-plane). Hence, the scattering configuration 
+is defined as z(xx)𝑧̅, and the corresponding polarization vector of incident and scattered light are 𝑒𝑖⃗⃗  
+= 𝑒𝑠
+⃗⃗⃗  = (cos ω, sin ω, 0). For the second configuration (Fig. 2 (b)), the incident laser (ki) is parallel 
+to a-axis and electric polarization vector (𝑒𝑖)  is making an angle θ with the b-axis (in bc-plane). 
+Correspondingly, the scattering configuration is defined as x(yy)𝑥̅ and the polarization vector of 
+incident and scattered light are 𝑒𝑖⃗⃗  = 𝑒𝑠
+⃗⃗⃗  = (0, cos θ, sin θ). Being isotropic in ab-plane, Bi2Se3 crystal 
+does not have any changes in intensity along a and b axes while the anisotropic light-matter 
+interactions along c axis and the details of Raman tensor is not reported in the literature.  
+ 
+FIG. 3.  Angle dependent polarized Raman spectra (a-b) and corresponding polarized Raman 
+colour plot with the rotation of the Bi2Se3 sample in parallel configuration of polarized 
+incident (ei) and scattered (es) light along ab as well as bc-plane. Colour scale on the right 
+side shows the intensity variation of Raman modes. 
+ 
+Polarized Raman spectra with the rotation of crystal along both ab(/bc)-plane and 
+corresponding colour plot is shown in Fig. 3. The intensity of 𝐴1𝑔
+1  (𝐴1𝑔
+2 ) and 𝐸𝑔
+1 (𝐸𝑔
+2) modes are not 
+changing along ab-plane (Fig. 3 (a)), whereas a periodic alteration has been observed along bc-
+
+(a)
+linensity (ab.anits)
+ab-plane
+0.0
+Intensity (arb.units)
+150
+61
+006
+12
+600
+59.6
+300
+01.9
+30
+60
+06
+120
+150
+180
+210
+204060
+80100120146160180
+4.00
+Ramanshift(cm
+o (deg)
+(b)
+bc-plane
+Intensity (ab.nits)
+Intensity (arb.units)
+200
+2700
+2139
+1800
+1T
+900
+30
+60
+90
+120150
+180210
+Ramanshift(cm)
+0 (deg)
+10012140106-189Rawat and Soni et. al. 2023 
+6 
+ 
+plane (Fig. 3 (b)). The results indicate that there is an existence of anisotropy along the bc-plane as 
+compared to ab-plane, which can be examined clearly from polar plots. According to classical 
+treatment of Raman tensor, the inelastic process can be explained by the scattering from an extended 
+medium, where the variations of the polarization can be expressed as a derivative of the 
+susceptibility of the materials.[21] The contribution of such spatial symmetry to the Raman 
+scattering intensity (I) can be expressed as ⟨𝑒𝑖|Ʈ|𝑒𝑠⟩2, where Ʈ is the Raman tensor for a given 
+mode. [24] Thus, the elements of Raman tensor of 𝐴1𝑔 and double degenerate 𝐸𝑔 modes can be 
+represented as: 
+Ʈ (𝐴1𝑔) = [
+ƞ𝑒𝑖∅ƞ
+0
+0
+0
+ƞ𝑒𝑖∅ƞ
+0
+0
+0
+𝛽𝑒𝑖∅𝛽
+], 
+Ʈ (𝐸𝑔) = [
+𝛾𝑒𝑖∅𝛾
+0
+0
+0
+−𝛾𝑒𝑖∅𝛾
+𝛿𝑒𝑖∅𝛿
+0
+𝛿𝑒𝑖∅𝛿
+0
+] ; [
+0
+−𝛾𝑒𝑖∅𝛾
+−𝛿𝑒𝑖∅𝛿
+−𝛾𝑒𝑖∅𝛾
+0
+0
+−𝛿𝑒𝑖∅𝛿
+0
+0
+], 
+Here the values corresponding to ƞ, β, γ, and δ indicate the amplitudes whereas ∅ƞ, ∅𝛽, ∅𝛾, and ∅𝛿 
+are the complex phases of the elements of Raman tensor. [21] Additionally, the magnitude of each 
+tensor element is related with the specific mode and the crystal symmetry of the material. The 
+calculated intensities for the estimation of the Ʈ (𝐸𝑔) has contributions from both the doubly 
+degenerate 𝐸𝑔 modes, thus added altogether. Using the Raman selection rule, |⟨𝑒𝑖|Ʈ∗|𝑒𝑠⟩|2, under 
+both ab(/bc)-plane, the scattering intensity of all modes have been calculated (Table I), which 
+clearly showed the distinct strength of interaction of polarized light with the crystal’s axes. 
+[5,6,26,27]  
+TABLE I. Mathematically derived intensity of modes using Raman selection rules. 
+S.no.   Configuration          Raman scattering intensity 
+1.              ab-plane                    𝑰𝑨𝟏𝒈
+||
+(ki||c-axis) = |ƞ|𝟐     
+                                                  𝑰𝑬𝒈
+||  (ki||c-axis) = |𝜸|𝟐     
+2.              bc-plane                    𝑰𝑨𝟏𝒈
+||
+(ki||a-axis) = |ƞ|𝟐𝒔𝒊𝒏𝟒𝜽 + |𝜷|𝟐𝒄𝒐𝒔𝟒𝜽 +
+𝟏
+𝟐 |ƞ||𝜷|𝒔𝒊𝒏𝟐(𝟐𝜽)𝒄𝒐𝒔𝝋ƞ𝜷 
+                                              𝑰𝑬𝒈
+||  (ki||a-axis) = |𝜸|𝟐𝒄𝒐𝒔𝟒𝜽 + |𝜹|𝟐𝒔𝒊𝒏𝟐𝟐𝜽 − |𝜹||𝜸| 𝐬𝐢𝐧(𝟐𝜽) 𝒄𝒐𝒔𝟐𝜽  𝟐𝒄𝒐𝒔𝝋𝜸𝜹 
+ 
+
+Rawat and Soni et. al. 2023 
+7 
+ 
+ 
+FIG. 4. Intensities of polar plots for 𝐴1𝑔
+1 , 𝐴1𝑔
+2 ,  𝐸𝑔
+1,  𝐸𝑔
+2 modes in ab-plane (a-b), and in bc-plane 
+(c-f). Here, solid symbols and green line represent the experimental data fitting of the data using 
+equation in Table I, respectively.  
+ 
+Further, the understanding of the isotropic behaviour along ab-plane of the intensity of 𝐴1𝑔 
+and 𝐸𝑔 modes are depicted as circular shapes of the polar intensity plots (Fig. 4 (a-b)). On the other 
+hand, the shape of polar plots for 𝐴1𝑔  (Fig. 4 (c-d)) and 𝐸𝑔 (Fig. 4 (e-f)) modes along bc-plane are 
+different from ab-plane showing the anisotropy of the light matter interaction along crystallographic 
+orientation. The intensities of all modes are stronger along bc-plane in comparison to the ab-plane, 
+which advocates the higher differential polarizability along bc-plane. Similar observations on the 
+anisotropic light-matter interaction in bc-plane have been reported for Graphene, hBN, 2H- MoSe2, 
+MoS2.[5,6,28] Fascinatingly, the out of plane modes at ~ 71 cm-1 and ~ 173 cm-1, (Fig. 4 (c-d)), 
+have 𝐴1𝑔 symmetry but showing considerably different polar pattern at 90o and 270o rotations. The 
+anomalous polarization dependence of the Raman intensities appeared  because of the difference in 
+Raman scattering cross-section through the second-order susceptibility or the electron–phonon 
+interactions.[21]  
+To understand the discrepancy, the microscopic quantum description of Raman tensor has 
+been adopted, which involved the electron-phonon interaction in addition to the electron-photon. 
+[29] Here, the total Raman intensity is described by the product of both the electron-photon and 
+
+in ab-plane
+inbc-plane
+(a)
+90
+Ai
+fcj
+120
+120
+06
+60
+(e)
+120
+90
+1200
+60
+AT
+3600
+AI
+300
+50
+(Sun
+800
+150
+30
+2400
+150
+30
+200
+150
+30
+400
+1200
+100
+Intensity(arb.
+0480
+10
+0180
+40
+0/180
+400
+1200
+100
+008
+210
+330
+2400
+210
+330
+200
+210
+330
+1200
+240
+300
+3600
+240
+300
+240
+270
+270
+270
+300
+(b)
+90
+E
+(p)
+()
+120
+60
+06
+120
+60
+2400
+120
+90
+360F
+a
+E
+1800/
+(sun
+240
+150
+1800
+1200
+150
+30
+1200
+150
+F
+30
+120
+600
+Intensity(arb.
+600
+0180
+0480
+0180
+120
+600
+600
+240
+210
+330
+1200
+210
+1200
+1800
+210
+330
+360
+240
+300
+1800
+270
+240
+270
+300
+2400 L
+240
+270
+300Rawat and Soni et. al. 2023 
+8 
+ 
+electron-phonon interactions. Hence, the Raman tensor (Ʈ𝑖𝑗
+𝑘 ) associated with all modes can be given 
+by:  
+Ʈ𝑖𝑗
+𝑘 = 1
+𝑉  ∑
+∑
+⟨𝛹𝑣(𝑞 )|𝑒 𝑠. ∇⃗⃗ |𝛹𝑐′(𝑞 )⟩ ⟨𝛹𝑐′(𝑞 )|𝐻𝑒𝑝
+𝑘 |𝛹𝑐(𝑞 )⟩⟨𝛹𝑐(𝑞 )|𝑒 𝑖. 𝛻⃗ |𝛹𝑣(𝑞 )⟩
+(𝐸𝐿 − 𝐸𝑐𝑣(𝑞 ) − 𝑖𝛤𝑐)(𝐸𝐿 −  ћ𝜔𝑝ℎ
+𝑘  − 𝐸𝑐′𝑣(𝑞 ) − 𝑖𝛤𝑐′)
+𝑞′
+𝑣,𝑐,𝑐′
+ 
+Here, the numerator consists of the product of three matrix elements, (i) the electron-phonon (e-ph) 
+matrix elements (⟨𝛹𝑐′(𝑞 )|𝐻𝑒𝑝
+𝑘 |𝛹𝑐(𝑞 )⟩) and two electron-photon matrix elements for incident and 
+scattered light (ii) (⟨𝛹𝑐(𝑞 )|𝑒 𝑖. 𝛻⃗ |𝛹𝑣(𝑞 )⟩, (iii) ⟨𝛹𝑣(𝑞 )|𝑒 𝑠. ∇⃗⃗ |𝛹𝑐′(𝑞 )⟩), where 𝑒 𝑖 and  𝑒 𝑠 are the 
+polarization vectors of incident and scattered light, respectively.[29] The summation is over the 
+electronics branches in conduction (𝑐, 𝑐’) and valance (𝑣) bands along with all wave vectors with 
+first Brillouin zone. 𝛤𝑐 and 𝛤𝑐′ are the broadening factor associated with the lifetime of photo-excited 
+states. The inclusion of e-ph matrix element gives the major differences among both the out of plane 
+𝐴1𝑔  modes. Thus, different patterns of polar plots for 𝐴1𝑔
+1 , and 𝐴1𝑔
+2  modes indicate electron-phonon 
+interactions in Bi2Se3, similar to the observations in other anisotropic layered chalcogenides like 
+WS2, ReS2, GaTe, PdSe2 and black phosphorus.[21,29-31] In contrast to the 𝐴1𝑔 modes, the polar 
+plots of 𝐸𝑔 modes show four-lobbed polar pattern (Fig. 4 (e-f)) with the rotation of the crystal, 
+which indicates the maximum strength of anisotropic nature in bc-plane. To understand the 
+behaviour of polar plots related to 𝐸𝑔 modes, the spectra have been captured by controlling the 
+polarization of incident light (ei). This configuration is done by rotating half wave plate from 0o to 
+360o while keeping sample stage and analyzer fixed (Fig. S3 of supplemental materials. [22]) Here, 
+the intensity of both 𝐴1𝑔 modes (Fig. S3 (a-b) of supplemental materials [22]) showed analogous 
+polar pattern with polarization angle, whereas 𝐸𝑔 modes (Fig. S3 (c) of supplemental materials [22]) 
+exhibited a low dependency on the rotation of the half wave plate. This discrepancy of the 𝐸𝑔 modes 
+between the rotation of crystallographic axis and incident laser suggest the anisotropic behaviour 
+along bc-plane. [5,6] Anisotropic light-matter interaction has been understood by estimating the 
+amplitude and phase difference of Raman tensor’s element, which mainly contain the information 
+of differential polarizability along different orientation. To estimate the Raman tensor elements of 
+all modes, we have fitted the experimental data (in Fig 4) using the intensity’s expressions given in 
+Table I and the obtained details are presented in Table II. 
+
+Rawat and Soni et. al. 2023 
+9 
+ 
+TABLE II. Estimated Raman tensor elements obtained from the fitting of experimental data (Fig 4). 
+   Modes                                       Raman tensor                               
+                           ab-plane                              bc-plane 
+ 
+    𝑨𝟏𝒈
+𝟏                      [
+𝟑𝟎
+𝟎
+𝟎
+𝟎
+𝟑𝟎
+𝟎
+𝟎
+𝟎
+𝜷
+]                                  [
+𝟑𝟓
+𝟎
+𝟎
+𝟎
+𝟑𝟓
+𝟎
+𝟎
+𝟎
+𝟓𝟕𝒆𝒊𝟎.𝟑𝟕𝝅
+]                   
+ 
+    𝑨𝟏𝒈
+𝟐                      [
+𝟏𝟕
+𝟎
+𝟎
+𝟎
+𝟏𝟕
+𝟎
+𝟎
+𝟎
+𝜷
+]                                  [
+𝟐𝟏
+𝟎
+𝟎
+𝟎
+𝟐𝟏
+𝟎
+𝟎
+𝟎
+𝟒𝟏𝒆𝒊𝟎.𝟐𝟒𝝅
+] 
+ 
+    𝑬𝒈𝟏                      [
+𝟖
+−𝟖
+𝛅
+−𝟖
+−𝟖
+ 𝛅
+𝛅
+𝛅
+ 𝟎
+]                                 [
+𝟖
+−𝟖
+−𝟏𝟑𝒆𝒊𝟎.𝟑𝟗𝝅
+−𝟖
+−𝟖
+𝟏𝟑𝒆𝒊𝟎.𝟑𝟗𝝅
+−𝟏𝟑𝒆𝒊𝟎.𝟑𝟗𝝅
+𝟏𝟑𝒆𝒊𝟎.𝟑𝟗𝝅
+𝟎
+] 
+    
+     𝑬𝒈𝟐                     [
+𝟏𝟔
+−𝟏𝟔
+𝛅
+−𝟏𝟔
+−𝟏𝟔
+ 𝛅
+𝛅
+𝛅
+  𝟎
+]                            [
+𝟏𝟒
+−𝟏𝟒
+−𝟑𝟖𝒆𝒊𝟎.𝟑𝟐𝝅
+−𝟏𝟒
+−𝟏𝟒
+𝟑𝟖𝒆𝒊𝟎.𝟑𝟐𝝅
+−𝟑𝟖𝒆𝒊𝟎.𝟑𝟐𝝅
+𝟑𝟖𝒆𝒊𝟎.𝟑𝟐𝝅
+𝟎
+]    
+ 
+ 
+In ab-plane, all modes show isotropic behaviour (Fig 4a and 4b), hence for Ʈ (𝐴1𝑔) and Ʈ (𝐸𝑔), the 
+component of Raman tensor, ƞ (𝐴1𝑔
+1 ~ 30 and 𝐴1𝑔
+2  ~ 17)  and γ (𝐸𝑔
+1~ 8 and 𝐸𝑔
+2 ~ 16), have been 
+evaluated from the fitting of polar plots. As the propagation vector ki of incident light is along the 
+c-axis, there is no polarization along c-axis, thus, 𝛽 for out of plane 𝐴1𝑔 mode is not evaluated while 
+𝛽 is zero for in-plane 𝐸𝑔 modes. Here, the phase factor (∅ƞ) is zero due to isotropic responses in ab-
+plane. On the other hand, in bc-plane (Fig. 4c and 4d), the component of Raman tensor, ƞ (𝐴1𝑔
+1 ~ 35 
+and 𝐴1𝑔
+2  ~ 21) and 𝛽 (𝐴1𝑔
+1 ~ 57 and 𝐴1𝑔
+2  ~ 41)  have been evaluated and the phase factor between ƞ 
+and 𝛽 (∅ƞ𝛽) is ~ 67.3o (0.37𝜋) for (𝐴1𝑔
+1 ) and ~ 44o  (0.24𝜋) for (𝐴1𝑔
+2 ), which is arising due to the 
+anisotropic responses. Additionally, the elements of Raman tensor for in-plane modes are 𝛾 (𝐸𝑔
+1 ~ 
+8 and 𝐸𝑔
+2  ~ 14) and  𝛿 (𝐸𝑔
+1 ~ 13 and 𝐸𝑔
+2 ~ 38) and the phase factor between 𝛾 and 𝛿 (∅𝛾𝛿) is ~ 71o 
+(0.39𝜋) for (𝐸𝑔
+1) and ~ 58.4o  (0.32𝜋) for (𝐸𝑔
+2). Overall, for out of plane 𝐴1𝑔 modes, 𝛽 > ƞ, (57 > 
+35 for 𝐴1𝑔
+1   and 41 > 21 for 𝐴1𝑔
+2 ), which indicates that differential polarizability is significantly 
+higher and anisotropic along c-axis (schematic Fig 1e). By comparing the tensor matrices of out of 
+plane modes, it is clearly evident that symmetric stretching (𝐴1𝑔
+1 ) induces larger dipole moment 
+(higher polarizability) than anti- symmetric stretching  (𝐴1𝑔
+2 ) and the situation is completely 
+otherwise for in-plane modes 𝐸𝑔
+1  and 𝐸𝑔
+2 as confirmed by the smaller magnitude of Raman tensor 
+elements in Table II. For both the ab- and bc-plane, the comparison of relative magnitude of Raman 
+tensor elements for of 𝐴1𝑔
+1  (|ƞ𝑏𝑐−𝑝𝑙𝑎𝑛𝑒 ƞ𝑎𝑏−𝑝𝑙𝑎𝑛𝑒
+⁄
+|~ 1.16) and 𝐸𝑔
+2  (|𝛾𝑏𝑐−𝑝𝑙𝑎𝑛𝑒 𝛾𝑎𝑏−𝑝𝑙𝑎𝑛𝑒
+⁄
+|~ 1.14), 
+
+Rawat and Soni et. al. 2023 
+10 
+ 
+which authenticate the estimated elements of the Raman tensor. [6] Comparing the APRS estimated 
+Raman tensor elements with studies on MoSe2, MoS2, WSe2, PdTe2, it is clear that the laser 
+polarization dependence Raman spectra demonstrates the anisotropic light-matter interactions in 
+Bi2Se3.  
+In Summary, the Raman tensor for all modes of single crystal Bi2Se3 corresponds to 𝐸𝑔
+1 ~ 
+37 cm-1, 𝐴1𝑔
+1  ~70 cm-1, 𝐸𝑔2 ~ 129 cm-1, and 𝐴1𝑔
+2  ~ 172 cm-1 have been systematically studied by APRS 
+measurements along both ab(/bc)-plane under parallel polarization (𝑒𝑖 ∥ 𝑒𝑠) scattering 
+configuration. We have estimated the amplitude and phase difference of the tensor elements by 
+fitting the experimental results with the intensity expression obtained by applying Raman selection 
+rule. The different shapes of polar plot of the similar vibrational symmetry (𝐴1𝑔) represents the 
+different interaction of electrons with phonons, which provide the evidence of electron-phonon 
+coupling. Among two different orientations (ab(/bc)-plane) of single crystal, strong polarization 
+dependence has been observed along bc-plane for both 𝐴1𝑔 and 𝐸𝑔 modes, which is showing the 
+anisotropic light matter interaction in Bi2Se3.  
+Acknowledgement 
+We would like to thank IIT Mandi for the instruments and research facilities. A.S would like to 
+acknowledge DST-SERB for funding (Grant No. CRG/2018/002197).  
+References 
+ 
+[1] 
+C. Grazianetti, C. Martella, and E. Cinquanta, Optical Materials: X 12, 100088 (2021). 
+[2] 
+Y. Xu et al., Advanced Optical Materials 6, 1800444 (2018). 
+[3] 
+Y. Xia et al., Nature Physics 5, 398 (2009). 
+[4] 
+J. Singh, Optical properties of condensed matter and applications (John Wiley & Sons, 
+2006), Vol. 6. 
+[5] 
+M. Jin et al., The Journal of Physical Chemistry Letters 11, 4311 (2020). 
+[6] 
+Y. Ding et al., Optics Letters 45 (2020). 
+[7] 
+L. Pi et al., Advanced Functional Materials 29, 1904932 (2019). 
+[8] 
+N. K. Singh et al., Physical Review B 105, 045134 (2022). 
+[9] 
+J. P. Heremans, Nature Physics 11, 990 (2015). 
+[10] 
+Z. Ren et al., Physical Review B 82, 241306 (2010). 
+[11] 
+N. K. Singh, A. Kashyap, and A. Soni, Applied Physics Letters 119, 223903 (2021). 
+[12] 
+J. E. Moore, Nature 464, 194 (2010). 
+
+Rawat and Soni et. al. 2023 
+11 
+ 
+[13] 
+J. Pandey and A. Soni, Physical Review Research 2, 033118 (2020). 
+[14] 
+S. Acharya, J. Pandey, and A. Soni, Applied Physics Letters 109, 133904 (2016). 
+[15] 
+W. Richter and C. R. Becker, physica status solidi b 84, 619 (1977). 
+[16] 
+Y. Kim et al., Applied Physics Letters 100, 071907 (2012). 
+[17] 
+A. C. Ferrari and D. M. Basko, Nature nanotechnology 8, 235 (2013). 
+[18] 
+S. R. Park et al., Physical Review Letters 108, 046805 (2012). 
+[19] 
+S. Sharma et al., Physical Review B 105, 115120 (2022). 
+[20] 
+M. Z. Hasan and C. L. Kane, Reviews of Modern Physics 82, 3045 (2010). 
+[21] 
+J. Kim, J.-U. Lee, and H. Cheong, Journal of Physics: Condensed Matter 32, 343001 
+(2020). 
+[22] 
+See Supplementary Material..... for Synthesis, characterization and details of Raman 
+tensor. 
+[23] 
+K. Mazumder and P. M. Shirage, Journal of Alloys and Compounds 888, 161492 (2021). 
+[24] 
+J. Zhang et al., Nano Letters 11, 2407 (2011). 
+[25] 
+A. Soni et al., Nano Letters 12, 1203 (2012). 
+[26] 
+Y. Ding et al., The Journal of Physical Chemistry Letters 11, 10094 (2020). 
+[27] 
+J. R. Ferraro, K. Nakamoto, and C. W. Brown, in Introductory Raman Spectroscopy, 
+edited by J. R. Ferraro, K. Nakamoto, and C. W. Brown (Academic Press, San Diego, 2003), pp. 
+1. 
+[28] 
+J. Ribeiro-Soares et al., Physical Review B 90, 115438 (2014). 
+[29] 
+G. C. Resende et al., 2D Materials 8, 025002 (2020). 
+[30] 
+Y. Ding et al., Science China Materials 63, 1848 (2020). 
+[31] 
+S. Huang et al., ACS Nano 10, 8964 (2016). 
+ 
+ 
+ 
+
+1 
+ 
+Supplemental Material 
+Anisotropic Light-Matter Interactions in Single Crystal 
+Topological Insulator Bismuth Selenide  
+Divya Rawat, Aditya Singh, Niraj Kumar Singh and Ajay Soni* 
+School of Physical Sciences, Indian Institute of Technology Mandi, Mandi, 175005, HP India 
+*Author to whom correspondence should be addressed: ajay@iitmandi.ac.in 
+ 
+In this supplemental file, we are providing the details of the synthesis, characterization techniques 
+and selected data complementing the main text. 
+ 
+(a) Synthesis and characterization details. 
+Single crystal of Bi2Se3 was synthesized using dual zone vertical Bridgman furnace, by taking a 
+stoichiometric amounts of bismuth ingot and selenium shots (both 99.999% pure) in a quartz 
+ampoule, which was then vacuum sealed at 10-5 mbar. The ampoule was kept in a box furnace at 
+1123 K for 15 hr for homogenization followed by hanging it in Bridgman furnace. The temperature 
+of the hot zone and cold zone were kept at 1003 K and 953 K, respectively. The translation rate of 
+the motor for the vertical motion of quartz tube from hot zone to cold zone was fixed at 2 mm/hr. 
+X-ray diffraction (XRD) was carried out using rotating anode Rigaku SmartLab diffractometer 
+equipped with CuKα radiation (λ = 1.5406 Å) and in Bragg-Brentano geometry. Rietveld 
+refinement of the Powder-XRD pattern was done to determine the crystal structure, lattice 
+parameter, and phase purity. Resistance measurement was performed in the temperature range of 
+2 to 300 K using Quantum Design make physical properties measurement system (PPMS). Raman 
+spectroscopy measurements were carried out using a Horiba LabRAM HR Evolution Raman 
+spectrometer having 532 nm laser excitation, 1800 grooves/mm with the help of a Peltier cooled 
+(CCD) detector. Ultra-low frequency filters were used to access low-frequency spectra, very close 
+to laser line. To control the polarization state, a (λ/2) half-waveplate and an analyzer were used 
+before objective lens and spectrometer to select the desired polarization component of the incident 
+
+2 
+ 
+and scattered light, respectively. To study the light-matter interaction on the crystallographic axis 
+of Bi2Se3, the sample was kept on the stage rotating from ~ 0o to ~ 360o with a step of ~ 20o. The 
+linearly polarized laser was directed on the sample and the scattered radiation was collected to the 
+detector in backscattering geometry.   
+(b) Rietveld refinement analysis. 
+ 
+ 
+FIG. S1:- Rietveld refined XRD pattern of single crystal Bi2Se3. Black closed circle represents 
+the experimental data point, Solid red line represents the refined data. 
+ 
+The as-synthesized Bi2Se3 crystal was ground into fine powder for XRD analysis. The phase purity 
+of Bi2Se3 sample has been confirmed by Rietveld refinements of the powder XRD pattern. [1]The 
+Fig. S1 shows the Rietveld refined XRD data. Goodness of fitting was showed by the extracting 
+parameter, χ2 ~ 2.9.  
+ 
+ 
+
+Observed
+Simulated
+Intensity (arb.units.)
+Difference
+Braggposition
+10
+20
+30
+40
+50
+60
+70
+80
+90
+20 (deg)3 
+ 
+c) Resistance data of single crystal Bi2Se3: 
+ 
+ 
+FIG.S2:- Four-probe resistance measurement with the variation of the temperature. 
+ 
+The electronic transport of the Bi2Se3 has been examined by the four probe resistance (R) and the 
+temperature dependence is consistent with the behavior of degenerate semiconductors. The 
+longitudinal resistance (R) is fitted using a phenomenological model: R = R0 + λe−θ /T + $T2, where 
+the λ and $ appear for phonon scattering and electron-electron scattering, respectively.[2,3]  The 
+fitting parameter are evaluated and λ ~ 12× 10−3and $ ~ 1.74 × 10−7 𝐾−2, where smaller value of 
+$ suggests negligible electron-electron scattering in Bi2Se3. The residual resistance ratio (RRR ~ 
+2.11) shows a high quality of the single crystal. 
+(d) APRS spectra of Bi2Se3 in bc-plane with the rotation of polarization vector of incident 
+light while keeping the sample fixed. 
+
+0.035
+Experimental data pount
+9
+Fittingdata
+0.020
+0
+50
+100
+150
+200
+250
+Temperature (K)4 
+ 
+ 
+FIG. S3:- (a) APRS spectra and Polar plot of (b) 𝐴1𝑔
+1  (c) 𝐴1𝑔
+2  (d) 𝐸𝑔
+2  of Bi2Se3 single crystal 
+with the rotation of half-wave plate by keeping the sample fixed in bc-plane. Solid symbols 
+represent the experimental data point. 
+APRS measurements has been performed in in parallel configuration (𝑒𝑖 ∥ 𝑒𝑠), where polarization 
+vector of incident light has varied by rotating the half-wave plate, while keeping the stage of 
+sample and analyzer fixed. Here, the intensity of both 𝐴1𝑔 modes showed expected two-lobed 
+analogous polar pattern polar pattern with polarization angle.  𝐸𝑔 mode showed a low dependency 
+on the rotation of the half wave plate, showed isotropic interaction on the rotation of polarization 
+vector of incident light. [4] 
+References 
+[1] 
+N. K. Singh, A. Kashyap, and A. Soni, Applied Physics Letters 119, 223903 (2021). 
+[2] 
+T. Kino, T. Endo, and S. Kawata, Journal of the Physical Society of Japan 36, 698 (1974). 
+[3] 
+N. K. Singh et al., Physical Review B 105, 045134 (2022). 
+[4] 
+J. Kim, J.-U. Lee, and H. Cheong, Journal of Physics: Condensed Matter 32, 343001 (2020). 
+
+(b)
+120
+90
+(a)
+Ata
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+(arb.units)
+E
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+E:
+(c)
+00
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+40°
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+1803
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+(d)
+150
+10
+1400
+180
+1800
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KNAyT4oBgHgl3EQff_hn/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf,len=374
+page_content='Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2023  1  Anisotropic Light-Matter Interactions in Single Crystal  Topological Insulator Bismuth Selenide  Divya Rawat, Aditya Singh, Niraj Kumar Singh and Ajay Soni*  School of Physical Sciences, Indian Institute of Technology Mandi, Mandi, 175005, HP India  Author to whom correspondence should be addressed: ajay@iitmandi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='in  Anisotropy of light-matter interactions in materials give remarkable information about the  phonons and their interactions with electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' We report the angle-resolved polarized Raman  spectroscopy of single-crystal of Bi2Se3 to obtain the elements of Raman tensor for understanding  the strength of polarization along different crystallographic orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Intensity variation in the  polar plots corresponding to 𝐸𝑔  1 ~ 37 cm-1,  𝐴1𝑔  1  ~71 cm-1, 𝐸𝑔  2 ~ 130 cm-1, and 𝐴1𝑔  2  ~ 173 cm-1  suggests the higher differential polarizability along cross-plane (bc-plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The polar patterns and  the differences in elements of the Raman tensor provides the evidence of the fundamental electron-  phonon and anisotropic light matter interactions in Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  Keywords: Bismuth Selenide, Anisotropic behaviour, Polarization Raman spectroscopy, Raman  tensor, Electron-phonon interactions  Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2023  2  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  INTRODUCTION  Light-matter interaction helps to understand the many body physics and fundamentals of  the electron and phonon coupling in materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [1,2] Exploring the optical properties can provide  significant understanding of the (an)-isotropic interaction of light along with the electronic  susceptibility and permittivity (dielectric constant) of the materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [3,4] Generally, the electric  field vector (𝐸⃗ ) of the incident and the scattered light are related through a complex matrix, known  as Raman tensor (Ʈ) associated with the polarizability (α) of materials along three crystallographic  orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [5] Recently, several layered materials such as MoS2 [6], WS2 , MoSe2 [5], PdTe2 [7]  have been studied using Raman spectroscopy by controlling the polarization vector of incident and  scattered light, to understand the dynamics of phonons along the different orientation of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  Layered chalcogenide materials have been known for their anisotropic carrier relaxation times,  which mainly arises due to their intriguing crystal structures and inherent anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [8,9]  Additionally, the Raman studies on ternary chalcogenides, Bi2GeTe4, Sb2SnTe4 have shown that  electronic topological properties can also be coupled with phonons, which has been shown by the  anomalous thermal behaviour of the Raman modes associated with bonds involved heavy elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  [8] Though several  chalcogenide quantum materials have been explored extensively for their exotic  electronic phenomena such as Shubnikov-de Haas quantum oscillations, [10] weak  (anti)localization [11], thermoelectricity, superconductivity, charge-density waves and topological  quantum insulating properties, yet the coupling of their topological electrons with phonons is less  explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [12-14] Bi2Se3 is one of the layered chalcogenides which has a fascinating layered crystal  structure of five atoms (quintuple layers) stacked with van der Waals (vdWs) gaps and a crystal  unit cell is composed of three quintuple layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [15] Primarily, the topological studies on Bi2Se3 has  a focus on investigating surface and bulk electronic structures using magneto-transport and angle-  resolved photoemission spectroscopy studies, phonon dispersion, [16-19], but there are  imperceptible reports on the anisotropic response of the inelastic light scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Since the  topological quantum phenomena are associated with electrons, electron-phonon and electron-  photon interactions [3,20], thus the investigation of the anisotropy of the electron-phonon-photon  interaction, dynamics of phonon and evaluation of Raman-tensor are very important to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' In  this regard, the polarized Raman spectroscopy can provide a significant information about the light  sensitive responses of single crystals along various orientations by controlling the polarization of  both the incident and scattered photons to acquire the evidences of electron-phonon interactions  and anisotropic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [21] In this work, we have discussed the angle resolved polarized Raman  spectroscopy (APRS) to corroborate the interaction between the polarized light (𝑘𝑖) and the  Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2023  3  crystallographic orientation of the single crystal Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The isotropic and anisotropic behaviour of  phonons are studied with the rotation of crystal along two different configurations in ab-plane  (𝑘𝑖||c-axis) and bc-plane (𝑘𝑖||a-axis), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The observed anisotropic behaviour and  polarizability of in-plane (𝐸𝑔) and out-of-plane (𝐴1𝑔) modes are quantified from the Raman tensor’s  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Our results open the opportunities to understand the role of anisotropic light-matter and  electron-phonon interactions by both classical as well as quantum treatment of the Raman tensors  obtained from the APRS analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The experimental details of synthesis and characterization of the  single crystal are mentioned in supplemental materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [22]  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' (a) Electron microscopy image of the fractured cross section of layered Bi2Se3, (b)  Powder X-ray diffraction pattern of single crystal showing the typical orientation along the  c-axis, (inset: photograph of the grown sample), (c) Schematic of the crystal structure  comprises of quintuple layers stacked with a weak Van der Waals gap, (d) Normalized  Raman spectra and (e) Schematic of the atomic displacements of the 𝐸𝑔, and 𝐴1𝑔 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  The layered nature of the grown Bi2Se3 is shown in FESEM image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 1 (a)) and the  XRD  pattern in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 1 (b), which confirms the orientation of the grown sample along c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [23]  Rietveld  refinement of the XRD pattern of powdered Bi2Se3 provides the lattice parameters a =b ~ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='13 Å,  c ~ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='63 Å, and unit cell volume (V) ~ 425 Å3, (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' S1 of supplemental materials [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The  residual resistance ratio (RRR ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='11) has been evaluated from the low temperature resistance  measurement (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' S2 of supplemental materials [22]), which shows a generate electron transport  in a high quality of single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [22] Bi2Se3 crystallizes in a rhombohedral crystal structure with  (0)  (o)  (d)  ntensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='units)  2)  (b)  20  160  200  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='units)  Ramanshift (cm*)  (e)  600  (  20 (deg)Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2023  4  space group R3̅m (166), which is comprised of quintuple layers (SeI-Bi-SeII-Bi-SeI) separated by  weak vdW gap represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Here, SeI and SeII represents the different chemical  environment of Se atoms in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [24,25] The primitive unit cell of Bi2Se3 has fifteen zone-  center vibrational modes, three acoustic and twelve optical, which can be represented by: Г =  2𝐸𝑔 + 2𝐴1𝑔 + 2𝐸𝑢 + 2𝐴1𝑢, where 𝐴1𝑔 and doubly degenerate 𝐸𝑔 are Raman active modes,  whereas 2𝐴1𝑢, 2𝐸𝑢 are the infra-red active modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [24] The normalized room temperature Raman  spectra, having modes at ~ 37 cm-1 (𝐸𝑔  1), ~ 71 cm-1 (𝐴1𝑔  1 ), ~ 130 cm-1 (𝐸𝑔  2), and ~ 173 cm-1 (𝐴1𝑔  2 ), is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 1(d) and the corresponding schematics of atomic displacements are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The modes 𝐴1𝑔  1  (𝐴1𝑔  2 ) and 𝐸𝑔  1 (𝐸𝑔  2) have a different polarizability as they involve the out-of-  plane and in-plane displacements in symmetric (anti-symmetric) stretching, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Thus,  angle-resolved polarized spectra (APRS) is an important tool to provide the detailed information  on the interaction of the light along the different orientations of the crystal for estimation of  elements of Raman tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Schematic representation of the two configurations used for APRS studies on Bi2Se3  crystal, where polarized laser (ki) incidents along (a) c-axis (on ab-plane) and (b) normal to c-  axis (bc -plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Here, ω and θ correspond to the angle between electric polarization vector (𝑒𝑖)  of incident light with a-axis (in ab-plane) and b-axis (in bc-plane), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  (a)  (inab-plane)  (b)  (in bc-plane)  532nm Laser  532nmLaser  kill = (c-axis)  E(e)  Sn-  E(e)  D  x(a-axis)  y(b-axis)  xisRawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2023  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2 represents the two configurations used for the APRS measurements, where  crystallographic axes a, b, and c are taken as equivalent to x, y, and z axes of rotating stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' For the  first configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2 (a)), the incident laser (ki) is parallel to the c-axis and electric polarization  vector (𝑒𝑖)  is making an angle ω with the a-axis (in ab-plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Hence, the scattering configuration  is defined as z(xx)𝑧̅, and the corresponding polarization vector of incident and scattered light are 𝑒𝑖⃗⃗  = 𝑒𝑠  ⃗⃗⃗  = (cos ω, sin ω, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' For the second configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2 (b)), the incident laser (ki) is parallel  to a-axis and electric polarization vector (𝑒𝑖)  is making an angle θ with the b-axis (in bc-plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  Correspondingly, the scattering configuration is defined as x(yy)𝑥̅ and the polarization vector of  incident and scattered light are 𝑒𝑖⃗⃗  = 𝑒𝑠  ⃗⃗⃗  = (0, cos θ, sin θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Being isotropic in ab-plane, Bi2Se3 crystal  does not have any changes in intensity along a and b axes while the anisotropic light-matter  interactions along c axis and the details of Raman tensor is not reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Angle dependent polarized Raman spectra (a-b) and corresponding polarized Raman  colour plot with the rotation of the Bi2Se3 sample in parallel configuration of polarized  incident (ei) and scattered (es) light along ab as well as bc-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Colour scale on the right  side shows the intensity variation of Raman modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  Polarized Raman spectra with the rotation of crystal along both ab(/bc)-plane and  corresponding colour plot is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The intensity of 𝐴1𝑔  1  (𝐴1𝑔  2 ) and 𝐸𝑔  1 (𝐸𝑔  2) modes are not  changing along ab-plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 3 (a)), whereas a periodic alteration has been observed along bc-  (a)  linensity (ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='anits)  ab-plane  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='0  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='units)  150  61  006  12  600  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='6  300  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='9  30  60  06  120  150  180  210  204060  80100120146160180  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='00  Ramanshift(cm  o (deg)  (b)  bc-plane  Intensity (ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='nits)  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='units)  200  2700  2139  1800  1T  900  30  60  90  120150  180210  Ramanshift(cm)  0 (deg)  10012140106-189Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2023  6  plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 3 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The results indicate that there is an existence of anisotropy along the bc-plane as  compared to ab-plane, which can be examined clearly from polar plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' According to classical  treatment of Raman tensor, the inelastic process can be explained by the scattering from an extended  medium, where the variations of the polarization can be expressed as a derivative of the  susceptibility of the materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [21] The contribution of such spatial symmetry to the Raman  scattering intensity (I) can be expressed as ⟨𝑒𝑖|Ʈ|𝑒𝑠⟩2, where Ʈ is the Raman tensor for a given  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [24] Thus, the elements of Raman tensor of 𝐴1𝑔 and double degenerate 𝐸𝑔 modes can be  represented as:  Ʈ (𝐴1𝑔) = [  ƞ𝑒𝑖∅ƞ  0  0  0  ƞ𝑒𝑖∅ƞ  0  0  0  𝛽𝑒𝑖∅𝛽  ],  Ʈ (𝐸𝑔) = [  𝛾𝑒𝑖∅𝛾  0  0  0  −𝛾𝑒𝑖∅𝛾  𝛿𝑒𝑖∅𝛿  0  𝛿𝑒𝑖∅𝛿  0  ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [  0  −𝛾𝑒𝑖∅𝛾  −𝛿𝑒𝑖∅𝛿  −𝛾𝑒𝑖∅𝛾  0  0  −𝛿𝑒𝑖∅𝛿  0  0  ],  Here the values corresponding to ƞ, β, γ, and δ indicate the amplitudes whereas ∅ƞ, ∅𝛽, ∅𝛾, and ∅𝛿  are the complex phases of the elements of Raman tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [21] Additionally, the magnitude of each  tensor element is related with the specific mode and the crystal symmetry of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The  calculated intensities for the estimation of the Ʈ (𝐸𝑔) has contributions from both the doubly  degenerate 𝐸𝑔 modes, thus added altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Using the Raman selection rule, |⟨𝑒𝑖|Ʈ∗|𝑒𝑠⟩|2, under  both ab(/bc)-plane, the scattering intensity of all modes have been calculated (Table I), which  clearly showed the distinct strength of interaction of polarized light with the crystal’s axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  [5,6,26,27]  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Mathematically derived intensity of modes using Raman selection rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Configuration          Raman scattering intensity  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='              ab-plane                    𝑰𝑨𝟏𝒈  ||  (ki||c-axis) = |ƞ|𝟐  𝑰𝑬𝒈  ||  (ki||c-axis) = |𝜸|𝟐  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='              bc-plane                    𝑰𝑨𝟏𝒈  ||  (ki||a-axis) = |ƞ|𝟐𝒔𝒊𝒏𝟒𝜽 + |𝜷|𝟐𝒄𝒐𝒔𝟒𝜽 +  𝟏  𝟐 |ƞ||𝜷|𝒔𝒊𝒏𝟐(𝟐𝜽)𝒄𝒐𝒔𝝋ƞ𝜷  𝑰𝑬𝒈  ||  (ki||a-axis) = |𝜸|𝟐𝒄𝒐𝒔𝟒𝜽 + |𝜹|𝟐𝒔𝒊𝒏𝟐𝟐𝜽 − |𝜹||𝜸| 𝐬𝐢𝐧(𝟐𝜽) 𝒄𝒐𝒔𝟐𝜽 \uf0b4 𝟐𝒄𝒐𝒔𝝋𝜸𝜹  Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2023  7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Intensities of polar plots for 𝐴1𝑔  1 , 𝐴1𝑔  2 ,  𝐸𝑔  1,  𝐸𝑔  2 modes in ab-plane (a-b), and in bc-plane  (c-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Here, solid symbols and green line represent the experimental data fitting of the data using  equation in Table I, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  Further, the understanding of the isotropic behaviour along ab-plane of the intensity of 𝐴1𝑔  and 𝐸𝑔 modes are depicted as circular shapes of the polar intensity plots (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 4 (a-b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' On the other  hand, the shape of polar plots for 𝐴1𝑔  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 4 (c-d)) and 𝐸𝑔 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 4 (e-f)) modes along bc-plane are  different from ab-plane showing the anisotropy of the light matter interaction along crystallographic  orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The intensities of all modes are stronger along bc-plane in comparison to the ab-plane,  which advocates the higher differential polarizability along bc-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Similar observations on the  anisotropic light-matter interaction in bc-plane have been reported for Graphene, hBN, 2H- MoSe2,  MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [5,6,28] Fascinatingly, the out of plane modes at ~ 71 cm-1 and ~ 173 cm-1, (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 4 (c-d)),  have 𝐴1𝑔 symmetry but showing considerably different polar pattern at 90o and 270o rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The  anomalous polarization dependence of the Raman intensities appeared  because of the difference in  Raman scattering cross-section through the second-order susceptibility or the electron–phonon  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [21]  To understand the discrepancy, the microscopic quantum description of Raman tensor has  been adopted, which involved the electron-phonon interaction in addition to the electron-photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  [29] Here, the total Raman intensity is described by the product of both the electron-photon and  in ab-plane  inbc-plane  (a)  90  Ai  fcj  120  120  06  60  (e)  120  90  1200  60  AT  3600  AI  300  50  (Sun  800  150  30  2400  150  30  200  150  30  400  1200  100  Intensity(arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  0480  10  0180  40  0/180  400  1200  100  008  210  330  2400  210  330  200  210  330  1200  240  300  3600  240  300  240  270  270  270  300  (b)  90  E  (p)  ()  120  60  06  120  60  2400  120  90  360F  a  E  1800/  (sun  240  150  1800  1200  150  30  1200  150  F  30  120  600  Intensity(arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  600  0180  0480  0180  120  600  600  240  210  330  1200  210  1200  1800  210  330  360  240  300  1800  270  240  270  300  2400 L  240  270  300Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2023  8  electron-phonon interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Hence, the Raman tensor (Ʈ𝑖𝑗  𝑘 ) associated with all modes can be given  by:  Ʈ𝑖𝑗  𝑘 = 1  𝑉  ∑  ∑  ⟨𝛹𝑣(𝑞 )|𝑒 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' ∇⃗⃗ |𝛹𝑐′(𝑞 )⟩ ⟨𝛹𝑐′(𝑞 )|𝐻𝑒𝑝  𝑘 |𝛹𝑐(𝑞 )⟩⟨𝛹𝑐(𝑞 )|𝑒 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 𝛻⃗ |𝛹𝑣(𝑞 )⟩  (𝐸𝐿 − 𝐸𝑐𝑣(𝑞 ) − 𝑖𝛤𝑐)(𝐸𝐿 −  ћ𝜔𝑝ℎ  𝑘  − 𝐸𝑐′𝑣(𝑞 ) − 𝑖𝛤𝑐′)  𝑞′  𝑣,𝑐,𝑐′  Here, the numerator consists of the product of three matrix elements, (i) the electron-phonon (e-ph)  matrix elements (⟨𝛹𝑐′(𝑞 )|𝐻𝑒𝑝  𝑘 |𝛹𝑐(𝑞 )⟩) and two electron-photon matrix elements for incident and  scattered light (ii) (⟨𝛹𝑐(𝑞 )|𝑒 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 𝛻⃗ |𝛹𝑣(𝑞 )⟩, (iii) ⟨𝛹𝑣(𝑞 )|𝑒 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' ∇⃗⃗ |𝛹𝑐′(𝑞 )⟩), where 𝑒 𝑖 and  𝑒 𝑠 are the  polarization vectors of incident and scattered light, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [29] The summation is over the  electronics branches in conduction (𝑐, 𝑐’) and valance (𝑣) bands along with all wave vectors with  first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 𝛤𝑐 and 𝛤𝑐′ are the broadening factor associated with the lifetime of photo-excited  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The inclusion of e-ph matrix element gives the major differences among both the out of plane  𝐴1𝑔  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Thus, different patterns of polar plots for 𝐴1𝑔  1 , and 𝐴1𝑔  2  modes indicate electron-phonon  interactions in Bi2Se3, similar to the observations in other anisotropic layered chalcogenides like  WS2, ReS2, GaTe, PdSe2 and black phosphorus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [21,29-31] In contrast to the 𝐴1𝑔 modes, the polar  plots of 𝐸𝑔 modes show four-lobbed polar pattern (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 4 (e-f)) with the rotation of the crystal,  which indicates the maximum strength of anisotropic nature in bc-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' To understand the  behaviour of polar plots related to 𝐸𝑔 modes, the spectra have been captured by controlling the  polarization of incident light (ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' This configuration is done by rotating half wave plate from 0o to  360o while keeping sample stage and analyzer fixed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' S3 of supplemental materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [22]) Here,  the intensity of both 𝐴1𝑔 modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' S3 (a-b) of supplemental materials [22]) showed analogous  polar pattern with polarization angle, whereas 𝐸𝑔 modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' S3 (c) of supplemental materials [22])  exhibited a low dependency on the rotation of the half wave plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' This discrepancy of the 𝐸𝑔 modes  between the rotation of crystallographic axis and incident laser suggest the anisotropic behaviour  along bc-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [5,6] Anisotropic light-matter interaction has been understood by estimating the  amplitude and phase difference of Raman tensor’s element, which mainly contain the information  of differential polarizability along different orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' To estimate the Raman tensor elements of  all modes, we have fitted the experimental data (in Fig 4) using the intensity’s expressions given in  Table I and the obtained details are presented in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2023  9  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Estimated Raman tensor elements obtained from the fitting of experimental data (Fig 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  Modes                                       Raman tensor  ab-plane                              bc-plane  𝑨𝟏𝒈  𝟏                      [  𝟑𝟎  𝟎  𝟎  𝟎  𝟑𝟎  𝟎  𝟎  𝟎  𝜷  ]                                  [  𝟑𝟓  𝟎  𝟎  𝟎  𝟑𝟓  𝟎  𝟎  𝟎  𝟓𝟕𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='𝟑𝟕𝝅  ]  𝑨𝟏𝒈  𝟐                      [  𝟏𝟕  𝟎  𝟎  𝟎  𝟏𝟕  𝟎  𝟎  𝟎  𝜷  ]                                  [  𝟐𝟏  𝟎  𝟎  𝟎  𝟐𝟏  𝟎  𝟎  𝟎  𝟒𝟏𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='𝟐𝟒𝝅  ]  𝑬𝒈𝟏                      [  𝟖  −𝟖  𝛅  −𝟖  −𝟖  𝛅  𝛅  𝛅  𝟎  ]                                 [  𝟖  −𝟖  −𝟏𝟑𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='𝟑𝟗𝝅  −𝟖  −𝟖  𝟏𝟑𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='𝟑𝟗𝝅  −𝟏𝟑𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='𝟑𝟗𝝅  𝟏𝟑𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='𝟑𝟗𝝅  𝟎  ]  𝑬𝒈𝟐                     [  𝟏𝟔  −𝟏𝟔  𝛅  −𝟏𝟔  −𝟏𝟔  𝛅  𝛅  𝛅  𝟎  ]                            [  𝟏𝟒  −𝟏𝟒  −𝟑𝟖𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='𝟑𝟐𝝅  −𝟏𝟒  −��𝟒  𝟑𝟖𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='𝟑𝟐𝝅  −𝟑𝟖𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='𝟑𝟐𝝅  𝟑𝟖𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='𝟑𝟐𝝅  𝟎  ]  In ab-plane, all modes show isotropic behaviour (Fig 4a and 4b), hence for Ʈ (𝐴1𝑔) and Ʈ (𝐸𝑔), the  component of Raman tensor, ƞ (𝐴1𝑔  1 ~ 30 and 𝐴1𝑔  2  ~ 17)  and γ (𝐸𝑔  1~ 8 and 𝐸𝑔  2 ~ 16), have been  evaluated from the fitting of polar plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' As the propagation vector ki of incident light is along the  c-axis, there is no polarization along c-axis, thus, 𝛽 for out of plane 𝐴1𝑔 mode is not evaluated while  𝛽 is zero for in-plane 𝐸𝑔 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Here, the phase factor (∅ƞ) is zero due to isotropic responses in ab-  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' On the other hand, in bc-plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 4c and 4d), the component of Raman tensor, ƞ (𝐴1𝑔  1 ~ 35  and 𝐴1𝑔  2  ~ 21) and 𝛽 (𝐴1𝑔  1 ~ 57 and 𝐴1𝑔  2  ~ 41)  have been evaluated and the phase factor between ƞ  and 𝛽 (∅ƞ𝛽) is ~ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='3o (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='37𝜋) for (𝐴1𝑔  1 ) and ~ 44o  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='24𝜋) for (𝐴1𝑔  2 ), which is arising due to the  anisotropic responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Additionally, the elements of Raman tensor for in-plane modes are 𝛾 (𝐸𝑔  1 ~  8 and 𝐸𝑔  2  ~ 14) and  𝛿 (𝐸𝑔  1 ~ 13 and 𝐸𝑔  2 ~ 38) and the phase factor between 𝛾 and 𝛿 (∅𝛾𝛿) is ~ 71o  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='39𝜋) for (𝐸𝑔  1) and ~ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='4o  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='32𝜋) for (𝐸𝑔  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Overall, for out of plane 𝐴1𝑔 modes, 𝛽 > ƞ, (57 >  35 for 𝐴1𝑔  1   and 41 > 21 for 𝐴1𝑔  2 ), which indicates that differential polarizability is significantly  higher and anisotropic along c-axis (schematic Fig 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' By comparing the tensor matrices of out of  plane modes, it is clearly evident that symmetric stretching (𝐴1𝑔  1 ) induces larger dipole moment  (higher polarizability) than anti- symmetric stretching  (𝐴1𝑔  2 ) and the situation is completely  otherwise for in-plane modes 𝐸𝑔  1  and 𝐸𝑔  2 as confirmed by the smaller magnitude of Raman tensor  elements in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' For both the ab- and bc-plane, the comparison of relative magnitude of Raman  tensor elements for of 𝐴1𝑔  1  (|ƞ𝑏𝑐−𝑝𝑙𝑎𝑛𝑒 ƞ𝑎𝑏−𝑝𝑙𝑎𝑛𝑒  ⁄  |~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='16) and 𝐸𝑔  2  (|𝛾𝑏𝑐−𝑝𝑙𝑎𝑛𝑒 𝛾𝑎𝑏−𝑝𝑙𝑎𝑛𝑒  ⁄  |~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='14),  Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 2023  10  which authenticate the estimated elements of the Raman tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [6] Comparing the APRS estimated  Raman tensor elements with studies on MoSe2, MoS2, WSe2, PdTe2, it is clear that the laser  polarization dependence Raman spectra demonstrates the anisotropic light-matter interactions in  Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  In Summary, the Raman tensor for all modes of single crystal Bi2Se3 corresponds to 𝐸𝑔  1 ~  37 cm-1, 𝐴1𝑔  1  ~70 cm-1, 𝐸𝑔2 ~ 129 cm-1, and 𝐴1𝑔  2  ~ 172 cm-1 have been systematically studied by APRS  measurements along both ab(/bc)-plane under parallel polarization (𝑒𝑖 ∥ 𝑒𝑠) scattering  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' We have estimated the amplitude and phase difference of the tensor elements by  fitting the experimental results with the intensity expression obtained by applying Raman selection  rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The different shapes of polar plot of the similar vibrational symmetry (𝐴1𝑔) represents the  different interaction of electrons with phonons, which provide the evidence of electron-phonon  coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Among two different orientations (ab(/bc)-plane) of single crystal, strong polarization  dependence has been observed along bc-plane for both 𝐴1𝑔 and 𝐸𝑔 modes, which is showing the  anisotropic light matter interaction in Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  Acknowledgement  We would like to thank IIT Mandi for the instruments and research facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
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+page_content='  1  Supplemental Material  Anisotropic Light-Matter Interactions in Single Crystal  Topological Insulator Bismuth Selenide  Divya Rawat, Aditya Singh, Niraj Kumar Singh and Ajay Soni*  School of Physical Sciences, Indian Institute of Technology Mandi, Mandi, 175005, HP India  Author to whom correspondence should be addressed: ajay@iitmandi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='in  In this supplemental file, we are providing the details of the synthesis, characterization techniques  and selected data complementing the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  (a) Synthesis and characterization details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  Single crystal of Bi2Se3 was synthesized using dual zone vertical Bridgman furnace, by taking a  stoichiometric amounts of bismuth ingot and selenium shots (both 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='999% pure) in a quartz  ampoule, which was then vacuum sealed at 10-5 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The ampoule was kept in a box furnace at  1123 K for 15 hr for homogenization followed by hanging it in Bridgman furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The temperature  of the hot zone and cold zone were kept at 1003 K and 953 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The translation rate of  the motor for the vertical motion of quartz tube from hot zone to cold zone was fixed at 2 mm/hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  X-ray diffraction (XRD) was carried out using rotating anode Rigaku SmartLab diffractometer  equipped with CuKα radiation (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='5406 Å) and in Bragg-Brentano geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Rietveld  refinement of the Powder-XRD pattern was done to determine the crystal structure, lattice  parameter, and phase purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Resistance measurement was performed in the temperature range of  2 to 300 K using Quantum Design make physical properties measurement system (PPMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Raman  spectroscopy measurements were carried out using a Horiba LabRAM HR Evolution Raman  spectrometer having 532 nm laser excitation, 1800 grooves/mm with the help of a Peltier cooled  (CCD) detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Ultra-low frequency filters were used to access low-frequency spectra, very close  to laser line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' To control the polarization state, a (λ/2) half-waveplate and an analyzer were used  before objective lens and spectrometer to select the desired polarization component of the incident  2  and scattered light, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' To study the light-matter interaction on the crystallographic axis  of Bi2Se3, the sample was kept on the stage rotating from ~ 0o to ~ 360o with a step of ~ 20o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The  linearly polarized laser was directed on the sample and the scattered radiation was collected to the  detector in backscattering geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  (b) Rietveld refinement analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' S1:- Rietveld refined XRD pattern of single crystal Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Black closed circle represents  the experimental data point, Solid red line represents the refined data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  The as-synthesized Bi2Se3 crystal was ground into fine powder for XRD analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The phase purity  of Bi2Se3 sample has been confirmed by Rietveld refinements of the powder XRD pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [1]The  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' S1 shows the Rietveld refined XRD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Goodness of fitting was showed by the extracting  parameter, χ2 ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  Observed  Simulated  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=')  Difference  Braggposition  10  20  30  40  50  60  70  80  90  20 (deg)3  c) Resistance data of single crystal Bi2Se3:  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='S2:- Four-probe resistance measurement with the variation of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  The electronic transport of the Bi2Se3 has been examined by the four probe resistance (R) and the  temperature dependence is consistent with the behavior of degenerate semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The  longitudinal resistance (R) is fitted using a phenomenological model: R = R0 + λe−θ /T + $T2, where  the λ and $ appear for phonon scattering and electron-electron scattering, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [2,3]  The  fitting parameter are evaluated and λ ~ 12× 10−3and $ ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='74 × 10−7 𝐾−2, where smaller value of  $ suggests negligible electron-electron scattering in Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' The residual resistance ratio (RRR ~  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='11) shows a high quality of the single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  (d) APRS spectra of Bi2Se3 in bc-plane with the rotation of polarization vector of incident  light while keeping the sample fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='035  Experimental data pount  9  Fittingdata  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='020  0  50  100  150  200  250  Temperature (K)4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' S3:- (a) APRS spectra and Polar plot of (b) 𝐴1𝑔  1  (c) 𝐴1𝑔  2  (d) 𝐸𝑔  2  of Bi2Se3 single crystal  with the rotation of half-wave plate by keeping the sample fixed in bc-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Solid symbols  represent the experimental data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  APRS measurements has been performed in in parallel configuration (𝑒𝑖 ∥ 𝑒𝑠), where polarization  vector of incident light has varied by rotating the half-wave plate, while keeping the stage of  sample and analyzer fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Here, the intensity of both 𝐴1𝑔 modes showed expected two-lobed  analogous polar pattern polar pattern with polarization angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' 𝐸𝑔 mode showed a low dependency  on the rotation of the half wave plate, showed isotropic interaction on the rotation of polarization  vector of incident light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' [4]  References  [1]  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Singh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Kashyap, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Soni, Applied Physics Letters 119, 223903 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  [2]  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Kino, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Endo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Kawata, Journal of the Physical Society of Japan 36, 698 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  [3]  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=', Physical Review B 105, 045134 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  [4]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Lee, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content=' Cheong, Journal of Physics: Condensed Matter 32, 343001 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='  (b)  120  90  (a)  Ata  00  150  180  210  330  240  270  300  (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}
+page_content='units)  E  120  90  E:  (c)  00  00  150  40°  Intensity  1803  800  210  900  240  300  1000  120  90  (d)  150  10  1400  180  1800  210  330  30  60  90  120  150  180  240  270  300  Ramanshift(cm1)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'}

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