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09E5T4oBgHgl3EQfqw9o/content/tmp_files/2301.05710v1.pdf.txt

@@ -0,0 +1,2148 @@
+arXiv:2301.05710v1  [gr-qc]  13 Jan 2023
+Perusing Buchbinder–Lyakhovich canonical formalism for Higher-Order
+Theories of Gravity.
+Dalia Saha†, Abhik Kumar Sanyal‡
+January 18, 2023
+† Dept. of Physics, University of Kalyani, West Bengal, India - 741235.
+†,‡ Dept. of Physics, Jangipur College, Murshidabad, West Bengal, India - 742213.
+Abstract
+Ostrogradsky’s, Dirac’s and Horowitz’s techniques of higher order theories of gravity produce identical
+phase-space structures. The problem is manifested in the case of Gauss-Bonnet-dilatonic coupled action in the
+presence of higher-order term, in which case, classical correspondence can’t be established. Here, we explore yet
+another technique developed by Buchbinder and his collaborators (BL) long back and show that it also suffers
+from the same disease. However, expressing the action in terms of the three-space curvature, and removing “the
+total derivative terms”, if Horowitz’s formalism or even Dirac’s constraint analysis is pursued, all pathologies
+disappear. Here we show that the same is true for BL formalism, which appears to be the simplest of all the
+techniques, to handle.
+Keywords: Higher Order theory; Canonical Formulation.
+1
+Introduction
+Canonical formulation of higher-order theories was developed by Ostrogradsky almost two centuries back [1, 2].
+However, it did not draw much attention, since other than toy mechanical models, practically no such physical
+theories were persuaded at that time. Exactly a century elapsed, when it was applied to a physically motivated
+problem, such as fourth order harmonic oscillator [3]. The real physical problem in this context appeared for
+the first time, while a renormalized quantum theory of gravity was attempted to formulate [4]. Higher-derivative
+theory of gravity is usually considered as a model of quantum gravity. The reason being, Einstein-Hilbert ac-
+tion is supplemented by curvature squared terms (R2, RµνRµν ) to ensure renormalizability [4] and asymptotic
+freedom [5–7]. Unfortunately, curvature-squared gravity theories have been found to suffer from the unresolved
+problem of physical unitarity in perturbative analysis, which is usual for higher-derivative theories. However,
+possibilities to overcome this difficulty were also discussed in some literatures [6, 8] and references therein. It is
+also ascertained that curvature squared gravity would arise as a low-energy effective theory derived from super-
+string theory in D = 10 dimensions [9–11]. Over the last couple of decades, higher order theories of gravity e.g.,
+F(R), F(G), F(R, T ) etc, theories, (R, G, T being the Ricci scalar, the Gauss-Bonnet term, and the torsion
+term respectively) have drawn much attention in search of alternatives to dark energy. Nonetheless, it is always
+suggestive to test viability of such modified theories of gravity in different contexts. In the context of the very
+early universe, a canonical formulation is required as a precursor, particularly to study quantum cosmology.
+Since Ostrogradsky’s technique does not apply in the degenerate case of singular Lagrangian, for which the
+Hessian determinant vanishes, Dirac’s constraint analysis [12] may be applied for the purpose. Nonetheless, a host
+of theories have been formulated over decades to bypass the constraint analysis. One of these in this direction
+was originally proposed by Boulware [13], and later reformulated by Horowitz’ [14], in particular in the context of
+higher-order theory of gravity. Since the canonical formulation of higher order theories requires an extra degree
+of freedom, in Horowitz’s formalism apart from the scale factor (‘a′ in the Robertson-Walker minisuperspace) an
+auxiliary variable is introduced by taking derivative of the action (say A) with respect to the highest derivative of
+1Electronic address:
+† daliasahamandal1983@gmail.com
+‡ sanyal ak@yahoo.com
+1
+
+the field variable present (Q = ∂A
+∂¨a ). In the end, the auxiliary variable is replaced by the basic variable (extrinsic
+curvature tensor) through a canonical transformation. The important finding in this regard is as follows: all the
+three formalisms, viz, Ostrogradsky’s (once degeneracy has been removed), Dirac’s and Horowitz’s formalisms,
+produce an identical phase-space structure [15]. Meanwhile, certain pathologies with Horowitz’ formalism have
+been identified. For example, it was noticed that Horowitz’s formalism can even be applied in the case of linear
+gravity theory (Einstein-Hilbert action) leading to wrong quantum dynamics [16–18], as well as some superfluous
+total derivative terms are eliminated [18, 19], which neither may be obtained from the variational principle, nor
+having any connection with Gibbons-Hawking–York term [20,21], nor any of its modified versions, associated with
+higher-order gravity. Further, the coupling parameter, in the case of the “non-minimally coupled scalar tensor the-
+ory of gravity associated with higher order term”, has not been found to play any particular role, since its derivative
+does not appear in the Hamiltonian [22]. The same is true for the “Dilatonic coupled Gauss-Bonnet-theory in
+the presence of higher order term”, where additionally, the classical correspondence with quantum counterpart,
+could not be established [22]. In view of such an uncanny situation, yet another technique was developed, called
+the “modified Horowitz’s formalism” (MHF), which was successfully applied to different modified higher-order
+theories of gravity, to explore the evolution of the very early universe [15,17–19,22–32]. In the MHF, the action
+is expressed in terms of the three-space curvature (instead of the scale factor), “the total derivative terms” are
+removed by integrating the action by parts, and Horowitz’s formalism (the introduction of the auxiliary variable
+etc.) was followed, thereafter.
+To be very specific, let us consider the following isotropic and homogeneous Robertson–Walker (RW) metric:
+ds2 = −N 2(t) dt2 + a2(t)
+�
+dr2
+1 − kr2 + r2(dθ2 + sin2θdφ2)
+�
+,
+(1)
+for which the degeneracy in the Lagrangian disappears if the gauge (N ) is fixed a priori, in which case, Ostrograd-
+sky’s technique applies as well. Once such degeneracy is removed, it is observed that Ostrogradsky’s technique
+produces the same Hamiltonian, obtained following Horowitz’s as well as Dirac’s formalism [15]. Therefore, it
+certainly follows that both the Ostrogradsky’s and Dirac’s formalism implicitly suffer from the same problem,
+in disguise, as was noticed in Horowitz’s technique, as discussed above. Therefore in the MHF, instead of the
+scale factor, the action is expressed in terms of the basic variable hij — the three space metric from the very
+beginning—so that redundant total derivative terms do not appear [18, 19]. Thereafter, all the total derivative
+terms are integrated out by parts, which become cancelled by the supplementary boundary (Gibbons–Hawking—
+York and modified Gibbons–Hawking—York) terms. Subsequently, the auxiliary variable is introduced following
+Horowitz’s proposal. In the end, the auxiliary variable is replaced by the other basic variable Kij — the extrinsic
+curvature tensor. In this process, the unwanted problems that appeared following Horowitz’s formalism disappear,
+while it produces a different Hamiltonian altogether. We mention that although both Hamiltonians (obtained
+following the MHF and Ostrogradky’s, Dirac’s and Horowitz’s formalisms) are related through the canonical
+transformation, they indeed produce different dynamics in the quantum domain. It is also important to mention
+that it is not possible to carry over the classical canonical transformations to the quantum domain for higher-order
+theories, due to the non-linearity. The MHF leads to an effective Hermitian Hamiltonian, a standard quantum
+mechanical probabilistic interpretation, and a viable semiclassical treatment, which exhibit oscillation of the wave
+function about the classical de-Sitter solution. As a result, the classical correspondence is established. In this
+regard, the MHF may be considered as the most-viable technique to handle the higher-order theories. It has later
+been established that, if the action is expressed in terms of the three-space metric (hij ) from the very beginning
+and the total derivative terms are addressed, Dirac’s constraint analysis [12] also produces the Hamiltonian iden-
+tical to that of the MHF [22,28–30].
+Amongst other techniques, Hawking-Luttrell technique [33] has limited application, since conformal trans-
+formation is not possible in general [19], Schmidt’s technique [34] is identical to the Horowitz’s formalism in
+disguise [17]. However, there is yet another technique developed in the 80’s by Buchbinder and his collabora-
+tors [35–39], which did not receive much attention. Querella [40] only noticed that although at a first glance, the
+general formalism developed by Buchbinder and his collaborators (BL) appears to be satisfactory, nevertheless it
+has pitfalls. BL formalism is our current concern. Here, we test this abstract theoretical settings of BL formalism
+in simple minisuperspace model to explore the pitfall, if any. The underlying essence of this formalism is to bypass
+Dirac’s constrained analysis, very much like Horowitz’s technique, but instead of introducing auxiliary variable,
+here the program is initiated with the basic variables {hij, Kij}, the three-space curvature and the extrinsic
+curvature tensors respectively from the very beginning. In our present attempt to explore the outcome of this
+2
+
+technique, we discover that the formalism leads to identical phase space structure as was found in the case of
+Ostrogradsky’s/Dirac’s/Horowitz’s formalism.
+This paper is organized as follows. In the following section, we study scalar tensor theory of gravity (both the
+minimal and non-minimal cases), and Gauss-Bonnet-Dilatonic coupled action being supplemented by the scalar
+curvature squared (R2 ) term, following BL formalism. In Section ??, we explore the fact that once total derivative
+terms are taken care of, the Hamiltonian does not differ from MHF. Section ?? discusses its physical application,
+in connection with some earlier work. Section ??, concludes our work.
+2
+BL Formalism in Three Different Higher Order Theories
+In view of the very importance of higher-order curvature invariant terms required to construct a renormalizable
+quantum theory of gravity when the curvature is extremely strong, a unique canonical formulation of the Einstein–
+Hilbert action being supplemented by higher-order curvature invariant terms, is therefore necessary. Here, we shall
+consider three different cases, minimally and non-minimally coupled scalar-tensor theory of gravity supplemented
+by R2 term, and the scalar-tensor theory of gravity being supplemented by R2 and Gauss-Bonnet terms. In the
+Robertson-Walker minisuperspace (1) under consideration, the Ricci scalar and the Gauss-Bonnet terms are
+R =
+6
+N 2
+�
+¨a
+a + ˙a2
+a2 + N 2 k
+a2 −
+˙N ˙a
+Na
+�
+.
+(2)
+G = R2 − 4RµνRµν + RαβµνRαβµν =
+24
+N 3a3
+�
+N¨a − ˙N ˙a
+� � ˙a2
+N 2 + k
+�
+.
+(3)
+respectively. For the sake of comparison with earlier results, we express actions in terms of the three space metric,
+instead of the scale factor, as its importance has been mentioned already, and will be explicitly shown at the
+beginning of Section ??. Since construction of higher-order theory to its canonical form requires an additional
+degree of freedom, hence, in addition to the three-space metric hij , the extrinsic curvature tensor Kij is treated
+as basic variable, as already stated. We therefore choose the basic variables hij = zδij = a2δij , so that Kij =
+−
+˙hij
+2N = − a ˙a
+N δij = −
+˙z
+2N δij . In terms of z = a2, the Ricci scalar and the Gauss-Bonnet terms take the following
+forms,
+R =
+6
+N 2
+�
+¨z
+2z + N 2 k
+z − 1
+2
+˙N ˙z
+Nz
+�
+,
+(4)
+G = 12
+N 2
+�
+¨z
+z − ˙z2
+2z2 −
+˙N ˙z
+Nz
+� �
+˙z2
+4N 2z2 + k
+z
+�
+,
+(5)
+respectively. It is noteworthy that since,
+RµνRµν = 12
+N 4
+�
+¨a2
+a2 + ˙a2¨a
+a3 + ˙a4
+a4 − 2
+˙N ˙a¨a
+Na2 −
+˙N ˙a3
+Na3 +
+˙N 2 ˙a2
+N 2a2 + k N 2¨a
+a3
++ 2k N 2 ˙a2
+a4
+− k N ˙N ˙a
+a3
++ k2 N 4
+a4
+�
+,
+(6)
+therefore,
+RµνRµν − 1
+3R2 = −
+� 12
+Na3
+� d
+dt
+�1
+3
+˙a3
+N 3 + k ˙a
+N
+�
+,
+(7)
+and as a result,
+� �
+RµνRµν − 1
+3R2
+� √−gd4x = −12C
+� � d
+dt
+�1
+3
+˙a3
+N 3 + k ˙a
+N
+��
+dt
+(8)
+3
+
+is a total derivative term. Thus, RµνRµν term is redundant in RW metric, once R2 term is taken (the constant C
+appears due to the integration of the three space). Hence, to scrutinize the BL formalism presented by Buchbinder
+and his collaborators in RW minisuperspace model (1), we consider scalar-tensor theories of gravity and also
+Gauss-Bonnet-Dilatonic coupled gravity theory, being associated with scalar curvature squared term R2.
+2.1
+Minimal coupling:
+Let us start with the following minimally coupled case,
+A1 =
+� √−g
+�
+αR + βR2 − 1
+2φ,µφ,ν − V (φ)
+�
+d4x + αΣR + βΣR2.
+(9)
+In the above, α =
+1
+16πG , β is a constant coupling parameter, αΣR = 2α
+�
+∂V K
+√
+hd3x is the Gibbons-
+Hawking–York boundary term [21] associated with Einstein–Hilbert sector of the above action, and βΣR2 =
+4β
+�
+∂V RK
+√
+hd3x is its modified version corresponding to R2 term, while, K is the trace of the extrinsic curva-
+ture tensor Kij . Note that, both the counter terms are required under the condition δR = 0, at the boundary.
+Instead, if the condition δKij = 0 is chosen at the boundary, the counter terms are not required, as in the case
+of Horowitz’s formalism [14], since both the boundary terms appearing under metric variation vanish. However,
+in the case of Ostrogradsky’s technique [1] and Dirac constrained analysis [12], boundary terms are not taken
+care of. This is true for BL formalism too, as we shall see shortly. Nevertheless, the modified Horowitz’s for-
+malism [17–19, 23–25] fixes δhij = 0 = δR at the boundary, and hence requires supplementary boundary terms.
+We have demonstrated earlier that proper attention to all the boundary terms is paid in modified Horowitz’s
+formalism (MHF). As a result, it presents a different phase space structure of the Hamiltonian for a particular
+action being supplemented by higher-order terms. Nonetheless, it is related to the others under a suitable set
+of canonical transformation [15]. Although, as mentioned, such transformations cannot be carried over in the
+quantum domain, due to non-linearity. So, it’s indeed required to check if BL formalism also produces the same.
+The action (9) in the RW minisuperspace model (1) may be written in terms of the basic variable hij = zδij , as
+A1 =
+� �
+3α√z
+� ¨z
+N −
+˙N ˙z
+N 2 + 2kN
+�
++ 9β
+√z
+� ¨z2
+N 3 − 2 ˙N ˙z¨z
+N 4
++
+˙N 2 ˙z2
+N 5
+− 4k ˙N ˙z
+N 2
++ 4k¨z
+N + 4k2N
+�
++ z
+3
+2
+� ˙φ2
+2N − V N
+��
+dt + αΣR + βΣR2.
+(10)
+The (0
+0) component of the field equation in terms of the scale factor ‘a′ takes the following form
+6α
+a2
+� ˙a2
+N 2 + k
+�
++ 36β
+a2N 4
+�
+2˙a...a − 2˙a2 ¨N
+N − ¨a2 − 4˙a¨a
+˙N
+N + 2˙a2 ¨a
+a + 5˙a2 ˙N 2
+N 2 − 2 ˙a3 ˙N
+aN
+− 3 ˙a4
+a2 − 2kN 2 ˙a2
+a2 + k2N 4
+a2
+�
+−
+� ˙φ2
+2N 2 + V
+�
+= 0,
+(11)
+which contains term upto third derivative. This is the energy constraint equation (E = 0), and when expressed
+in terms of the phase space variables, becomes the Hamiltonian constraint equation, (due to diffeomorphic invari-
+ance) of the theory under consideration. This we aim at, following the formalism presented by Buchbinder and
+his collaborators (BL).
+The action (9) has already been expressed in terms of the basic variable {hij}, instead of the scale factor.
+Canonical formulation of higher order theories requires additional degree of freedom, and the only choice is the
+extrinsic curvature tensor {Kij}.
+In contrast to Horowitz’s formalism, where apart from {hij} an auxiliary
+variable is introduced and at the end the Hamiltonian is expressed in terms of the basic variables {hij, Kij},
+in BL formalism, these basic variables are associated from the very beginning. In Robertson-Walker metric, the
+extrinsic curvature tensor is expressed as,
+Kij = −
+˙hij
+2N = −2a˙a
+2N δij = − ˙z
+2N δij = −qij say.
+(12)
+4
+
+Since there is only one independent component, so instead of qij , the new generalized coordinate is chosen to be
+its trace viz,
+q = 3 ˙z
+2N ,
+i.e. qij = q
+3δij.
+(13)
+To express the action in terms of velocities, we choose,
+v ≡ ˙q, vφ ≡ ˙φ.
+(14)
+The scalar curvature (4) therefore takes the following form,
+R = 2 ˙q
+Nz + 6k
+z ≡ Rq =
+2
+Nz (v + 3Nk),
+(15)
+and action (10) can be expressed as,
+A1q =
+� �
+2α√z(v + 3kN) +
+4β
+N√z (v + 3kN)2 + z
+3
+2
+�
+v2
+φ
+2N − NV
+��
+dt,
+(16)
+while the Lagrangian density is,
+L1q = 2α√z(v + 3kN) +
+4β
+N√z (v + 3kN)2 + z
+3
+2
+�
+v2
+φ
+2N − NV
+�
+.
+(17)
+Note that the boundary terms remain intact in the action as well as in the point Lagrangian. Canonical momenta
+are
+pq = ∂Lq
+∂v = 2α√z +
+8β
+N√z (v + 3kN), pN = ∂L1q
+∂vN
+= 0, pz = ∂Lq
+∂vz
+= 0 and pφ = ∂Lq
+∂vφ
+= z
+3
+2 vφ
+N
+.
+(18)
+Clearly, there exists two primary constraints C ≡ pN ≈ 0, and D ≡ pz ≈ 0. Therefore, Dirac constraint analysis
+appears to be essential. However, here is a wonderful twist in the BL formalism. For example, one can express
+the modified Lagrangian density as,
+L∗
+1 = L1q + pq ( ˙q − v) + pN
+�
+˙N − vN
+�
++ pz
+�
+˙z − 2Nq
+3
+�
++ pφ
+�
+˙φ − vφ
+�
+,
+(19)
+and equivalently, the Hamiltonian density as,
+H∗
+1 = pq ˙q + pN ˙N + pφ ˙φ + pz ˙z − L∗
+1 = pqv + pNvN + pφvφ + pz
+2Nq
+3
+− L1q.
+(20)
+As a consequence, one can immediately find that the primary constraint D ≡ pz ≈ 0 disappears. Further, since
+N is a non-dynamical Lagrange multiplier, hence the constraint C vanishes strongly. Therefore, one arrives at,
+H∗
+1 = pqv + CvN + pφvφ + pz
+2qN
+3
+− L1q = CvN + pqv + pφvφ + pz
+2qN
+3
+− L1q = CvN + NHm
+BL,
+(21)
+where,
+NHm
+BL = pqv + pφvφ + 2N
+3 qpz − L1q
+= pqv + pφvφ + 2N
+3 qpz − 2α√z(v + 3kN) −
+4β
+N√z (v + 3kN)2 − z
+3
+2
+�
+v2
+φ
+2N + NV
+�
+.
+(22)
+5
+
+In the above, m in the superscript stands for minimally coupled theory. Now upon substituting v and vφ from
+the definition of momentum (18), we obtain,
+NHm
+BL = N
+�2q
+3 pz +
+√z
+16β p2
+q −
+�αz
+4β + 3k
+�
+pq + α2
+4β z
+3
+2 +
+1
+2z
+3
+2 p2
+φ + V z
+3
+2
+�
+,
+(23)
+so that the canonical Hamiltonian finally reads as,
+Hm
+BL = 2q
+3 pz +
+√z
+16β p2
+q −
+�αz
+4β + 3k
+�
+pq + α2
+4β z
+3
+2 +
+1
+2z
+3
+2 p2
+φ + V z
+3
+2 .
+(24)
+The action (10) may also be cast in the canonical form,
+A1q =
+� �
+˙zpz + ˙qpq + ˙φpφ − NHBL
+�
+dt d3x
+=
+� �
+˙hijπij + ˙KijΠij + ˙φpφ − NHBL
+�
+dt d3x,
+(25)
+where, πij and Πij are momenta canonically conjugate to hij and Kij respectively. For the sake of comparison,
+let us make the following canonical transformation:
+q → 3
+2x;
+pq → 2
+3px,
+(26)
+to express the above Hamiltonian (24) as:
+Hm
+BL = xpz +
+√z
+36β p2
+x −
+�αz
+6β + 2k
+�
+px + α2z
+3
+2
+4β
++
+p2
+φ
+2z
+3
+2 + V z
+3
+2 .
+(27)
+It is revealed that the above Hamiltonian (27) is exactly the one obtained earlier, following Ostrogradsky, Dirac
+as well as Horowitz’s formalisms [15]. Note that, very much like the Ostrogradsky’s and Dirac’s formalisms here
+also, once the formalism is initiated, i.e. (R is expressed in terms of {hij, Kij} (15) as well as the action (16)
+and the point Lagrangian (17)), there remains no option to integrate the action by parts. As a result, even the
+Gibbons-Hawking–York term [20,21] which is physically meaningful, being associated with the entropy of the black
+hole, along with its higher-order counterpart, also remain obscure. On the contrary, following Modified Horowitz’s
+Formalism (MHF), where boundary terms are taken care of, we earlier obtained [15]
+Hm
+MHF = xpz +
+√z
+36β p2
+x + 3αx2
+2√z − 18βkx2
+z
+3
+2
+− 36βk2
+√z
+− 6kα√z +
+p2
+φ
+2z
+3
+2 + V z
+3
+2 .
+(28)
+Although the two (27) and (28) exactly match under the following set of canonical transformations,
+pz → pz − 18kβx
+z
+3
+2
++ 3αx
+2√z ,
+z → z,
+px → px + 36kβ
+√z − 3α√z,
+x → x,
+pφ → pφ,
+φ → φ.
+and apparently there is no contradiction between the two, note the essential difference: linear term in the mo-
+mentum (px ), which is very much present in (27), remains absent from the Hamiltonian (28). As a result, the
+two Hamiltonians (27) and (28) induce completely different quantum dynamics, since in the quantum domain, as
+mentioned, canonical transformation cannot be carried over due to non-linearity.
+6
+
+2.2
+Non-minimally coupled case
+We find that the two different Hamiltonians (27) and (28) render two different quantum descriptions of the
+same classical model. Although, some of the essential features (Gibbons-Hawking-York term and its higher order
+counterpart) are absent from the Hamiltonian (27), it is not clear, which one gives correct quantum description
+of the theory. Further, there may exist a unitary transformation (we have not found it though) relating the two
+Hamiltonian operators. Therefore, to inspect the situation more deeply, we consider the non-minimally coupled
+case next, whose action
+A2 =
+� √−g d4x
+�
+f(φ)R + βR2 − 1
+2φ,µφ,ν − V (φ)
+�
++ f(φ)ΣR + BΣR2,
+(29)
+may be expressed in the RW metric (1) as
+A2 =
+� �
+3f(φ)√z
+� ¨z
+N −
+˙N ˙z
+N 2 + 2kN
+�
++ 9β
+√z
+� ¨z2
+N 3 − 2 ˙N ˙z¨z
+N 4
++
+˙N 2 ˙z2
+N 5
+− 4k ˙N ˙z
+N 2
++ 4k¨z
+N + 4k2N
+�
++ z
+3
+2
+� ˙φ2
+2N − V N
+��
+dt + f(φ)ΣR + BΣR2,
+(30)
+where, as already mentioned, the supplementary boundary terms are required when MHF is taken into account. In
+the above, we consider an arbitrary functional coupling parameter f(φ). Pursuing the same procedure as above,
+one finally arrives at the following Hamiltonian:
+HnmBL = xpz +
+√z
+36β p2
+x −
+�
+f(φ) z
+6β + 2k
+�
+px + f 2(φ)z
+3
+2
+4β +
+p2
+φ
+2z
+3
+2 + V z
+3
+2 ,
+(31)
+which is again identical to the one found following Dirac formalism and may be found following Ostrogradsky’s
+and Horowitz’s techniques as well [22]. In the superscript nm stands for non-minimal coupling. The action (30)
+may also be cast in the canonical form as in (25). On the contrary, following MHF, one finds [22]
+HnmMHF = xpz +
+√z
+36β p2
+x + 3f(φ)
+� x2
+2√z − 2k√z
+�
+− 18kβ
+√z
+�x2
+z + 2k
+�
++
+p2
+φ
+2z
+3
+2
++ 3xf ′(φ)pφ
+z
++ 9f ′(φ)2x2
+2√z
++ V z
+3
+2 .
+(32)
+However, under the following set of canonical transformations,
+pz → pz − 18βkx
+z
+3
+2
++ 3f(φ)x
+2√z ,
+z → z,
+px → px + 36β k
+√z + 3f(φ)√z,
+x → x,
+pφ → pφ + 3f ′(φ)x√z,
+φ → φ,
+the two Hamiltonians (31) and (32) match again [22]. Nevertheless, here the difference is predominant and explicit.
+Note that f ′(φ) term does not appear in (31), while it is coupled to pφ in (32). This coupled (f ′(φ)pφ ) term
+requires operator ordering in the quantum domain, which is different for different form of f(φ). Hence even if the
+two Hamiltonians are related through unitary transformation, such transformation would be different for different
+form of f(φ).
+7
+
+2.3
+Einstein-Gauss-Bonnet-Dilatonic action in the presence of higher order term
+Although, it is clear that two different quantum descriptions follow from the same classical action using different
+techniques, it is still abstruse to select the correct description. Therefore, we next consider Einstein-Gauss-Bonnet-
+Dilatonic coupled action in the presence of higher order curvature invariant term. Gauss-Bonnet (GB) term arises
+quite naturally as the leading order of the α′ expansion of heterotic superstring theory, where α′ is the inverse
+string tension [41–46]. Several interesting features of GB term have been explored in the past and appear in
+the literature [47–67]. However, Gauss–Bonnet term is topological invariant in 4-dimensions, and so to get its
+contribution in the field equations, a dynamic dilatonic scalar coupling is required. It is worth mentioning that,
+in string induced gravity near initial singularity, GB coupling with scalar field plays a very crucial role for the
+occurrence of nonsingular cosmology [68,69]. The particular hallmark of GB term is the fact that, despite being
+formed from a combination of higher order curvature invariant terms (G = R2 − 4RµνRµν + RαβµνRαβµν) (3),
+it ends up only with second order field equations, avoiding Ostrogradsky’s instability, and equivalently, ghost
+degrees of freedom. Nonetheless, such a wonderful feature ultimately leads to a serious pathology of ‘Branched
+Hamiltonian’, which has no unique resolution till date [70–72].
+Nevertheless, it has been revealed that, the
+pathology may be bypassed upon supplementing the action with higher order curvature invariant term [24, 25].
+We therefore consider the following action,
+A3 =
+� √−g d4x
+�
+αR + βR2 + γ(φ)G − 1
+2φ,µφ,ν − V (φ)
+�
++ αΣR + βΣR2 + γ(φ)ΣG.
+(33)
+In the above, Gauss-Bonnet term G , is coupled with γ(φ), while V (φ) is the dilatonic potential. Further, the
+symbol K stands for K = K3 − 3KKijKij + 2KijKikKk
+j , where, K is the trace of the extrinsic curvature
+tensor Kij , and γ(φ)ΣG = 4γ(φ)
+�
+∂V
+�
+2GijKij + K
+3
+� √
+hd3x is the supplementary boundary term associated with
+Gauss-Bonnet sector. The (0
+0) component of the Einstein’s field equation in terms of the scale factor here reads
+as,
+6α
+a2
+� ˙a2
+N 2 + k
+�
++ 36β
+a2N 4
+�
+2˙a...a − 2˙a2 ¨N
+N − ¨a2 − 4˙a¨a
+˙N
+N + 2˙a2 ¨a
+a + 5˙a2 ˙N 2
+N 2 − 2 ˙a3 ˙N
+aN
+− 3 ˙a4
+a2 − 2kN 2 ˙a2
+a2 + k2N 4
+a2
+�
++ 24γ′ ˙a ˙φ
+N 2a3
+� ˙a2
+N 2 + k
+�
+−
+� ˙φ2
+2N 2 + V
+�
+= 0.
+(34)
+The action (33) in terms of the basic variable (hij = a2δij = zδij ) may be expressed as,
+A3 =
+� �
+3α√z
+�
+¨z
+N −
+˙N ˙z
+N 2 + 2kN
+�
++ 9β
+√z
+�
+¨z2
+N 3 − 2 ˙N ˙z¨z
+N 4
++
+˙N 2 ˙z2
+N 5
+− 4k ˙N ˙z
+N 2
++ 4k¨z
+N + 4k2N
+�
++ 3γ(φ)
+N√z
+�
+˙z2¨z
+N 2z + 4k¨z −
+˙z4
+2N 2z2 −
+˙N ˙z3
+N 3z − 2k ˙z2
+z
+− 4k ˙N ˙z
+N
+�
++ z
+3
+2
+� 1
+2N
+˙φ2 − V N
+� �
+dt
++ αΣR + βΣR2 + γ(φ)ΣG,
+(35)
+where the additional supplementary boundary term γ(φ)ΣG = −γ(φ)
+˙z
+N√z
+�
+˙z2
+N 2z + 12k
+�
+, is required in the case of
+MHF. Inserting the other basic variable (Kij = − q
+3δij ) and considering ˙q = v (13), the action (35) finally may
+be expressed as,
+A3q =
+� �
+2α√z(v + 3kN) +
+4β
+N√z (v + 3kN)2 + 8γ(φ)
+√z
+�vq2
+9z − q4N
+27z2 + kv − kNq2
+3z
+�
++ z
+3
+2
+�
+v2
+φ
+2N − NV
+� �
+dt + αΣR + βΣR2 + Λ(φ)ΣG.
+(36)
+Thus, the Lagrangian density takes the following form,
+L3q = 2α√z(v + 3kN) +
+4β
+N√z (v + 3kN)2 + 8γ(φ)
+√z
+�vq2
+9z − q4N
+27z2 + kv − kNq2
+3z
+�
++ z
+3
+2
+� 1
+2N v2
+φ − V N
+�
+,
+(37)
+8
+
+where, boundary terms are not taken care of. The canonical momenta are
+pq = ∂Lq
+∂v = 2α√z +
+8β
+N√z (v + 3kN) + 8γ(φ)
+√z
+� q2
+9z + k
+�
+,
+pN = ∂L3q
+∂vN
+= 0,
+pφ = ∂Lq
+∂vφ
+= z
+3
+2 vφ
+N
+,
+and,
+pz = ∂L3q
+∂vz
+= 0.
+(38)
+Clearly, there exists two primary constraints C ≡ pN ≈ 0, and D ≡ pz ≈ 0, which are usually handled by Dirac
+constraint analysis. However, as mentioned, such analysis is not at all required in the BL formalism. For example,
+one can express the modified Lagrangian density as,
+L∗
+3 = L3q + pq ( ˙q − v) + pN
+�
+˙N − vN
+�
++ pz
+�
+˙z − 2Nq
+3
+�
++ pφ
+�
+˙φ − vφ
+�
+,
+(39)
+so that the corresponding Hamiltonian density takes the following form,
+H∗
+3 = pq ˙q + pN ˙N + pφ ˙φ + pz ˙z − L∗
+3 = pqv + pNvN + pφvφ + 2Nq
+3
+pz − L3q.
+(40)
+As a result, the primary constraint D ≡ pz ≈ 0 disappears and one obtains,
+H∗
+3 = pqv + CvN + pφvφ + pz
+2qN
+3
+− L3q = CvN +
+�
+pqv + pφvφ + 2qN
+3
+pz − L3q
+�
+= CvN + NHGB
+BL. (41)
+In the superscript GB stands for Hamiltonian in connection with Einstein-Gauss-Bonnet-Dilatonic coupling.
+Note that the constraint C ≡ pN strongly vanishes, since the lapse function N is simply a Lagrange multiplier.
+Therefore,
+NHGBBL = pqv + pφvφ + 2qN
+3
+pz − L3q
+= pqv + pφvφ + 2qN
+3
+pz − 2α√z(v + 3kN) −
+4β
+N√z (v + 3kN)2
+− 8γ(φ)
+√z
+�vq2
+9z − q4N
+27z2 + kv − kNq2
+3z
+�
+− z
+3
+2
+� 1
+2N v2
+φ − V N
+�
+.
+(42)
+Now upon substituting v from the definition of momentum (38), one obtains,
+NHGBBL =N
+�2qpz
+3
++
+√zp2
+q
+16β − pq
+�αz
+4β + 3k
+�
++ α2z
+3
+2
+4β
+− pq
+� γq2
+9βz + γk
+β
+�
++ 2αγ
+β
+� q2
+9√z + k√z
+�
++ 4γq4
+27z
+3
+2
+� γ
+3β + 2
+�
++ 8γkq2
+3z
+3
+2
+� γ
+3β + 2
+�
++ 12γk2
+√z
+� γ
+3β + 2
+�
++
+p2
+φ
+2z
+3
+2 + V z
+3
+2
+�
+.
+(43)
+The canonical Hamiltonian therefore finally reads as,
+HGB
+BL =2qpz
+3
++
+√zp2
+q
+16β − pq
+�αz
+4β + 3k
+�
++ α2z
+3
+2
+4β
+− pq
+� γq2
+9βz + γk
+β
+�
++ 2αγ
+β
+� q2
+9√z + k√z
+�
++ 4γq4
+27z
+3
+2
+� γ
+3β + 2
+�
++ 8γkq2
+3z
+3
+2
+� γ
+3β + 2
+�
++ 12γk2
+√z
+� γ
+3β + 2
+�
++
+p2
+φ
+2z
+3
+2 + V z
+3
+2 .
+(44)
+Again, for the sake of comparison, let us make the canonical transformation q → 3
+2x; pq → 2
+3px (26), to express
+the above Hamiltonian (44) in the following form,
+HGBBL = xpz +
+√zp2
+x
+36β + α2z
+3
+2
+4β
+−
+�αz
+6β + γx2
+6βz + 2kγ
+3β + 2k
+�
+px +
+p2
+φ
+2z
+3
+2 +
+� γ2
+4βz
+5
+2 + 3γ
+2z
+5
+2
+�
+x4
++
+� αγ
+2β√z + 12kγ
+z
+3
+2
++ 2kγ2
+βz
+3
+2
+�
+x2 + 2αkγ√z
+β
++ 24k2γ
+√z
++ 4k2γ2
+β√z + V z
+3
+2 ,
+(45)
+9
+
+and notice that, it is similar to the one already found, following Dirac formalism and may be found following
+Ostrogradsky’s and Horowitz’s techniques as well [22]. The action (35) may also be cast in the canonical form
+with respect to the basic variables as,
+A3q =
+� �
+˙zpz + ˙qpq + ˙φvφ − NHBL
+�
+dt d3x =
+� �
+˙hijπij + ˙KijΠij + ˙φvφ − NHMHF
+�
+dt d3x,
+(46)
+where, πij and Πij are momenta canonically conjugate to hij and Kij respectively. Hence, everything appears
+to be consistent. On the contrary, although following MHF, we found [22]
+HGB
+MHF = xpz +
+√zp2
+x
+36β + 3α
+� x2
+2√z − 2k√z
+�
+− 18kβ
+√z
+�x2
+z + 2k
+�
++
+� x6
+2z
+9
+2 + 12kx4
+z
+7
+2
++ 72k2x2
+z
+5
+2
+�
+γ′2
++
+�x3
+z3 + 12kx
+z2
+�
+γ′pφ +
+p2
+φ
+2Z
+3
+2 + V z
+3
+2 ,
+(47)
+nonetheless, under the following set of canonical transformations,
+pz → pz − 18βkx
+z
+3
+2
++ 3αx
+2√z − 6kγ(φ)x
+z
+3
+2
+− 3γ(φ)x3
+2z
+5
+2
+,
+z → z,
+px → px + 36β k
+√z + 3α√z + 3γ(φ)x2
+z
+3
+2
++ 12kΛ
+√z ,
+x → x,
+pφ → pφ − γ′(φ)x3
+z
+3
+2
+− 12kγ′(φ)x
+√z
+,
+φ → φ,
+(48)
+the two Hamiltonians (45) and (47) match again [22].
+Apparently therefore, there is absolutely no problem.
+Nevertheless note that, the Hamiltonian (47) contains a term (γ′(φ)pφ ), which is absent from (45). Now, during
+canonical quantization the presence of this term requires operator ordering, which is different for different form
+of γ(φ). As a result, even if the two may be related through unitary transformation, such transformation would
+be different for different form of γ(φ). Thus, there does not exist a unique unitary transformation. In a nutshell,
+we repeat that the two Hamiltonians (45) and (47) induce two different descriptions in the quantum domain, and
+apparently, there is no way to choose one to be the correct.
+3
+The role of divergent terms:
+The very first important point to mention is, in all the formalisms the scale factor is treated as the basic variable,
+while we initiate our program treating three three-space curvature, instead. To explain the reason behind this
+choice, let us consider curvature squared action, A =
+�
+βR2d4x, as an example. Under variation, it gives a total
+derivative term σ = −4β
+�
+RK
+√
+h d3x, as mentioned earlier, where K is the trace of the extrinsic curvature
+tensor Kij . A counter term (−σ), known by the name modified Gibbons-Hawking–York term [20, 21], must be
+added to the action in case, instead of δ ˙q, δR is kept fixed at the boundary, as in MHF. In the RW (1) metric
+under consideration, the action reads as,
+A = 36β
+� �
+a¨a2 + 2˙a2¨a + 2k¨a + ˙a4
+a + 2k ˙a2
+a
++ k2
+a
+�
+dt
+�
+d3x.
+(49)
+Under integration by parts, we end up with,
+A = C
+� �
+a¨a2 + ˙a4
+a + 2k ˙a2
+a
++ k2
+a
+�
+dt + C
+�2
+3 ˙a3 + 2k ˙a
+�
+.
+(50)
+where, C = 36β
+�
+d3x. Now following Horowitz’s program, we introduce an auxiliary variable Q = ∂A
+∂¨a = 2Ca¨a,
+judiciously into the action in the following manner, such that it may be cast in canonical form,
+A =
+� �
+Q¨a − Q2
+4Ca + C
+� ˙a4
+a + 2k ˙a2
+a
++ k2
+a
+��
+dt + C
+�2
+3 ˙a3 + 2k ˙a
+�
+.
+(51)
+10
+
+Integrating the action again by parts we find
+A =
+�
+− ˙Q˙a − Q2
+4Ca + C
+� ˙a4
+a + 2k ˙a2
+a
++ k2
+a
+��
++ C
+�Q˙a
+C + 2
+3 ˙a3 + 2k ˙a
+�
+.
+(52)
+The action is canonical, since the Hessian determinant is non-zero. It is trivial to check that the above action
+gives correct field equations, but the left out total derivative term may be expressed as,
+σ′ = −4β
+�
+RK
+√
+h d3x + 16β
+�
+K
+√
+h
+� ˙a2
+a2
+�
+d3x,
+(53)
+and as a result σ ̸= σ′ . Thus some redundant total derivative terms are pulled out in the process, which has severe
+consequence in the quantum domain. On the contrary, if we start with, z = a2, the action reads as,
+A = C
+� � ¨z2
+4√z + k¨z
+√z + k2
+√z
+�
+dt = C
+� � ¨z2
+4√z +
+2
+√z
+�
++ C k ˙z
+√z ,
+(54)
+where the last expression is found under integration by parts. Now following Horowitz’s program, we find the
+auxiliary variable as Q = ∂A
+∂¨z = C
+¨z
+2√z , which is again judiciously introduced in the action as,
+A =
+� �
+Q¨z −
+√zQ2
+C
++ C
+� k¨z
+√z + k2
+√z
+��
+dt + C k ˙z
+√z .
+(55)
+Finally, performing integration by parts again, one obtains,
+A =
+� �
+− ˙Q ˙z −
+√zQ2
+C
++ C
+� k¨z
+√z + k2
+√z
+��
+dt + C
+�Q ˙z
+C + k ˙z
+√z
+�
+,
+(56)
+The action is again canonical, the Euler-Lagrange equations here again lead to the appropriate field equations,
+while one can express the total derivative term as σ. In a nut-shell, although total derivative terms do not affect
+the classical field equations, for non-linear theories such as gravity, such terms tell upon the quantum dynam-
+ics. Therefore, to establish consistency in every respect, hij should be treated as the basic variable, instead of
+the scale factor. This is essentially the so-called MHF, which finally requires to replace the auxiliary variable
+by the the second basic variable, viz., the extrinsic curvature tensor Kij = −a˙a = − ˙z = x(say), in the Hamiltonian.
+Next, we observe that the phase-space structures obtained following BL formalism although are identical to
+the Ostrogradsky/Dirac/Horowitz’s formalism, they all differ from the MHF upto a canonical transformation. We
+quote from [22] the general argument in connection with the total derivative terms, which runs as; “it is just
+the change of the variables in the wave function and the phase transformation, plus the change of the integra-
+tion measure, and the transformation of the momenta respecting the change of the measure, and so a unitary
+transformation relates the two”. It’s possible (we have not found though) that each pair of quantum equations
+cast from {(27) and (28)}; {(31) and (32)}; {(45) and (47)}, are related by unitary transformation. However,
+it was also mentioned [22] that different forms of coupling parameter yield different quantum dynamics in the
+case of MHF, due to the presence of a coupling term (f ′(φ)pφ ) for non-minimal coupled case, and (γ′(φ)pφ ) for
+the Gauss-Bonnet-Dilaton coupled case, in the Hamiltonian. Thus, different unitary transformations (if exist)
+are required to relate the last two pairs. Such coupling as well as the derivative of coupling parameter remain
+absent in other formalisms. In a nutshell, unitary transformation relating each pair is not unique. Further, the
+semiclassical wave functions found for all the three cases studied here, exhibit different pre-factors and exponents
+for each pair [22]. This generates different probability amplitude and the evolution of the wave function while
+entering the classical domain.
+Finally, it is important to note that, if the coupling parameter f(φ) is treated as constant in Subsection ??,
+the Hamiltonian (32) merely reduces to (28), while the Hamiltonian (31) reduces to (27). Hence the question is:
+which of the two should be treated as the correct quantum description of the models under consideration? In
+this connection we mention that a serious problem arises with Ostrogradsky/Dirac/Horowitz as well as with BL
+11
+
+formalisms when considering Gauss-Bonnet-Dilaton induced action. To be specific, in Subsection ?? if γ(φ) is
+treated as a constant, then the contribution of Gauss-Bonnet term disappears from the Hamiltonian (47), and
+it reduces to (28).
+Indeed, it should since as mentioned, Gauss-Bonnet term is topologically invariant in 4-
+dimensions, and so without functional coupling, it does not contribute to the field equations and the Hamiltonian
+as well. On the contrary, a constant γ does not affect the form of the Hamiltonian (45), and it does not reduce
+to (27). This means, if we had started with a constant γ from the very beginning, all the terms appearing with
+γ in (45) would have been absent, and the end result would be (27). While, after constructing the Hamiltonian
+with arbitrary γ = γ(φ), if we set it equal to a constant, then its contribution remains present, and we obtain
+a different Hamiltonian, altogether. Clearly this is wrong. Hence, we realize that boundary terms indeed play
+a crucial role while constructing the phase-space structure of non-linear theories. In fact, if boundary terms are
+taken into account from the very beginning, treating hij as the basic variable, then Horowitz’s formalism reduces
+to the MHF, as already demonstrated. It was also noticed that if Dirac algorithm is applied after integrating the
+action by parts, then it also yields Hamiltonian identical to MHF [22]. It is therefore suggestive to test the same
+for BL formalism too. In this section we shall first integrate actions by parts to get rid of the total derivative
+terms and follow the BL formalism thereafter, to explore the outcome.
+3.1
+Scalar-tensor theory: minimal coupling;
+Upon integrating the action (30) by parts, we obtain
+A1 =
+� 
+− 3α ˙z2
+2N√z + 6αkN√z +
+9β
+N√z
+
+
+
+�
+¨z
+N −
+˙N ˙z
+N 2
+�2
++ 2k ˙z2
+z
++ 4k2N 2
+
+
+ + z
+3
+2
+� ˙φ2
+2N − NV
+�
+ dt.
+(57)
+Replacing ˙z by 2N
+3 q in view of (13), the above action may be cast as,
+A1q =
+� �
+−2
+3αN q2
+√z + 6αkN√z +
+9β
+N√z
+�4
+9 ˙q2 + 8kN 2q2
+9z
++ 4k2N 2
+�
++ z
+3
+2
+� ˙φ2
+2N − NV
+��
+dt.
+(58)
+Note that the action (58) cannot be expressed only in terms of velocities, due to the explicit presence of q unlike
+(16). However, similar situation arrived at, in the case of Gauss-Bonnet-Dilaton case, and so it doesn’t matter.
+The canonical momenta are the following:
+pq =
+8β
+N√z ˙q;
+pφ = z
+3
+2
+N
+˙φ;
+pz = 0 = pN.
+(59)
+Dirac constraint analysis appears to be inevitable, since the action is singular. However as mentioned, the lapse
+function N being the Lagrange multiplier, the constraint strongly vanishes, so that one can ignore it without
+loss of generality. Still, another primary constraint pz = 0 is apparent. Nonetheless, as already noticed, in BL
+formalism, Dirac analysis may be bypassed despite the presence of the constraint pz = 0 in the following manner.
+The Lagrangian density is:
+L1q = −2
+3αN q2
+√z + 6αkN√z +
+9β
+N√z
+�4
+9 ˙q2 + 8kN 2q2
+9z
++ 4k2N 2
+�
++ z
+3
+2
+� ˙φ2
+2N − NV
+�
+,
+(60)
+and hence the Hamiltonian reads as,
+NHm
+MBL = pq ˙q + pz ˙z + pφ ˙φ − L1q
+= N√z
+16β p2
+q + 2
+3Nqpz + N
+2z
+3
+2 p2
+φ + 2αNq2
+3√z
+− 6αkN√z − 8βkNq2
+z
+3
+2
+− 36βk2N
+√z
++ NV z
+3
+2 ,
+(61)
+where we have used (59) and replaced ˙z by 2N
+3 q, in view of (13), and the suffix {MBL} now stands for ‘Modified
+Buchbinder-Lyakhovich’ formalism. Finally as before, for the sake of comparison, if we perform the canonical
+12
+
+transformation q → 3
+2x,
+and
+pq → 2
+3px, then the above Hamiltonian (61) may be expressed in the following
+form,
+Hm
+MBL = xpz +
+√z
+36β p2
+x +
+p2
+φ
+2z
+3
+2 + 3α
+2√z (x2 − 4kz) − 18βk
+z
+3
+2
+(x2 + 2kz) + V z
+3
+2 ,
+(62)
+which is identical to Hm
+MHF presented in (28).
+3.2
+Scalar-tensor theory: non-minimal coupling;
+Here again, upon integrating the action (30) by parts we obtain,
+A2 =
+� �
+− 3f ˙z2
+2N√z − 3f ′ ˙φ ˙z√z
+N
++ 6fkN√z +
+9β
+N√z
+�� ¨z
+N −
+˙N ˙z
+N 2
+�2
++ 2k ˙z2
+z
++ 4k2N 2�
++ z
+3
+2
+� ˙φ2
+2N − NV
+��
+dt. (63)
+Now, replacing ˙z by 2N
+3 q in view of (13), the above action (63) may be cast as,
+A2 =
+� �
+−2
+3fN q2
+√z − 2f ′√zq ˙φ + 6fkN√z +
+9β
+N√z
+�4
+9 ˙q2 + 8kN 2q2
+9z
++ 4k2N 2
+�
++ z
+3
+2
+� ˙φ2
+2N − NV
+��
+dt. (64)
+Canonical momenta may therefore be found as,
+pq =
+8β
+N√z ˙q,
+pφ = −2f ′q√z + z
+3
+2
+N
+˙φ,
+pN = 0 = pz.
+(65)
+As before, leaving out the constraint associate with the lapse function, and replacing ˙z = 2N
+3 q in view of (13),
+the Hamiltonian may be cast as,
+NHnmMBL = pq ˙q + pz ˙z + pφ ˙φ − L
+= N
+� √z
+16β p2
+q + 2
+3qpz +
+p2
+φ
+2z
+3
+2 + 2f ′
+z qpφ + 2fq2
+3√z + 2f ′2q2
+√z
+− 6kf√z − 8βkq2
+z
+3
+2
+− 36βk2
+√z
++ V z
+3
+2
+�
+.
+(66)
+Finally, applying the canonical transformation relations q → 3
+2x, and pq → 2
+3px, we obtain
+HnmMBL = xpz +
+√z
+36β p2
+x +
+p2
+φ
+2z
+3
+2 + 3x
+z f ′pφ + 3f
+2√z (x2 − 4kz) − 18βk
+z
+3
+2
+(x2 + 2kz) + 9x2f ′2
+2√z
++ V z
+3
+2 .
+(67)
+Clearly, HnmMBL ∼= HnmMHF presented in (32).
+3.3
+Einstein-Gauss-Bonnet-Dilatonic action
+Eventually, in order to construct the correct Hamiltonian in connection with the Einstein-Gauss-Bonnet-Dilatonic
+action (35), let us integrate it by parts to obtain,
+A3 =
+� �
+α
+�
+−
+3 ˙z2
+2N√z + 6kN√z
+�
++
+9β
+N√z
+�� ¨z
+N −
+˙N ˙z
+N 2
+�2
++ 2k ˙z2
+z
++ 4k2N 2�
+− γ′(φ) ˙z ˙φ
+N√z
+� ˙z2
+N 2z + 12k
+�
++ z
+3
+2
+� ˙φ2
+2N − NV
+��
+dt.
+(68)
+13
+
+As before, replacing ˙z by 2N
+3 q in view of (13), the above action may be cast as,
+A3q =
+� �
+− 2
+3αN q2
+√z + 6αkN√z +
+9β
+N√z
+�4
+9 ˙q2 + 8kN 2q2
+9z
++ 4k2N 2
+�
+− 2qγ′(φ) ˙φ
+3√z
+�4q2
+9z + 12k
+�
++ z
+3
+2
+� ˙φ2
+2N − NV
+� �
+dt.
+(69)
+Canonical momenta are now found as,
+pq =
+8β
+N√z ˙q,
+pφ = −2qγ′(φ)
+3√z
+�4q2
+9z + 12k
+�
++ z
+3
+2
+N
+˙φ,
+pN = 0 = pz.
+(70)
+As always, leaving out the constraint associated with the lapse function, and replacing ˙z = 2N
+3 q in view of (13),
+the Hamiltonian may be cast as,
+NHGB
+MBL =pq ˙q + pz ˙z + pφ ˙φ − L
+= N
+� √z
+16β p2
+q + 2
+3qpz +
+p2
+φ
+2z
+3
+2 + 2αq2
+3√z − 6kα√z − 8βkq2
+z
+3
+2
+− 36βk2
+√z
++ 2qγ′(φ)pφ
+3z2
+�4q2
+9z + 12k
+�
++ 2q2γ′2(φ)
+9z
+5
+2
+�4q2
+9z + 12k
+�2
++ V z
+3
+2
+�
+,
+(71)
+Finally, the set of canonical transformations q → 3
+2x, and pq → 2
+3px, allows one to express the Hamiltonian (71)
+as,
+HGBMBL =xpz +
+√z
+36β p2
+x +
+p2
+φ
+2z
+3
+2 + 3α
+2√z (x2 − 4kz) − 18βk
+z
+3
+2
+(x2 + 2kz)
++ xγ′(φ)
+z2
+�x2
+z + 12k
+�
+pφ + γ′2(φ)x2
+2z
+5
+2
+�x2
+z + 12k
+�2
++ V z
+3
+2 .
+(72)
+As a result one finds, HGBMBL ∼= HGBMHF presented in (47). It is important to note that in the process of
+constructing the Hamiltonian starting from a divergent free action, the pathology discussed in regard of canonical
+formulation of Einstein-Gauss-Bonnet-Dilatonic action in the presence of higher-order term is also removed.
+4
+Application
+It is mentioned in the introduction that canonical formulation is a precursor to canonical quantization. In the
+absence of a viable quantum theory of gravity, it is suggestive to canonically quantize the cosmological equa-
+tion and study quantum cosmology to extract some ethos of pre-Planck era. For example, one can explore the
+Euclidean wormhole solution. Nonetheless, ‘cosmological inflationary scenario’ has been developed since 1980,
+to solve horizon, flatness (fine tuning), structure formation and monopole problems, singlehandedly. Short-lived
+(10−36 −10−26)s. inflation, occurred just after Planck’s era and falls within the periphery of ‘quantum field theory
+in curved space-time’. To be more specific, ‘inflation is a quantum theory of perturbations on the top of the
+classical background’, so that the energy scale of the background remains much below Planck’s scale. Nonetheless
+in this context, Hartle [73] prescribed that, most of the important physics may still be extracted from the classical
+action provided, the semiclassical wave-function is strongly peaked. The reason being, in that case correlation
+between the geometrical and matter degrees of freedom is established, and hence the emergence of classical trajec-
+tories (i.e. the universe) is expected. Hence, quantization and an appropriate semiclassical approximation must
+be treated as a forerunner to study inflation.
+Canonical quantization and the semiclassical wave-function in connection with the Hamiltonian (67) for non-
+minimally coupled higher order theory had been presented in [26], which reduces to the minimally coupled case
+14
+
+when the coupling parameter becomes constant [19]. The Hamiltonian operator was found to be hermitian, stan-
+dard probabilistic interpretation holds, and the semiclassical wave-functions was found to be oscillatory about the
+classical inflationary solution. Inflation has been studied and the parameters are found with excellent agreement
+with the observational constraints [74,75]. Gravitational perturbation has also been studied.
+In [29] again, the quantum counterpart of the Hamiltonian (72) in connection with Einstein-Gauss-Bonnet-
+Dilatonic coupled action has been presented.
+Hermiticity of the Hamiltonian operator has been established,
+probabilistic interpretation is explored, and the semiclassical wave-function is found to be oscillatory about a
+classical inflationary solution. Finally, we have studied inflation and found that the inflationary parameters more-
+or-less satisfy observational constraints [74,75]. In a nut-shell, the results obtained in [29] are the following.
+iℏ∂Ψ
+∂σ =
+�
+− ℏ2φ
+54β0x
+� ∂2
+∂x2 + n
+x
+∂
+∂x
+�
+−
+ℏ2
+3xσ
+4
+3
+∂2
+∂φ2 + 2iℏα0
+σ
+� 1
+φ2
+∂
+∂φ − 1
+φ3
+�
+− 2iℏγ0x2
+3σ
+7
+3
+�
+2φ ∂
+∂φ + 1
+�
++ Ve
+�
+Ψ
+= �HeΨ,
+(73)
+where, the proper volume, σ = z
+3
+2 = a3 plays the role of internal time parameter, and n is the operator ordering
+index. In the above equation, �He is the effective hermitian Hamiltonian operator, while the the effective potential
+Ve is given by,
+Ve = 3α2
+0x
+σ
+2
+3 φ4 − 4α0γ0x3
+σ2φ
++ 4γ2
+0x5φ2
+3σ
+10
+3
++ α0x
+σ
+2
+3 φ
++ λ2σ
+2
+3 φ2
+3x
++ 2σ
+2
+3 ΛM 2
+P
+x
+.
+(74)
+The effective Hamiltonian operator is found to be hermitian for n = −1, which selects the operator ordering
+parameter from physical consideration. Standard quantum mechanical probability interpretation also holds. Under
+a suitable (WKB) semiclassical approximation, the wave-function has been found to be,
+Ψ = Ψ0e
+i
+ℏ
+�
+− 6α0λz2
+a0φ0 +16γ0a2
+0φ2
+0λ3√z
+�
+,
+(75)
+which exhibits oscillatory behaviour about the classical inflationary solution a = a0eλt , where, α0, φ0, γ0 are
+constants. We have also presented several sets of inflationary parameters in [29], which depict that the spectral
+index of scalar perturbation and the scalar to tensor ratio lie within the range 0.967 ≤ ns ≤ 0.979 and 0.056 ≤
+r ≤ 0.089 respectively, showing reasonably good agreement with the recently released data [74,75]. The number
+of e-folding also remains within the acceptable range 46 < N < 73, which is sufficient to solve the horizon and
+flatness problems.
+5
+Concluding remarks
+Although initiated two centuries back, canonical formulation of higher-order theory of gravity is particularly
+non-trivial. In fact, only after probing Dilatonic coupled Gauss-Bonnet action, it is learnt that divergent terms
+play a vital role to formulate correct quantum dynamics of non-linear gravity theory. The scheme is therefore
+first, to express the action in terms of the basic variable hij , otherwise if expressed in terms of the scale factor,
+as commonly done, some unwanted divergent terms are removed in the process of integration by parts, which are
+unaccredited by the variational principle. Next, unless divergent terms are taken care of, the Hamiltonian is found
+to be different, which is related through canonical transformation though, such transformation cannot be carried
+over in the quantum domain due to non-linearity. It is shown that in the case of Einstein-Gauss-Bonnet-Dilatonic
+coupled action in 4-dimension that, unless the action is divergent free, an erroneous Hamiltonian is constructed,
+since it does not reflect the topological invariance of the theory. This proves the importance of divergent terms in
+higher order theories. In this respect the difference of BL formalism with MHF is apparent. In fact BL formalism
+produces identical Hamiltonian as obtained earlier following Ostrogradsky’s, Dirac’s or Horowitz’s formalisms.
+However, MHF is essentially the Horowitz formalism, after expressing the action in terms of the three space
+curvature and taking care of the total derivative terms under integration by parts. It was shown that following the
+same route if Dirac’s algorithm is applied, the Hamiltonian becomes identical to the one found following MHF,
+15
+
+and one obtains unique quantum description. Here, we reveal that the same is true with BL formalism. In fact,
+BL formalism not only bypasses constraint analysis, as in the case of Horowitz’s formalism, it also does not require
+auxiliary variable to cast the action in canonical form, which is a bit intricate. In a straightforward manner, it
+establishes diffeomorphic invariance, and therefore is the easiest technique to handle higher-order theories.
+References
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+Acad. St. Petersbourg Ser. VI, 1850, 4, 385-517.
+[2] Whittaker, E.T. A treatise on the analytical dynamics of particles and rigid bodies (1904, Cambridge Uni-
+versity Press, Cambridge, England).
+[3] Pais, A.; Uhlenbeck, G.E. On field theories with non-localized action, Phys. Rev. D 1950, 79, 145.
+[4] Stelle, K.S. Renormalization of higher-derivative quantum gravity, Phys. Rev. D 1977, 16, 953.
+[5] Fradkin, E.S.; Tseytlin, A. A. Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B
+1982, 201, 469.
+[6] Tomboulis, E.T. Renormalization and asymptotic freedom in quantum gravity, in ‘Quantum Theory of Grav-
+ity’ 1984, ed: Christensen, S.M. Bristol: Adam Hilger, pZ5l.
+[7] Buchbinder, I.L.; Kalashnikov, O.K.; Shapiro, I.L.; Vologadsky, V.B.; Wolfengaut, yu yu The stability of
+asymptotic freedom in grand unified models coupled to R2 gravity, Phys. Lett. B 1989, 216, 127.
+[8] Antoniadis, I.; Tomboulis, E.T. Gauge invariance and unitarity in higher-derivative quantum gravity, Phys.
+Rev. D 1986, 33, 2756.
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+Exploring the Use of WebAssembly in HPC
+Mohak Chadha, Nils Krueger, Jophin John,
+Anshul Jindal, Michael Gerndt
+{firstname.lastname}@tum.de
+Chair of Computer Architecture and Parallel Systems,
+Technische Universität München, Germany
+Shajulin Benedict
+shajulin@iiitkottayam.ac.in
+Department of Computer Science and Engg., Indian
+Institute of Information Technology Kottayam, Kerala
+Abstract
+Containerization approaches based on namespaces offered
+by the Linux kernel have seen an increasing popularity in the
+HPC community both as a means to isolate applications and
+as a format to package and distribute them. However, their
+adoption and usage in HPC systems faces several challenges.
+These include difficulties in unprivileged running and build-
+ing of scientific application container images directly on HPC
+resources, increasing heterogeneity of HPC architectures, and
+access to specialized networking libraries available only on
+HPC systems. These challenges of container-based HPC appli-
+cation development closely align with the several advantages
+that a new universal intermediate binary format called We-
+bAssembly (Wasm) has to offer. These include a lightweight
+userspace isolation mechanism and portability across oper-
+ating systems and processor architectures. In this paper, we
+explore the usage of Wasm as a distribution format for MPI-
+based HPC applications. To this end, we present MPIWasm, a
+novel Wasm embedder for MPI-based HPC applications that
+enables high-performance execution of Wasm code, has low-
+overhead for MPI calls, and supports high-performance net-
+working interconnects present on HPC systems. We evaluate
+the performance and overhead of MPIWasm on a production
+HPC system and AWS Graviton2 nodes using standardized
+HPC benchmarks. Results from our experiments demonstrate
+that MPIWasm delivers competitive native application per-
+formance across all scenarios. Moreover, we observe that
+Wasm binaries are 139.5x smaller on average as compared
+to the statically-linked binaries for the different standardized
+benchmarks.
+CCS Concepts: • Software and its engineering → Process
+management.
+Keywords: WebAssembly, Wasmer, Wasm, MPI, HPC
+Permission to make digital or hard copies of all or part of this work for
+personal or classroom use is granted without fee provided that copies are not
+made or distributed for profit or commercial advantage and that copies bear
+this notice and the full citation on the first page. Copyrights for components
+of this work owned by others than ACM must be honored. Abstracting with
+credit is permitted. To copy otherwise, or republish, to post on servers or to
+redistribute to lists, requires prior specific permission and/or a fee. Request
+permissions from permissions@acm.org.
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+© 2023 Association for Computing Machinery.
+ACM ISBN 979-8-4007-0015-6/23/02...$15.00
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+ACM Reference Format:
+Mohak Chadha, Nils Krueger, Jophin John, Anshul Jindal, Michael
+Gerndt and Shajulin Benedict. 2023. Exploring the Use of We-
+bAssembly in HPC. In The 28th ACM SIGPLAN Annual Sympo-
+sium on Principles and Practice of Parallel Programming (PPoPP
+’23), February 25-March 1, 2023, Montreal, QC, Canada. ACM,
+New York, NY, USA, 16 pages. https://doi.org/10.1145/nnnnnnn.
+nnnnnnn
+1
+Introduction
+Linux containers, due to their portability and high availability,
+have become the de-facto standard for developing, testing,
+and deploying a wide range of applications from enterprise
+to web services in cloud environments [29]. This is because
+containers enable users to package their application along
+with its custom software dependencies as a single unit into
+easy-to-deploy images. Motivated by their popularity in the
+cloud, containers have also seen a growing interest in the HPC
+community [30, 74, 90]. For HPC systems, containers provide
+flexibility to users and allow them to define custom software
+stacks, i.e., user-defined software stack (UDSS) for their large-
+scale scientific applications. Moreover, they enable easy, reli-
+able, and verifiable environments that can be reproduced in
+the future. To this end, several HPC-focused containerization
+solutions, such as Charliecloud [72], Shifter [49], Singular-
+ity [58], Podman [48], and Sarus [33] have been introduced.
+In contrast to previous approaches, this paper investigates us-
+ing a new novel technology called WebAssembly (Wasm) [52],
+dubbed as an alternative to Linux containers [81], for packag-
+ing and distributing HPC applications.
+Despite their increasing popularity, the adoption and us-
+age of containers in HPC systems is still significantly lim-
+ited [36]. This can be attributed to the several challenges
+commonly faced by users in running and building container
+images for their applications on HPC systems. For execut-
+ing containers, most containerization solutions require root
+privileges which are not possible for normal HPC users due
+to shared filesystems and their UNIX permissions in HPC.
+While HPC-focused containerization solutions such as Sin-
+gularity [58] and Podman [48] support rootless-containers
+through fakeroot [61], their current implementations do
+not support distributed filesystems such as GPFS commonly
+found on HPC systems [70, 80]. Moreover, as argued by [71],
+building Open Container Initiative (OCI) [69] compliant con-
+tainer images on HPC resources by unprivileged (normal)
+arXiv:2301.03982v1  [cs.DC]  10 Jan 2023
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Mohak Chadha et al.
+users where the applications will eventually run is signifi-
+cantly hard and requires support from the supercomputing
+center. This is because most container building solutions such
+as Docker [43] also require root privileges. As a result, most
+users use their own local systems for building/developing their
+application container images and then transfer the built image
+to a login/front-end node of an HPC system for execution.
+However, this scenario leads to several problems in container-
+based HPC application development. First, HPC nodes are
+becoming more heterogeneous [11] with different processor
+architectures such as x86_64 or aarch64 and have special-
+ized accelerators such as GPUs. As application performance
+is critical in HPC, compiling an application using the specific
+microarchitectural features of a particular processor is signifi-
+cantly important. While building container images for mul-
+tiple platforms either by cross-compiling HPC applications
+or by emulation with QEMU is possible with plugins such as
+build-x [44], it is not widely supported by HPC application
+build procedures and requires the presence of specific Linux
+kernel features (binfmt_misc [62]). Moreover, testing and
+developing HPC applications offers insights only on the target
+system. In addition, most container images can range from
+several MiBs to several GiBs. As a result, frequent network
+transfers from the local to the HPC system can be cumber-
+some. Second, building HPC applications requires access to
+specialized networking libraries and licenses to compilers
+that are not available on the local user systems. Finally, while
+different containerization solutions have almost no impact on
+the performance of the containerized application [72, 75, 84],
+building high-performant HPC application container images
+is non-trivial, involves a steep learning curve, and requires
+knowledge about specific MPI library versions (e.g., Open-
+MPI [10] 4.0) and high performance network interconnect
+hardware (e.g., Intel OmniPath [4]) and libraries (e.g., Intel
+Performance Scaled Messaging [5]) present on the target sys-
+tem. These challenges of container-based HPC application
+development closely align with the several advantages and
+core problems that Wasm [52] aims to solve.
+Wasm is a low-level, statically typed universal binary in-
+struction format for memory-safe, sandboxed execution in a
+virtual machine. It offers portability across modern proces-
+sor architectures and operating systems, fast execution, and
+a low-level memory model [52]. Although originally meant
+for execution in Web browsers, due to its simplicity and gen-
+erality, Wasm has seen widespread adoption and usage in
+non-Web domains such as serverless computing [78], edge
+computing [47, 53], and Internet of Things [51]. It does not
+require garbage collection and is designed to be a universal
+compilation target with mature support for programming lan-
+guages with an LLVM [59] front-end such as C, C++, C#,
+and Rust [32, 35, 40, 45, 77].
+Figure 1 demonstrates a general workflow for using Wasm
+in HPC. Developers can compile their HPC applications to
+Wasm once on their local systems ahead-of-time (AoT) and
+HPC Application
+WebAssembly
+x86_64
+aarch64
+WebAssembly embedder
+Compile
+Figure 1. An HPC application can be compiled to WebAssem-
+bly and distributed to multiple platforms where it can be
+executed efficiently by a supporting WebAssembly embedder.
+distribute it across multiple platforms instead of distribut-
+ing source code or building application containers. Typically,
+Wasm binaries have a smaller size as compared to native
+x86_64 binaries [52, 56, 91]. Following this, the resulting
+binary can be executed on any platform using a standalone
+Wasm embedder [52]. The Wasm embedder serves two major
+purposes. First, it provides an isolated execution environ-
+ment for running a Wasm binary on a platform. In contrast to
+container-based approaches that utilize different Linux names-
+paces [71] for isolation and security, Wasm provides light-
+weight isolation at the application level based on software
+fault isolation (SFI) [85] and control flow integrity (§2.2).
+Second, it is responsible for compiling Wasm binaries to
+native machine code, either by using Just-in-Time (JIT) en-
+gines at the time of execution, or AoT by using the same
+JIT engines or AoT compilers. Note that, Wasm binaries
+can be executed by normal users and are completely unpriv-
+ileged. Several open-source standalone embedders such as
+Wasmer [87], Wasmtime [37], and Wasm3 [79] are currently
+available. However, none of them support the execution of
+HPC applications.
+As the first step towards bringing Wasm to the HPC ecosys-
+tem, we only focus on MPI-based [6] HPC applications in
+this paper. We chose MPI due to its understanding and in-
+fluence in the HPC community [34]. Towards this, our key
+contributions are:
+• We implement and present MPIWasm, a novel Wasm
+embedder for MPI-based HPC applications based on
+Wasmer [87]. MPIWasm enables high performance exe-
+cution of Wasm code, has low-overhead for MPI calls
+through zero-copy memory operations, and supports
+high-performance networking interconnects such as
+Intel OmniPath [4].
+• We demonstrate with extensive experiments the low-
+overhead and performance of MPIWasm using standard-
+ized HPC benchmarks on a production HPC system and
+AWS Graviton2 [1] nodes based on the x86_64 and the
+aarch64 architectures respectively.
+• We elaborate on the different possible future directions
+for using Wasm in the HPC ecosystem.
+
+</>WA口口Exploring the Use of WebAssembly in HPC
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+The rest of this paper is structured as follows. §2 provides a
+detailed overview on Wasm. In §3, we describe our embedder
+MPIWasm in detail. Our experimental results are presented
+in §4. In §5, we describe the different possible directions
+for using Wasm in the HPC ecosystem. §6 describes some
+previous approaches related to our work. In §7, we conclude
+the paper and present an outlook.
+2
+A primer on WebAssembly
+2.1
+WebAssembly Overview
+WebAssembly (Wasm) was introduced in 2015 as an alter-
+native to JavaScript for web-browser based applications. It
+superseded asm.js [67], a previous attempt by Mozilla which
+focused on a subset of Javascript code that can be optimized
+AoT.
+When an application is compiled to Wasm, the resulting
+binary is called a module. Wasm modules contain function
+definitions, declarations of global variables, tables, and a lin-
+ear memory address space. All of the application code in
+Wasm is organized in functions. The conceptual machine in
+Wasm is stack-based and does not contain registers, there-
+fore all instructions pop their operands from the stack of
+the machine. However, since application control flow is an
+explicit part of the module and Wasm operations are typed,
+it is possible to statically predict the layout of the stack at
+any point in the program which allows compilers to trans-
+late the stack semantics to a register-based instruction set.
+Similar to other higher-level programming languages, Wasm
+allows the definition of global variables that are not scoped
+to a specific function or block. Tables in Wasm modules are
+used for storing references to functions [52]. The Wasm ISA
+currently supports only four data types for variables: (i) i32,
+32-bit integers, (ii) i64, 64-bit integers, (iii) f32, 32-bit IEEE
+754 floating point numbers, and (iv) f64 64-bit IEEE 754
+floating point numbers. For constructing, complex types a
+combination of these basic types is commonly used.
+Wasm provides the capability for data and code to be shared
+between the module and its embedder using the import/export
+system. All of the function definitions that can occur in a
+Wasm module can be imported from the embedder instead of
+being defined within it. Similarly, function definitions that are
+present in the module can be exported so that the embedder
+can utilize them (§2.3).
+2.2
+WebAssembly Security and Sandboxing Model
+Wasm utilizes software fault isolation techniques (SFI) [85]
+to sandbox the executing Wasm module. By default, a Wasm
+module cannot interact with the host system or perform I/O
+operations of any kind. Any system interaction that is to be
+initiated by the Wasm module’s code must be done through
+the functions imported from the embedder (§2.1). As a result,
+the embedder can act both as a translation layer and as an
+arbiter to enforce isolation requirements. As a translator, it
+1
+(type (;1;) (func (param i32) (result i32)))
+2
+...
+3
+(type (;5;) (func (param i32 i32) (result i32)))
+4
+...
+5
+(type (;14;) (func (param i32 i32 i32 i32 i32 i32)
+6
+(result i32)))
+7
+(type (;15;) (func (param i32 i32 i32 i32) (result i32 )))
+8
+...
+9
+(import "wasi_snapshot_preview1" "path_open"
+10
+(func $__wasi_path_open (type 22)))
+11
+(import "wasi_snapshot_preview1" "fd_close"
+12
+(func $__wasi_fd_close (type 1)))
+13
+(import "wasi_snapshot_preview1" "fd_seek"
+14
+(func $__wasi_fd_seek (type 23)))
+15
+(import "wasi_snapshot_preview1" "fd_read"
+16
+(func $__wasi_fd_read (type 15)))
+17
+(import "wasi_snapshot_preview1" "proc_exit"
+18
+(func $__wasi_proc_exit (type 0)))
+19
+...
+20
+(export "_start" (func $_start ))
+21
+(export "memory" (memory 0))
+Listing 1. Example representation of a compiled C++ appli-
+cation’s Wasm module using the WASI-SDK in WebAssembly
+text format (WAT) [68]. Ellipses signify sections that are
+omitted for brevity.
+is possible for the embedder to provide a common interface
+to the Wasm module even though the underlying system may
+have different native interfaces, while as an arbiter it is possi-
+ble for the embedder to restrict access of the Wasm module
+to system resources based on an application-level security
+policy. For instance, it is possible for the embedder to allow
+file I/O only to files that reside in a specific directory to iso-
+late the Wasm module from the rest of the filesystem. While
+in principle similar to kernel-level system call filtering tech-
+niques such as Seccomp-BPF [83] on Linux, performing such
+filtering on the application level allows to define semantically
+more meaningful policies.
+In Wasm, all memory access is confined to a module’s lin-
+ear memory which is separate from the code space. Currently,
+the Wasm specification [52] supports 32-bit addresses to in-
+dex the memory that a module has access to. While this limits
+a single module’s memory to 4GiB, it also enables hardware
+accelerated bound checks of memory accesses at runtime [42].
+If an embedder is a process with a 64-bit memory address
+space, it can safely execute an untrusted Wasm module in its
+memory space without requiring additional isolation by re-
+serving a continuous range of virtual memory for the module
+to use. Not all pages in this range need to be mapped to phys-
+ical memory, it is sufficient to only map the required number
+of pages to fit the amount of memory used by the module at
+a given point in time. This ensures that a Wasm module can
+only operate in its own execution environment and cannot
+corrupt the memory of the embedder, since any out-of-bounds
+memory access will result in a page fault which can then be
+handled by it. Moreover, since the memory instructions in
+Wasm’s specification [52] work with offsets, it is not possible
+to read and write to arbitrary memory locations in Wasm.
+In the assembly produced by C programs, where a func-
+tion call is expressed as a jump instruction to the address of
+the function’s first instruction, a typical exploit is to change
+this address to take control of the program’s control flow.
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Mohak Chadha et al.
+1
+typedef int MPI_Comm;
+2
+typedef int MPI_Datatype;
+3
+...
+4
+int MPI_Init(int* argc , char *** argv);
+5
+int MPI_Finalize(void);
+6
+int MPI_Send(
+7
+const void* buf , int count , MPI_Datatype datatype ,
+8
+int dest , int tag , MPI_Comm comm
+9
+);
+10
+int MPI_Recv(
+11
+void* buf , int count , MPI_Datatype datatype ,
+12
+int source , int tag , MPI_Comm comm , MPI_Status* status
+13
+);
+Listing 2. Excerpt of the custom MPIWasm mpi.h header
+file.
+However, such exploits are not possible with Wasm since it
+features control flow integrity by enforcing structured pro-
+gram control flow. This is because of two reasons. First, in
+Wasm, a function is represented as an index in a table (§2.1)
+which adds an additional level of indirection to express the
+function address. Second, the Wasm specification prevents
+constructing arbitrary memory addresses [42] and the sepa-
+ration of the embedder and the module’s memory prevents
+overwriting function instructions.
+2.3
+WebAssembly System Interface
+Since Wasm was originally designed for web browsers, a sys-
+tem interface that targets POSIX environments and enables
+execution of Wasm modules on them was not part of the orig-
+inal specification [52]. To overcome this, the WebAssembly
+System Interface (WASI) specification [89] was designed.
+WASI specifies the interface an embedder needs to implement
+to execute most POSIX applications. Embedders that imple-
+ment the WASI specification will be able to run any generic
+application compiled with the WASI-SDK [28]. The WASI-SDK
+includes the clang compiler and its own C library based on
+musl libc that call WASI systemcalls imported from the em-
+bedder instead of relying on Linux systemcalls [22]. Note that,
+due to the ubiquity of glibc [26] on Linux systems, some ap-
+plications have come to depend on glibc-specific functions or
+behavior. Such applications will require modifications before
+they can be compiled to a WASI-compliant Wasm module.
+Listing 1 shows a compiled Wasm module of a C++ appli-
+cation using the WASI-SDK in the WebAssembly text format
+(WAT). WAT is a human readable format that enables de-
+velopers to examine the source code of a Wasm module. It
+can be observed that the module contains several functions
+with integers as parameter and return types (Lines 1-7) (§2.1),
+imports WASI functions (Lines 9-18), and exports its _start
+(main function) and memory (Lines 20-21). Exporting these
+two definitions allows the embedder that executes this module
+to call its entrypoint function and to read from and write to
+the module’s linear memory. While the import statements on
+Lines 9-16 enable the Wasm module to open and read from
+a file, the function proc_exit is used by the embedder to
+handle the termination of the application, e.g., by deallocat-
+ing the memory reserved for the module. For the module to
+1
+(import "env" "MPI_Init" (
+2
+func $MPI_Init (param i32 i32) (result i32)
+3
+))
+4
+(import "env" "MPI_Finalize" (func $MPI_Finalize (result i32 )))
+5
+(import "env" "MPI_Send" (
+6
+func $MPI_Send (param i32 i32 i32 i32 i32 i32) (result i32)
+7
+))
+8
+(import "env" "MPI_Recv" (
+9
+func $MPI_Recv (param i32 i32 i32 i32 i32 i32 i32)
+10
+(result i32)
+11
+))
+Listing 3. WAT representation of module imports that corre-
+spond to the functions shown in Listing 2.
+execute, the imported functions need to be implemented by
+the embedder.
+3
+MPIWasm
+In this section, we describe MPIWasm, our embedder for
+executing Wasm modules that utilize functions from the MPI
+standard in detail.
+3.1
+Overview
+The purpose of MPIWasm is to support the execution of MPI
+applications compiled to Wasm on HPC systems. To facili-
+tate its adoption and suitability in HPC environments, it (i)
+supports high-performance execution of MPI-based HPC ap-
+plications compiled to Wasm (§3.3), (ii) has low-overhead
+for MPI calls through zero-copy memory operations (§3.6),
+and (iii) supports high-performance interconnects such as
+Infiniband [64] and Intel OmniPath [4]. These network inter-
+connects are utilized by MPI libraries on HPC systems for
+high-performance inter-rank communication. To enable the
+immediate support for network interconnects present on mod-
+ern HPC systems, MPIWasm links against the MPI library on
+the target HPC system at runtime and provides a translation
+layer between the Wasm module and the host1 MPI library.
+As a result, the developer doesn’t need to be aware about
+the particular networking libraries or network interconnects
+present on the target HPC system. Depending on the partic-
+ular host MPI library such as OpenMPI [10] or MPICH [8],
+MPIWasm needs to be built separately. Both of these libraries
+are currently supported by MPIWasm.
+Our embedder currently supports the execution of MPI
+applications written in C/C++ and conforming to the MPI-2.2
+standard [65]. Integrating the support for MPI-3.1 [66] is of
+our interest for the future but is out of scope for this work.
+We chose to focus on C/C++ applications due to the stability
+and maturity of the Wasm backend in the LLVM/Clang [59]
+project since llvm-8. As the base for MPIWasm, we use the
+open-source Wasm embedder called Wasmer [87]. Wasmer
+supports the execution of Wasm modules on three major plat-
+forms, i.e., Linux, Windows, and macOS, and supports both
+x86_64 and aarch64 instruction set architectures. Moreover,
+it implements the WASI specification (§2.3) and provides
+1We use the term target and host interchangeably for the system on which
+the Wasm module is executing.
+
+Exploring the Use of WebAssembly in HPC
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+ergonomic mechanisms to define additional functions that
+are provided to the module. This dynamic extension of the
+embedder’s functionality enables the addition of MPI func-
+tions to the functionality it provides to the Wasm module. For
+implementing MPIWasm, we use the Rust programming lan-
+guage. This is because of two reasons. First, it provides high
+performance comparable to C/C++ with memory-safety [82].
+Second, it has extensive support and documentation for em-
+bedding Wasmer and using it as a library.
+3.2
+Compiling C/C++ MPI applications to Wasm
+Most MPI applications expect POSIX functionality to be
+available in their execution environment, for instance the abil-
+ity to read from and write to file descriptors. WASI (§2.3)
+defines the WebAssembly exports that enable Wasm mod-
+ules that target it to call most of the functions defined in the
+C standard libraries shipped on POSIX systems. Towards
+this, the WASI-SDK [28] combines the clang compiler and
+the wasi-libc C library to enable the compilation of C/C++
+applications that only make use of POSIX functions and no
+additional libraries to Wasm. The compilation of C/C++ MPI
+applications is not supported by the stock WASI-SDK. To this
+end, we implement a custom mpi.h MPI header file and add
+it to the WASI-SDK. The header file includes the definitions
+for the different MPI types such as MPI_Op, MPI_Comm, and
+MPI_Datatype and the definition for the MPI_Status struc-
+ture. Moreover, it defines the signatures for the MPI functions
+according to the MPI-2.2 [65] standard. An excerpt from the
+header file is shown in Listing 2. It is a reduced version of
+a traditional header file found with MPI libraries with most
+types defined as integers (§3.6). By combining our header file
+with the WASI-SDK, a C/C++ MPI application conforming to
+the MPI-2.2 standard can be compiled to Wasm. Moreover,
+to facilitate the ease-of-use and enable adoption, we imple-
+ment a custom python-based tool that simplifies the entire
+compilation process for MPI applications. Listing 3 shows
+the different MPI-specific imports present in a Wasm module
+corresponding to the functions shown in Listing 2. MPIWasm
+provides definitions for these imports to enable the execution
+of MPI-based HPC applications. In addition, it supports the
+WASI specification which enables the POSIX functionality
+for MPI applications.
+3.3
+Executing Wasm Code with High Performance
+There exist several strategies for executing Wasm modules.
+These include using an interpreter [79], Ahead-of-Time (AoT)
+compilation [37], or Just-in-Time (JIT) compilation [37].
+However, for HPC systems the most useful approach is trans-
+lating the Wasm instructions (Wasm ISA) to the native instruc-
+tion set of the host machine before the application is executed,
+i.e., AoT. Towards this, MPIwasm builds on the code genera-
+tion infrastructure provided by Wasmer [87]. Wasmer currently
+supports three compiler backends, i.e., Singlepass [86],
+Cranelift [3], and LLVM [59]. The SinglePass compiler
+Table 1. Comparing compile duration and performance for
+the different compiler backends supported by Wasmer [87]
+for the HPCG [73] Wasm module. The Wasm module was
+generated using our WASI-SDK (§3.3). The Wasm module is
+executed using MPIWasm on an x86_64 system.
+Compiler
+Compile Duration (ms)
+Single-Core Performance (GFLOP/s)
+Singlepass [86]
+52
+0.3769
+Cranelift [3]
+150
+1.3240
+LLVM [59]
+2811
+1.5426
+is designed to emit machine code in linear time and does
+not perform many code optimizations. The Cranelift com-
+piler is completely based on Rust and is similar to LLVM.
+With Cranelift, the WASM instructions are first translated
+to the intermediate representation (IR) of Cranelift, i.e.,
+(Cranelift-IR) which are then translated to the native in-
+struction set of the host machine by taking microarchitecture-
+specific optimizations into account. On the other hand, with
+LLVM the Wasm ISA is first translated to LLVM-IR followed
+by the generation of native machine code. Cranelift-IR is
+similar to LLVM-IR but at a lower level of abstraction which
+hinders mid-level code optimizations2. At the end of the com-
+pilation process, all three compilers produce a shared ob-
+ject, which can be loaded with a fast dlopen call using the
+libloading library [27].
+Table 1 shows a comparison of the compile-time and run-
+time performance of the three different compilers supported
+by Wasmer for the HPCG benchmark. While LLVM is the slow-
+est to compile the Wasm module, it also results in the fastest
+runtime performance for the HPCG application. As a result,
+we chose LLVM as the compiler backend in MPIWasm. To
+offset the longer compilation times required by LLVM as com-
+pared to the other two compilers, we implement a caching
+mechanism for the generated machine code. Our caching
+mechanism builds on the FileSystemCache [46] provided
+by Wasmer. In our implementation, we generate a hash for
+each Wasm module using the Blake-3 hash function [2].
+Moreover, we store the generated shared object from LLVM
+as the generated hash in the local filesystem. As a result, any
+changes to the Wasm module lead to the generation of a new
+hash which triggers the recompilation of the module. To this
+end, repeated execution of the same application on a system
+with MPIWasm will not lead to recompilation overhead for
+execution.
+3.4
+Filesystem Isolation with MPIWasm
+Since in Wasm all system interactions by the application have
+to be performed by calling functions implemented by the
+embedder (§2.2,§2.3), it enables the embedder to place addi-
+tional restrictions on their use and to employ checks on the
+arguments supplied to them. In Wasmer, all exported functions
+2A more detailed discussion between Cranelift-IR and LLVM-IR can be
+found here [25].
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Mohak Chadha et al.
+Figure 2. Memory address space of MPIWasm with an instan-
+tiated Wasm module. All memory access instructions to the
+Wasm module’s linear address space are given offsets relative
+to the base address.
+that handle file I/O perform their own permission handling
+that is separate from the one employed by the OS. This in-
+process indirection of filesystem accesses allows Wasmer to
+present a virtual directory tree to the Wasm module that only
+contains directories that the module is allowed to access. In
+addition, access rights to individual directories can be more
+restrictive than the permissions granted to the user that is exe-
+cuting the embedder. For instance, a user can have read and
+write access to their home directory and all of its subdirecto-
+ries, but grant read-only access to one specific subdirectory
+to a Wasm module executed by the embedder. MPIWasm ex-
+poses this isolation functionality with its -d flag that grants
+read-write access to the given directory to the Wasm module.
+Note that the full absolute path to the exposed directories
+is not presented to the Wasm module. In the virtual direc-
+tory tree presented to it, all of the subdirectories it has been
+given access to are direct children of the root directory. This
+approach to mapping directory paths avoids exposure of in-
+formation contained in the full path to the directories, such as
+a username in the case of a home directory.
+3.5
+Translating from Wasm to Host Memory Address
+A major part of the Wasm security model is the separation
+of the host and the module’s linear memory address space
+(§2.2). Since it is the responsibility of the Wasm embedder to
+uphold capability restrictions, protecting it’s data structures
+from unintended or malicious access by the modules’ code is
+significantly important. However, this separation presents a
+challenge for supporting MPI applications, because the MPI
+API is based on the library being able to read and write di-
+rectly to the memory of the application. The executing Wasm-
+based MPI application can only provide memory addresses
+in its own linear memory address space, while the target MPI
+library requires addresses in the host memory address space.
+For executing Wasm modules, MPIWasm reserves a part
+of its own address space for use by the Wasm module. As a
+result, every byte contained in this range can be addressed
+either with a memory address in the module’s memory space
+or with a memory address in the embedder’s (host’s) memory
+space. Moreover, while instantiating the module’s linear mem-
+ory, MPIWasm records its base address. Following this, it is
+possible to convert an address from the linear address space of
+the Wasm module to the embedder’s address space and vice-
+versa by treating the address in the linear address space as
+an offset relative to the module’s base address. This is shown
+in Figure 2. In particular, MPIWasm directly converts 32-bit
+Wasm pointers that refer to the module’s linear address space
+to 64-bit pointers that refer to the embedder’s address space
+and vice-versa. To this end, MPIWasm directly utilizes the
+MPI library present on the host system without copying any
+data from the module’s address space to a different location,
+i.e., it supports zero-copy memory operations.
+Our mechanism for memory address translation does not
+violate memory-safety because: (i) a malicious Wasm module
+cannot violate control flow integrity (§2.2) and (ii) since the
+size of the linear memory is always known, MPIWasm can
+perform runtime bound checks for all memory accesses. As a
+result, a module cannot access the memory of the embedder
+or the memory of the underlying operating system unless
+explicitly given access to it.
+3.6
+Translating MPI Datatypes
+MPI is implemented as a library with the most common being
+OpenMPI [10], MPICH [8], and MVAPICH [9]. Hence, it
+does not guarantee an Application Binary Interface (ABI)
+and interoperability between libraries. This means that chang-
+ing the MPI implementation requires recompilation of the
+entire application code. One of the reasons for ABI incom-
+patibility is that the MPI standard does not specify explicit
+types for its datatypes such as MPI_Op and their implemen-
+tation is completely up to the MPI library. However, since
+Wasm modules are designed to be portable not just between
+the different MPI libraries but also between different CPU
+architectures, it becomes necessary to add an abstraction be-
+tween the datatypes used by the host’s MPI library and the
+datatypes exposed to the Wasm module by MPIWasm. An
+abstraction is possible since most MPI datatypes such as
+MPI_Comm, MPI_Datatype and MPI_Op are opaque to the ap-
+plication and only used as arguments to MPI functions. MPI-
+Wasm defines most MPI datatypes as 32-bit integers from
+the perspective of the Wasm module (Listing 2) and transpar-
+ently translates these datatypes to the host equivalents (§3.7).
+We use integers as datatypes since MPIWasm internally uses
+IDs to identify data structures that it creates on behalf of the
+module in order to communicate with the host MPI library.
+3.7
+Implementing MPI Functions in MPIWasm
+Wasm imports are referred to by namespace and name of
+the definition to import. By default, any symbols that are
+not defined while compiling C/C++ applications to Wasm
+will be resolved by making them imports of the module in
+the env namespace. This is also demonstrated in Listing 3
+with the function imports related to the MPI standard. MPI-
+Wasm provides definitions for all these functions with the
+same name as the original MPI function and exports them
+in the env namespace. For implementing these functions, we
+
+0x0
+OxFFFF_FFFFExploring the Use of WebAssembly in HPC
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+combine the memory address and MPI datatype translations
+as described in §3.5 and §3.6 respectively. Towards this, we
+maintain a structure called Env that stores the global state
+required by these translations. This structure includes infor-
+mation about the memory allocated to the Wasm module, it’s
+base pointer (§3.5) and information about the different used
+datatypes such as MPI_Comm by the module. For directly utiliz-
+ing the host MPI library, we use the project rsmpi [7] in MPI-
+Wasm. rsmpi provides MPI bindings for Rust and supports
+OpenMPI [10] and MPICH [8]. It utilizes the rust-bindgen
+project to generate foreign function interfaces tailored to spe-
+cific MPI libraries. Each MPI function in MPIWasm directly
+calls the equivalent function in rsmpi with the appropriate
+arguments.
+While for most functions in the MPI-2.2 standard MPI-
+Wasm directly defers the execution to the host MPI library,
+the implementation of the MPI functions MPI_Alloc_mem and
+MPI_Free_mem is done differently. With these functions, it is
+possible to allocate memory for use with other MPI functions.
+When a Wasm module calls MPI_Alloc_mem, it expects a 32-
+bit memory address in the module’s address space, while call-
+ing the MPI_Alloc_mem function of the host MPI library re-
+turns a 64-bit memory address in the embedder’s memory ad-
+dress space which is not inside the chunk of memory reserved
+for the Wasm module. To overcome this, MPIWasm only
+supports MPI_Alloc_mem and MPI_Free_mem if the Wasm
+module defines and exports the functions malloc and free.
+When MPI_Alloc_mem is called, MPIWasm simply invokes
+the exported malloc and receives a suitable 32-bit module
+memory address. This address can then be used as the return
+value for MPI_Alloc_mem. We implement MPI_Free_mem in
+a similar way.
+3.8
+Limitations
+The Wasm specification currently assumes little-endian byte
+order for multi-byte values [52] in execution environments.
+By giving direct access to the Wasm module’s memory to the
+host MPI library, we assume that the byte order of values in
+the module address space and embedder address space is the
+same. As a result, MPIWasm does not support big-endian CPU
+architectures. This is not a disadvantage since most processor
+architectures in HPC systems are little-endian. Moreover, due
+to the current linear 32-bit memory space for a Wasm module,
+HPC applications compiled to Wasm cannot have more than
+4GiB of memory. The support for 64-bit memory addresses
+is an important milestone for the Wasm specification and is
+highlighted in the Wasm Memory64 proposal [20], but is out
+of scope for this work.
+4
+Experimental Results
+In this section, we present performance results for our embed-
+der MPIWasm across different processor architectures. For
+1
+mpirun -np <number -of-processes > ./ mpiWasm mpi -app.wasm <args >
+Listing 4. Executing MPI applications compiled to Wasm
+with MPIWasm.
+Table 2. Comparing the size of native dynamically-linked,
+statically-linked, and Wasm binaries for the different MPI
+applications. The native applications are compiled for the
+x86_64 architecture.
+Application
+Native Size Dynamic (KiB)
+Native Size Static (MiB)
+Wasm Size (KiB)
+Intel MPI Benchmarks [55].
+1087
+27
+893
+HPCG [73].
+164
+26
+722
+IOR [60].
+364
+16
+315.32
+IS [31].
+36
+15
+57.88
+DT [31].
+40
+15
+49.51
+all our experiments, we follow best practices while reporting
+results [54].
+4.1
+System Description
+For analyzing the performance of our implemented Wasm
+embedder, we use two systems. First, a production HPC clus-
+ter located at our institute, i.e., SuperMUC-NG. Second, an
+AWS EC2 virtual machine (VM) instance with the Gravi-
+ton2 processor [1]. Our HPC cluster contains eight islands
+comprising a total of 6480 compute nodes based on the Intel
+Skylake-SP architecture. Each compute node has two sockets,
+comprising two Intel Xeon Platinum 8174 processors, with
+24 cores each and a total of 96GiB of main memory. The
+nominal operating core frequency for each core is 3.10 GHz.
+Hyper-Threading and Turbo Boost are disabled on the system.
+The internal interconnect on our system is a fast Intel Omni-
+Path [4] network with a bandwidth of 100 Gbit/s. Moreover,
+our cluster provides a general parallel filesystem based on
+the Lenovo DSS-G for IBM Spectrum Scale [19] with an
+aggregate bandwidth of 200 GiB/s. For our experiments, we
+use up to 128 nodes of the HPC system, i.e., 6144 cores. On
+the other hand, the AWS Graviton2 processor based on the
+64-bit ARMv8-A Neoverse-N1 [24] architecture consists of 32
+cores each with a nominal frequency of 2.50 GHz and a total
+main memory of 64GiB. We limit our experiments to one
+node for the Graviton2 processor.
+4.2
+HPC Benchmarks
+For our experiments with MPIWasm, we use the Intel MPI
+Benchmarks [55], two benchmarks from the the NASA Ad-
+vanced Supercomputing (NAS) Parallel Benchmark (NPB)
+suite [31], the IOR benchmark [60], and the High Perfor-
+mance Compute Gradient (HPCG) benchmark [73].
+The Intel MPI benchmarks perform a set of MPI perfor-
+mance measurements for point-to-point and global communi-
+cation operations for a range of message sizes. We use them
+since they characterize the performance of a cluster and are
+an indication of the efficiency of the used MPI implementa-
+tion. The NPB suite includes a set of benchmarks that aim
+to evaluate the overall performance of HPC clusters. Due to
+the support for compiling Fortran to Wasm being in the early
+stages (§5), only the Integer Sort (IS) and Data Transfer (DT)
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Mohak Chadha et al.
+20
+22
+24
+26
+28
+210
+1
+1.5
+Bytes
+Iteration Time (usec)
+PingPong (time) ≤ 1024 Bytes
+Native
+WASM
+212 214 216 218 220 222
+101
+102
+Bytes
+Iteration Time (usec)
+PingPong (time) > 1024 Bytes
+Native
+WASM
+(a) PingPong.
+20
+22
+24
+26
+28
+210
+2
+4
+Bytes
+Iteration Time (usec)
+Sendrecv 6144 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212 214 216 218 220 222
+101
+102
+103
+104
+Bytes
+Iteration Time (usec)
+Sendrecv 6144 Ranks (time) > 1024 Bytes
+Native
+WASM
+(b) SendRecv.
+20
+22
+24
+26
+28
+210
+0
+20
+40
+Bytes
+Iteration Time (usec)
+Bcast 6144 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212 214 216 218 220 222
+100
+102
+104
+Bytes
+Iteration Time (usec)
+Bcast 6144 Ranks (time) > 1024 Bytes
+Native
+WASM
+(c) Broadcast.
+22
+24
+26
+28
+210
+40
+60
+Bytes
+Iteration Time (usec)
+Allreduce 6144 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212 214 216 218 220 222
+102
+104
+Bytes
+Iteration Time (usec)
+Allreduce 6144 Ranks (time) > 1024 Bytes
+Native
+WASM
+(d) AllReduce.
+20
+22
+24
+26
+28
+210
+0
+1
+2
+·104
+Bytes
+Iteration Time (usec)
+Allgather 6144 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212
+214
+216 217
+105
+106
+Bytes
+Iteration Time (usec)
+Allgather 6144 Ranks (time) > 1024 Bytes
+Native
+WASM
+(e) AllGather.
+20
+22
+24
+26
+28
+210
+0
+1
+2
+·105
+Bytes
+Iteration Time (usec)
+Alltoall 6144 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212
+214
+216
+106
+Bytes
+Iteration Time (usec)
+Alltoall 6144 Ranks (time) > 1024 Bytes
+Native
+WASM
+(f) Alltoall.
+22
+24
+26
+28
+210
+0
+10
+20
+30
+Bytes
+Iteration Time (usec)
+Reduce 768 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212 214 216 218 220 222
+100
+102
+104
+Bytes
+Iteration Time (usec)
+Reduce 768 Ranks (time) > 1024 Bytes
+Native
+WASM
+22
+24
+26
+28
+210
+0
+20
+40
+60
+Bytes
+Iteration Time (usec)
+Reduce 6144 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212 214 216 218 220 222
+100
+102
+104
+Bytes
+Iteration Time (usec)
+Reduce 6144 Ranks (time) > 1024 Bytes
+Native
+WASM
+(g) Reduce.
+20
+22
+24
+26
+28
+210
+0
+50
+100
+150
+Bytes
+Iteration Time (usec)
+Gather 768 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212
+214
+216217218219220
+100
+102
+104
+Bytes
+Iteration Time (usec)
+Gather 768 Ranks (time) > 1024 Bytes
+Native
+WASM
+20
+22
+24
+26
+28
+210
+0
+200
+400
+600
+Bytes
+Iteration Time (usec)
+Gather 6144 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212
+214
+216 217
+100
+102
+104
+Bytes
+Iteration Time (usec)
+Gather 6144 Ranks (time) > 1024 Bytes
+Native
+WASM
+(h) Gather.
+20
+22
+24
+26
+28
+210
+0
+100
+200
+300
+Bytes
+Iteration Time (usec)
+Scatter 768 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212
+214
+216217218219220
+101
+103
+105
+Bytes
+Iteration Time (usec)
+Scatter 768 Ranks (time) > 1024 Bytes
+Native
+WASM
+20
+22
+24
+26
+28
+210
+0
+1,000
+2,000
+Bytes
+Iteration Time (usec)
+Scatter 6144 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212
+214
+216 217
+101
+103
+105
+Bytes
+Iteration Time (usec)
+Scatter 6144 Ranks (time) > 1024 Bytes
+Native
+WASM
+(i) Scatter.
+Figure 3. Performance comparison of the Intel MPI benchmarks for MPIWasm and their native execution on our HPC system.
+20
+22
+24
+26
+28
+210
+0.4
+0.6
+0.8
+1
+Bytes
+Iteration Time (usec)
+PingPong (time) ≤ 1024 Bytes
+Native
+WASM
+212 214 216 218 220 222
+100
+101
+102
+Bytes
+Iteration Time (usec)
+PingPong (time) > 1024 Bytes
+Native
+WASM
+(a) PingPong.
+20
+22
+24
+26
+28
+210
+0.5
+1
+1.5
+Bytes
+Iteration Time (usec)
+Sendrecv 32 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212 214 216 218 220 222
+100
+101
+102
+103
+Bytes
+Iteration Time (usec)
+Sendrecv 32 Ranks (time) > 1024 Bytes
+Native
+WASM
+(b) SendRecv.
+22
+24
+26
+28
+210
+2
+4
+6
+8
+Bytes
+Iteration Time (usec)
+Allreduce 32 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212 214 216 218 220 222
+101
+102
+103
+104
+Bytes
+Iteration Time (usec)
+Allreduce 32 Ranks (time) > 1024 Bytes
+Native
+WASM
+(c) AllReduce.
+20
+22
+24
+26
+28
+210
+0
+10
+20
+Bytes
+Iteration Time (usec)
+Allgather 32 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212 214 216 218 220 222
+102
+103
+104
+105
+Bytes
+Iteration Time (usec)
+Allgather 32 Ranks (time) > 1024 Bytes
+Native
+WASM
+(d) AllGather.
+20
+22
+24
+26
+28
+210
+20
+40
+Bytes
+Iteration Time (usec)
+Alltoall 32 Ranks (time) ≤ 1024 Bytes
+Native
+WASM
+212 214 216 218 220 222
+102
+104
+Bytes
+Iteration Time (usec)
+Alltoall 32 Ranks (time) > 1024 Bytes
+Native
+WASM
+(e) Alltoall.
+12 4
+8
+16
+32
+0
+10
+20
+Ranks
+GFLOP/s
+HPCG GFLOPS
+Native
+WASM
+12 4
+8
+16
+32
+0
+50
+100
+150
+Ranks
+GB/s
+HPCG Bandwidth
+Native
+WASM
+(f) HPCG
+Figure 4. Performance comparison of selected Intel MPI benchmarks and HPCG for MPIWasm against their native execution on
+the AWS Graviton2 Processor.
+benchmarks from this suite were used since they are written
+in pure C. The IS benchmark performs bucketed parallel sort-
+ing of integers across all participating processes, while the
+DT benchmark tests the communication and the performance
+of 64-bit floating point operations of a HPC cluster by send-
+ing data through a topology of nodes. We use the topologies
+Black-Hole (bh), White-Hole (wh), and Shuffle (sh) for
+the DT benchmark. For our experiments, we use the classes
+C and B for the IS and DT benchmarks respectively. The IOR
+Benchmark measures the filesystem I/O performance avail-
+able to MPI processes. It supports multiple backends that
+utilize different APIs to perform system I/O. For our experi-
+ments with MPIWasm, we use the POSIX API backend since
+the POSIX filesystem APIs are included in the WASI specifi-
+cation (§2.3, §3.2). The HPCG benchmark aims to evaluate
+the real-world performance of HPC systems by solving a sys-
+tem of linear equations with the conjugate gradient method.
+For our experiments, we use the default available problem
+
+Exploring the Use of WebAssembly in HPC
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+size for HPCG. Note that, in our experiments we use the ver-
+sions 2019 Update 6 and 3.3.1 for the Intel MPI and NAS
+parallel benchmarks respectively.
+4.3
+Experiment Setup
+For all our experiments, we execute the benchmarks in a pure-
+MPI configuration without shared memory parallelization
+with OpenMP, as it is currently not supported by MPIWasm.
+We use OpenMPI-4.0 as the MPI library since it is available
+on our HPC system and can be easily installed on the AWS
+Graviton2 nodes. For compiling the native applications on
+our HPC system (§4.1), we use the clang-11 compiler, while
+for the AWS Graviton2 node, we use the gcc7.1 compiler.
+In both cases, the applications were compiled with the -O3
+optimization flag. For compiling the different benchmarks
+to Wasm, we use the clang-11 compiler along with our cus-
+tomized WASI-SDK with -O3 -msimd128 flags for both test
+systems (§3.2). The -msimd128 flag enables the generation
+of SIMD instructions in Wasm. We compile the applications
+to Wasm only once on our local systems and execute them
+directly with MPIWasm on the different test systems. We
+build MPIWasm on our local system for the different plat-
+forms, i.e., x86_64 and aarch64 with OpenMPI to generate
+bindings for rsmpi (§3.7). Following this, we directly exe-
+cute the applications compiled to Wasm on the test systems
+as shown in Listing 4. Each MPI rank corresponds to one
+instance of the embedder with it’s own Wasm module. The
+native applications were executed directly using mpirun.
+4.4
+Comparing Wasm Binary Size
+Table 2 shows the comparison between the absolute binary
+sizes for the different applications. The static versions of the
+binaries are generated by supplying the -static flag to the
+clang-11 compiler (§4.3) and linking the different applica-
+tions with the static versions of the required libraries such
+as libmpi.a, libopen-rte.a, and libz.a. To this end, we
+made necessary changes to the Make [50] and CMake [39]
+files used by the different applications (§4.2). While the stack-
+based instruction set and compact binary format give Wasm
+the potential to produce smaller binaries for the same appli-
+cations as compared to the native dynamically-linked bina-
+ries [52], three out of five applications that we used had a
+bigger binary size when compiled to WebAssembly in com-
+parison to the equivalent dynamically-linked native binary.
+While Wasm can benefit from a smaller representation on
+a function-by-function basis, in practice dynamically-linked
+native binaries can offset that advantage by being able to rely
+on commonly used libraries to be present on the system. For
+instance, a native binary can dynamically link against glibc,
+while a Wasm binary must statically include functions from
+wasi-libc (§2.3). However, in contrast to containers, Wasm
+binaries are significantly smaller making them more feasible
+for application distribution in HPC environments. In addi-
+tion, Wasm binaries are 139.5x smaller on average than the
+statically-linked binaries of the different applications. This is
+because the linker, i.e., lld copies all library routines from
+the different libraries used by an application into the binary
+during static linking.
+4.5
+Benchmarking MPIWasm
+Figure 3 and Figure 4 show the iteration times for the different
+Intel MPI benchmarks for their native execution as compared
+to their execution with MPIWasm on our HPC system and
+the AWS Graviton2 processor respectively. For execution
+with MPIWasm, the iteration times don’t include the time
+required for compiling the Wasm modules to native machine
+code (§3.3). To avoid repetition, we omit some results for the
+Graviton2 processor. Error bars in the graphs represent mini-
+mum and maximum values for iteration timings as reported
+by the Intel MPI Benchmarks, while points in the graphs
+represent the average timings as reported by the benchmarks.
+For the PingPong benchmark using MPIWasm leads to a
+geometric mean (GM) average slowdown of 0.05x for the
+x86_64 system and a GM average speedup of 1.01x for the
+aarch64 system across all message sizes (Figures 3a, 4a). We
+calculate this value by dividing the metric t_avg_us reported
+by the benchmarks for their native execution by the value
+reported for execution with MPIWasm, followed by a GM of
+the obtained values. For computing slowdown, we subtract
+one from the obtained GM value. We observe a maximum
+bandwidth of 12.80 GiB/s and 10.98 GiB/s for the native exe-
+cution of the PingPong benchmark on the HPC and Graviton2
+processor respectively. On the other hand, with MPIWasm,
+we observe a maximum bandwidth of 13.44 GiB/s and 10.61
+GiB/s on the two systems. For the SendRecv benchmark,
+we observe a GM average slowdown of 0.06x and 0.07x
+with MPIWasm across all message sizes on the x86_64 and
+aarch64 systems respectively (Figures 3b, 4b). For the native
+version of the benchmark, we observe a maximum bandwidth
+of 7.24 GiB/s and 11.01 GiB/s on the two systems, while
+with MPIWasm, we observe a maximum bandwidth of 7.50
+GiB/s and 10.83 GiB/s. For the collective communication
+Broadcast routine, we observe an average GM slowdown of
+0.13x with MPIWasm across all message sizes for 128 nodes
+as shown in Figure 3c. We observe an average GM slow-
+down of 0.06x and 0.10x with MPIWasm across all message
+sizes for the collective communication AllReduce routine
+as shown in Figures 3d and 4c. For AllGather with MPI-
+Wasm, we observe an average GM slowdown of 0.06x and
+0.09x across all message sizes for the HPC system and Gravi-
+ton2 processor respectively (Figure 3e, 4d). Similarly, for the
+Alltoall collective communication routine, we observe an
+average GM slowdown of 0.10x for the two systems across all
+message sizes with MPIWasm as shown in Figures 3f and 4e.
+For 16 nodes of our HPC system, we observe an average GM
+slowdown of 0.12x, 0.14x, and 0.05x across message sizes
+for the routines Reduce, Gather, and Scatter as shown in
+Figures 3g, 3h, and 3i. On the other hand, for 128 nodes,
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Mohak Chadha et al.
+64128
+256
+512
+1,024
+0.4
+0.6
+0.8
+1
+1.2
+·104
+Ranks
+Mop/s
+IS Total Operations/s
+Native
+WASM
+bh
+wh
+sh
+0
+500
+1,000
+1,500
+Topology
+MB/s
+DT Total Throughput
+Native
+WASM w/o SIMD
+WASM w SIMD
+(a) NPB.
+1
+4
+8
+12
+16
+2.4
+2.6
+2.8
+3
+3.2
+·104
+Block Size (MiB)
+MiB/s
+IOR Bandwidth (Read)
+Native
+WASM
+1
+4
+8
+12
+16
+3
+4
+·104
+Block Size (MiB)
+MiB/s
+IOR Bandwidth (Write)
+Native
+WASM
+(b) IOR.
+48 16
+48
+96
+144
+0
+20
+40
+60
+80
+Ranks
+GFLOP/s
+HPCG GFLOPS
+Native
+WASM
+48 16
+48
+96
+144
+0
+200
+400
+600
+Ranks
+GB/s
+HPCG Bandwidth
+Native
+WASM
+192 768 1,536
+3,072
+6,144
+0
+2,000
+4,000
+Ranks
+GFLOP/s
+HPCG GFLOPS
+Native
+WASM
+192 768 1,536
+3,072
+6,144
+0
+1
+2
+3
+·104
+Ranks
+GB/s
+HPCG Bandwidth
+Native
+WASM
+(c) HPCG.
+Figure 5. Performance comparison of standardized HPC benchmarks for MPIWasm against their native execution on our HPC
+system.
+we observe an average GM slowdown of 0.05x, 0.10x, and
+0.08x for the three routines. The results for testing MPIWasm
+with the Intel MPI Benchmarks compiled to Wasm demon-
+strate that neither the mechanism for calling host functions
+in Wasmer [87] nor the translation layer implemented in MPI-
+Wasm induce significant overhead for MPI communication
+(§3.5,§3.6,§3.7). We expand on the translation overhead in
+MPIWasm in §4.6. Overall, our results indicate that MPI-
+Wasm delivers close to native performance for the different
+MPI routines on both x86_64 and aarch64 architectures.
+The performance of MPIWasm on our HPC system for
+the IS and DT benchmarks is shown in Figure 5a. For the IS
+benchmark with MPIWasm, we observe 8260 average mega
+operations per second across all processes as compared to
+8546 average mega operations per second for the native exe-
+cution. For the DT benchmark with different topologies (§4.2),
+execution with MPIWasm leads to decreased throughput as
+compared to the native execution. The DT benchmark per-
+forms a significant number of pairwise comparison opera-
+tions which benefit greatly from vectorization with SIMD
+instructions. To demonstrate the effect of SIMD for the DT
+benchmark, we compile it to Wasm by disabling and enabling
+the generation of SIMD instructions. The Wasm version of the
+DT benchmark with SIMD leads to 1.36x better throughput
+as compared to the Wasm version without SIMD (Figure 5a).
+The difference in performance as compared to the native ver-
+sion of the DT benchmark can be attributed to the support
+for only 128-bit SIMD instructions in the Wasm specifica-
+tion [52] as compared to 512-bit SIMD instructions present
+in modern Intel processors [38, 57, 76] (§3.3). Support for
+higher-width SIMD in Wasm is an important milestone in its
+road-map but out of scope for this work (§5).
+Figure 5b shows total aggregated read and write band-
+width available to all MPI processes for the IOR benchmark.
+Points in the graph represent the average bandwidth reported
+by the benchmark, while error bars in the graph represent
+the maximum and minimum bandwidth observed over all
+iterations of the benchmark with the same block size. With
+four nodes of our HPC system, the upper bound for achiev-
+able bandwidth in our setup (§4.1) with IOR is 400 GBit/s
+(≈ 47684 MiB/s). With MPIWasm, we observe similar read
+(29411 MiB/s) and write (40206 MiB/s) bandwidth averaged
+across all block sizes as compared to the native execution
+of the benchmark. Testing the filesystem I/O performance
+of the MPIWasm demonstrates that the userspace permission
+handling and virtual directory tree implemented by Wasmer
+to provide filesystem isolation (§3.4) has no significant im-
+pact on the achievable bandwidth when performing I/O with
+the POSIX filesystem API. For the HPCG benchmark, we ob-
+serve similar performance when executed with MPIWasm as
+compared to it’s native execution on the HPC system and
+the Graviton2 processor up to 192 MPI processes (Figures 5c
+and 4f). On increasing the number of processes, the native exe-
+cution of the HPCG benchmark outperforms the execution with
+MPIWasm as shown in Figure 5c. For 6144 MPI processes, we
+observe a 14% reduction in GFLOP/s on execution with MPI-
+Wasm. This behavior can be attributed to the significantly fre-
+quent amount of communication required by the HPCG bench-
+mark [63]. HPCG repeatedly uses the Allreduce routine to
+reduce a single variable of size double over all MPI processes
+to finalize vector-vector dot operations. With increasing num-
+ber of processes, the number of times the Allreduce routine
+is called also increases. For instance, executing HPCG with
+768 processes results in four times more calls to Allreduce
+as compared to the execution with 192 processes. As a re-
+sult, the repeated datatype translations in MPIWasm increase
+the cost for invoking the collective communication routine
+leading to performance degradation (§4.6).
+4.6
+Analyzing Datatype Translation Overhead
+To measure the datatype translation overhead in MPIWasm,
+we implement a custom PingPong application that sends/re-
+ceives messages of varying sizes between two processes and
+iterates over the different MPI datatypes, i.e., BYTE, CHAR,
+INT, FLOAT, DOUBLE, and LONG. Following this, we instru-
+ment the Send routine in MPIWasm to determine the latency
+for translating the different datatypes. Finally, we execute the
+application on our HPC system. Figure 6 shows the trans-
+lation overhead for different datatypes and message sizes
+in MPIWasm. We observe an average overhead of 85.44ns,
+
+Exploring the Use of WebAssembly in HPC
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+8
+64
+256
+1024
+32768
+262144 1048576 2097152 4194304
+Bytes
+0
+50
+100
+150
+200
+250
+300
+Translation Time (ns)
+MPI_BYTE
+MPI_CHAR
+MPI_INT
+MPI_FLOAT
+MPI_DOUBLE
+MPI_LONG
+Figure 6. Comparing the datatype translation overhead in
+MPIWasm.
+84.72ns, 99.78ns, 96.32ns, 103.35ns, and 104.79ns across all
+message sizes for the MPI datatypes BYTE, CHAR, INT, FLOAT,
+DOUBLE, and LONG respectively. We observe an increase in the
+translation overhead for message sizes greater than 256KB.
+This can be attributed to an increased latency for acquiring
+read locks from the Env structure that maintains the global
+state for translations in MPIWasm (§3.7).
+5
+Into the Future: Wasm and HPC
+In this section, we highlight and discuss the different exten-
+sions proposed by the Wasm community to the current Wasm
+specification [52] that can be implemented in an embedder
+for HPC applications to enhance performance and portability.
+Controlled Threading for Wasm modules. The Wasm
+Threads proposal [16] lays the foundation for utilizing Wasm
+for multithreaded algorithms. It enables Wasm modules to de-
+fine shared memories, informing the embedder that the mod-
+ule expects the memory to be accessed by multiple threads.
+To enable safe multithreaded access to shared memory, the
+proposal also defines atomic Wasm instructions that can be
+used to implement locks and atomic data structures in the
+functions of the module. To enable HPC applications to make
+use of the functionality added by the threads proposal, an API
+that allows Wasm modules to create additional threads on its
+own needs to be added to the embedder. Implementing the
+POSIX threads [15] and OpenMP [41] APIs in the embed-
+der would enable compatibility with the threading code in
+existing HPC applications.
+Wasm Extended SIMD. The current Wasm SIMD pro-
+posal [52] only specifies 128-bit SIMD instructions, while
+modern CPUs support higher-width-SIMD, for instance the
+AVX-512 instruction set extensions for x86_64, which speci-
+fies 512-bit SIMD instructions. Towards this, the Wasm Flex-
+ible Vector proposal [13] aims to provide support for SIMD
+instructions that are wider than 128-bit. Moreover, the Wasm
+relaxed SIMD instructions [21] aim to make it possible to uti-
+lize hardware SIMD instructions that are not well defined, i.e.,
+they differ in rounding behavior from the Wasm specification.
+Implementing these proposals in the embedder would allow
+compiled Wasm modules for HPC applications that contain
+vectorizable code to make better use of SIMD instructions
+available in modern CPU architectures.
+20
+22
+24
+26
+28
+210
+2
+4
+6
+Bytes
+Iteration Time (usec)
+PingPong (time) ≤ 1024 Bytes
+MPIWasm
+Faasm
+212 214 216 218 220 222
+100
+101
+102
+103
+Bytes
+Iteration Time (usec)
+PingPong (time) > 1024 Bytes
+MPIWasm
+Faasm
+Figure 7. Comparing the performance of MPIWasm and
+Faasm [78].
+Wasm PGAS. Partitioned Global Address Space (PGAS)
+is a programming model for parallel distributed memory ap-
+plications that introduces a memory address space that spans
+the local memory of multiple processes. With a memory ad-
+dress from this global address space a process can read from
+and write to the memory of other processes. Since Wasm
+already specifies the concept of defining and importing mem-
+ories, the embedder could be extended to provide non-local
+memory to the Wasm module. To support this use-case, the
+Wasm Multi-Memory proposal [14] needs to be implemented,
+which allows a Wasm module to define or import more than
+one memory.
+Dynamic Linking of Wasm Modules. While there is ex-
+isting work on establishing an ABI for dynamic linking be-
+tween Wasm modules [12], it has not been standardized yet.
+Supporting dynamic linking would significantly decrease the
+size of Wasm binaries for more complex applications as they
+would no longer need to statically link parts of wasi-libc.
+For HPC, it would enable commonly used libraries such as
+BLAS to be provided by MPIWasm. Combining dynamic link-
+ing with efforts to provide repositories for Wasm modules
+such as WAPM [88] could enable automatic dependency man-
+agement for Wasm applications.
+Compiling Fortran applications to Wasm Currently, the
+support for compiling Fortran-based applications to Wasm is
+very nascent with only one known attempt based on Dragon-
+egg [17]. However, the implementation of the Memory64
+proposal [20] should enable the usage of existing Fortran
+LLVM compilers such as F18 for easily compiling Fortran-
+based applications to Wasm [18].
+Wasm and Accelerators The module execution hints pro-
+posal [23] highlights the changes required in the Wasm speci-
+fication to enable the support for executing Wasm modules
+on hardware accelerators such as GPUs. Implementing the
+proposal in the embedder would enable compatibility with
+existing GPU-based HPC applications.
+6
+Related Work
+Solutions for Packaging and distributing HPC applica-
+tions. Recently several HPC-focused tools such as Char-
+liecloud [72] and Singularity [58] have been introduced for
+distributing HPC applications through containerization. In
+contrast, we utilize the universal binary instruction format
+Wasm to package and distribute HPC applications. Moreover,
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Mohak Chadha et al.
+while containerization requires building HPC application con-
+tainers for different platforms, HPC applications can be com-
+piled once to Wasm and executed on any platform using a
+supporting Wasm embedder.
+MPI and WebAssembly. To the best of our knowledge,
+Faasm [78] is the only compute platform that enables the exe-
+cution of MPI applications compiled to Wasm. It is based on
+a gRPC-based distributed messaging library called Faabric
+and contains a Wasm runtime, a workload scheduler, and
+a distributed state store. In order to run an application on
+Faasm, it needs to be compiled to Wasm and uploaded to the
+shared function storage. Following this, the application can
+be invoked using events such as HTTP requests. For support-
+ing MPI applications, Faasm implements only a subset of
+the MPI-1 specification on top of its messaging library and
+its own workload scheduler. Moreover, it also does not sup-
+port user-defined communicators required by the Intel MPI
+benchmarks (§4.1). In contrast, we take an inverted approach,
+where MPIWasm builds on top of existing native MPI libraries
+and provides a way for Wasm modules to call functions from
+them efficiently. Figure 7 shows the performance comparison
+between MPIWasm and Faasm for the PingPong benchmark
+(§4.2). With MPIWasm, we achieve a GM average speedup
+of 4.28x across all message sizes as compared to Faasm.
+7
+Conclusion and Future Work
+In this paper, we took the first step towards bringing We-
+bAssembly to the HPC ecosystem and presented MPIWasm, a
+Wasm embedder that enables high performance execution of
+MPI applications compiled to Wasm across different proces-
+sor architectures. In the future, we plan to extend MPIWasm
+to support acceralators such as GPUs found on HPC sys-
+tems [23].
+8
+Acknowledgement
+We thank our shepherd Milind Chabbi for his help in prepar-
+ing the final version of this paper. Furthermore, we thank
+the anonymous reviewers for their insightful comments and
+valuable feedback that significantly improved the quality of
+this paper. This work was supported by the funding of the
+German Federal Ministry of Education and Research (BMBF)
+in the scope of the Software Campus program.
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+A
+Artifact Appendix
+A.1
+Description
+MPIWasm is an embedder for MPI-based HPC applications
+based on Wasmer [87]. It enables the high performance execu-
+tion of these applications compiled to WebAssembly (Wasm)
+and serves two purposes:
+1. Delivering close to native application performance, i.e.,
+when applications are executed directly on a host ma-
+chine without using Wasm.
+2. Enabling the distribution of MPI-based HPC applica-
+tions as Wasm binaries.
+Our artifact contains the source code for MPIWasm, toolchain
+for compiling C/C++ based MPI applications to Wasm, the
+Wasm binaries for the different standardized HPC bench-
+marks used in this paper, pre-built versions of our embedder
+for different operating systems, and scripts for parsing experi-
+ment data and generating plots. The artifact is available at:
+https://doi.org/10.5281/zenodo.7468121
+or
+https://github.com/kky-fury/MPIWasm
+A.2
+Getting Started
+For testing our Wasm embedder for executing MPI appli-
+cations compiled to WebAssembly, we provide a pre-built
+docker image for the linux/amd64 platform with all the re-
+quired dependencies.
+1
+sudo
+docker
+run − i t
+kkyfury / ppoppae : v2
+/ bin / bash
+2
+#Executing the HPCG benchmark compiled to Wasm
+3
+mpirun −−allow −run −as − r o o t −np 4
+. / t a r g e t / r e l e a s e / embedder
+\
+4
+examples / xhpcg . wasm
+5
+#Executing the IntelMPI benchmarks compiled to Wasm
+6
+mpirun −−allow −run −as − r o o t −np 4
+. / t a r g e t / r e l e a s e / embedder
+\
+7
+examples / imb . wasm
+Listing 5. Getting started with MPIWasm.
+Towards this, a user can follow the steps described in Listing 5.
+Following this, MPIWasm should successfully execute the
+HPCG and IntelMPI benchmarks. We provide sample output
+for the two benchmarks in the provided artifact. The user can
+increase/decrease the number of processes (-np) for executing
+the benchmarks. However, depending on the system where
+
+Exploring the Use of WebAssembly in HPC
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+the container is executing, the user might need to provide the
+-oversubscribe flag to mpirun.
+A.3
+Running Experiments with MPIWasm
+This section describes how to run experiments with our em-
+bedder to obtain plots similar to the ones in this paper.
+A.3.1
+Running small-scale experiments. To run small-scale
+experiments inside the docker container, we provide an end-
+to-end script. This script:
+1. Executes the HPCG, IS, and IntelMPI benchmarks for
+their native execution and when they are executed using
+MPIWasm.
+2. Parses the obtained results and generates the relevant
+plots.
+1
+sudo
+docker
+run − i t
+kkyfury / ppoppae : v2
+/ bin / bash
+2
+cd
+run_experiments
+3
+. / runme . sh
+Listing 6. Running small-scale experiments with MPIWasm.
+For running the experiments, the user can follow the steps de-
+scribed in Listing 6. The script can take around 10-15 minutes
+to finish execution. After completion, you can see the gener-
+ated data in the run_experiments/experiment_data folder. The
+generated plots can be found in the run_experiments/Plots
+folder. We provide sample plots for the different benchmarks
+in our artifact. However, on executing benchmarks inside
+the container, the performance difference between the native
+execution of the application and MPIWasm can be around
+8-12%.
+A.3.2
+Running large-scale experiments on an HPC sys-
+tem. For running large-scale experiments with our embedder,
+a user needs to do the following:
+1. Build a version of the embedder for your HPC sys-
+tem depending on the particular architecture, operating
+system, and the MPI library on the system. MPIWasm
+currently supports the OpenMPI library with limited sup-
+port for MPICH and MVAPICH. We provide examples for
+building MPIWasm for different operating systems and
+architectures in our artifact.
+2. Execute the MPI applications using the built embedder
+on the HPC system. This can be done via submitting
+jobs to a RJMS software on an HPC system such as
+SLURM [92]. We provide sample job scripts for our HPC
+system, i.e., SuperMUC-NG that uses SLURM in our
+artifact.
+3. After executing the applications, the user can utilize
+the different parsers provided in our artifact to parse
+the benchmark data. Following this, the results can
+be visualized using the plotting helper provided in the
+artifact.
+A.4
+Compiling C/C++ applications to Wasm
+We have setup a docker container with the required depen-
+dencies for compiling different MPI applications conformant
+to the MPI-2.2 standard to Wasm. The artifact also includes
+HPCG, IntelMPI, and IS benchmarks as examples.
+1
+sudo
+docker
+run − i t
+kkyfury / w a s i t o o l c h a i n : v1
+/ bin / bash
+2
+#Compiling HPCG
+3
+cd
+/ work / example / hpcg −benchmark
+4
+. / wasi −cmake . sh
+5
+cd cmake− build −wasi
+6
+make
+Listing 7. Compiling applications to Wasm.
+Listing 7 describes the steps a user can follow for compiling
+the HPCG benchmark to Wasm. For steps to compile the other
+benchmarks to Wasm, please look at the base Readme.md
+file provided with the artifact. All the different applications
+compiled to Wasm that we used in this paper are also present
+in the artifact.
+A.5
+Using MPIWasm
+For detailed usage instructions, please look at the base Readme.md
+file provided with the artifact.
+A.6
+Modifying MPIWasm
+For modifying our embedder, we recommend using our pro-
+vided docker-compose file in the artifact. This docker-compose
+file mounts the volume with the embedder’s source code in-
+side the container. As a result, any changes to it’s source code
+will be reflected inside it. For our embedder, we currently
+support the following operating systems:
+1. CentOS-8.2
+2. Opensuse-15-1
+3. Ubuntu-20-04
+4. MacOS-monterey
+1
+cd wasi −mpi− r s
+2
+docker
+compose run
+centos −8−2
+3
+cargo
+b u i l d −− r e l e a s e
+4
+#After the build process , you can see the built embedder in
+5
+#
+the /target/release/ folder.
+Listing 8. Building MPIWasm.
+Listing 8 describes the steps for building MPIWasm for
+CentOS-8.2 after modifications. The instructions for building
+the embedder for other operating systems are provided in the
+base Readme.md file inside the artifact. After the build process,
+the embedder can be copied to the user’s local filesystem
+using the docker-cp command as shown in Listing 9.
+1
+docker
+cp < c o n t a i n e r −id > : / s / t a r g e t / r e l e a s e / embedder
+\
+2
+< d e s t i n a t i o n −path −user − f i l e s y s t e m >
+Listing 9. Copying MPIWasm.
+We provide provide the base image dockerfiles for the dif-
+ferent supported operating systems inside the artifact. These
+example dockerfiles can be easily extended to support other
+different linux distributions.
+
+PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
+Mohak Chadha et al.
+A.7
+Support for aarch64
+Our embedder also supports execution on linux/arm64 plat-
+forms. We provide pre-built versions of our embedder for
+arm64 for the different supported operating systems in the
+artifact.
+A.7.1
+Building images for aarch64. If the user is building
+the docker image on an x86_64 system then docker-buildx
+is required. Note that, in this case, building the image might
+take around 12 hours.
+1
+sudo
+docker
+buildx
+c r e a t e
+−−name mybuilder −−use −− b o o t s t r a p
+2
+cd wasi −mpi− r s / . g i t l a b / c i / images /
+3
+sudo
+docker
+buildx
+b u i l d −−push −f
+ubuntu −20 −04. D o c k e r f i l e
+\
+4
+−− p l at f or m
+l i n u x / arm64
+\
+5
+− t
+kkyfury / ubuntumodifiedbase : v1
+.
+6
+cd
+. . / . . / . . /
+7
+sudo
+docker
+buildx
+b u i l d −−push −f
+D o c k e r f i l e
+\
+8
+−− p l at f or m
+l i n u x / arm64
+\
+9
+− t
+kkyfury / embedderarm : v1
+.
+Listing 10. Building MPIWasm for aarch64 on x86_64.
+Listing 10 describes the steps required for building MPI-
+Wasm for arm systems with docker-buildx. The user should
+change the docker image tags according to their docker reg-
+istry account, i.e., replace kkyfury with your registry user-
+name. On the other hand, if the user is using an aardch64
+system then please follow the instructions described in §A.6.
+

69E4T4oBgHgl3EQfCAuy/content/tmp_files/2301.04857v1.pdf.txt

@@ -0,0 +1,1309 @@
+Neural Spline Search for Quantile Probabilistic Modeling
+Ruoxi Sun1*, Chun-Liang Li1*, Sercan Ö. Arık1, Michael W. Dusenberry2, Chen-Yu Lee1, Tomas
+Pfister 1
+1Google Cloud AI 2Google Research, Brain Team
+{ruoxis, chunliang, soarik, dusenberrymw, chenyulee, tpfister}@google.com
+Abstract
+Accurate estimation of output quantiles is crucial in many use
+cases, where it is desired to model the range of possibility.
+Modeling target distribution at arbitrary quantile levels and at
+arbitrary input attribute levels are important to offer a compre-
+hensive picture of the data, and requires the quantile function
+to be expressive enough. The quantile function describing the
+target distribution using quantile levels is critical for quantile
+regression. Althought various parametric forms for the distri-
+butions (that the quantile function specifies) can be adopted,
+an everlasting problem is selecting the most appropriate one
+that can properly approximate the data distributions. In this
+paper, we propose a non-parametric and data-driven approach,
+Neural Spline Search (NSS), to represent the observed data
+distribution without parametric assumptions. NSS is flexible
+and expressive for modeling data distributions by transform-
+ing the inputs with a series of monotonic spline regressions
+guided by symbolic operators. We demonstrate that NSS out-
+performs previous methods on synthetic, real-world regression
+and time-series forecasting tasks.
+Introduction
+For many machine learning applications, modeling the pre-
+diction intervals (e.g. estimating the ranges all individual
+predictions observation fall), beyond point estimates, is cru-
+cial (Salinas et al. 2020; Wen et al. 2017; Tagasovska and
+Lopez-Paz 2019; Gasthaus et al. 2019; Pearce et al. 2018).
+The prediction intervals can help with decision making for
+retail sales optimization (Simchi-Levi et al. 2008), medi-
+cal diagnoses (Begoli, Bhattacharya, and Kusnezov 2019;
+Mhaskar, Pereverzyev, and van der Walt 2017; Jiang et al.
+2012), information safety (Smith, Dinev, and Xu 2011), fi-
+nancial investment management (Engle 1982), robotics and
+control (Buckman et al. 2018), autonomous transformation
+(Xu et al. 2014) and many others.
+To estimate prediction intervals, we would need to estimate
+different levels of quantiles for the target distribution using
+quantile regression (Koenker and Regression 2005; Wald-
+mann 2018). A real-world challenge is to select the paramet-
+ric forms of target distributions, which is specified by the
+quantile function (also known as the inverse CDF function),
+*These authors contributed equally.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+X
+2
+1
+0
+1
+2
+Y
+X=X0
+P(Y|X)
+True
+Quantile 25%
+Quantile 75%
+Figure 1: Modeling multiple quantiles at different
+condition-levels with a universal quantile function. The
+goal is to model target data distribution y at any arbitrary
+quantile level and attribute level X, using one versatile quan-
+tile function. Gray dots are observed data points, while green
+and blue lines indicate 25% and 75% quantile levels. The
+data distribution y varies at different levels of X, say variance
+of y increases when X is away from zero. Red dots are data
+points at X = X0, p(Y |X0)).
+to properly align with observed data distribution. Different
+choices for the target distribution (Gaussian, Poisson, Neg-
+ative Binomial, Student-t etc.) may yield different quantile
+predictions, and misalignment of the assumption with the real
+distribution may hinder the performance of the model. There-
+fore, such heuristic or empirical hand-picking based paramet-
+ric assumptions for the distribution can be sub-optimal. An
+approach based on learning from the data in an automated
+way, would be highly desirable, from both foundational and
+practical perspectives.
+For learnable parametric modeling, one challenge is how to
+model all quantiles for all input attributes level in a com-
+putationally efficient way. First, modeling an any arbitrary
+quantile, as opposed to a couple of pre-defined quantile levels,
+offers a more comprehensive view on the target distribution,
+and provides convenience to use the quantile model (e.g. no
+need to re-train the model when quantiles at testing are dif-
+ferent from the ones at training). Second, real-world data can
+have complex distributions beyond what simple assumptions
+can model. Fig. 1 shows different input attribute X levels
+arXiv:2301.04857v1  [cs.AI]  12 Jan 2023
+
+0
+1
+2
+3
+4
+5
+Y|X
+0.000
+0.005
+0.010
+0.015
+0.020
+Probability Density
+PDF(Y|x)
+0
+1
+2
+3
+4
+5
+Y|X
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Probability Density
+CDF(Y|x)
+Figure 2: An example target distribution with a complex
+shape, in PDF and CDF space. Black lines are observed tar-
+get distributions, in the form of mixture of the other three dis-
+tributions shown with color. Fitting the black line accurately
+would be extremely difficult for most of the commonly-used
+single parametric splines, motivating for the use of learnable
+spline family composed of multiple splines.
+have different dependency dynamics with target y level (i.e.
+the variance of y increases when X apart from 0). Fig. 2
+shows that the observed distribution cannot trivially fit well
+with one single distribution. Therefore, in order to model all
+quantiles at all X, we need a quantile function with a com-
+plexity that does not increase significantly with number of
+input attributes and the number of quantiles. This necessitates
+a versatile and highly-expressive quantile function.
+There has been many efforts on improving various aspects
+of quantile regression. Gasthaus et al. (2019) proposes linear
+spline interpolation between knots in the inverse CDF space
+to model the target distribution in time-series forecasting
+setup. This is proposed to avoid the assumption on paramet-
+ric form of the target distribution. Park et al. (2022) and Moon
+et al. (2021) focus on learning a valid quantile function with-
+out quantile crossing (e.g. quantiles violate monotonically
+increasing property), via special design of the neural network
+architecture or first-order inequality constraint optimization.
+Despite being distribution agnostic, these approaches for de-
+scribing the target distribution (specified by quantile function)
+are restricted to one function family (e.g. linear spline), which
+may limit the expressiveness to represent the target distribu-
+tion. In this paper, with the goal of designing an expressive
+quantile function for various quantiles and input levels, we
+propose a data-driven approach Neural Spline Search (NSS),
+which transforms the inputs with a series of monotonic spline
+regressions guided by symbolic operators. The contributions
+of our paper can be summarized as:
+1. We propose an efficient search space and mechanism to
+find an expressive quantile function to model the data
+distribution, avoiding specifying a parametric form of the
+observed distribution as prior.
+2. We propose a novel approach to generate an expressive
+quantile function using a combination of different distri-
+butions and operators guided by symbolic operators.
+3. The proposed method can be incorporated into other tasks
+(including but not limited to time series forecasting) as
+their quantile function.
+4. We demonstrate significant accuracy improvements across
+numerous regression or time series forecasting tasks. For
+example, on UCI benchmarks, we show 3.5%-7.0% im-
+provement compared to next best methods.
+Related Work
+Quantile regression is used to estimate the target distribu-
+tion at different quantile levels. The α-quantile estimator
+is the solution when minimizing quantile loss at level α
+(Koenker and Bassett Jr 1978). Another quantile regression
+related loss is continuous ranked probability score (CRPS)
+(Gneiting and Raftery 2007), which is the averaging over all
+quantile levels, instead of one single quantile.
+Neural network quantile forecasting. To model sequential
+dependency of time series, several forecasting models pro-
+pose a hidden state-emission framework ((Salinas et al. 2020;
+Wen et al. 2017; Gasthaus et al. 2019; de Bézenac et al. 2020;
+Wang et al. 2019)), where the dynamics of hidden states
+are modeled by auto-regressive recurrent neural works (e.g.
+LSTM), which takes previous hidden states and current ob-
+servations as input and outputs current observation. Different
+from modeling the likelihood with parametric distributions
+(e.g. Gaussian (Salinas et al. 2020)), emission models for
+quantile estimation is to learn the parameters of quantile
+function. The overall framework is optimized by employing
+a quantile (Wen et al. 2017) or CRPS (Gasthaus et al. 2019)
+loss.
+Symbolic regression has shown great success in many fields,
+including program synthesis (Parisotto et al. 2016), mathe-
+matical expressions extraction (Cranmer et al. 2020), physics-
+based learning (Li et al. 2019; Petersen et al. 2019). As the
+search space is enormous and scaled exponentially with the
+length of operators, symbolic regression rule operators are
+usually set to be a small number and are learned by Monte
+Carlo Tree Search guided evolutionary strategies (Li et al.
+2019) or reinforcement learning (Petersen et al. 2019).
+Methods
+Learning quantile function in quantile regression
+Let the input data attributes X and the target variable y are
+jointly distributed as p(X, y). The conditional cumulative
+distribution function (CDF) is F(Y = y|X) = P(Y ≤
+y|X). The quantile function, which is also called the inverse
+CDF function, takes quantile level as inputs and returns a
+threshold value Y below which random draws from the given
+CDF would fall quantile percent of the time. Specifically, the
+α-th quantile function of y|X = x is denoted as:
+q(α, x) = F −1
+y|X=x(α) = inf{y : F(y|X = x) ≥ α}
+(1)
+Here we can think the quantile function is to perform a
+transformation on a uniform-distributed random variable
+α ∼ U(0, 1) to the target distribution p(y|X). Quantile
+function is able to fully specify a distribution. So specifying
+the quantile function is describing the target distribution
+p(y|X).
+Quantile regression estimates different conditional quantile
+levels of the target variable given a certain level of input
+
+P(y|X)
+Inverse CDF
+alpha
+y
+P(y|X)
+Inverse CDF
+alpha
+y
+P(y|X)
+Inverse CDF
+alpha
+y
+P(y|X)
+Inverse CDF
+alpha
+y
+Spline Basis
+P(y|X)
+Inverse CDF
+alpha
+y
+S
+C
++
+P(y|X)
+Inverse CDF
+alpha
+y
+Spline Basis
+P(y|X)
+Inverse CDF
+alpha
+y
+....
+P(y|X)
+Inverse CDF
+alpha
+y
+P(y|X)
+Inverse CDF
+alpha
+y
+...
+....
+....
+P(y|X)
+Inverse CDF
+alpha
+y
+P(y|X)
+Inverse CDF
+alpha
+y
+....
+....
+NSS-sum
+Initial distribution
+target distribution
+Spline Basis
+Spline Basis
+NSS-chain
+P(y|X)
+Inverse CDF
+alpha
+y
+Operators
++
+Sum
+S
+Scale
+C
+Chaining
+....
+....
+Figure 3: Overview of Neural Spline Search (NSS). Modeling the target data distribution can be done by learning the quantile
+function (e.g. inverse CDF), which maps a [0, 1]-variable (quantile) to a target value y. Unlike parametric methods which specify
+a distribution family and learn the parameters, NSS can generate the target distribution through a set of transformations on the
+inverse CDF space (quantile space), where the transformation is guided by a series of operators. Here, the bottom gray box shows
+possible operators (denoted as circles), including but not limited to summation (“+”), scale (“S”), and chaining (“C”). The basis
+splines are shown with color-shaded squares. The initial distribution is a uniform distribution, as shown in the leftmost panel
+(blue shaded), and the target distribution is the rightmost distribution (purple shaded). There is no obvious parametric distribution
+to achieve this transformation. Therefore, NSS is used to search for the suitable transformation through simple operators. In
+the first row of the middle panel, we show operators for NSS-sum, where the initial uniform distribution is transformed by
+the red- and the yellow-shaded splines (e.g. c-spline) through sum (“+”) and scale (“S”) operators. The second row shows the
+chaining transformation of the initial distribution, where the orange and cyan splines are used to transform the initial spline. The
+parameters of the splines are learned by a neural network. In general, the operators and transformations in NSS are not limited to
+two splines (we represent them as the gray splines next to the yellow and cyan shaded splines).
+attributes, as opposed to regression, which estimates the con-
+ditional mean of the target variable. In quantile regression,
+a particular quantile level α of the conditional distribution
+of y given X = x, q(α, x) is estimated by minimizing the
+pinball loss ρ (or quantile loss), as the the quantile function
+q is shown to be the minimizer of the expected pinball loss
+(Koenker and Bassett Jr 1978):
+ρα(y, q) = (y − q)(α − 1(y < q)),
+(2)
+q(α, x) = arg min
+q
+Ey[ρα(y, q)].
+(3)
+where 1 is the indicator function. One shortcoming of pinball
+loss is only measuring the loss at a single quantile level,
+which hinders the estimated q for a global picture of the
+distribution (i.e. other α levels). On contrast, the continuous
+ranked probability score (CRPS) considers all quantile levels
+by integrating the pinball loss over α = [0, 1] (Matheson and
+Winkler 1976; Gneiting and Raftery 2007).
+CRPS(y, q) =
+� 1
+0
+2ρα(y, q)dα
+(4)
+As a proper scoring rule (Gneiting and Raftery 2007), CRPS
+is minimized when the quantile function is q = F. That is,
+F −1
+y
+= arg min
+q
+Ey[CRPS(y, q)].
+(5)
+Please refer (Koenker and Regression 2005) for detailed
+proof.
+Improving the expressiveness of quantile function
+Fig. 2 demonstrate the need of an expressive quantile func-
+tion for modeling target distribution. Inspired from neural
+architecture search (NAS) (Elsken, Metzen, and Hutter 2019),
+we propose an approach to search for the suitable combina-
+tion of distributions. The search is over different operations
+and basis distributions. We first introduce parametrization of
+quantile function, and the two non-parametric spline-based
+distributions.
+Parameterizing quantile functions
+We propose to param-
+eterize the quantile function qθ(α, x) using a deep neural
+network with parameters θ. The quantile function is aimed
+to be accurate for any quantile levels α and input attributes
+level X = x. X is high dimensional in real data, not as the
+one dimensional in the toy examples in Fig. 1 and Fig. 2.
+C-spline distribution
+The c-spline (yα = qcsplie
+θ
+(α, x))
+describes the CDF (Fig. 2, Right Panel) of a probability
+distribution Fy|X by setting K anchor points (denoted as
+knots) on the CDF curve and performing linear interpolation
+to fill in the gap between the knots. Specifically, the knots
+split CDF curve into bins and c-spline learns the width wi
+and height hi of bins by neural networks NN that depend on
+the input attributes level X = x.
+{wi, hi}K = NNθ(x)
+yα = r({wi, hi}K, α)
+∀α ∈ [0 : 1]
+
+where hi and wi are non-negative delta values imposed by
+non-negative activation (i.e. Relu or Sigmoid), and the loca-
+tion of each bin (e.g. Y|X) is Li = �i
+k=0 wk and quantile
+level αi = �i
+k=0 hk. The accumulation sum design is to en-
+sure that quantile function is monotically increasing and there
+is no quantile crossing. r is a function to convert knots to
+output of quantile function: for quantile level αi that is on the
+knots, we can directly read from li , for quantile levels that
+are off the knots, quantile values can be computed through
+linear algebra operations on the two nearby knots r(α) =
+�
+li + (α−αi)(lj−li)
+αj−αi
+,
+if αi ≤ α ≤ αj
+0 ≤ i, j ≤ K
+lk,
+if hk = α
+P-spline distribution
+The difference between p-spline
+from c-spline is having anchor knots in PDF space, instead
+of CDF space. Similarly with C-spline, P-spline also per-
+form linear interpolation over knots, and the quantile level is
+achieved by integration over pdf via polynomial operations.
+Neural Spline Search (NSS)
+We describe our proposed method, Neural Spline Search
+(NSS), which is overviewed in Fig. 3. Similar to symbolic
+regression (Parisotto et al. 2016; Li et al. 2019), NSS effec-
+tively searches in the space of discrete symbolic operators
+and distribution space for a candidate that can better fit the
+target data distribution. Specifically, let T(O, S, k) denote the
+space of all transformations, via operators O on all distribu-
+tion S with a maximum sequence length k. NSS aims to find
+the function f(x) selecting operators and distributions in the
+space T such that {f(x) ∈ T(O, S, k) : ℓ(f(x), xtrain) ≤ δ
+}, where ℓ denotes loss function CRPS, xtrain is training
+data and δ is the acceptance threshold. Given the large search
+space composed of combinations of numerous splines and
+operators, we restrict to use spline-based distribution as the
+basis distribution, and limit the operator search space to sum-
+mation and chaining operations upon the transformation basis
+spline regressions. Note that this work can be easily extend
+to other operations and distributions, which we leave to fu-
+ture work. We describe the following NSS transformations as
+they are observed to work well consistently across different
+datasets: NSS with summation (NSS-sum) and NSS with
+chaining (NSS-chain). Algorithm 1 and Fig. 4(b)
+NSS-sum
+NSS-sum performs transformations using the scale and sum-
+mation operators. We represent this scenario with two splines:
+Spline 1: c-spline and Spline 2: p-spline, and two operators:
+scale O1 : O(a) = λa and summation O2 : O(a, b) : a + b;
+therefore, the overall transformation is (Spline 1-Operator 1) -
+(Spline 2-Operator 2), which yields: f = c-spline + λ p-spline.
+Essentially, NSS-sum performs weighted sum of different
+splines. The motivation behind is that c-spline with fewer
+parameters can be more robust against overfitting, whereas
+p-spline increases the expressiveness of the splines.
+NSS-chain
+Another proposed NSS design is NSS-chain. We focus on
+the chaining operator due to its expressiveness. This design
+is inspired by the success of normalizing flow (Rezende and
+Mohamed 2015), where a sequence of bijector transforms is
+utilized to transform distributions. Different from normaliz-
+ing flow which has practical applicability challenges, NSS-
+chain only requires the forward pass of the transformation,
+not the inverse as normalizing flow does. This significantly
+reduces the computational complexity and broadens the fea-
+sibility of transformations. As mentioned, quantile function
+takes input attributes level (X) to predict the target value (y)
+at quantile level (α).
+y = qθ(X, α),
+(6)
+where X ∈ Rm and α ∈ [0, 1]. We present two designs to
+chain different transformations (see Fig. 4 (a)). We note that
+chaining of transformation is not limited to the two designs.
+Algorithm 1: Neural Spline Search
+Operators = {+, ×, Scale, Chain, ...}
+Splines = {c-spline, p-spline, Gaussian, Cauchy ...}
+Data: Quantile level α ∈ [0, 1], N data points
+{X ∈ Rd, y ∈ R1}N, d ≥ 1, with chain depth k.
+Transform indicates the transformation using the
+input spline Sθ and operator O.
+Result: p(y|X) and F −1
+y|X(α)
+k ← 1;
+while k ≤ K do
+Select O = {Oi}no ∈ Operators ;
+Select S = {Sj}ns ∈ Splines ;
+θ ← MLP(X) ;
+ypred ← Transform(Sθ, O, α);
+if α NSS-chain then
+Normalize ypred to [0, 1] as y′
+pred ;
+α ← y′
+pred;
+else
+X ← Y
+▷ if X-NSS-chain ;
+end
+k ← k + 1;
+end
+• α-chaining
+The α-chaining is when we consider the condition level
+(X) unchanged during the chain of transformation, and
+the output of each transformation is a scaled version of
+quantile level for the next transformation. In particular,
+after each transformation, we normalize the output y to
+be in the range [0, 1], and then the normalized output is
+re-input as the new α to the next transformation. This is
+repeated until the maximum depth is reached. This design
+is more similar with normalizing ��ow methods.
+y = qθK(X, ...fn(qθ2(X, fn(qθ1(X, α)))))
+(7)
+θk for k=1,2,..K are parameters for different splines in
+K-length chain. fn is the normalization function.
+• X-chaining
+X-chaining is when we consider quantile level α level is
+unchanged during chaining, as each transformation learns
+
+alpha
+P(y|X)
+y_alpha
+Inverse CDF
+alpha
+y
+        MLP 
+X
+Spline's parameters
+alpha
+P(y|X)
+y_alpha
+Inverse CDF
+alpha
+y
+        MLP 
+X
+Spline's parameters
+x-chaining
+alpha
+P(y|X)
+y_alpha
+Inverse CDF
+alpha
+y
+        MLP 
+X
+Spline's parameters
+alpha-chaining
+NNS Chain
+Figure 4: (a) Illustration of NSS-chain methods. The dia-
+gram demonstrates chaining for NSS-chain. Left: α-chaining.
+The output y of the spline, after re-scaling to [0, 1], is re-
+inputted to the quantile spline at quantile level α. Right: X-
+chaining. The output y is instead re-inputted to the quantile
+spline as X. Both rely on input attributes X.
+a suitable condition level (or feature) for next iteration.
+Similarly with α-chaining in the iterative manner, except
+that the output y of each transformation is projected to
+generate X for the next iteration of Eq. 6.
+y = qθK(...qθ2(qθ1(X, α), α), α)
+(8)
+The advantage of this approach, compared tp α-chaining,
+is that we keep quantile levels α unchanged, and re-
+normalizing output is not needed.
+Remarks on NSS: . (1) why a simple spline-based algo-
+rithm, e.g. C-spline, is not enough? Although in theory
+spline-based algorithms can represent any arbitrary distribu-
+tions with sufficiently high number of knots K, in practice,
+we find a large K often lead to unstable training, as also
+studied in (Park et al. 2022). In contrast, we find the combina-
+tion (combined or chained) over a relatively restricted splines
+are more robust in capturing the overall of the target distribu-
+tion (2) Include both spline-based distribution and classic
+parametric distribution In addition to spline-based distribu-
+tion, we also encourage incorporating parametric distribution
+(e.g. Gaussian) as basis distribution for NSS, especially when
+prior knowledge (say Gaussian noise) is available. Because, it
+is challenging for spline based methods to reconstruct Gaus-
+sian distribution even with infinite number of knots; and , the
+benefits of combining the two are the parametric distribution
+offers advantage of classic statistics and robust to noise, and
+the non-parametric spline offers flexibility.
+Training
+Once we select the operators and splines, the parameters of
+the splines are trained in an end-to-end way by optimizing
+CRPS (Eq. 4). Specifically, during training, we fit parameters
+by optimizing over with the empirical mean of CRPS over N
+data points:
+θ∗ = arg min
+θ
+1/N
+N
+�
+i=1
+Ey[CRPS(y, qθ(Xi, α))].
+(9)
+Algorithm 2 overviews the training of NSS for spline parame-
+ter selection. Because of the form of the transformations, the
+analytical solution of CRPS integration is intractable. Thus,
+we use a Monte Carlo estimation for the CRPS loss. In par-
+ticular, we sample m number of α values from the range of
+[0, 1] and average them for the corresponding pinball loss.
+Algorithm 2: Training with CRPS
+Data: N data points {Xi ∈ Rd, yi ∈ R1}N
+i=1, m
+quantile levels, T transformation, which
+takes selected splines Sselect and selected
+operators Oselect from NSS. lr is learning
+rate.
+Result: Neural network weights θ
+e ← 1;
+while e ≤ Nepoch do
+f = Transform(Sselect, Oselect) ℓ ← 0 ;
+for α in [0, 1
+m, 2
+m, ..1] do
+ypred
+α
+= fθ(X, α) ;
+ℓ ← ℓ + pinball_loss (ypred
+α
+, y, α)
+end
+CRPS = ℓ/m ;
+θ ← θ − lr · ∇θ CRPS ;
+e ← e + 1;
+end
+Experiments
+Comparison methods
+QD (Pearce et al. 2018) generates prediction intervals (PIs)
+for estimating uncertainty for regression tasks with the as-
+sumption that high-quality PIs should be as narrow as possi-
+ble. Deep Quantile Aggregation (Kim et al. 2021) proposes
+weighted ensembling strategies where aggregation weights
+vary over both individual models and feature values plus
+(pairs of) quantile levels. The monotonization layer in the
+network is applied to avoid crossing of quantile estimates.
+RQspline (Durkan et al. 2019) proposes a fully-differentiable
+module based on monotonic rational-quadratic splines, which
+enhances the flexibility of coupling and autoregressive trans-
+forms while retaining analytic invertibility. Global-Coarse
+(Ratcliff 1979) provides an analysis of distribution statis-
+tics of group reaction time distributions. MLE (NB) and
+Mix. MLE are Negative Binomial and mixture likelihood
+based methods (Awasthi et al. 2021). C-spline is proposed in
+(Gasthaus et al. 2019), where C-spline is used as the quantile
+function in time-series forecasting.
+Metrics
+For point predictions, we focus on the following metrics:
+Mean absolute error (MAE): 1
+n
+�n
+t=1 |Tt − Pt| where Tt and
+Pt are true and predicted value; Mean Absolute Percentage
+Error (MAPE): 1
+n
+�n
+t=1 | Tt−Pt
+Tt
+|. Weighted Average Percent-
+age Error (WAPE):
+�n
+t=1 |Tt−Pt|
+�n
+t=1 |Tt|
+; and Root Mean Square
+Error (RMSE):
+� �N
+t (Tt−Pt)2
+n
+. For quantile predictions, we
+use the Pinball Loss (Eq. 2), with 50%-th, Q50; 90%-th, Q90;
+and 10%-th Q10 quantiles.
+
+Methods
+Boston
+Concrete
+kin8nm
+Power
+Protein
+Wine
+Gaussian
+0.0754
+0.0564
+0.048
+0.0449
+0.2116
+0.0978
+QD
+0.5003
+0.4150
+0.3945
+0.3688
+0.6689
+0.4456
+RQspline
+0.0917
+0.0622
+0.0479
+0.0485
+0.2153
+0.0912
+p-sline
+0.0778
+0.0570
+0.0444
+0.0453
+—
+0.0966
+c-spline
+0.0806
+0.0543
+0.0430
+0.0447
+0.2002
+0.0947
+NSS-X-chain
+0.0787
+0.0588
+0.0430
+0.0448
+0.2052
+0.0962
+NSS-α-chain
+0.0846
+0.0568
+0.0417
+0.0448
+0.2067
+0.0976
+NSS-sum
+0.0709
+0.0512
+0.0414
+0.0442
+0.1949
+0.0957
+Gain percentage
+12.0%
+17.7%
+3.7%
+1.1%
+2.6%
+-
+Table 1: Mean Absolute Error (MAE) on UCI benchmarks. Test performance of the proposed method (NSS) and existing
+methods on UCI benchmarks. We use the 50th quantile estimator as our estimates. The dash indicates unavailability. The shaded
+area is the proposed methods. Bold is the top one. Lower is better. Gaussian: Gaussian kernel; QD is quantity-driven methods
+proposed in (Pearce et al. 2018); RQ spline proposed in (Durkan et al. 2019); c-spline proposed in (Gasthaus et al. 2019). Boston,
+Concrete, Power is short for Boston Housing, Concrete Strength, Power Plant. Gain percentage is computed as (best nss - best
+baseline)/best baseline.
+Training
+For simplicity, the proposed NSS methods use depth-
+2
+splines,
+which
+contain
+{(c-spline,
+p-spline),
+(c-
+spline,
+p-spline),
+(c-spline,
+c-spline),
+(p-spline,
+p-
+spline)}. NSS-sum is tuned with λ in the range of
+[0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9].
+NSS-chain
+nor-
+malizing of y in α chaining can be achieved by applying
+sigmoid layer or scaling by max value. As splines are
+monotonically-increasing functions, the spline value y with
+α = 0 is the minimum value of y and α = 1 yields the
+maximum value of y. Scale is yscale =
+y−ymin
+ymax−ymin . We use
+a batch size=128 and a learning rate of 0.005 for 100 epochs.
+Results
+To demonstrate the effectiveness of proposed methods, we
+conduct experiments on synthetic, real-world tabular regres-
+sion, and time series forecasting datasets.
+Synthetic data
+Dataset. We generate 2000 data points (X
+∈ R1 and
+y ∈ R1), where X is in the range of [−2, 2] and y has Gaus-
+sian distribution y ∼ N(0.3 sin(3x), 0.2x2), where sin is the
+sinusodial function. We construct the validation and test sets
+to come from the same distribution. Unlike real-world data,
+the synthetic data would have known quantile levels, that can
+be used for evaluating the accuracy of quantile estimates. We
+make the task more challenging by setting a data-dependent
+variance for the Gaussian noise to evaluate the ability of learn-
+ing condition-specific quantile values. Fig. 5 shows that the
+proposed NSS-chain and NSS-sum can capture the true under-
+lying quantiles, whereas QD (Pearce et al. 2018) struggles on
+the varying variance locations (e.g. around x = 0). The upper
+and lower black lines are the predicted 2.5%-th and 97.5%-th
+quantiles for the observed data (e.g. red dots), shown along
+with the ground truth quantiles (e.g. shaded red area). The
+results indicate that more expressive NSS transformations
+are superior in more challenging scenarios, where true data
+points are distributed differently (e.g., distributions depend
+on the value of the inputs"). Fig. 6 shows the calibration plot
+of the predicted vs. true distributions at different quantile
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+x
+2
+1
+0
+1
+2
+y
+QD
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+x
+2
+1
+0
+1
+2
+y
+NSS-SUM.
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+x
+2
+1
+0
+1
+2
+y
+NSS-CHAIN.
+Figure 5: NSS on Synthetic data. We compare the per-
+formance of proposed NSS against existing methods QD
+(Pearce et al. 2018). The red dots are observed data points,
+shaded red area is the ground truth 2.5% and 97.5% quantile
+levels, and the dark black lines are the predicted 2.5% and
+97.5% quantile levels.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True percentile
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Predicted percentile
+QD
+NSS-SUM
+NSS-CHAIN
+2
+0
+2
+X
+y
+X=0.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True percentile
+Calibration plot
+QD
+NSS-SUM
+NSS-CHAIN
+2
+0
+2
+X
+y
+X=1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True percentile
+QD
+NSS-SUM
+NSS-CHAIN
+2
+0
+2
+X
+y
+X=1.5
+Figure 6: Calibration plots. Predicted vs. ground truth per-
+centiles at condition levels: X=0.5, 1.0 and 1.5. The perfect
+calibration would correspond to the diagonal (red dotted)
+line.
+levels. Here, we show the true percentile p as the fraction of
+data in the dataset such that the p percentile of the predictive
+distribution is larger than the ground truth data. The perfect
+prediction would be the diagonal line. Fig. 6 indicates that
+the proposed methods NSS-sum and NSS-chain can capture
+the proposed true distribution at various levels by close to the
+red line, whereas QD does not fit as well.
+Real-world tabular regression
+We use UCI benchmarks (Asuncion and Newman 2007) that
+contain tabular data from diverse domains (e.g. real estate
+and physics). Following (Salem, Langseth, and Ramampiaro
+2020), the datasets are normalized with z-score standardiza-
+
+Methods
+Boston
+Concrete
+kin8nm
+Power
+Protein
+Wine
+Gaussian
+0.0276
+0.0203
+0.0171
+0.0158
+0.0725
+0.0357
+Global-Coarse∗
+0.0745
+0.0596
+0.0681
+0.0473
+0.1321
+—
+Deep Quantile Aggregation∗
+0.0754
+0.0541
+0.0684
+0.0441
+0.1253
+—
+QD
+0.1212
+0.1076
+0.1004
+0.0972
+0.1547
+0.1164
+RQspline
+0.0458
+0.0418
+0.0203
+0.0189
+0.0863
+0.0424
+p-sline
+0.0308
+0.0211
+0.016
+0.0160
+—
+0.0358
+c-spline
+0.0312
+0.0198
+0.0157
+0.0159
+0.0688
+0.0351
+NSS-X-chain
+0.0311
+0.0216
+0.0165
+0.0162
+0.0707
+0.0358
+NSS-α-chain
+0.0322
+0.0208
+0.0151
+0.0159
+0.0726
+0.0363
+NSS-sum
+0.0265
+0.0191
+0.0152
+0.0157
+0.0674
+0.0357
+Gain percentage
+4.0%
+3.5%
+3.8%
+0.6%
+7.0%
+-
+Table 2: Average pinball loss on UCI benchmarks. The test pinball loss (the lower, the better) is over 99 quantile levels,
+α = {0.01, 0.02, ...0.99}. The compared methods are Global-Coarse proposed in (Ratcliff 1979); QD (Pearce et al. 2018); Deep
+Quantile Aggregation (DQA) (Kim et al. 2021); RQspline (Durkan et al. 2019); ∗ indicates entries are from (Kim et al. 2021)
+(under the same experiment setup).
+Methods
+MAPE
+WAPE
+RMSE
+Q50
+Q90
+Q10
+MLE (NB)
+0.44434
+0.27240
+7.70958
+0.27240
+0.10907
+0.15275
+Mix MLE
+0.44839
+0.26838
+7.22556
+0.26838
+0.10293
+0.14508
+c-spline
+0.44672
+0.26635
+7.06332
+0.26635
+0.10238
+0.14241
+p-spline
+0.44912
+0.26834
+7.14643
+0.26834
+0.10343
+0.14333
+NSS-sum
+0.44501
+0.26545
+6.96697
+0.26545
+0.10238
+0.14266
+NSS-chain
+0.44883
+0.26420
+6.91726
+0.26420
+0.10243
+0.14149
+Table 3: Performance comparisons for time series forecasting on M5. Different evaluation metrics are included in this table
+for M5. Detailed descriptions of the metrics are in Sec . Qk indicates the pinball loss of k-th quantile. e.g. Q50 is the pinball loss
+of 50th quantile. Lower is better.
+tion.
+We evaluate the accuracy for both point predictions and quan-
+tiles. As the point predictions, we use the 50th quantile es-
+timator as our estimates. Table 1 shows that the proposed
+NSS methods outperform the other existing methods on most
+datasets in mean absolute error (MAE). In mean square error
+(MSE), the results are provided in Appendix Table 4. We
+observe that the NSS-sum performs better than NSS-chain.
+For quantile metrics, we use the pinball loss (Eq. 2) over
+100 quantile levels α = {0.01, 0.02, ...0.99} in Table 2. The
+results indicate that NSS consistently outperforms other al-
+ternatives across different UCI benchmarks. In pinball loss,
+NSS-sum performs better than NSS-chain. We attribute the
+superiority of NSS-sum for regression to make balance be-
+tween different transformation, which is helpful in explaining
+the variance in the data.
+Retail demand forecasting
+For time series forecasting, we focus on the M5 dataset,
+which contains time-varying sales data for retail goods, along
+with other relevant covariates like price, promotions, day
+of the week, special events etc. It represents an important
+real-world scenario, where the accurate estimation of the
+output distribution is crucial, as retailers use them to optimize
+prices or promotions.
+The time series forecasting experiments are conducted by
+performing one-step ahead prediction, yielding predictions
+in an autoregressive way. Table 3 shows the results of
+our method compared to other alternatives. We observe
+consistent outperformance of NSS in various forecasting
+evaluation metrics. Different from regression tasks, we
+observe that NSS-chain is better than NSS-sum, indicating
+its benefit in capturing time-dependent relationship.
+Remarks on NSS-sum vs NSS-chain. The results show that
+NSS-sum is superior on regression, while NSS-chain has
+advantages on time series forecasting. The observations may
+indicate NSS-sum is suitable for more constrained tasks (e.g.
+regression, one time step time series-forecasting), where be-
+ing moderately expressive would suffice. NSS-sum is also
+more robust and easier to train. On the other hand, NSS-chain
+may be more expressive, which is beneficial to fit tasks re-
+quires more complex distributions at different time steps of
+the time series, but for individual step NSS-chain is not as
+accurate as NSS-sum in fitting the distribution.
+Conclusion
+We propose a novel approach for modeling uncertainty. The
+proposed Neural Spline Search (NSS) method employs a se-
+ries of monotonic spline regression transformations, guided
+by symbolic operators. We demonstrate the effectiveness of
+NSS for superior modeling of output distributions, on both
+synthetic and real-world datasets. We leave the extensions to
+different operators and splines, including parametric distribu-
+tion transformations to future work.
+
+.
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+Appendix
+Mean square error for UCI dataset
+
+Methods
+Bost House
+Concr Stren
+kin8nm
+Power plant
+Protein
+Wine
+Gaussian
+0.0105
+0.0054
+0.0042
+0.0032
+0.0648
+0.0164
+(Salem, Langseth, and Ramampiaro 2020)∗
+0.1120
+0.0560
+0.0600
+0.0420
+0.3100
+0.5970
+QD
+0.2705
+0.1839
+0.1613
+0.1393
+0.5277
+0.2164
+RQspline
+0.0255
+0.0070
+0.0040
+0.0037
+0.0809
+0.0195
+p-sline
+0.0136
+0.0058
+0.0032
+0.0032
+—
+0.0162
+c-spline
+0.0162
+0.0050
+0.0031
+0.0032
+0.0757
+0.0159
+NSS-X-chain
+0.0128
+0.0056
+0.0031
+0.0032
+0.0751
+0.0164
+NSS-α-chain
+0.0184
+0.0058
+0.0029
+0.0032
+0.0760
+0.0169
+NSS-sum
+0.0112
+0.0046
+0.0029
+0.0032
+0.0711
+0.0160
+Table 4: Mean Square Error of UCI datasets
+

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+Identification of lung nodules CT scan using YOLOv5 based on
+convolution neural network
+Haytham Al Ewaidat
+ID 1,*, Youness El Brag
+ID 2
+1Jordan University of Science and Technology, Faculty of Applied Medical Sciences, Department of Allied Medical
+Sciences-Radiologic Technology, Irbid, Jordan, 22110
+2Abdelmalek Essaˆadi University of Science and Technology, Faculty of Multi-Disciplinary Larache, Department of
+Computer Sciences, ksar el kebir , Morocco, 92150
+Correspondence author: Dr Haytham Al Ewaidat, Department of Allied Medical Sciences-Radiologic Technology,
+Faculty of Applied Medical Sciences, Jordan University of Science and Technology. PO Box 3030, Irbid 22110,
+Jordan Tel: (+962)27201000-26939; Fax: (+962)27201087; E-mail: haewaidat@just.edu.jo
+arXiv:2301.02166v1  [eess.IV]  31 Dec 2022
+
+Abstract
+Purpose: The lung nodules localization in CT scan images is the most difficult task due to the complexity of the
+arbitrariness of shape, size, and texture of lung nodules. This is a challenge to be faced when coming to developing
+different solutions to improve detection systems. the deep learning approach showed promising results by using
+convolutional neural network (CNN), especially for image recognition and it’s one of the most used algorithm in
+computer vision.
+Approach: we use (CNN) building blocks based on YOLOv5 (you only look once) to learn the features representations
+for nodule detection labels, in this paper, we introduce a method for detecting lung cancer localization. Chest X-rays
+and low-dose computed tomography are also possible screening methods, When it comes to recognizing nodules
+in radiography, computer-aided diagnostic (CAD) system based on (CNN) have demonstrated their worth. One-
+stage detector YOLOv5 trained on 280 annotated CT SCAN from a public dataset LIDC-IDRI based on segmented
+pulmonary nodules.
+Results: we analyze the predictions performance of the lung nodule locations, and demarcates the relevant CT scan
+regions. In lung nodule localization the accuracy is measured as mean average precision (mAP). the mAP takes into
+account how well the bounding boxes are fitting the labels as well as how accurate the predicted classes for those
+bounding boxes, the accuracy we got 92.27% .
+Conclusion: this study was to identify the nodule that were developing in the lungs of the participants. It was difficult
+to find information on lung nodules in medical literature,
+Keywords: computer-aided diagnostic, deep learning, Convolutional Neural Networks ,Lung Nodule.
+*Address all correspondence to Haytham Al Ewaidat , haewaidat@just.edu.jo
+1
+introduction
+As far as noninvasive therapy and clinical assessment are concerned, medical image analysis offers
+a tremendous advantage. X-rays, CTs, MRIs, and ultrasounds are utilized to make precise diag-
+noses based on the obtained restorative images. By using attractive fields, CT can capture pictures
+on film in medical imaging. One-of-a-kind lung cancer is responsible for 1.61 million fatalities per
+year. Most of the cases of lung cancer in Indonesia are observed in the MIoT centers. If the tumor
+is identified early, the survival percentage is better then. It’s not an easy task to find lung cancer
+in its early stages. Approximately 80% of cancer patients are diagnosed at the core or accelerated
+phase of the disease. Lung cancer is the second most common cancer among men and the tenth
+most common among women worldwide. After breast and colorectal cancer, lung cancer is the
+thirdly most common cancer among women. Features extraction in image processing is one of the
+simplest and most efficient dimensionality reduction approaches. The non-invasive nature of CT
+imaging is one of its most notable characteristics. It’s surprising to see angles increasing when
+compared to other imaging modalities.
+Computed tomography imaging is the best technique for examining lung disorders. CT scans, on
+the other hand, have a high probability of false-positive results and are associated with cancer-
+causing radiation exposure. When compared to standard-dose CT, low-dose CT utilizes a lot less
+radiation contact power. The findings reveal that the detection sensitivity of low-dose and standard-
+dose CT images is not significantly different. A well know database the National Lung Screening
+Trial database shows that cancer-related fatalities were considerably decreased in the group that
+was subjected to low-dose CT scans rather than chest radiography. The sensitivity of lung nodule
+1
+
+identification may be improved by the use of more detailed anatomical information, and better
+image registration methods. As a result, the datasets have grown enormously. Up to 500 seg-
+ments/slice may be generated from a single scan, depending on how thick the slice is. A single
+slice is examined by a competent radiologist in 2–3.5 minutes. A radiologist’s workload keeps
+rising while screening a Ct for the presence of a suspicious nodule. The detection sensitivity of
+nodules is influenced by a variety of factors, including the size, location, form, nearby structures,
+edges, and density, in addition to the CT slice section thickness.
+Only 68 percent of lung cancer nodules are properly identified when only one radiologist doctor
+views the scan, and up to 82% of the time when two radiologists check the scan, according to
+the study results. Early diagnosis of malignant lung nodules by radiologists is a tough, time-
+consuming, and laborious process in and of itself. The radiologist needs a lot of time to carefully
+screen a large number of images, but this method is prone to mistake when looking for microscopic
+nodules.
+An aid for radiologists is required in this case to speed up readings, catch any missing nodules,
+and enable improved localization. A primary goal of computer-aided detection systems was to
+minimize radiologists’ labor and boost the detection rate of nodules. Newer CAD systems, on
+the other hand, can distinguish between benign, and malignant nodules, which is helpful in the
+screening process. CAD systems frequently beat professional radiologists in nodule identification
+and localization tasks because of recent breakthroughs in deep learning models, particularly in
+image processing. CAD systems, on the other hand, have an FP rate of 1–8.2 per scan and a
+detection range of 38–100%, according to different studies. As a result of their likeness to one
+other, benign, and malignant nodule remain a difficult challenge to solve.
+During the screening process, a variety of mistakes might occur. For example, if a scan fails to
+capture or recognize a specific region of the lesion or fails to distinguish between benign, and
+malignant lesions in a patient’s body, the patient may be at risk of misdiagnosis. Most people
+die as a result of misdiagnoses and delays in treatment because of these mistakes. In radiology,
+over 4% of reports include diagnostic mistakes on a daily, and about 30% of aberrant radiological
+diagnoses are ignored. Early-stage lung nodules may be detected and classified more accurately
+using different methodologies such as deep learning.
+Lung nodule identification using deep learning with a specific methodology is presented in this
+research. lung CT images, physiological symptoms, and clinical indicators, the suggested ap-
+proach reduces false-positive findings and eventually prevents invasive procedures. YOLOv5 is
+used which has convolutional networks were built to identify and classify nodules. For nodule
+identification. Nodule identification and classification using the publicly accessible data set LIDC-
+IDRI surpasses state-of-the-art deep learning techniques. Using a variety of techniques, we were
+able to reduce the number of false positives in the learning algorithm.
+Lung nodule computer-aided detection (CAD) systems were originally developed in the late 1980s,
+but the processing resources required for sophisticated image analysis methods at the time made
+these efforts unattractive. For image analysis, and decision support systems based on computers,
+the graphics processing unit and convolutional neural networks revolutionized their performance.
+Some of the most important lung nodule identification and classification approaches have been
+suggested by researchers in deep learning based medical images analysis models. For lung nodule
+2
+
+classification, Yutong Xie et al1 . proposed a method that utilizes Texture, Shape, and Deep Model
+learned Data at the choice level.
+Nodule heterogeneity may be shown with the use of this algorithm’s GLCM-based surface de-
+scriptor, Fourier-shape descriptor, and a DCNN. Based on CNNs Chougrad et al.2 studied the
+classification of breast cancer using a CAD framework. Transfer learning, on the other hand, takes
+just a small number of medical images to train a system. With the use of the transfer learning
+approach, the CNNs were taught to their fullest potential. In terms of accuracy, CNN came out on
+top with a score of 98.94 percent. Using the wavelet transform and principal component analysis,
+Heba Mohsen et al3 developed a DNN classifier for brain tumor classification. A technique of reg-
+ularized linear discriminant analysis was developed in 2015 by Sharma et al,4 and it used a regular-
+ization parameter to perform a standard cross-validation methodology. An appropriate collection
+of characteristics is needed to evaluate medical data for illness prediction. Several evolutionary
+algorithms have been used to find the best possible traits. Gravitational search and Elephant Herd
+optimizations have recently been used to choose the best features.5 Another deep learning-based
+model created by Kuruvilla, and Gunavathi, K. in 2014, an ANN-based cancer classification for
+CT scans. Development of the statistical model used to classify the data was completed. Compared
+to feed-forward networks feed-forward backpropagation networks are more accurate, according to
+research. Classifier accuracy may be improved even more by using the skewness feature.6
+Lung cancer detection categorization is becoming more and more popular due to the rapid advance-
+ment of pattern recognition and image processing methods. Textural evaluation of thin-section CT
+images has been used in the literature to help distinguish various obstructive lung disorders. At-
+tenuation distribution statistics, acquisition-length parameter, and co-occurrence descriptor are all
+included in 13-dimensional vectors of local textures information developed by Chabat et al.7 A
+Bayesian classifier is used for feature segmentation. These five scalar metrics, max, entropy, en-
+ergy, contrast, and homogeneity were recovered per each co-occurrence matrix to minimize the
+feature vector’s dimensionality. The textural characteristics of Solitary Pulmonary Nodules dis-
+covered by CT have been described and assessed by Yanjie Zhu et al.8 It took 300 generations for
+67 characteristics to be retrieved, however, only 25 features were picked. SVM-based classifiers
+are used for classification. For Interstitial Lung Disease, Sang Cheol Park and colleagues9 used a
+genetic algorithm to identify the best picture attributes (ILD). Hiram et al10 used the frequency do-
+main, and SVM with RBF to classify lung nodule classifications. Solitary pulmonary nodules may
+be automatically detected using an algorithm provided by Hong et al.11 True nodules are identified
+and labeled on original images using an SVM classifier. The LIDC-IDRI images database was
+used by Antonio et al,12 to classify lung nodules. Ecological taxonomic diversity and taxonomic
+distinctness indexes are used for classification using SVM.13 Results show a 98.11% accuracy rate.
+The mesh grid region growth approach was used in CT to select and analyze just the pixels that
+were most likely to be relevant to the diagnosis. The ILD status of all unselected pixels was deter-
+mined to be negative. To recognize lung cancer cells, Zhi-Hua et al14 presented Neural Ensemble-
+based Detection (NED), which makes use of an artificial neural network ensemble. Using this
+technology, it is possible to accurately identify cancer cells. An algorithm developed by Hui Chen
+et al,15 uses a Neural Networks Ensemble to construct the categorization of a lung nodule on a thin
+section CT image (NNE). A model suggested by Aggarwal, Furquan, and Kalra16 is characterized
+by normal lung architecture by which segmentation is done using the best possible thresholds. Ge-
+3
+
+ometric, statistical, and grey level properties are used to extract features. Classification is done
+using LDA. The accuracy is 84%, the sensitivity is 97%, and the specificity is 53%. An inference-
+based approach to identify lung cancer nodules has been developed by Roy, Sirohi, and Patle.17 To
+improve contrast, this technique employs grey transformations using an active contour model, the
+image is segmented. Training the classifier is done by extracting features such as area, mean, major
+axis, and minor axis length. Overall, the system’s accuracy rate is 94.12%. This approach has a
+disadvantage in that it does not distinguish between benign cancers and those that are malignant.
+Authors have used wavelet feature descriptors to classify lung nodules.18 One and two-level de-
+compositions of wavelet transformations are used in this example. A total of 19 characteristics are
+derived from each wavelet sub-band. SVM is used to distinguish between CT images that include
+malignant nodules and those that do not.
+2
+Material and Methods
+In this section, we introduce our methods for Lung Nodules localization We use a one-stage-
+method based on YOLOv5 detection , the methodology has been split into the following Subsec-
+tions to explain the whole process of our method .
+2.1
+Dataset
+For this research, the dataset has been collected from LIDC-IDRI. In LIDC-IDRI image collection,
+thoracic CT scans with marked-up annotated lesions are included. For the development, training,
+and assessment of computer-assisted diagnostic (CAD) approaches for the detection and diagnosis
+of lung cancer is a worldwide web-accessible resource One example of a public-private partnership
+founded on consensus-based decision-making is this collaboration between the National Cancer
+Institute, the Foundation for the National Institute of Health, and Food and Drug Administration
+(FDA), which was spearheaded by NCI and supported by the FDA. This data collection, which
+includes 239 Ct images for training and 41 images for validation. is a subset of the original dataset.
+Some of the samples are given below in the following Fig.1.
+Fig. 1: Samples from dataset LIDC-IDRI Lung Cancer
+2.2
+Pre-Processing Data
+Real-world data tends to be fragmentary, noisy, and inconclusive. This may lead to low-quality data
+collection, which in turn can lead to low-quality models. Data Preprocessing offers procedures that
+4
+
+LIDC-IDRI-0001
+LightSpeed Plus
+1-January-2000
+ST:2.50SL
+ST
+LittleEndianExplicit
+Images:1/1
+400mA120.00kV
+Series:
+3000566
+WL:
+-600WW:1600LIDC-IDRI-0003
+LightSpeed16
+1-January-2008
+ST:2.50S
+T
+LittleEndianExplicit
+Images:1/1
+300mA120.00kV
+Series:
+3000611
+WL:
+-600WW:1600may properly organize the data for better comprehension in the deep learning process to solve these
+challenges. Data Preprocessing steps that have been used in this research study are given in the
+following Fig.2.
+Fig. 2: Preprocessing steps for images
+2.3
+Model architecture
+As discussed in the introduction in this research YOLOv5 model is used for feature extraction and
+detection of lung nodules in CT scans. Let have a brief discussion about Yolov5 and its architecture.
+2.3.1
+YOLOv5 for lung nodules localization
+the whole structure of Yolov4 Optimal speed and accuracy of object detection19 is shown in Fig.3
+and YOLOv5 illustration representation shown in Fig.4. the YOLO family of models consists of
+three main components to every single-stage object detector, and YOLOv5 has its own three main
+modules
+Fig. 3: Overview of YOLOv5 building blocks model architecture
+(1) Backbone Figure 3:it’s mostly used to extract the elements of the most significant feature
+from the images that have been provided. Cross Stage Partial Networks(CSP) is the back-
+bone of YOLOv5’s feature extraction, which uses them to extract an image’s most informa-
+tive details
+5
+
+input image
+output
+image
+Gray
+Noise
+Edge
+Filter
+scale
+Removal
+DetectionBackbone: CSPDarknet
+Neck: PANet
+Head: Yolo Layer
+BottleNeckCSP
+Concat
+BottleNeckCSP
+Convlx1
+input image
+UpSample
+Conv3×3 S2
+Conv1×1
+Concat
+BottleNeckCSP
+Final
+BottleNeckCSP
+Concat
+BottleNeckCSP
+Output
+Convl×1
+UpSample
+Conv3x3 S2
+Convl×1
+Concat
+SPP
+BottleNeckCSP
+BottleNeckCSP
+Convl×1
+CSP
+Cross Stage Partial Netword
+Conv
+ Convolutional Layer
+ SPP
+ Spatial pyramid pooling
+Concat
+ Concatenate Function(2) Neck Figure 3: it used to create feature pyramids. Feature pyramids aid models in generaliz-
+ing successfully when it comes to object scaling. It aids in the identification of the same item
+at various scales and dimensions. Feature pyramids are quite valuable and can help models
+perform effectively on data that has never been examined. It’s not only FPN, BiFPN, and
+PANet that are used in feature pyramid models
+(3) Head Figure 3: it has layers that generate predictions from anchor boxes on features and
+generated final output vectors with probabilities, object classes scores, and bounding boxes.,
+YOLOv5 uses the following choices for training20
+Fig. 4: Model detection can be considered a regression problem. The image is divided into S * S grids in
+which bounding boxes are predicted for each grid cell, along with their confidence value
+2.3.2
+Training Model
+During the training and validation process, a total of 270 CT Scan images are used of which 239
+CT Scans are used for training and 41 are used for validations. For training, the Google Colab is
+used which is an online platform for training models. Which provides 16GB GPU free for training.
+The batch size was kept to 16 and the number of epochs was kept to 100. Splitting of data can be
+seen in Fig.5 .
+6
+
+Head
+Bounding Boxes + confidence Score
+Backbone
+Neck
+images
+Extraction of
+Elaboration in
+informative
+Featuer
+Labels
+features
+Pyramids
+S x S Grid on input
+image
+Bounding Boxes + confidence Score
+ Localization of Lung Nodule
+Class Probability
+MapFig. 5: Dataset Splitting Diagram CT Scan images
+3
+Results
+the model had initial leverage to train faster and predict the location of lung nodules and demarcates
+the relevant CT scan regions. before diving into the analysis of the results is necessary to explain
+the statistical machine learning knowledge behind those results, the explanations have been split
+into the following Subsections to explain the whole analysis of the method we use.
+3.1
+Evaluation Metrics
+In this section, we describe Charts of evaluation metrics that got from our experiment. It is known
+to us that, in the computer Aide system, the main part is detecting the object inside the image.
+Common metrics for measuring the performance of classification algorithms such as YOLOv5
+that are based on CNN include, Recall, precision, F-score, mAP, PR curve, F1 curve , IOU,21
+overlapping error, and boundary-based evaluation, the evaluation metrics we used is the mean
+Average Precision (mAP),22 the precision, and F1-Curve. We will briefly explain them in the
+following part. According to the theory of the statistical machine learning , precision is a two-
+category statistical indicator whose formula is .
+Precision : measures how accurate is our predictions was. the percentage of our predictions are
+correct as shown in Fig.8,and following equation1.
+Precision =
+TP
+TP + FP
+(1)
+Recall: measures how much of the true bbox were correctly predicted as shown in the following
+equation.2.
+Recall =
+TP
+TP + FN
+(2)
+7
+
+Total Data
+Distrubtion CT Scan
+270Samples
+Traning
+Validation
+images 239 Samples
+images 41 Samplesmoreover, it is necessary to know TP, FP, and FN in the localization Nodules task.
+(1) True positive (TP): IoU>[formula] (in this work, [formula] takes 0.2) the number of Local-
+ization frames (the same Ground Truth is only calculated once23)
+(2) False positive (FP): the number of check boxes for IoU<=[formula] or the number of re-
+dundant check boxes that detect the same Ground Truth
+(3) False negative (FN): the number of Ground Truths not detected
+the IoU is a measures of the degree of overlap between two boundaries. We use that to measure
+how much our predicted frame overlaps with the ground truth (the actual ground frame) ,the IOU
+is shown with Fig.7 as follows, and the formula is as following Fig.6:
+Fig. 6: Graphical representation of the Intersection over Union (IoU=0.2) calculation on a narrow-band
+imaging. The light blue rectangle represents the ground truth bounding box, while the red rectangle repre-
+sents the model prediction. The IoU is calculated by dividing the overlap area by the total area of union
+after getting familiar with these definitions of statistical learning formulas, we introduce the mAP
+(mean Average Precision). The mAP compares the ground-truth bounding box to the detected box
+and returns a score. The higher the score, the more accurate the model is in its detections.
+F1-score is defined as the harmonic average of precision and recall as shown in figure 10a:
+F1 Score = 2 ∗ Precision ∗ Recall
+Precision + Recall
+(3)
+8
+
+LIDC-IDRI-0003
+LightSpeed16
+1-January-20e6
+Ground truth
+intersection
+0
+area of overlap
+Prediction
+Iou =
+area of union
+Ground truth
+Ground truth
+Prediction
+ST:2.58
+Prediction
+ittleEndianExplicit
+ges: 1/1
+300mA120.00kl
+WL:-600hW:1606(a) the Predicted location of a
+Single Nodule
+(b) the Predicted location of
+tow Nodules in region
+Fig. 7: Example of output Results
+3.2
+Experiment’s Setting
+the set Hyper-parameters of our fine tuning model are shown in Table 1, Our experiment uses
+Pytorch framework deep learning on GPU Tesla K80 by Google open Platform Colab-research .
+Parameters
+Value
+Batch size
+16
+Image size
+416
+Epoch
+145
+Learning rate
+0.01
+Optimizer
+SGD
+Table 1: Parameters and their value.
+3.3
+Experiment’s Result and Analysis
+To check the model’s predictions, and generalizations a few evaluation parameters must be tracked
+during training and validation. There are several criteria to keep in mind while evaluating a box
+loss, Precision, and recall values. The variable box benefits from objectivity and categorization.
+Fig.9 shows all of the graphs that were used for this work. And Figure 10a shows the F1 indicator
+training process for a single category that we want to be detected. The F1 score tends to be 0 with
+increasing confidence . Training and validation box losses are reduced Fig.9, suggesting that the
+model is sound good. the mAP is the abbreviation of median accuracy performances. The high
+number indicates that this parameter is correct 92.27% as shown blow in Fig.8.
+9
+
+THORAXW/OCONTRAST
+LightSpeed1e
+1-January-2000 9:01:09
+Nodules 0.76
+ST:2.502
+ST
+LittleEndianExplicit
+Images:1/1
+265mA120.00kV
+Series:
+WL:
+-600WW:1600LIDC-IDRI-0011
+LightSpeed16
+1-January-200e
+Nodules 0.33
+Nodules 0.78
+ST:2.50SL:
+ST
+LittleEndianExplicit
+Images:1/1
+265mA120.00kV
+Series:
+3000559
+WL:
+-600wW:1600(a) mean Average Precision Evaluation
+(b) Precision Evaluation
+Fig. 8: the important Evaluation Metrics
+(a) the training confidence of object pres-
+ence loss
+(b) the validation confidence of object pres-
+ence loss
+(c) the training bounding box regression
+loss
+(d) the validation bounding box regression
+loss
+Fig. 9: Results of feature extraction training and validation
+Precision is needed to determine how accurate the model forecasts are 92.82% following the figure
+8. Only excellent results may be achieved by using the recall method. the model performance
+showed a good benefit of using Hyper-parameter tuning to make better Learning from data samples
+and generalize good knowledge from distribution can be seen the following Fig.9 . Due to the
+importance of both precision and recall, there is a precision-recall curve the shows the tradeoff
+between the precision and recall values for different thresholds. This curve helps to select the best
+threshold to maximize both metrics, tin the following Fig.10b
+10
+
+metrics/mAP 0.5
+0.8
+0.6
+0.4
+0.2
+Epoch
+0
+0
+20
+40
+60
+80
+100
+120
+140metrics/precision
+0.8
+0.6
+0.4
+0.2
+Epoch
+0
+0
+20
+40
+60
+80
+100
+120
+140train/obj_loss
+0.012
+0.01
+0.008
+0.006
+0.004
+0.002
+Epoch
+0
+0
+20
+40
+60
+80
+100
+120
+140val/obj_loss
+0.012
+0.01
+0.008
+0.006
+0.004
+0.002
+Epoch
+0
+20
+40
+60
+80
+100
+120
+0
+140train/box_loss
+0.12
+0.1
+0.08
+0.06
+0.04
+0.02
+Epoch
+0
+0
+20
+40
+60
+80
+100
+120
+140val/box_loss
+0.1
+0.08
+0.06
+0.04
+0.02
+Epoch
+0
+0
+20
+40
+60
+80
+100
+120
+140(a) F1 indicator training process for single
+category
+(b) Precision — Recall Curve of the valida-
+tion data
+Fig. 10: the important Evaluation Metrics
+4
+Discussion Ana Conclusion
+This research examined at how an AI model can help readers detect viewable lung cancer in Ct
+images. Residents identified more viewable lung cancer when AI was being used as a second
+reader.In this research, the dataset has been collected from LIDC-IDRI. In LIDC-IDRI image col-
+lection, thoracic CT scans with marked-up annotated lesions are included. Yolov5 model is used
+for feature extraction and detection of lung nodules in CT scans.During the training and validation
+process, a total of 270 CT Scan images are used of which 239 CT Scans are used for training and
+41 are used for validations. In this study, the model’s performance was assessed using accuracy,
+precision, and recall. The accuracy metric indicates how well the model recognised both positive
+and negative instances. The precision metric measures how well the model predicts both negative
+and positive cases. The model’s high accuracy, precision, and recall imply that it has a small error
+possibility. Our findings imply that the AI technique assists low experienced individuals in terms
+of recall while benefiting more-experienced audience in terms of precision. Previous research has
+revealed that inexperienced readers are more likely to overlook lung malignancies, particularly le-
+sions with a limited visibility score. In this research LIDC-IDRI dataset is used which have lung
+nodules in it. The purpose of this study was to identify the nodule that were developing in the
+lungs of the participants. It was difficult to find information on lung nodules in medical literature.
+Research in the medical field often use deep learning. Deep learning will be utilised to develop
+an algorithm with the support of previous medical imaging research, according to the findings of
+a literature review. Using over 270 CT images , we were able to classify and identify nodules
+using a deep learning algorithm. Using medical images analysis based on deep neural networks,
+this study found that as much as 92.27% of cancer could be detected. Nodules on radiographs are
+easier to see with its help. Using this technology in the future will help treat illnesses including
+brain tumours and breast cancer.
+5
+Disclosures
+The authors declare that they have no conflict of interest
+6
+Acknowledgments
+We would like to thank our respectful research assistant Moath Alawaqla, for his distinguished role
+of data collection.
+11
+
+1.0
+Nodules
+all classes 0.91 at 0.437
+0.8
+0.6
+0.4 
+0.2
+0.0 +
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Confidence1.0
+Nodules 0.923
+all classes 0.923 mAP@0.5
+0.8
+0.6
+ Precision
+0.4
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Recall7
+Funding
+This work supported by Jordan University of Science and Technology, Irbid-Jordan,
+References
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+

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+Parametric “Non-nested” Discriminants
+for Multiplicities of Univariate Polynomials
+Hoon Hong
+Department of Mathematics, North Carolina State University
+Box 8205, Raleigh, NC 27695, USA
+hong@ncsu.edu
+Jing Yang∗
+SMS–HCIC–School of Mathematics and Physics,
+Center for Applied Mathematics of Guangxi,
+Guangxi Minzu University, Nanning 530006, China
+yangjing0930@gmail.com
+Abstract
+We consider the problem of complex root classification, i.e., finding the conditions on the coefficients
+of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known
+that such conditions can be written as conjunctions of several polynomial equations and one inequation
+in the coefficients. Those polynomials in the coefficients are called discriminants for multiplicities. It
+is well known that discriminants can be obtained by using repeated parametric gcd’s. The resulting
+discriminants are usually nested determinants, that is, determinants of matrices whose entries are deter-
+minants, and so son. In this paper, we give a new type of discriminants which are not based on repeated
+gcd’s. The new discriminants are simpler in that they are non-nested determinants and have smaller
+maximum degrees.
+1
+Introduction
+In this paper, we consider the problem of complex root classification, i.e., finding the conditions on the
+coefficients of a polynomial over the complex field C for every potential multiplicity structure its complex
+roots may have. For example, consider a quintic polynomial F = a5x5 +a4x4 +a3x3 +a2x2 +a1x+a0 where
+ai’s take values over C. We would like to find conditions C0, C1, . . . , C6 on a = (a0, . . . , a5) such that
+multiplicity structure of F =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(1, 1, 1, 1, 1)
+if
+C0 (a) holds
+(2, 1, 1, 1)
+if
+C1 (a) holds
+(2, 2, 1)
+if
+C2 (a) holds
+(3, 1, 1)
+if
+C3 (a) holds
+(3, 2)
+if
+C4 (a) holds
+(4, 1)
+if
+C5 (a) holds
+(5)
+if
+C6 (a) holds
+In general, the problem is stated as follows:
+Problem: For every µ = (µ1, . . . , µm) such that µ1 ≥ . . . ≥ µm > 0 and µ1 +· · ·+µm = n, find a condition
+on the coefficients of a polynomial Fover C of degree n such that the multiplicity structure of F is µ.
+∗Corresponding author.
+1
+arXiv:2301.00315v1  [cs.SC]  1 Jan 2023
+
+The problem is important because many tasks in mathematics, science and engineering can be reduced to
+the problem. Due to its importance, the problem and several related problems have been already carefully
+studied [4, 6, 7, 8, 9, 11].
+The problem can be viewed as a generalization of a well known problem of finding a condition on
+coefficients such that the polynomial has a given number of distinct roots.
+This subproblem has been
+extensively studied. For instance, the subdiscriminant theory provides a complete solution to the subproblem:
+a univariate polynomial of degree n has m distinct roots if and only if its 0-th, . . ., (n − m − 1)-th psd’s
+(i.e., principal subdiscriminant coefficient) vanish and the (n−m)-th psd does not. For details, see standard
+textbooks on computational algebra (e.g., [1]).
+In [11], Yang, Hou and Zeng gave an algorithm to generate conditions for discriminating different mul-
+tiplicity structures of a univariate polynomial (referred as YHZ’s condition hereinafter) by making use of
+repeated gcd computation for parametric polynomials [2, 3, 10]. It is based on a similar idea adopted by
+Gonzalez-Vega et al. [4] for solving the real root classification and quantifier elimination problems by using
+Sturm-Habicht sequences. The conditions produced by these methods are conjunctions of several polynomial
+equations and one inequation on the coefficients. Those polynomials in he coefficients are called discrimi-
+nants for multiplicities. The maximum degree of the discriminants grow exponentially in the degree of F.
+Furthermore, each discriminant is a “nested” determinant, that is, it is a determinant of a matrix whose
+entries are again determinants and so on.
+In [6], the authors developed a new type of multiplicity discriminants to distinguish different multiplicities
+when the number of distinct roots is fixed. The main idea is to convert the multiplicity condition expressed
+as a permanent inequation in roots into a sum of determinants in coefficients. In order to generate conditions
+for all the possible multiplicity structures of a univariate polynomial, one may first use subdiscriminants in
+classical resultant theory to decide the number of distinct complex roots and then add one more inequation
+to discriminate different multiplicity structures with the same number of distinct roots. In the new condition,
+the maximum degree of the discriminants grow linearly in the degree of F, which makes the size of discrim-
+inants significantly smaller. However, the form of resulting discriminants is a sum of many determinants,
+which makes the further analysis (reasoning) difficult.
+The main contribution in this paper is to provide a new type of discriminants, which are non-nested
+determinants and whose max degrees are smaller than those in the previous methods.
+The method is
+based on a significantly different theory and techniques from the previous methods (which are essentially
+based on repeated parametric gcd or subdiscriminant theory). The new condition is given by a newly devised
+multiplicity discriminant in coefficients for every potential multiplicity vector of a given degree, which can be
+viewed as a generalization of subdiscriminant theory to higher order derivatives. To build up the connection
+between the new discriminants and multiple roots, we first convert it into the ratio of two determinants
+in terms of generic roots (without considering the multiplicities). Then by making use of the connection
+between divided difference with multiple nodes and the derivatives of higher orders at the nodes, we integrate
+the multiplicity information into the expression and convert it into an expression in terms of multiple roots.
+After careful manipulation, it is shown that the new discriminant can capture the multiplicity information.
+The paper is structured as follows. In Section 2, we first present the problem to be solved in a formal
+way. In Section 3, we give a precise statement of the main result of the paper (Theorem 9). Then a proof
+of Theorem 9 is provided in Section 4. The proof is long thus we divide the proof into three subsections
+which are interesting on their own.
+In Section 5, we compare the form and size of polynomials in the
+multiplicity-discriminant condition in Theorem 9 and those given by previous works.
+2
+Problem
+Definition 1 (Multiplicity vector). Let F ∈ C [x] with m distinct complex roots, say r1, . . . , rm, with mul-
+tiplicities µ1, . . . , µm respectively. Without losing generality, we assume that µ1 ≥ · · · ≥ µm > 0. Then the
+multiplicity vector of F, written as mult (F), is defined by
+mult (F) = (µ1, . . . , µm)
+2
+
+Example 2. Let F = x5 − 5x4 + 7x3 + x2 − 8x + 4. Then mult (F) = (2, 2, 1), since it can be verified that
+F = (x − 1)2 (x + 1)1 (x − 2)2. Note that the multiplicity vector is a partition of 5, which is the degree of F.
+Definition 3 (Potential multiplicity vectors). Let n be a positive integer. Let M(n) stand for the set of all
+the potential multiplicity vectors of polynomials of degree n, equivalently, the set of all partitions of n, that
+is,
+M(n) = {(µ1, . . . , µm) : µ1 + · · · + µm = n, µ1 ≥ · · · ≥ µm > 0}
+Example 4. M (5) = { (1, 1, 1, 1, 1) ,
+(2, 1, 1, 1) , (2, 2, 1) , (3, 1, 1) , (3, 2) , (4, 1) , (5) }.
+Problem 5 (Parametric multiplicity problem). The parametric multiplicity problem is stated as:
+In : n, a positive integer standing for the polynomial of degree n with parametric coefficients a, that is,
+F =
+n
+�
+i=0
+aixi where an ̸= 0
+Out: For each µ ∈ M(n), find a condition Cµ on a such that mult (F) = µ.
+3
+Main Result
+Definition 6 (Determinant polynomial). Consider a vector of univariate polynomials
+P =
+�
+��
+P0
+...
+Pk
+�
+�� ∈ C[x]k+1
+where deg Pi ≤ k and Pi = �
+0≤j≤k aijxj. The coefficient matrix of P, written as C (P) , is defined by
+C (P) = coef (P) =
+�
+��
+coef (P0)
+...
+coef (Pk)
+�
+�� =
+�
+��
+a0k
+· · ·
+a00
+...
+...
+akk
+· · ·
+ak0
+�
+��
+The determinant polynomial of P, written as dp (P) , is defined by
+dp (P) = |C (P) |
+Definition 7 (Multiplicity Discriminant). Let F = �n
+i=0 aixi where an ̸= 0. Let γ = (γ1, . . . , γs) ∈ M (n).
+The the γ-discriminant of F, written as D (γ) , is defined by
+D (γ) = 1
+an
+dp
+�
+��������������������
+F (0)xγ0−1
+...
+F (0)x0
+F (1)xγ1−1
+...
+F (1)x0
+...
+F (s)xγs−1
+...
+F (s)x0
+�
+��������������������
+where γ0 is the smallest so that the above matrix is square. It is straightforward to show that γ0 = γ1 − 1.
+3
+
+Example 8. Let n = 5 and F = �n
+i=0 aixi and an ̸= 0. Then
+D (5)
+=
+dp
+�
+�������������
+F (0)x3
+F (0)x2
+F (0)x1
+F (0)x0
+F (1)x4
+F (1)x3
+F (1)x2
+F (1)x1
+F (1)x0
+�
+�������������
+=
+1
+a5
+������������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+������������������
+D (4, 1)
+=
+dp
+�
+�����������
+F (0)x2
+F (0)x1
+F (0)x0
+F (1)x3
+F (1)x2
+F (1)x1
+F (1)x0
+F (2)x0
+�
+�����������
+=
+1
+a5
+����������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5 · 4a5
+4 · 3a4
+3 · 2a3
+2 · 1a2
+����������������
+D (3, 2)
+=
+dp
+�
+���������
+F (0)x1
+F (0)x0
+F (1)x2
+F (1)x1
+F (1)x0
+F (2)x1
+F (2)x0
+�
+���������
+=
+1
+a5
+��������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5 · 4a5
+4 · 3a4
+3 · 2a3
+2 · 1a2
+5 · 4a5
+4 · 3a4
+3 · 2a3
+2 · 1a2
+��������������
+D (3, 1, 1)
+=
+dp
+�
+���������
+F (0)x1
+F (0)x0
+F (1)x2
+F (1)x1
+F (1)x0
+F (2)x0
+F (3)x0
+�
+���������
+=
+1
+a5
+��������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5 · 4a5
+4 · 3a4
+3 · 2a3
+2 · 1a2
+5 · 4 · 3a5
+4 · 3 · 2a4
+3 · 2 · 1a3
+��������������
+D (2, 2, 1)
+=
+dp
+�
+�������
+F (0)x0
+F (1)x1
+F (1)x0
+F (2)x1
+F (2)x0
+F (3)x0
+�
+�������
+=
+1
+a5
+������������
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5 · 4a5
+4 · 3a4
+3 · 2a3
+2 · 1a2
+5 · 4a5
+4 · 3a4
+3 · 2a3
+2 · 1a2
+5 · 4 · 3a5
+4 · 3 · 2a4
+3 · 2 · 1a3
+������������
+D (2, 1, 1, 1)
+=
+dp
+�
+�������
+F (0)x0
+F (1)x1
+F (1)x0
+F (2)x0
+F (3)x0
+F (4)x0
+�
+�������
+=
+1
+a5
+������������
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+1a1
+5a5
+4a4
+3a3
+2a2
+1a1
+5 · 4a5
+4 · 3a4
+3 · 2a3
+2 · 1a2
+5 · 4 · 3a5
+4 · 3 · 2a4
+3 · 2 · 1a3
+5 · 4 · 3 · 2a5
+4 · 3 · 2 · 1a4
+������������
+D (1, 1, 1, 1, 1)
+=
+dp
+�
+�����
+F (1)x0
+F (2)x0
+F (3)x0
+F (4)x0
+F (5)x0
+�
+�����
+=
+1
+a5
+����������
+5a5
+4a4
+3a3
+2a2
+1a1
+5 · 4a5
+4 · 3a4
+3 · 2a3
+2 · 1a2
+5 · 4 · 3a5
+4 · 3 · 2a4
+3 · 2 · 1a3
+5 · 4 · 3 · 2a5
+4 · 3 · 2 · 1a4
+5 · 4 · 3 · 2 · 1a5
+����������
+4
+
+Note that the last one D (1, 1, 1, 1, 1) = 5544332211a4
+5. Since a5 ̸= 0, we see that D (1, 1, 1, 1, 1) ̸= 0.
+Theorem 9 (Main Result). Let F = �n
+i=0 aixi
+where an ̸= 0. Let M(n) =
+�
+µ0, µ1, . . . , µp
+�
+where the
+entries are ordered in the lexicographically increasing order, that is, µ0 ≺lex µ1 ≺lex · · · ≺lex µp. Then we
+have the following conditions for the multiplicity vectors.
+mult(F) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+µ0
+if
+D
+�
+µp
+�
+̸= 0
+µ1
+else if
+D
+�
+µp−1
+�
+̸= 0
+...
+...
+...
+̸= 0
+µp
+else if
+D (µ0)
+̸= 0
+Equivalently,
+mult(F) = µi
+⇐⇒
+D
+�
+µp
+�
+= · · · = D
+�
+µp−i−1
+�
+= 0 ∧ D
+�
+µp−i
+�
+̸= 0
+Example 10. We have the following condition for each multiplicity vector for degree 5.
+mult(F) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(1, 1, 1, 1, 1)
+if
+D (5)
+̸= 0
+(2, 1, 1, 1)
+else if
+D (4, 1)
+̸= 0
+(2, 2, 1)
+else if
+D (3, 2)
+̸= 0
+(3, 1, 1)
+else if
+D (3, 1, 1)
+̸= 0
+(3, 2)
+else if
+D (2, 2, 1)
+̸= 0
+(4, 1)
+else if
+D (2, 1, 1, 1)
+̸= 0
+(5)
+else if
+D (1, 1, 1, 1, 1)
+̸= 0
+Equivalently, for instance,
+mult(F) = (2, 2, 1)
+⇐⇒
+D (5) = D (4, 1) = 0 ∧ D (3, 2) ̸= 0
+Remark 11.
+1. Note that µ0 = (1, . . . , 1) and
+D (µ0) = 1
+an
+���������
+nan
+· · ·
+1a1
+n (n − 1) an
+· · ·
+2 · 1a2
+...
+...
+n (n − 1) · · · 1an
+���������
+=
+n
+�
+i=1
+ii · an−1
+n
+̸= 0
+Hence the last condition is always satisfied and there is no need to check the condition.
+2. Note that µi and µp−i are conjugates of each other.
+4
+Proof of the Main Theorem
+Here is a high level view of the proof.
+We start with converting D (µ) into the equivalent symmetric
+polynomials in generic roots (though displayed as a ratio of two determinants) which is easier to embed the
+multiplicity information. Then by making use of the connection between divided difference with multiple
+nodes and the derivatives of higher orders at the nodes, we convert the expression in generic roots to that
+in distinct roots with multiplicity information integrated. The theorem will be proved by eliminating the
+entries in the determinantal expression obtained from the second stage which may vanish under the given
+multiplicity structure.
+5
+
+4.1
+Multiplicity discriminant in terms of roots
+We first understand what the multiplicity discriminants look like in terms of roots. .
+Notation 12. V (α1, . . . , αn) :=
+�������
+αn−1
+1
+· · ·
+αn−1
+n
+...
+...
+α0
+1
+· · ·
+α0
+n
+�������
+Lemma 13 (Multiplicity discriminant in generic roots). Let F = an(x−α1) · · · (x−αn) and γ = (γ1, . . . , γs) ∈
+M(n). Then
+D(γ) =
+aγ1−2
+n
+·
+������������������
+F (1)(α1)αγ1−1
+1
+· · ·
+F (1)(αn)αγ1−1
+n
+...
+...
+F (1)(α1)α0
+1
+· · ·
+F (1)(αn)α0
+n
+...
+...
+F (s)(α1)αγs−1
+1
+· · ·
+F (s)(αn)αγs−1
+n
+...
+...
+F (s)(α1)α0
+1
+· · ·
+F (s)(αn)α0
+n
+������������������
+V (α1, . . . , αn)
+(1)
+Proof.
+1. Since γ1 ≥ · · · ≥ γs and γ0 = γ1 − 1, we have
+deg(F (0)xn−2) > · · · > deg(F (0)xγ1−1) > max(F (0)xγ0−1, F (1)xγ1−1, . . . , F (s)xγs−1)
+Thus
+D(γ) = 1
+an
+dp
+�
+��������������������
+F (0)xγ1−2
+...
+F (0)x0
+F (1)xγ1−1
+...
+F (1)x0
+...
+F (s)xγs−1
+...
+F (s)x0
+�
+��������������������
+= 1
+an
+· aγ1−n
+n
+dp
+�
+���������������������������
+F (0)xn−2
+...
+F (0)xγ1−1
+F (0)xγ1−2
+...
+F (0)x0
+F (1)xγ1−1
+...
+F (1)x0
+...
+F (s)xγs−1
+...
+F (s)x0
+�
+���������������������������
+= aγ1−n−1
+n
+dp
+�
+��������������������
+F (0)xn−2
+...
+F (0)x0
+F (1)xγ1−1
+...
+F (1)x0
+...
+F (s)xγs−1
+...
+F (s)x0
+�
+��������������������
+2. Now we recall the following result from [6] which is the key for proving the lemma. Let G1, . . . , Gn ∈
+C [x]2n−2 where C [x]2n−2 consists of all the polynomials in x with degree no greater than 2n−2. Then
+dp
+�
+���������
+F (0)xn−2
+...
+F (0)x0
+G1
+...
+Gn
+�
+���������
+=
+an−1
+n
+·
+�������
+G1(α1)
+· · ·
+G1(αn)
+...
+...
+Gn(α1)
+· · ·
+Gn (αn)
+�������
+V (α1, . . . , αn)
+(2)
+6
+
+3. After specializing G1, . . . , Gn in (2) with F (1)xγ1−1, . . . , F (1)x0, . . . , F (s)xγs−1, . . . , F (s)x0, respectively,
+we have
+D(γ) = aγ1−n−1
+n
+·
+an−1
+n
+·
+�����������������
+�
+F (1)xγ1−1�
+(α1)
+· · ·
+�
+F (1)xγ1−1�
+(αn)
+...
+...
+�
+F (1)x0�
+(α1)
+· · ·
+�
+F (1)x0�
+(αn)
+...
+...
+�
+F (s)xγs−1�
+(α1)
+· · ·
+�
+F (s)xγs−1�
+(αn)
+...
+...
+�
+F (s)x0�
+(α1)
+· · ·
+�
+F (s)x0�
+(αn)
+�����������������
+V (α1, . . . , αn)
+which can be easily simplified into (1).
+Remark 14. It is very important to note that the right hand side is a polynomial function in α1, . . . , αn,
+even though written as a rational function, since the numerator is exactly divisible by the denominator.
+Hence the above definition should be read as follows:
+1. Treating α1, . . . , αn as distinct indeterminates, carry out the exact division obtaining a polynomial.
+2. Treating α1, . . . , αn as numbers, evaluate the resulting polynomial.
+Lemma 15 (Multiplicity discriminant in multiple roots). Let F be of degree n with m distinct roots
+r1, . . . , rm, of multiplicities µ1, . . . , µm, that is µ1 + · · · + µm = n. Let γ = (γ1, . . . , γs) ∈ Γ(n). Then
+we have
+D(γ) =
+c ·
+�����������������
+(F (1)xγ1−1)(0)(r1) · · · (F (1)xγ1−1)(µ1−1)(r1) · · · · · · (F (1)xγ1−1)(0)(rm) · · · (F (1)xγ1−1)(µm−1)(rm)
+...
+...
+...
+...
+(F (1)x0)(0)(r1)
+· · · (F (1)x0)(µ1−1)(r1)
+· · · · · · (F (1)x0)(0)(rm)
+· · · (F (1)x0)(µm−1)(rm)
+...
+...
+...
+...
+(F (s)xγs−1)(0)(r1) · · · (F (s)xγs−1)(µ1−1)(r1) · · · · · · (F (s)xγs−1)(0)(rm) · · · (F (s)xγs−1)(µm−1)(rm)
+...
+...
+...
+...
+(F (s)x0)(0)(r1)
+· · · (F (s)x0)(µ1−1)(r1)
+· · · · · · (F (s)x0)(0)(rm)
+· · · (F (s)x0)(µm−1)(rm)
+�����������������
+�
+1≤i<j≤m
+(ri − rj)µiµj
+(3)
+where c = ±1
+� ��m
+i=1
+�µi−1
+j=0 j!
+�
+· aγ1−2
+n
+.
+Proof.
+1. Let F = an(x − α1) · · · (x − αn). When α1, . . . , αn are treated as numbers, without loss of generality,
+we may assume that α1, . . . , αn are grouped into m sets as follows:
+S1 :=
+{α1
+· · ·
+· · ·
+· · ·
+· · ·
+αµ1}
+S2 :=
+{αµ1+1
+· · ·
+· · ·
+· · ·
+αµ1+µ2}
+...
+Sm :=
+{αµ1+···+µm−1+1
+· · ·
+αµ1+···+µm}
+where elements in Si are all equal to ri.
+7
+
+2. Recall that
+D(γ) = aγ1−2
+n
+·
+���������������������
+(F (1)xγ1−1)(α1)
+· · ·
+(F (1)xγ1−1)(αn)
+...
+...
+(F (1)x0)(α1)
+· · ·
+(F (1)x0)(αn)
+...
+...
+...
+...
+(F (s)xγs−1)(α1)
+· · ·
+(F (s)xγs−1)(αn)
+...
+...
+(F (s)x0)(α1)
+· · ·
+(F (s)x0)(αn)
+���������������������
+�
+V (α1, . . . , αn)
+Next we will treat α1, . . . , αn as indeterminates and carry out the exact division so that difference
+between the collapsed αi’s do not appear in the denominator.
+3. For the sake of simplicity, we use the follow shorthand notion:
+F :=
+�
+F (1)xγ1−1, . . . , F (1)x0, . . . , F (s)xγs−1, . . . , F (s)x0�T
+4. Let P[x1, . . . , xi] denote the (i − 1)th divided difference of P ∈ C[x] at x1, . . . , xi and let
+F [α1, . . . , αi] :=
+�
+(F (1)xγ1−1)[α1, . . . , αi], . . . , (F (1)x0)[α1, . . . , αi],
+. . . . . . , (F (s)xγs−1)[α1, . . . , αi], . . . , (F (s)x0)[α1, . . . , αi]
+�T
+5. It follows that
+D(γ) = aγ1−2
+n
+·
+�� F (α1)
+· · ·
+F (αn)
+��
+V (α1, . . . , αn)
+= aγ1−2
+n
+·
+�� F [α1]
+· · ·
+F [αµ1]
+F [αµ1+1]
+· · ·
+F [αn]
+��
+�
+αi,αj∈S1
+j−i>0
+(αi − αj)
+�
+αi,αj /∈S1
+j−i>0
+(αi − αj)
+�
+αi∈S1
+αj /∈S1
+(αi − αj)
+= ±aγ1−2
+n
+·
+�� F [α1]
+F [α1, α2]
+· · ·
+F [αµ1−1, αµ1]
+F [αµ1+1]
+· · ·
+F [αn]
+��
+�
+αi,αj∈S1
+j−i>1
+(αi − αj)
+�
+αi,αj /∈S1
+j−i>0
+(αi − αj)
+�
+αi∈S1
+αj /∈S1
+(αi − αj)
+= ±aγ1−2
+n
+·
+�� F [α1]
+F [α1, α2]
+F [α1, α2, α3]
+· · ·
+F [αµ1−2,µ1−1, αµ1]
+F [αµ1+1]
+· · ·
+F [αn]
+��
+�
+αi,αj∈S1
+j−i>2
+(αi − αj)
+�
+αi,αj /∈S1
+j−i>0
+(αi − αj)
+�
+αi∈S1
+αj /∈S1
+(αi − αj)
+...
+= ±aγ1−2
+n
+·
+�� F [α1]
+F [α1, α2]
+· · ·
+F [α1, . . . , αµ1]
+F (αµ1+1)
+· · ·
+F (αn)
+��
+�
+αi,αj /∈S1
+j−i>0
+(αi − αj)
+�
+αi∈S1
+αj /∈S1
+(αi − αj)
+8
+
+6. Repeating the procedure for αj’s in each Si for i = 2, . . . , m successively, we get
+D(γ) = ±aγ1−2
+n
+·
+�� F [α1]
+· · ·
+F [α1, . . . , αµ1]
+· · ·
+· · ·
+F [αµ1+···+µm−1+1]
+· · ·
+F [αµ1+···+µm−1+1, . . . , αn]
+��
+�
+1≤i<j≤m
+�
+αp∈Si
+αq∈Sj
+(αp − αq)
+7. Now we substitute α1 = · · · = αµ1 = r1, . . . , αµ1+···+µm−1+1 = · · · = αn = rm into D(γ) and obtain
+D(γ) = ±aγ1−2
+n
+·
+�� F [r1]
+· · ·
+F [r1, . . . , r1]
+· · ·
+· · ·
+F [rm]
+· · ·
+F [rm, . . . , rm]
+��
+�
+1≤i<j≤m
+(ri − rj)µiµj
+(4)
+8. Recall that for any given polynomial P ∈ C[x],
+P[ri, . . . , ri
+�
+��
+�
+k ri’s
+] = P (k−1)(ri)
+(k − 1)!
+Hence
+F [ri, . . . , ri
+�
+��
+�
+k ri’s
+] =
+�(F (1)xγ1−1)(k−1)(ri)
+(k − 1)!
+, . . . , (F (1)x0)(k−1)(ri)
+(k − 1)!
+, . . . , (F (s)xγs−1)(k−1)(ri)
+(k − 1)!
+, . . . , (F (s)x0)(k−1)(ri)
+(k − 1)!
+�T
+(5)
+9. Substituting (5) into (4) , we have
+D(γ) = ±aγ1−2
+n
+·
+��������������������
+(F (1)xγ1−1)(0)(r1)
+0!
+· · ·
+(F (1)xγ1−1)(µ1−1)(r1)
+(µ1−1)!
+· · · · · ·
+(F (1)xγ1−1)(0)(rm)
+0!
+· · ·
+(F (1)xγ1−1)(µm−1)(rm)
+(µm−1)!
+...
+...
+...
+...
+(F (1)x0)(0)(r1)
+0!
+· · ·
+(F (1)x0)(µ1−1)(r1)
+(µ1−1)!
+· · · · · ·
+(F (1)x0)(0)(rm)
+0!
+· · ·
+(F (1)x0)(µm−1)(rm)
+(µm−1)!
+...
+...
+...
+...
+(F (s)xγs−1)(0)(r1)
+0!
+· · ·
+(F (s)xγs−1)(µ1−1)(r1)
+(µ1−1)!
+· · · · · ·
+(F (s)xγs−1)(0)(rm)
+0!
+· · ·
+(F (s)xγs−1)(µm−1)(rm)
+(µm−1)!
+...
+...
+...
+...
+(F (s)x0)(0)(r1)
+0!
+· · ·
+(F (s)x0)(µ1−1)(r1)
+(µ1−1)!
+· · · · · ·
+(F (s)x0)(0)(rm)
+0!
+· · ·
+(F (s)x0)(µm−1)(rm)
+(µm−1)!
+��������������������
+�
+1≤i<j≤m
+(ri − rj)µiµj
+=
+c ·
+�����������������
+(F (1)xγ1−1)(0)(r1) · · · (F (1)xγ1−1)(µ1−1)(r1) · · · · · · (F (1)xγ1−1)(0)(rm) · · · (F (1)xγ1−1)(µm−1)(rm)
+...
+...
+...
+...
+(F (1)x0)(0)(r1)
+· · · (F (1)x0)(µ1−1)(r1)
+· · · · · · (F (1)x0)(0)(rm)
+· · · (F (1)x0)(µm−1)(rm)
+...
+...
+...
+...
+(F (s)xγs−1)(0)(r1) · · · (F (s)xγs−1)(µ1−1)(r1) · · · · · · (F (s)xγs−1)(0)(rm) · · · (F (s)xγs−1)(µm−1)(rm)
+...
+...
+...
+...
+(F (s)x0)(0)(r1)
+· · · (F (s)x0)(µ1−1)(r1)
+· · · · · · (F (s)x0)(0)(rm)
+· · · (F (s)x0)(µm−1)(rm)
+�����������������
+�
+1≤i<j≤m
+(ri − rj)µiµj
+where c = ±1
+� ��m
+i=1
+�µi−1
+j=0 j!
+�
+· aγ1−2
+n
+.
+9
+
+4.2
+Connection between the multiplicity discriminants and multiplicity vectors
+We first decompile Theorem 9 and identify the two essential ingredients therein, which are re-stated as
+Lemmas 16 and 17 below. From now on, we will use γ to denote the conjugate of γ ∈ M(n).
+Lemma 16. Let mult(F) = µ. Then D(¯µ) ̸= 0.
+Proof. In order to convey the main underlying ideas effectively, we will show the proof for a particular case
+first. After that, we will generalize the ideas to arbitrary cases.
+Particular case: Consider the case n = 5 and mult(F) = µ = (3, 2).
+1. Assume that r1 and r2 are the two distinct roots with multiplicities 3 and 2, respectively. In other
+words, F = a5(x − r1)3(x − r2)2.
+2. Let γ = ¯µ. Then
+γ1 = #{µj : µj ≥ 1} = 2,
+γ2 = #{µj : µj ≥ 2} = 2,
+γ3 = #{µj : µj ≥ 3} = 1
+Thus γ = (2, 2, 1).
+3. By Lemma 15,
+D(γ) =
+c ·
+����������
+(F (1)x1)(0)(r1)
+(F (1)x1)(1)(r1)
+(F (1)x1)(2)(r1)
+(F (1)x1)(0)(r2)
+(F (1)x1)(1)(r2)
+(F (1)x0)(0)(r1)
+(F (1)x0)(1)(r1)
+(F (1)x0)(2)(r1)
+(F (1)x0)(0)(r2)
+(F (1)x0)(1)(r2)
+(F (2)x1)(0)(r1)
+(F (2)x1)(1)(r1)
+(F (2)x1)(2)(r1)
+(F (2)x1)(0)(r2)
+(F (2)x1)(1)(r2)
+(F (2)x0)(0)(r1)
+(F (2)x0)(1)(r1)
+(F (2)x0)(2)(r1)
+(F (2)x0)(0)(r2)
+(F (2)x0)(1)(r2)
+(F (3)x0)(0)(r1)
+(F (3)x0)(1)(r1)
+(F (3)x0)(2)(r1)
+(F (3)x0)(0)(r2)
+(F (3)x0)(1)(r2)
+����������
+(r1 − r2)6
+where
+c = ±(0! · 1! · 2!) · (0! · 1!) · a0
+5 = ±2
+4. Since
+F (i)(r1)
+� = 0,
+for i = 0, 1, 2
+̸= 0,
+for i = 3
+F (i)(r2)
+� = 0,
+for i = 0, 1
+̸= 0,
+for i = 2
+we immediately know
+(F (1)x1)(0)(r1) = 0 (F (1)x1)(1)(r1) = 0 (F (1)x1)(2)(r1) = F (3)(r1)r1
+1 (F (1)x1)(0)(r2) = 0 (F (1)x1)(1)(r2) = F (2)(r2)r1
+2
+(F (1)x0)(0)(r1) = 0 (F (1)x0)(1)(r1) = 0 (F (1)x0)(2)(r1) = F (3)(r1)r0
+1 (F (1)x0)(0)(r2) = 0 (F (1)x0)(1)(r2) = F (2)(r2)r0
+2
+(F (2)x1)(0)(r1) = 0 (F (2)x1)(1)(r1) = F (3)(r1)r1
+1
+(F (2)x1)(0)(r2) = F (2)(r2)r1
+2
+(F (2)x0)(0)(r1) = 0 (F (2)x0)(1)(r1) = F (3)(r1)r0
+1
+(F (2)x0)(0)(r2) = F (2)(r2)r0
+2
+(F (3)x0)(0)(r1) = F (3)(r1)r0
+1
+5. Therefore,
+D(γ) =
+c ·
+����������
+0
+0
+F (3)(r1)r1
+1
+0
+F (2)(r2)r1
+2
+0
+0
+F (3)(r1)r0
+1
+0
+F (2)(r2)r0
+2
+0
+F (3)(r1)r1
+1
+·
+F (2)(r2)r1
+2
+·
+0
+F (3)(r1)r0
+1
+·
+F (2)(r2)r0
+2
+·
+F (3)(r1)r0
+1
+·
+·
+·
+·
+����������
+(r1 − r2)6
+10
+
+6. By rearranging the columns of the determinant in the numerator, we have
+D(γ) =
+c ·
+����������
+F (3)(r1)r1
+1
+F (2)(r2)r1
+2
+F (3)(r1)r0
+1
+F (2)(r2)r0
+2
+F (3)(r1)r1
+1
+F (2)(r2)r1
+2
+·
+·
+F (3)(r1)r0
+1
+F (2)(r2)r0
+2
+·
+·
+F (3)(r1)r0
+1
+·
+·
+·
+·
+����������
+(r1 − r2)6
+=
+c ·
+������
+M1
+M2
+·
+M3
+·
+·
+������
+(r1 − r2)6
+where
+M1 =
+� F (3)(r1)r1
+1
+F (2)(r2)r1
+2
+F (3)(r1)r0
+1
+F (2)(r2)r0
+2
+�
+M2 =
+� F (3)(r1)r1
+1
+F (2)(r2)r1
+2
+F (3)(r1)r0
+1
+F (2)(r2)r0
+2
+�
+M3 =
+�
+F (3)(r1)r0
+1
+�
+7. Obviously,
+D(γ) = ±c · |M1| · |M2| · |M3|
+(r1 − r2)6
+We only need to show that Mi ̸= 0 for i = 1, 2, 3. The claim follows from the following observations:
+|M1| = F (3)(r1)F (2)(r2)V (r1, r2) ̸= 0
+|M2| = F (3)(r1)F (2)(r2)V (r1, r2) ̸= 0
+|M3| = F (3)(r1)V (r1) ̸= 0
+The proof is completed.
+Arbitrary case. Now we generalize the above ideas to arbitrary cases.
+1. Let µ = (µ1, . . . , µm). Assume that r1, . . . , rm are the m distinct roots with multiplicities µ1, . . . , µm
+respectively. In other words, F = an(x − r1)µ1 · · · (x − rm)µm.
+2. Let γ = ¯µ = (γ1, . . . , γs). In other words, γi = #{µj : µj ≥ i}. Note that s = µ1 since µ1 ≥ · · · ≥ µm.
+3. Recall that
+D(γ) =
+c ·
+�����������������
+(F (1)xγ1−1)(0)(r1)
+· · · (F (1)xγ1−1)(µ1−1)(r1)
+· · · · · · (F (1)xγ1−1)(0)(rm)
+· · · (F (1)xγ1−1)(µm−1)(rm)
+...
+...
+...
+...
+(F (1)x0)(0)(r1)
+· · · (F (1)x0)(µ1−1)(r1)
+· · · · · · (F (1)x0)(0)(rm)
+· · · (F (1)x0)(µm−1)(rm)
+...
+...
+...
+...
+(F (µ1)xγµ1−1)(0)(r1) · · · (F (µ1)xγµ1−1)(µ1−1)(r1) · · · · · · (F (µ1)xγµ1−1)(0)(rm) · · · (F (µ1)xγµ1−1)(µm−1)(rm)
+...
+...
+...
+...
+(F (µ1)x0)(0)(r1)
+· · · (F (µ1)x0)(µ1−1)(r1)
+· · · · · · (F (µ1)x0)(0)(rm)
+· · · (F (µ1)x0)(µm−1)(rm)
+�����������������
+�
+1≤i<j≤m
+(ri − rj)µiµj
+(6)
+where c = ±1
+� ��m
+i=1
+�µi−1
+j=0 j!
+�
+· aγ1−2
+n
+.
+11
+
+4. Since for k = 1, . . . , m,
+F (i)(rk)
+� = 0,
+for i < µk
+̸= 0,
+for i = µk
+we immediately know
+(F (1)xj)(i)(rk)
+� = 0,
+for i < µk − 1
+= F (µk)(rk)rj
+k,
+for i = µk − 1
+where µk − 1 ≥ 0
+...
+(F (ℓ)xj)(i)(rk)
+� = 0,
+for i < µk − ℓ
+= F (µk)(rk)rj
+k,
+for i = µk − ℓ
+where µk − ℓ ≥ 0
+(7)
+...
+(F (µ1)xj)(i)(rk)
+� = 0,
+for i < µk − µ1
+= F (µk)(rk)rj
+k,
+for i = µk − µ1
+where µk − µ1 ≥ 0
+5. Plugging (7) into (6), we have
+D(γ) =
+c ·
+������������������������������
+0
+· · ·
+0
+F (µ1)(r1)rγ1−1
+1
+· · ·
+0
+· · ·
+0
+F (µi)(ri)rγ1−1
+i
+· · ·
+...
+...
+...
+...
+...
+...
+0
+· · ·
+0
+F (µ1)(r1)r0
+1
+· · ·
+0
+· · ·
+0
+F (µi)(ri)r0
+i
+· · ·
+0
+· · · F (µ1)(r1)rγ2−1
+1
+·
+· · ·
+0
+· · · F (µi)(ri)rγ2−1
+i
+·
+· · ·
+...
+...
+...
+...
+...
+...
+0
+· · · F (µ1)(r1)r0
+1
+·
+· · ·
+0
+· · · F (µi)(ri)r0
+i
+·
+· · ·
+...
+...
+...
+...
+...
+...
+0
+·
+·
+· · · F (µi)(ri)r
+γµi −1
+i
+·
+·
+· · ·
+...
+...
+...
+...
+...
+...
+0
+·
+·
+· · · F (µi)(ri)r0
+i
+·
+·
+· · ·
+...
+...
+...
+...
+...
+...
+F (µ1)(r1)r
+γµ1 −1
+1
+· · ·
+·
+·
+· · ·
+·
+· · ·
+·
+·
+· · ·
+...
+...
+...
+...
+...
+...
+F (µ1)(r1)r0
+1
+·
+·
+· · ·
+·
+· · ·
+·
+·
+· · ·
+������������������������������
+�
+1≤i<j≤m
+(ri − rj)µiµj
+6. By rearranging the columns of the determinant in the numerator, we have
+D(γ) = ±
+c ·
+�����������������������
+F (µ1)(r1)rγ1−1
+1
+· · · F (µm)(rm)rγ1−1
+m
+...
+...
+F (µ1)(r1)r0
+1
+· · · F (µm)(rm)r0
+m
+F (µ1)(r1)rγ2−1
+1
+· · · F (µγ2 )(rγ2)rγ2−1
+γ2
+·
+· · ·
+·
+...
+...
+...
+...
+F (µ1)(r1)r0
+1
+· · · F (µγ2 )(rγ2)r0
+γ2
+·
+· · ·
+·
+· · · · ·
+·
+· · ·
+·
+·
+· · ·
+·
+...
+...
+...
+...
+...
+...
+· · · · ·
+·
+· · ·
+·
+·
+· · ·
+·
+F (µ1)(r1)r
+γµ1 −1
+1
+· · · F
+(µγµ1
+)(rγµ1 )r
+γµ1 −1
+γµ1
+· · · · ·
+·
+· · ·
+·
+·
+· · ·
+·
+...
+...
+...
+...
+...
+...
+...
+...
+F (µ1)(r1)r0
+1
+· · · F
+(µγµ1
+)(rγµ1 )r0
+γµ1
+· · · · ·
+·
+· · ·
+·
+·
+· · ·
+·
+�����������������������
+�
+1≤i<j≤m
+(ri − rj)µiµj
+12
+
+= ±
+c ·
+�����������
+M1
+M2
+·
+...
+·
+Mµ1
+· · ·
+·
+·
+�����������
+�
+1≤i<j≤m
+(ri − rj)µiµj
+where
+Mi =
+�
+��
+F (µ1)(r1)rγi−1
+1
+· · ·
+F (µγi)(rγi)rγi−1
+γi
+...
+...
+F (µ1)(r1)r0
+1
+· · ·
+F (µγi)(rγi)r0
+γi
+�
+��
+for i = 1, . . . , µ1. Then
+D(γ) = ±
+c · |M1| · · ·
+��Mµ1
+��
+�
+1≤i<j≤m
+(ri − rj)µiµj
+7. It only remains to show that Mi ̸= 0. The claim follows from the following observations:
+|Mi| =
+�
+�
+γi
+�
+j=1
+F (µj)(rj)
+�
+� V (r1, . . . , rγi) ̸= 0 for i = 1, . . . , µ1
+The proof is completed.
+Lemma 17. Let mult(F) = µ. Then D(λ) = 0 for any λ such that ¯µ ≺lex λ.
+Proof. In order to convey the main underlying ideas, we will show the proof for a particular case first. After
+that, we will generalize the ideas to arbitrary cases.
+Particular case: Consider the case n = 5 and mult(F) = µ = (3, 2). Thus ¯µ = (2, 2, 1). Let λ = (3, 1, 1).
+Obviously, ¯µ ≺lex λ. We will show that D(λ) = 0.
+1. Assume that r1 and r2 are the two distinct roots with multiplicities 3 and 2, respectively. In other
+words, F = a5(x − r1)3(x − r2)2.
+2. By Lemma 15,
+D(λ) =
+c ·
+����������
+(F (1)x2)(0)(r1)
+(F (1)x2)(1)(r1)
+(F (1)x1)(2)(r1)
+(F (1)x2)(0)(r2)
+(F (1)x2)(1)(r2)
+(F (1)x1)(0)(r1)
+(F (1)x1)(1)(r1)
+(F (1)x1)(2)(r1)
+(F (1)x1)(0)(r2)
+(F (1)x1)(1)(r2)
+(F (1)x0)(0)(r1)
+(F (1)x0)(1)(r1)
+(F (1)x0)(2)(r1)
+(F (1)x0)(0)(r2)
+(F (1)x0)(1)(r2)
+(F (2)x0)(0)(r1)
+(F (2)x0)(1)(r1)
+(F (2)x0)(2)(r1)
+(F (2)x0)(0)(r2)
+(F (2)x0)(1)(r2)
+(F (3)x0)(0)(r1)
+(F (3)x0)(1)(r1)
+(F (3)x0)(2)(r1)
+(F (3)x0)(0)(r2)
+(F (3)x0)(1)(r2)
+����������
+(r1 − r2)6
+where
+c = ±(0! · 1! · 2!) · (0! · 1!) · a1
+5 = ±2a5
+3. Since
+F (i)(r1)
+� = 0,
+for i = 0, 1, 2
+̸= 0,
+for i = 3
+F (i)(r2)
+� = 0,
+for i = 0, 1
+̸= 0,
+for i = 2
+13
+
+we immediately know
+(F (1)x2)(0)(r1) = 0 (F (1)x2)(1)(r1) = 0 (F (1)x2)(2)(r1) = F (3)(r1)r2
+1 (F (1)x2)(0)(r2) = 0 (F (1)x2)(1)(r2) = F (2)(r2)r2
+2
+(F (1)x1)(0)(r1) = 0 (F (1)x1)(1)(r1) = 0 (F (1)x1)(2)(r1) = F (3)(r1)r1
+1 (F (1)x1)(0)(r2) = 0 (F (1)x1)(1)(r2) = F (2)(r2)r1
+2
+(F (1)x0)(0)(r1) = 0 (F (1)x0)(1)(r1) = 0 (F (1)x0)(2)(r1) = F (3)(r1)r0
+1 (F (1)x0)(0)(r2) = 0 (F (1)x0)(1)(r2) = F (2)(r2)r0
+2
+(F (2)x0)(0)(r1) = 0 (F (2)x0)(1)(r1) = F (3)(r1)r0
+1
+(F (2)x0)(0)(r2) = F (2)(r2)r0
+2
+(F (3)x0)(0)(r1) = F (3)(r1)r0
+1
+Therefore,
+D(λ) =
+c ·
+����������
+0
+0
+F (3)(r1)r2
+1
+0
+F (2)(r2)r2
+2
+0
+0
+F (3)(r1)r1
+1
+0
+F (2)(r2)r1
+2
+0
+0
+F (3)(r1)r0
+1
+0
+F (2)(r2)r0
+2
+0
+F (3)(r1)r0
+1
+·
+F (2)(r2)r0
+2
+·
+F (3)(r1)r0
+1
+·
+·
+·
+·
+����������
+(r1 − r2)6
+4. By rearranging the columns of the determinant in the numerator, we have
+D(λ) = ±
+c ·
+����������
+F (3)(r1)r2
+1
+F (2)(r2)r2
+2
+F (3)(r1)r1
+1
+F (2)(r2)r1
+2
+F (3)(r1)r0
+1
+F (2)(r2)r0
+2
+F (3)(r1)r0
+1
+F (2)(r2)r0
+2
+·
+·
+F (3)(r1)r0
+1
+·
+·
+·
+·
+����������
+(r1 − r2)6
+= ±
+c ·
+��������
+M1
+M2
+·
+M3
+·
+·
+��������
+(r1 − r2)6
+where
+M1 =
+�
+�
+F (3)(r1)r2
+1
+F (2)(r2)r2
+2
+F (3)(r1)r1
+1
+F (2)(r2)r1
+2
+F (3)(r1)r0
+1
+F (2)(r2)r2
+�
+�
+M2 =
+�
+F (3)(r1)r0
+1
+F (2)(r2)r2
+�
+M3 =
+�
+F (3)(r1)r0
+1
+�
+5. We repartition the columns so that the reverse diagonal consists of two square matrices and obtain the
+following:
+D(λ) =
+c ·
+����������
+F (3)(r1)r2
+1
+F (2)(r2)r2
+2
+F (3)(r1)r1
+1
+F (2)(r2)r1
+2
+F (3)(r1)r0
+1
+F (2)(r2)r0
+2
+F (3)(r1)r0
+1
+F (2)(r2)r0
+2
+·
+·
+F (3)(r1)r0
+1
+·
+·
+·
+·
+����������
+(r1 − r2)6
+=
+c ·
+����
+T
+B
+·
+����
+(r1 − r2)6
+14
+
+where the size of the square matrix T is λ1 = 3, namely,
+T =
+� 0
+M1
+�
+where 0 is the λ1 × (λ1 − γ1) matrix.
+6. Since λ1 − γ1 = 3 − 2 > 0, the first column of T is all zeros. Hence |T| = 0 and in turn D(λ) = 0
+Arbitrary case. Now we generalize the above ideas to arbitrary cases.
+1. Let µ = (µ1, . . . , µm).
+Assume that r1, . . . , rm are the m distinct roots of F with multiplicities
+µ1, . . . , µm respectively. In other words, F = an(x − r1)µ1 · · · (x − rm)µm.
+2. Let γ = ¯µ = (γ1, . . . , γs). By the definition of conjugate, γi = #{µj : µj ≥ i}. Note that s = µ1 since
+µ1 ≥ · · · ≥ µm.
+3. Consider λ = (λ1, . . . , λt) ∈ M(n) such that γ ≺lex λ. By Lemma 15, we have
+D(λ) =
+c ·
+�����������������
+(F (1)xλ1−1)(0)(r1) · · · (F (1)xλ1−1)(µ1−1)(r1) · · · · · · (F (1)xλ1−1)(0)(rm) · · · (F (1)xλ1−1)(µm−1)(rm)
+...
+...
+...
+...
+(F (1)x0)(0)(r1)
+· · · (F (1)x0)(µ1−1)(r1)
+· · · · · · (F (1)x0)(0)(rm)
+· · · (F (1)x0)(µm−1)(rm)
+...
+...
+...
+...
+(F (t)xλt−1)(0)(r1) · · · (F (t)xλt−1)(µ1−1)(r1) · · · · · · (F (t)xλt−1)(0)(rm) · · · (F (t)xλt−1)(µm−1)(rm)
+...
+...
+...
+...
+(F (t)x0)(0)(r1)
+· · · (F (t)x0)(µ1−1)(r1)
+· · · · · · (F (t)x0)(0)(rm)
+· · · (F (t)x0)(µm−1)(rm)
+�����������������
+�
+i<j(ri − rj)µiµj
+(8)
+where c = ±1
+� ��m
+i=1
+�µi−1
+j=0 j!
+�
+· aλ1−2
+n
+.
+4. Plugging (7) into (8), we have
+D(γ) =
+c ·
+������������������������������
+0
+· · ·
+0
+F (µ1)(r1)rλ1−1
+1
+· · ·
+0
+· · ·
+0
+F (µi)(ri)rλ1−1
+i
+· · ·
+...
+...
+...
+...
+...
+...
+0
+· · ·
+0
+F (µ1)(r1)r0
+1
+· · ·
+0
+· · ·
+0
+F (µi)(ri)r0
+i
+· · ·
+0
+· · · F (µ1)(r1)rλ2−1
+1
+·
+· · ·
+0
+· · · F (µi)(ri)rλ2−1
+i
+·
+· · ·
+...
+...
+...
+...
+...
+...
+0
+· · · F (µ1)(r1)r0
+1
+·
+· · ·
+0
+· · · F (µi)(ri)r0
+i
+·
+· · ·
+...
+...
+...
+...
+...
+...
+0
+·
+·
+· · · F (µi)(ri)r
+λµi −1
+i
+·
+·
+· · ·
+...
+...
+...
+...
+...
+...
+0
+·
+·
+· · · F (µi)(ri)r0
+i
+·
+·
+· · ·
+...
+...
+...
+...
+...
+...
+F (µ1)(r1)r
+λµ1 −1
+1
+· · ·
+·
+·
+· · ·
+·
+· · ·
+·
+·
+· · ·
+...
+...
+...
+...
+...
+...
+F (µ1)(r1)r0
+1
+·
+·
+· · ·
+·
+· · ·
+·
+·
+· · ·
+������������������������������
+�
+1≤i<j≤m
+(ri − rj)µiµj
+15
+
+5. By rearranging the columns of the determinant in the numerator, we have
+D(λ) = ±
+c ·
+�����������������������
+F (µ1)(r1)rλ1−1
+1
+· · · F (µγ1 )(rγ1)rλ1−1
+γ1
+...
+...
+F (µ1)(r1)r0
+1
+· · · F (µγ1 )(rγ1)r0
+γ1
+F (µ1)(r1)rλ2−1
+1
+· · · F (µγ2 )(rγ2)rλ2−1
+γ2
+·
+· · ·
+·
+...
+...
+...
+...
+F (µ1)(r1)r0
+1
+· · · F (µγ2 )(rγ2)r0
+γ2
+·
+· · ·
+·
+· · · · ·
+·
+· · ·
+·
+·
+· · ·
+·
+...
+...
+...
+...
+...
+...
+· · · · ·
+·
+· · ·
+·
+·
+· · ·
+·
+F (µ1)(r1)rλt−1
+1
+· · · F (µγs )(rγs)rλt−1
+γs
+· · · · ·
+·
+· · ·
+·
+·
+· · ·
+·
+...
+...
+...
+...
+...
+...
+...
+...
+F (µ1)(r1)r0
+1
+· · · F (µγs )(rγs)r0
+γs
+· · · · ·
+·
+· · ·
+·
+·
+· · ·
+·
+�����������������������
+�
+1≤i<j≤m
+(ri − rj)µiµj
+= ±
+c ·
+�����������
+M1
+M2
+·
+...
+·
+Mµ1
+· · ·
+·
+·
+�����������
+�
+1≤i<j≤m
+(ri − rj)µiµj
+where Mi is λi by γi.
+6. Since γ ≺lex λ, there exists ℓ such that γj = λj for j < ℓ and γℓ < λℓ. Thus
+γ1 + · · · + γℓ < λ1 + · · · + λℓ
+7. We repartition the numerator matrix so that the reverse diagonal consists of two square matrices T
+and B as follows.
+D(λ) = ±
+c ·
+����
+T
+B
+����
+�
+1≤i<j≤m
+(ri − rj)µiµj
+where the size of the square matrix T is λ1 + · · · + λℓ, namely,
+T =
+�
+��
+M1
+...
+...
+0
+Mℓ
+· · ·
+·
+�
+��
+where 0 is the γℓ × p and p = (λ1 + · · · + λℓ) − (γ1 + · · · + γℓ).
+8. Obviously,
+D(λ) = ±
+c · |T| · |B|
+�
+1≤i<j≤m
+(ri − rj)µiµj
+9. Since p > 0, the first column of T is all zeros. Hence |T| = 0, which implies that D(λ) = 0.
+16
+
+4.3
+Proof of Theorem 9
+Now we are ready to prove Theorem 9.
+Proof of Theorem 9.
+The result of Theorem 9 is equivalent to the following claim: let
+δ =
+max
+γ∈M(n)
+D(γ)̸=0
+γ
+where max is with respect to the lexicographic ordering ≺lex. Then mult(F) = δ.
+Next we will show the correctness of the claim.
+1. Assume that mult(F) = µ. We will show µ = δ by disproving µ ≺lex δ and δ ≺lex µ.
+2. If µ ≺lex δ, then δ ≺lex µ. By the condition for determining δ, we immediately have D(µ) = 0, leading
+to a contradiction with Lemma 16.
+3. If δ ≺lex µ, then µ
+≺lex δ. By Lemma 17, D(δ) = 0. However, it contradicts the condition for
+determining δ.
+4. Therefore, the only possibility is µ = µ′.
+5
+Comparison
+In this section, we compare the multiplicity discriminant condition given by Theorem 9 (mentioned as HY22
+hereinafter) and that given by a complex root version of YHZ’s condition [11] as well as the one given by
+the authors in [6, Theorem 6] (mentioned as HY21 hereinafter). In particular, we will make comparison on
+the forms and the maximum degrees of discriminants appearing in the conditions.
+5.1
+Form of discriminants
+We will illustrate the forms of conditions generated by the three methods for a fixed µ. For example, we
+consider the polynomial F = a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0 and µ = (2, 2, 1). The condition for F
+having the multiplicity structure µ is given as follows:
+1. YHZ’s condition: P1 = 0 ∧ P2 = 0 ∧ P3 ̸= 0 where
+P1 =
+������������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+������������������
+P2 =
+��������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+��������������
+17
+
+P3 =
+����������������������������������
+����������
+a5
+a4
+a3
+a2
+a1
+a5
+a4
+a3
+a2
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+5a5
+4a4
+3a3
+����������
+����������
+a5
+a4
+a3
+a2
+a0
+a5
+a4
+a3
+a1
+5a5
+4a4
+3a3
+2a2
+5a5
+4a4
+3a3
+a1
+5a5
+4a4
+2a2
+����������
+����������
+a5
+a4
+a3
+a2
+a5
+a4
+a3
+a0
+5a5
+4a4
+3a3
+2a2
+5a5
+4a4
+3a3
+5a5
+4a4
+a1
+����������
+2
+����������
+a5
+a4
+a3
+a2
+a1
+a5
+a4
+a3
+a2
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+5a5
+4a4
+3a3
+����������
+����������
+a5
+a4
+a3
+a2
+a0
+a5
+a4
+a3
+a1
+5a5
+4a4
+3a3
+2a2
+5a5
+4a4
+3a3
+a1
+5a5
+4a4
+2a2
+����������
+2
+����������
+a5
+a4
+a3
+a2
+a1
+a5
+a4
+a3
+a2
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+5a5
+4a4
+3a3
+����������
+����������
+a5
+a4
+a3
+a2
+a0
+a5
+a4
+a3
+a1
+5a5
+4a4
+3a3
+2a2
+5a5
+4a4
+3a3
+a1
+5a5
+4a4
+2a2
+����������
+����������������������������������
+2. HY21’s condition: Q1 = 0 ∧ Q2 = 0 ∧ Q3 ̸= 0 ∧ Q4 ̸= 0 where
+Q1 =
+������������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+������������������
+Q2 =
+��������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+��������������
+Q3 =
+����������
+a5
+a4
+a3
+a2
+a1
+a5
+a4
+a3
+a2
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+5a5
+4a4
+3a3
+����������
+Q4 =
+������������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+5a5
+4a4
+3a3
+2a2
+a1
+������������������
++
+������������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+5a5
+4a4
+3a3
+2a2
+a1
+10a5
+6a4
+3a3
+a2
+������������������
+18
+
+��������������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+5a5
+4a4
+3a3
+2a2
+a1
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+��������������������
++
+������������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+10a5
+6a4
+3a3
+a2
+5a5
+4a4
+3a3
+2a2
+a1
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+������������������
+������������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+a1
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+10a5
+6a4
+3a3
+a2
+������������������
+3. HY22’s condition: R1 = 0 ∧ R2 = 0 ∧ R3 ̸= 0 where
+R1 = 1
+a5
+������������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+������������������
+R2 = 1
+a5
+��������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+20a5
+12a4
+6a3
+2a2
+��������������
+R3 = 1
+a5
+��������������
+a5
+a4
+a3
+a2
+a1
+a0
+a5
+a4
+a3
+a2
+a1
+a0
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+5a5
+4a4
+3a3
+2a2
+a1
+20a5
+12a4
+6a3
+2a2
+20a5
+12a4
+6a3
+2a2
+��������������
+From the above conditions, we make the following observations which are also true in general.
+1. YHZ’s discriminant involves a nested determinant;
+2. HY21’s discriminant involves a sum of several determinants;
+3. HY22’s discriminant involves a non-nested determinant.
+19
+
+5.2
+Maximum degree of discriminants
+For the sake of simplicity, we use the following short-hands:
+• dYHZ : the maximum of the degrees of the polynomials appearing in YHZ’s conditions ([11])
+• dHY21 : the maximum of the degrees of the polynomials appearing in HY21’s conditions ([6] )
+• dHY22 : the maximum of the degrees of the polynomials appearing in the new conditions (Theorem 9).
+Lemma 18. Let dYHZ(µ),dHY21(µ) and dHY22(µ) denote the maximum degrees of the polynomials appear-
+ing in YHZ’s condition, HY21’s condition and HY22’s condition for a given µ = (µ1, . . . , µm) ∈ M(n),
+respectively. Then we have:
+1. Under some minor and reasonable assumption (see [5, Assumption 2]),
+dYHZ(µ) =
+�
+�
+�
+�
+�
+�
+�
+µ2−1
+�
+j=0
+(2 mj − 1)
+�
+�
+�
+1
+if
+µ1 = µ2
+1 +
+2
+2mµ2−1−1
+if
+µ1 = µ2 + 1
+(2 (µ1 − µ2) − 1)
+if
+µ1 > µ2 + 1
+≥
+2n + 3µ2 − 4µ2,
+for m > 1
+2n − 1,
+for m = 1
+where mi is the largest k such that µk > i;
+2. dHY21(µ) = 2n − 1;
+3. dHY22(µ) = 2n − 2.
+Proof.
+1. When m = 1, µ = (n). In this case, the condition for the polynomial having multiplicity structure µ
+is given by the 0-th,. . . ,(n − 1)-th subdiscriminants. Thus the maximum degree dYHZ(µ) is 2n − 1,
+achieved at the 0-th subdiscriminant.
+When m > 1, see [5, Appendix] for a detailed proof.
+2. Recall that HY21’s condition consists of two parts: (i) the 0-th,. . . ,(n − m)-th subdiscriminants whose
+highest degree is 2n − 1; (ii) the multiplicity discriminant given by
+�
+σ∈Sp
+dp
+�
+���������
+xn−µm−1F
+...
+x0F
+xn−1F(σ1)/σ1!
+...
+x0F (σn)/σn!
+�
+���������
+where p = (µ1, . . . , µ1
+�
+��
+�
+µ1
+, . . . , µm, . . . , µm
+�
+��
+�
+µm
+) and Sp is the set of all permutations of p. It is easy to see
+that the degree of the multiplicity discriminant is 2n − µm. Hence the maximum degree of the above
+discriminants is 2n − 1.
+20
+
+3. HY22’s condition only consists of the multiplicity discriminants given by
+D (γ) = 1
+an
+dp
+�
+��������������������
+F (0)xγ0−1
+...
+F (0)x0
+F (1)xγ1−1
+...
+F (1)x0
+...
+F (s)xγs−1
+...
+F (s)x0
+�
+��������������������
+where γ = (γ1, . . . , γs) ranges over (n) ≻lex · · · ≻lex µ. Note that the highest degree is achieved when
+γ = (n) and γ0 = γ1 − 1. In this case, the degree of D (γ) is 2n − 2.
+Remark 19. It is noted that in HY21’s condition, the multiplicity discriminant is always divisible by the
+leading coefficient an and thus with this division carried out, the degree can be made smaller by 1.
+By Lemma 18, the maximum degree in YHZ’s condition grows exponentially with respect to n while the
+maximum degrees in HY21 and HY22’s conditions grow linearly. Below we show a comparison with examples
+where n < 10.
+n
+dYHZ
+dHY21
+dHY22
+3
+5
+5
+4
+4
+9
+7
+6
+5
+15
+9
+8
+6
+27
+11
+10
+7
+45
+13
+12
+8
+81
+15
+14
+9
+135
+17
+16
+0
+20
+40
+60
+80
+100
+120
+140
+160
+3
+4
+5
+6
+7
+8
+9
+YHZ
+HY21
+HY22
+max deg
+n
+Acknowledgements. The second author’s work was supported by National Natural Science Foundation of
+China (Grant Nos.: 12261010 and 11801101).
+References
+[1] S. Basu, R. Pollack, and M.-F. Roy. Algorithms in real algebraic geometry. Springer-Verlag, Berlin-
+Heidelberg, 2006.
+[2] W. Brown and J. Traub.
+On Euclid’s algorithm and the theory of subresultants.
+Journal of the
+Association for Computing Machinery, 18:505–514, 1971.
+[3] G. Collins. Subresultants and Reduced Polynomial Remainder Sequences. Journal of the Association
+for Computing Machinery, 14:128–142, 1967.
+21
+
+[4] L. Gonz´alez-Vega, T. Recio, H. Lombardi, and M.-F. Roy. Sturm-Habicht Sequences, Determinants and
+Real Roots of Univariate Polynomials. In Quantifier Elimination and Cylindrical Algebraic Decomposi-
+tion. Texts and Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic
+Computation, Johannes-Kepler-University, Linz, Austria), pages 300–316. Springer, 1998.
+[5] H. Hong and J. Yang.
+A condition for multiplicity structure of univariate polynomials.
+CoRR,
+abs/2001.02388, 2020.
+[6] H. Hong and J. Yang. A condition for multiplicity structure of univariate polynomials. Journal of
+Symbolic Computation, 104:523–538, 2021.
+[7] S. Liang and D. J. Jeffrey. An algorithm for computing the complete root classification of a para-
+metric polynomial. In J. Calmet, T. Ida, and D. Wang, editors, Artificial Intelligence and Symbolic
+Computation, pages 116–130, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg.
+[8] S. Liang, D. J. Jeffrey, and M. M. Maza. The complete root classification of a parametric polynomial
+on an interval. In International Symposium on Symbolic and Algebraic Computation, 2008.
+[9] S. Liang and J. Zhang. A complete discrimination system for polynomials with complex coefficients and
+its automatic generation. Science in China Series E: Technological Sciences, 42:113–128, 1999.
+[10] R. Loos. Generalized Polynomial Remainder Sequences, pages 115–137. Springer Vienna, Vienna, 1983.
+[11] L. Yang, X. Hou, and Z. Zeng. A complete discrimination system for polynomials. Science in China
+(Series E), 39(6):628–646, 1996.
+22
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+Type II multiferroic order in two-dimensional transition metal halides from first principles
+spin-spiral calculations
+Joachim Sødequist and Thomas Olsen∗
+Computational Atomic-Scale Materials Design (CAMD), Department of Physics,
+Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
+(Dated: January 13, 2023)
+We present a computational search for spin spiral ground states in two-dimensional transition metal halides
+that are experimentally known as van der Waals bonded bulk materials. Such spin spirals break the rotational
+symmetry of the lattice and lead to polar ground states where the axis of polarization is strongly coupled to
+the magnetic order (type II multiferroics). We apply the generalized Bloch theorem in conjunction with non-
+collinear density functional theory calculations to find the spiralling vector that minimizes the energy and then
+include spin-orbit coupling to calculate the preferred orientation of the spin plane with respect to the spiral
+vector. We find a wide variety of magnetic orders ranging from ferromagnetic, stripy anti-ferromagnetic, 120◦
+non-collinear structures and incommensurate spin spirals. The latter two introduce polar axes and are found in
+the majority of materials considered here. The spontaneous polarization is calculated for the incommensurate
+spin spirals by performing full supercell relaxation including spinorbit coupling and the induced polarization
+is shown to be strongly dependent on the orientation of the spiral planes. We also test the effect of Hubbard
+corrections on the results and find that for most materials LDA+U results agree qualitatively with LDA. An
+exception is the Mn halides, which are found to exhibit incommensurate spin spiral ground states if Hubbard
+corrections are included whereas bare LDA yields a 120◦ non-collinear ground state.
+I.
+INTRODUCTION
+The recent discovery of ferromagnetic order in two-
+dimensional (2D) CrI3 [1] has initiated a vast interest in 2D
+magnetism [2–4]. Several other materials have subsequently
+been demonstrated to preserve magnetic order in the mono-
+layer limit when exfoliated from magnetic van der Waals
+bonded compounds and the family of 2D magnets is steadily
+growing. A crucial requirement for magnetic order to persist
+in the 2D limit is the presence of magnetic anisotropy that
+breaks the spin rotational symmetry that would otherwise ren-
+der magnetic order at finite temperatures impossible by the
+Mermin-Wagner theorem [5]. This is exemplified by the cases
+of 2D CrBr3 [6, 7] and CrCl3 [8, 9], which are isostructural
+to CrI3 and while the former remains ferromagnetic in the
+atomic limit due to easy-axis anisotropy (like CrI3) the lat-
+ter has a weak easy plane that forbids proper long range or-
+der. Other materials with persisting ferromagnetic order in
+the 2D limit include the metallic compounds Fe3/4/5GeTe2
+[10–12] and the anisotropic insulator CrSBr [13], which has
+an easy-axis aligned with the atomic plane. Finally, FePS3
+[14] and MnPS3 [15] constitute examples of in-plane anti-
+ferromagnets that preserve magnetic order in the monolayer
+limit due to easy-axis anisotropy, whereas the magnetic order
+is lost in monolayers of the isostructural easy-plane compound
+NiPS3 [16]. The 2D materials mentioned above all consti-
+tute examples of rather simple collinear magnets. However,
+the ground state of three-dimensional magnetic materials of-
+ten exhibit complicated non-collinear order that gives rise to a
+range of interesting properties [17]. Such materials, are so far
+largely lacking from the field of 2D magnetism and the discov-
+ery of new non-collinear 2D magnets would greatly enhance
+∗ tolsen@fysik.dtu.dk
+the possibilities of constructing versatile magnetic materials
+using 2D magnets as building blocks [18].
+The ground state of the classical isotropic Heisenberg
+model can be shown to be a planar spin spiral characterised by
+a propagation vector Q [19] and such spin configurations thus
+comprise a broad class of states that generalise the concept of
+ferromagnetism and anti-ferromagnetism. In fact, spin spiral
+order is rather common in layered van der Waals bonded ma-
+terials [20] and it is thus natural to investigate the ground state
+order of the corresponding monolayers for spin spiral order.
+Moreover, for non-bipartite magnetic lattices the concept of
+anti-ferromagnetism is not unique. This is exemplified by the
+abundant example of the triangular lattice where one may con-
+sider the cases of anti-aligned ferromagnetic stripes or 120◦
+non-collinear order, which can be represented as spin spirals
+of Q = (1/2,0) and Q = (1/3,1/3) respectively [21, 22]. The
+concept of spin spirals thus constitute a general framework for
+specifying the magnetic order, which may or may not be com-
+mensurate with the crystal lattice.
+Finite spin spiral vectors typically break symmetries inher-
+ent to the crystal lattice and may thus induce physical prop-
+erties that are predicted to be absent if one only considers the
+crystal symmetries. In particular, the spin spiral may yield a
+polar axis that lead to ferroelectric order [23]. Such materials
+are referred to as type II multiferroics and examples include
+MnWO4 [24], CoCr2O4 [25], LiCu2O2 [26] and LiCuVO4
+[27] as well as the triangular magnets CuFeO2 [28], CuCrO2
+[28], AgCrO2 [29] and MnI2 [30]. In addition to these ma-
+terials, 2D NiI2 has recently been shown to host a spin spiral
+ground state that induces a spontaneous polarization [31] and
+2D NiI2 thus comprises the first example of a 2D type II mul-
+tiferroic.
+The prediction of new materials with certain desired prop-
+erties can be vastly accelerated by first principles simulations.
+In general, the search for materials with spin spiral ground
+states is complicated by the fact that the magnetic order re-
+arXiv:2301.05107v1  [cond-mat.mtrl-sci]  12 Jan 2023
+
+2
+quires large super cells in the simulations. However, if one
+neglects spinorbit coupling, spin spirals of arbitrary wavevec-
+tors can be represented in the chemical unit cell by utilising
+the generalized Bloch theorem that encodes the spiral in the
+boundary conditions [32, 33]. This method has been applied
+in conjunction with density functional theory (DFT) to a wide
+range of materials and typically produces results that are in
+good agreement with experiments [34–38].
+In the present work we use DFT simulations in the frame-
+work of the generalized Bloch theorem to investigate the mag-
+netic ground state of monolayers derived from layered van der
+Waals magnets. We then calculate the preferred orientation of
+the spiral plane by adding a single component of the spinorbit
+coupling in the normal direction of various trial spiral planes.
+This yields a complete classification of the magnetic ground
+state for these materials under the assumption that higher or-
+der spin interactions can be neglected. On the other hand, the
+effect of higher order spin interactions can be quantified by
+deviations between spin spiral energies in the primitive unit
+cell and a minimal super cell. The results for all compounds
+are discussed and compared with existing knowledge from ex-
+periments on the parent bulk materials. Finally, we analyse the
+spontaneous polarization in all cases where an incommensu-
+rate ordering vector is predicted.
+The paper is organised as follows. In Sec. II we summarise
+the theory used to obtain spin spiral ground states based on the
+generalized Bloch theorem and briefly outline the implemen-
+tation. In Sec. III we present the results and summarise the
+magnetic ground states of all the investigated materials. Sec.
+IV provides a conclusion and outlook.
+II.
+THEORY
+A.
+Generalized Bloch’s Theorem
+The Heisenberg model plays a prominent role in the the-
+ory of magnetism and typically gives an accurate account of
+the fundamental magnetic excitations as well as the thermo-
+dynamic properties of a given material. In the isotropic case
+it can be written as
+H = −1
+2 ∑
+i j
+Ji jSi ·Sj,
+(1)
+where Si is the spin operator for site i and Ji j is the exchange
+coupling between sites i and j. In a classical treatment, the
+spin operators are replaced by vectors of fixed magnitude and
+it can be shown that the classical energy is minimised by a
+planar spin spiral [19]. Such a spin configuration is charac-
+terised by a wave vector Q, which is determined by the set of
+exchange parameters Ji j. The spin at site i is rotated by an an-
+gle Q · Ri with respect to the origin and the wave vector may
+or may not be commensurate with the lattice.
+In a first principles framework it is thus natural to search for
+planar spin spiral ground states that give rise to periodically
+modulated magnetisation densities satisfying
+mq(r+Ri) = Uq,Rimq(r).
+(2)
+Here Ri is a lattice vector (of the chemical unit cell) and Uq,Ri
+is a rotation matrix that rotates the magnetisation by an an-
+gle q · Ri around the normal of the spiral plane. In the ab-
+sence of spinorbit coupling we are free to perform a global
+rotation of the magnetisation density and we will fix the spi-
+ral plane to the xy-plane from hereon. In the framework of
+DFT, the magnetisation density (2) gives rise to an exchange-
+correlation magnetic field satisfying the same symmetry un-
+der translation. If spinorbit coupling is neglected the Kohn-
+Sham Hamiltonian thus commutes with the combined action
+of translation (by a lattice vector) and a rotation of spinors by
+the angle q·Ri. This implies that the Kohn-Sham eigenstates
+can be written as
+ψq,k(r) = eik·rU†
+q(r)
+�
+u↑
+q,k(r)
+u↓
+q,k(r)
+�
+(3)
+where u↑
+q,k(r) and u↓
+q,k(r) are periodic in the chemical unit cell
+and the spin rotation matrix is given by
+Uq(r) =
+�
+eiq·r/2
+0
+0
+e−iq·r/2
+�
+(4)
+This is known as the generalized Bloch Theorem (GBT) and
+the Kohn-Sham equations can then be written as
+HKS
+q,kuq,k = εq,kuq,k
+(5)
+where the generalized Bloch Hamiltonian:
+HKS
+q,k = e−ik·rUq(r)HKSU†
+q(r)eik·r
+(6)
+is periodic in the unit cell. Here k is the crystal momentum,
+q is the spiral wave vector and HKS is the Kohn-Sham Hamil-
+tonian, which couples to the spin degrees of freedom through
+the exchange-correlation magnetic field.
+In the present work, we will not consider constraints be-
+sides the boundary conditions defined by Eq. (2). For a given
+q we can thus obtain a unique total energy Eq and the mag-
+netic ordering vector is determined as the point where Eq has
+a minimum (denoted by Q) when evaluated over the entire
+Brillouin zone. However, if the chemical unit cell contains
+more than one magnetic atom there may be different local ex-
+trema corresponding to different intracell alignments of mag-
+netic moments. In order ensure that the correct ground state
+is obtained it is thus pertinent to perform a comparison be-
+tween calculations that are initialised with different relative
+magnetic moments. As a simple example of this, one may
+consider a honeycomb lattice of magnetic atoms where the
+ferromagnetic and anti-ferromagnetic configurations both cor-
+respond to q = 0, but are distinguished by different intracell
+orderings of the local magnetic moments. We will discuss this
+in the context of CrI3 in section III C.
+We also note that the true magnetic ground state is not nec-
+essarily representable by the ansatz (2) and one is therefore
+not guaranteed to find the ground state by searching for spin
+spirals based on the minimal unit cell. In figure 1 we show
+four examples of possible magnetic ground states of the tri-
+angular lattice. Three of these correspond to spin spirals of
+
+3
+Q = (1/3, 1/3)
+Q = (1/2, 0)
+Q = (0.14, 0.14)
+Q = (0, 1/2)
+(a)
+Γ
+M/S
+K
+X
+Y
+(b)
+FIG. 1. (a) Examples of magnetic structures in the triangular lattice. The Q = (1/3,1/3) (corresponding to the high symmetry point K)
+is the classical ground state in the isotropic Heisenberg model with nearest neighbour antiferromagnetic exchange and is degenerate with
+Q = (−1/3,−1/3). The stripy antiferromagnetic Q = (1/2,0) (corresponding to the high symmetry point M) is only found for CoI2 in
+the present study and is degenerate with Q = (0,1/2) and Q = (1/2,1/2). The incommensurate spiral with Q = (0.14,0.14) corresponds
+to the prediction of NiI2 in the present work. The rectangular cell with Q = (0,1/2) is a bicollinear antiferromagnet that corresponds to
+superpositions of (0, ±1/4) states in the primitive cell. (b) Brillouin zone of the hexagonal (blue) and rectangular (orange) unit cell. The high
+symmetry band paths used to sample the spiral ordering vectors are shown in black.
+the minimal unit cell while the fourth - a bicollinear antifer-
+romagnet - requires a larger unit cell. The bicollinear state
+may arise as a consequence of higher order exchange interac-
+tions, which tend to stabilize linear combinations of degener-
+ate single-q states.
+B.
+Spinorbit coupling
+In the presence of spinorbit coupling, the spin spiral plane
+will have a preferred orientation and the magnetic ground state
+is thus characterised by a normal vector ˆn0 of the spiral plane
+as well as the spiral vector Q. Spinorbit coupling is, however,
+incompatible with application of the GBT and has to be ap-
+proximated in a post processing step when working with the
+spin spiral representation in the chemical unit cell. It can be
+shown that first order perturbation theory only involves contri-
+butions from the spinorbit components orthogonal to the plane
+[39]
+⟨ψq,ˆn|L·S|ψq,ˆn⟩ = ⟨ψq,ˆn|(L· ˆn)(S· ˆn)|ψq,ˆn⟩,
+(7)
+and this term is thus expected to yield the most important con-
+tribution to the spinorbit coupling. Since (L · ˆn)(S · ˆn) com-
+mutes with a spin rotation around the axis ˆn, the spin spi-
+ral wavefunctions remain eigenstates when such a term is in-
+cluded in HKS. This approach was proposed by Sandratskii
+[40] and we will refer to it as the projected spinorbit coupling
+(PSO). For the spin spiral calculations in the present work we
+include spinorbit coupling non-selfconsistently by performing
+a full diagonalization of the HKS
+q,k including the PSO. The mag-
+netic ground state is then found by evaluating the total energy
+at all normal vectors ˆn, which will yield ˆn0 as the normal vec-
+tor that minimizes the energy.
+C.
+Computational Details
+The GBT has been implemented in the electronic structure
+software package GPAW [41], which is based on the projector
+augmented wave method (PAW) and plane waves. The im-
+plementation uses a fully non-collinear treatment within the
+local spin density approximation where both the interstitial
+and atom-centered PAW regions are handled non-collinearly.
+Spinorbit coupling is included non-selfconsistently [42] as de-
+scribed in Section II B. The implementation is described in de-
+tail in Appendix V A and benchmarked for fcc Fe in Appendix
+V B. We find good agreement with previous results from the
+literature and we also assert that results from spin spiral cal-
+culations within the GBT agree exactly with supercell calcu-
+lations without spinorbit in the case of bilayer CoPt. Finally,
+we compare the results of the PSO approximations with full
+inclusion of spinorbit coupling for both supercells and GBT
+spin spirals of the CoPt bilayer. We find exact agreement be-
+tween the PSO in the supercell and GBT spin spiral and the
+approximation only deviates slightly compared to full spinor-
+bit coupling for the supercell calculations.
+All calculations have been carried out with a plane wave
+cutoff of 800 eV, a k-point density of 14 ˚A and a Fermi smear-
+ing of 0.1 eV. The structures and initial magnetic moments
+are taken from the Computational Materials Database (C2DB)
+[43, 44].In order to find the value of Q, which describes the
+ground state magnetic order, we calculate Eq along a represen-
+tative path connecting high symmetry points in the Brillouin
+zone. While the true value of Q could be situated away from
+such high symmetry lines we deem this approach sufficient
+for the present study.
+III.
+RESULTS
+A comprehensive review on the magnetic properties of lay-
+ered transition metal halides was provided in Ref. [20]. Here
+we present spin spiral calculations and extract the magnetic
+properties of the corresponding monolayers. In addition to the
+magnetic moments, the properties are mainly characterised by
+a spiral ordering vector Q and the normal vector to the spin
+spiral plane ˆn0. The materials either have AB2 or AB3 stoi-
+chiometries and we will discuss these cases separately below.
+
+4
+Q
+Emin [meV] (θ,ϕ)
+Exp. IP order BW [meV] PSO BW [meV] mΓ [µB] ∆εQ [eV]
+TiBr2
+(1/3, 1/3)
+-78.12
+(90,90)
+-
+78.1
+0.6
+1.5
+0.0
+TiI2
+(1/3, 1/3)
+-44.33
+(90,90)
+-
+44.3
+1.0
+1.9
+0.0
+NiCl2
+(0.06, 0.06) -0.81
+(90,31)
+FM ∥
+45.2
+0.0
+2.0
+0.81
+NiBr2
+(0.11, 0.11) -8.62
+(44,0)
+FM ∥, HM
+50.7
+0.3
+2.0
+0.62
+NiI2
+(0.14, 0.14) -28.48
+(64,0)
+HM
+68.3
+4.1
+1.8
+0.28
+VCl2
+(1/3, 1/3)
+-60.07
+(90,0)
+120◦
+60.1
+0.1
+3.0
+0.96
+VBr2
+(1/3, 1/3)
+-36.21
+(90,18)
+120◦
+36.2
+0.1
+3.0
+0.9
+VI2
+(0.14, 0.14) -4.43
+(6,0)
+stripe
+9.8
+0.7
+3.0
+0.96
+MnCl2 (1/3, 1/3)
+-20.48
+(90,15)
+stripe or HM 20.5
+0.0
+5.0
+1.92
+MnBr2 (1/3, 1/3)
+-20.13
+(90,15)
+stripe ∥
+20.1
+0.1
+5.0
+1.76
+MnI2
+(1/3, 1/3)
+-21.32
+(0,0)
+HM
+21.3
+1.1
+5.0
+1.41
+FeCl2
+(0, 0)
+0.0
+(0, 0)∗
+FM ⊥
+115.2
+0.5∗
+4.0
+0.0
+FeBr2
+(0, 0)
+0.0
+(0, 0)∗
+FM ⊥
+81.3
+0.8∗
+4.0
+0.0
+FeI2
+(0, 0)
+0.0
+(0, 0)∗
+stripe ⊥
+36.5
+1.9∗
+4.0
+0.0
+CoCl2
+(0, 0)
+0.0
+(90,90)∗ FM ∥
+46.0
+1.2∗
+3.0
+0.0
+CoBr2 (0.03, 0.03) -0.04
+(0,0)
+FM ∥
+21.2
+0.1
+3.0
+0.0
+CoI2
+(1/2, 0)
+-20.95
+(90,90)
+HM
+41.7
+5.6
+1.2
+0.0
+TABLE I. Summary of magnetic properties of the AB2 compounds. The ground state ordering vector is denoted by Q and Emin is the ground
+state energy relative to the ferromagnetic state. The normal vector of the spiral plane is defined by the angles θ and ϕ (see text). We also
+display the experimental in-plane order of the parent layered compound (Exp. IP order). In addition we state the spin spiral band width
+BW, the magnetic moment per unit cell in the ferromagnetic state mΓ and the band gap at the ordering vector ∆εQ. For the case of NiI2, mΓ
+deviates from an integer value because the ferromagnetic state is metallic in LDA (whereas the spin spiral ground state has a gap). The cases
+of FeX2, CoCl2 and CoBr2 are half metals, which enforces integer magnetic moment despite the metallic ground state. The asterisks indicate
+ferromagnets where full spinorbit coupling was included and the angles then refer to the direction of the spins rather that the spiral plane
+normal vector.
+We have performed LDA and LDA+U calculations for all
+materials. In most cases, the Hubbard corrections does not
+make any qualitative difference although the spiral ordering
+vector does change slightly and we will not discuss these cal-
+culations further here. The Mn halides comprise an exception
+to this where LDA+U calculations differ significantly from
+those of bare LDA and the LDA+U calculations will be dis-
+cussed separately for these materials below.
+For the AB2 materials, we find 12 that exhibit a spiral or-
+der that breaks the crystal symmetry and yields a ferroelec-
+tric ground state. For six of these compounds we have calcu-
+lated the spontaneous polarization by performing full relax-
+ation (including self-consistent spinorbit coupling) in super-
+cells hosting the spiral order.
+A.
+Magnetic ground state of AB2 materials
+The AB2 materials all have space group P¯3m1 correspond-
+ing to monolayers of the CdI2 (or CdCl2) prototype. The mag-
+netic lattice is triangular and a few representative possibilities
+for the magnetic order is illustrated in figure 1. The magnetic
+properties of all the considered compounds are summarized
+in table I. In addition to the ordering vector Q we provide
+the angles θ and φ, which are the polar and azimuthal an-
+gles of ˆn0 with respect to the out-of-plane direction and the
+ordering vector respectively. It will be convenient to consider
+three limiting cases of the orientation of spin spiral planes:
+the proper screw (θ = 90,ϕ = 0), the out-of-plane cycloid
+(�� = 90,ϕ = 90) and the in-plane cycloid (θ = 0,ϕ = 0).
+We also provide the ground state energy relative to the fer-
+romagnetic configuration (Q = (0,0)), the band gap, the spin
+spiral band width, which reflects the strength of the magnetic
+interactions and the PSO band width, which is the energy dif-
+ference between the easy and hard orientations of the spiral
+plane. The magnetic moments are calculated as the total mo-
+ment in the unit cell using the ferromagnetic configurations
+without spinorbit interaction and thus yields an integer num-
+ber of Bohr magnetons for insulators. The magnitude of the
+local magnetic moments (obtained by integrating the magne-
+tization density over the PAW spheres) in the ground state are
+generally found to be very close to the moments in the ferro-
+magnetic configuration, unless explicitly mentioned. The spin
+spiral energy dispersions are provided for all AB2 materials in
+the supporting information. The different classes of materials
+are described in detail below.
+NiX2
+The nickel halides all have ground states with incommen-
+surate spiral vectors between Γ and K. Experimentally, both
+NiI2 and NiBr2 in bulk form have been determined to have in-
+commensurate spiral vectors [45–47] in qualitative agreement
+with the LDA results. The case of NiCl2, however, have been
+found to have ferromagnetic intra-layer order whereas we find
+a rather small spiral vector of Q = (0.06,0.06).
+In bulk NiI2 the experimental ordering vector Qexp =
+(0.1384,0,1.457) has an in-plane component in the ΓM-
+direction with a magnitude of roughly 1/7 of a recipro-
+cal lattice vector, while for the monolayer we find Q =
+(0.14,0.14,0), which is in the ΓK-direction. Evaluating the
+spin spiral energy in the entire Brillouin zone, however, re-
+veals a nearly degenerate ring encircling the Γ-point with a
+
+5
+Γ
+M
+K
+Γ
+−20
+0
+20
+40
+E(q) [meV]
+(a)
+K
+G
+M
+28
+18
+8
+0
+10
+20
+30
+40
+Energy [meV]
+(b)
+IP
+Screw
+OoP
+IP
+−44
+−42
+−40
+Esoc(θ, ϕ) [meV]
+θ
+ϕ
+θ
+(c)
+FIG. 2. Spin spiral energy of NiI2. Left: the spin spiral energy as a function of q without spinorbit coupling. Center: Spin spiral energy in
+evaluated in entire Brillouin zone. Right: spiral energy as a function of spiral plane orientation evaluated at the minimum Q = (0.14,0.14).
+The spiral plane orientation is parameterized in terms of the polar angle θ and azimuthal angle ϕ (measured from Q) of the spiral plane normal
+vector.
+radius of roughly 1/5 of a reciprocal lattice vector. The point
+qM = (0.21,0) thus comprises a very shallow saddle point
+with an energy that exceeds the minimum by merely 2 meV.
+This is illustrated in figure 2. We also show a scan of the
+spin spiral energy (within the PSO approximation) as a func-
+tion of orientation of the spin spiral plane on a path that con-
+nects the limiting cases of in-plane cycloid, out-of-plane cy-
+cloid and proper screw. An unconstrained spin spiral calcu-
+lation using the rectangular unit cell of figure 1 does not re-
+veal any new minima in the energy, which implies that the
+ground state is well represented by a single-q spiral and that
+higher order exchange interactions are neglectable in NiI2.
+The normal vector of the spiral makes an angle of 64◦ with
+the out-of-plane direction. This orientation is in good agree-
+ment with the experimental assignment of a proper screw
+(along Qexp = (0.1384,0,1.457)), which corresponds to a tilt
+of 55◦±10◦ with respect to the c-axis [47], but disagrees with
+the model proposed in Ref. [31] where the spiral was found
+to be a proper screw.
+At low temperatures NiBr2 has been reported to exhibit
+Qexp = (x,x,3/2) where x changes continuously from 0.027
+at 4.2 K to 0.09 at 22.8 K and then undergoes first order transi-
+tion at 24 K to intra-layer ferromagnetic order [48]. The struc-
+ture predicted here is close to the one observed in bulk at 22.8
+K. The discrepancy could be due to the magnetoelastic defor-
+mation [49] that has been associated with the modulation of
+the spiral vector. This effect could in principle be captured by
+relaxing the structure in supercell calculations, but the small
+wavelength spirals require prohibitively large supercells and
+are not easily captured by first principles methods. It is also
+highly likely that LDA is simply not accurate enough to de-
+scribe the intricate exchange interactions that define the true
+ground state in this material.
+Bulk NiCl2 is known to be an inter-layer antiferromag-
+net with ferromagnetically ordered layers [50]. We find the
+ground state to be a long wavelength incommensurate spin
+spiral with Q = (0.06,0.06), which is in rather close prox-
+imity to ferromagnetic order. The ground state energy is less
+than 1 meV lower than the ferromagnetic state, but we cannot
+say at present whether this is due to inaccuracies of LDA or if
+the true ground state indeed exhibits spiral magnetic order in
+the monolayer limit.
+VX2
+The three vanadium halides are insulators and whereas
+VCl2 and VBr2 are found to form Q = (1/3,1/3) spiral struc-
+tures, VI2 has an incommensurate ground state with Q =
+(0.14,0.14). The magnetic ground state of VCl2 and VBr2 is
+in good agreement with experiments on bulk materials where
+both have been found to exhibit out-of-plane 120◦ order [51].
+This structure is expected to arise from strong nearest neigh-
+bour anti-ferromagnetic interactions between the V atoms.
+The case of VI2 has a significantly smaller spiral band width,
+signalling weaker exchange interactions compared to VCl2
+and VBr2. A collinear energy mapping based on the Perdew-
+Burke-Ernzerhof (PBE) exchange-correlation functional [44]
+yields a weakly ferromagnetic nearest neighbour interaction
+for VI2 and strong anti-ferromagnetic interactions for VCl2
+and VBr2. This is in agreement with the present result, which
+indicate that the magnetic order of VI2 is not dominated by
+nearest neighbour interactions.
+Experimentally [52], the bulk VI2 magnetic order has been
+found to undergo a phase transition at 14.4 K from a 120◦ state
+to a bicollinear state with Q = (1/2,0), where the spins are
+perpendicular to Q and tilted by 29◦ from the z-axis. Such a
+bicollinear state implies that the true ground state is a double-
+q state stabilized by higher order spin interactions and cannot
+be represented as a spin spiral in the primitive unit cell. To
+check whether LDA predicts the experimental ground state we
+have therefore performed spiral calculations in the rectangular
+cell shown in figure 1. The result is shown in figure 3 along
+with the spiral calculation in the primitive cell and we do not
+find any new minima in the super cell calculation. We have
+initalized angles in the super cell caluculation such that they
+corresponds to bicollinear order and the angles are observed
+to relax to the single-q spin spiral of the primitive cell. It
+is likely that LDA is insufficient to capture the subtle higher
+order exchange interactions in this material, but it is possible
+that the monolayer simply has a magnetic order that differs
+
+6
+Γ
+K
+M
+Γ
+X
+S
+Y
+Γ
+S
+−4
+−2
+0
+2
+4
+E(q) [meV/uc]
+FIG. 3. Spin spiral energies of VI2 obtained from the primitive cell
+(black) and the rectangular super cell (blue). The dashed lines repeat
+the primitive cell results on the corresponding super cell path.
+from the individual layers in the bulk material.
+In the PSO approximation we find that VCl2 and VBr2 pre-
+fer out-of-plane spiral planes. The energy is rather insensitive
+to ϕ forming a nearly degenerate subspace of ground states
+with a slight preference of the proper screw. The ground state
+of VI2 is found to be close to the in-plane cycloid with a nor-
+mal vector to the spiral plane forming a 6◦ angle with Q. The
+spinorbit corrections in VI2 are also found to be the smallest
+compared to other iodine based transition metal halides stud-
+ied here and the ground state energy only deviates by 0.7 meV
+per unit cell from the out-of-plane cycloid, which constitutes
+the orientation of the spin plane with highest energy.
+MnX2
+The manganese halides are all found to form 120◦ ground
+states, which is in agreement with previous theoretical studies
+[53] using PBE. In contrast to the other insulators studied in
+the present work, however, we find that the results are qualita-
+tively sensitive to the inclusion of Hubbard corrections. This
+was also found in Ref. [54], where the sign of the nearest
+neighbour exchange coupling was shown to change sign when
+a Hubbard U parameter was included in the calculations. With
+U = 3.8 eV we find that all three compounds has spiral ground
+states with incommensurate spiral vector Q = (0.11,0.11,0).
+Moreover, spin spiral band width in the LDA+U calculations
+decrease by more than an order of magnitude compared to the
+bare LDA calculations.
+The experimental magnetic structure of the manganese
+halides are rather complicated, exhibiting several magnetic
+phase transitions in a range of 0.1 K below the initial order-
+ing temperature. In particular MnI2 (MnBr2) has been found
+to have three (two) complex non-collinear phases [55], and
+MnCl2 has two complex phases that are possibly collinear
+[56].
+The experimental ground state of bulk MnCl2 not unam-
+biguously known, but under the assumption of collinearity a
+possible ground state contains 15 Mn atoms in an extended
+stripy pattern [56]. Due to the weak and subtle nature of mag-
+netic interactions in the manganese compounds, however, it is
+not unlikely that the ground state in the monolayers can dif-
+fer from that of bulk. This is corroborated by an experimental
+study of MnCl2 intercalated by graphite where a helimagnetic
+ground state with Qexp = (0.153,0.153) was found [57]. This
+is rather close to our predicted ordering vector obtained from
+LDA+U.
+Experimentally, bulk MnBr2 is found to exhibit a stripy
+bicollinear uudd order at low temperatures [58]. The order
+cannot be represented by a spiral in the minimal cell, but re-
+quires calculations in rectangular unit cells with spiral order
+Q = (0,1/2) similar to VI2 discussed above. We have calcu-
+lated the high symmetry band path required to show this order
+and do not find any new minima. It is likely that the situation
+resembles MnCl2 where a single-q spiral has been observed
+for decoupled monolayers in agreement with our calculations.
+FeX2
+We find all the iron halides to have ferromagnetic ground
+states. For FeCl2 and FeBr2 this is in agreement with the
+experimentally determined magnetic order for the bulk com-
+pounds [59]. In contrast, FeI2 has been reported to exhibit a
+bicollinear antiferromagnetic ground state [60] similar to the
+case of MnBr2 discussed above. It is again possible that the
+ground state of the monolayer (calculated here) could differ
+from the magnetic ground state of the bulk compound as has
+been found for MnCl2.
+LDA predict the three compounds to be half metals, mean-
+ing that the majority spin bands are fully occupied and only
+the minority bands have states at the Fermi level. This en-
+forces an integer number of Bohr magnetons (four) per unit
+cell at any q-vector in the spin spiral calculations. Thus longi-
+tudinal fluctuations are expected to be strongly suppressed in
+iron halides and it is likely that these materials can be accu-
+rately modelled by Heisenberg Hamiltonians despite the itin-
+erant nature of the electronic structure.
+The projected spin orbit coupling is not applicable to
+collinear structures and we therefore include full spin orbit
+coupling, which is compatible with the Q = (0,0) ground
+state. We find that all the iron compounds have an out-of-
+plane easy axis, which is in agreement with experiments. The
+bandwidth provided in table I then simply corresponds to the
+magnetic anisotropy energy which is smallest for FeCl2 and
+increases for the heavier Br and I compounds as expected.
+CoX2
+We predict CoCl2 to have an in-plane ferromagnetic ground
+state in agreement with the experimentally determined mag-
+netic order of the bulk compound [59]. CoBr2 is found to have
+a long wavelength spin spiral with Q = (0.03,0.03). The spi-
+ral energy in the vicinity of the Γ-point is, however, extremely
+flat with almost vanishing curvature and the ground state en-
+ergy is merely 0.04 meV lower than the ferromagnetic state.
+We regard this as being in agreement with the experimental re-
+port of intra-layer ferromagnetic order in the bulk compound
+[59].
+
+7
+The case of CoI2 deviates substantially from the other two
+halides. CoCl2 and CoBr2 are half-metals with m = 3 µB per
+unit cell, whereas CoI2 is an ordinary metal with m ≈ 1.2 µB
+per unit cell. We find the magnetic ground state of CoI2 to
+be stripy anti-ferromagnetic with Q = (1/2,0), whereas ex-
+periments on the bulk compound have reported helimagnetic
+in-plane order with Qexp = (1/6,1/8,1/2) in the rectangular
+cell [61]. We note, however, that the calculated local mag-
+netic moments vary strongly with q (up to 0.5 µB) in the spin
+spiral calculations, which signals strong longitudinal fluctu-
+ations. This could imply that the material comprises a rather
+challenging case for DFT and LDA may be insufficient to treat
+this material properly.
+B.
+Spontaneous polarization of AB2 materials
+The materials in table I that exhibit spin spiral ground states
+are expected to introduce a polar axis due to spinorbit cou-
+pling and thus allow for spontaneous electric polarization.
+The stripy antiferromagnet with Q = (1/2,0) preserves a site-
+centered inversion center and remains non-polar. In addition,
+the case of Q = (1/3,1/3) with in-plane orientation of the spi-
+ral plane breaks inversion symmetry, but retains the three-fold
+rotational symmetry (up to translation of a lattice vector) and
+therefore cannot acquire components of in-plane polarization.
+To investigate the effect of symmetry breaking we have
+constructed 7×1 supercells of VI2 and the Ni halides and per-
+formed a full relaxation of the q = (1/7,0) spin spiral com-
+mensurate with the supercell. This is not exactly the spin spi-
+rals found as the ground state from LDA, but we will use these
+to get a rough estimate of the spontaneous polarization. We
+note that this is very close to the in-plane component of Qexp
+for bulk NiI2, which is found to be nearly degenerate with
+the predicted ground state (see figure 2). The other materi-
+als exhibit similar near-degeneracies, but the calculated polar-
+ization could be sensitive to which spiral ordering vector is
+used. We have chosen to focus on the incommensurate spi-
+rals, but note that all the Q = (1/3,1/3) materials of table II
+are expected to introduce a spontaneous polarization as well.
+Besides the incommensurate spirals we thus only include the
+cases of MnBr2 and MnI2 where the Q = (1/3,1/3) spirals
+may be represented in
+√
+3 ×
+√
+3 supercells. The former case
+represents an example of a proper screw while the latter is an
+in-plane cycloid. The experimental order in the Mn halides
+materials is complicated, and our LDA+U calculations yield
+an ordering vector that differs from that of LDA. However,
+here we mostly consider these examples for comparison and
+to check the symmetry constraints on the polarization in the
+Q = (1/3,1/3) spirals.
+In order to calculate the spontaneous polarization we relax
+the atomic positions in the super cells both with and without
+spinorbit coupling (included self-consistently) and calculate
+the 2D polarization from the Berry phase formula [44]. The
+results are summarized in Tab. II. We can separate the effect
+of relaxation from the pure electronic contribution by calcu-
+lating the polarization (including spin-orbit) of the structures
+that were relaxed without spinorbit coupling. These numbers
+are stated in brackets in table II as well as the total polarization
+(including relaxation) and the angles that define the orienta-
+tion of the spiral plane with respect to Q. The self-consistent
+calculations yield the optimal orientations of the spiral planes
+without the PSO approximations and it is reassuring that the
+orientation roughly coincides with the results of the GBT and
+the PSO approximation.
+The magnitude of polarization largely scales with the
+atomic number of ligands (as expected from the strength of
+spinorbit coupling) and the iodide compounds thus produce
+the largest polarization. The in-plane cycloid in MnI2 only
+give rise to out-of-plane polarization as expected from sym-
+metry and the Q = (1/3,1/3) proper screw in MnBr2 has po-
+larization that is strictly aligned with Q. The latter results is
+expected for any proper screw in the ΓK-direction because Q
+then coincides with a two-fold rotational axis and the ground
+state remains invariant under the combined action of this ro-
+tation and time-reversal symmetry. Since the polarization is
+not affected by time-reversal it must be aligned with the two-
+fold axis. The polarization vectors of the remaining materials
+(except for NiCl2) are roughly aligned with the intersection
+between the spiral plane and the atomic plane.
+It is interesting to note that the calculated magnitudes of to-
+tal polarization are 5-10 times larger than the prediction from
+the pure electronic contribution where the atoms were not re-
+laxed with spinorbit coupling. We also tried to calculate the
+polarization by using the Born effective charge tensors (with-
+out spin-orbit) and the atomic deviations from the centrosym-
+metric positions. However, this approximation severely un-
+derestimates the polarization and even produces the wrong
+sign of the polarization in the case of NiBr2 and NiI2. To
+obtain reliable values for the polarization it is thus crucial to
+include the relaxation effects and take the electronic contribu-
+tion properly into account (going beyond the Born effective
+charge approximation). In Ref. [31] a value of 141 fC/m was
+predicted in 2D NiI2 from the gKNB model [62] and this is
+comparable to the values found in table II without relaxation
+effects. When relaxation is included we find a magnitude of
+1.9 pC/m for NiI2, which is an order of magnitude larger com-
+pared to the previous prediction. The results are, however, not
+directly comparable since Ref. [31] considered a spiral along
+the ΓK direction whereas the present result is for a spiral along
+ΓM. We note that Ref. [31] finds the polarization to be aligned
+with Q in agreement with the symmetry considerations above.
+Finally, the values for the spontaneous polarization in table II
+may be compared with those of ordinary 2D ferroelectrics,
+which are typically on the order of a few hundred pC/m for
+in-plane ferroelectrics and a few pC/m for out-of-plane ferro-
+electrics [63] .
+In all of these type II multiferroics, the orientation of the
+induced polarization depends on the direction of the ordering
+vector, which may thus be switched by application of an ex-
+ternal electric field. We have checked explicitly that the sign
+of polarization is changed if we relax a right-handed instead
+of a left-handed spiral (corresponding to a reversed ordering
+vector). The small values of spontaneous polarization in these
+materials implies that rather modest electric fields are required
+for switching the ordering vector and thus comprise an in-
+
+8
+(θ,ϕ)
+P∥
+P⊥
+Pz
+VI2
+(11, 0)
+-0.6 (-31)
+290 (96)
+0.05 (0.11)
+NiCl2 (90, -30) -37 (-1.4)
+76 (15)
+3.5(-5.1)
+NiBr2 (69, -10)
+12 (-6)
+340 (32)
+26 (37)
+NiI2
+(70, 0)
+-8 (-48)
+1890 (400) -0.18 (12)
+MnBr2
+(90, 0)
+430 (38)
+0 (0.02)
+0 (0)
+MnI2
+(0, 0)
+0 (0.6)
+0.3 (-7)
+-260 (-105)
+TABLE II. Orientation of spin planes, and 2D polarization (in fC/m)
+of selected transition metal halides. P∥ denotes the polarization along
+Q, while P⊥ denotes the polarization in the atomic plane orthogonal
+to Q and Pz is the polarization orthogonal to the atomic plane. The
+numbers in brackets are the polarization values obtained prior to re-
+laxation of atomic positions. We have used 7×1 supercells for the V
+and Ni halides and
+√
+3×
+√
+3 supercells for the Mn halides. All calcu-
+lations are set up with left-handed spirals. The numbers in brackets
+state the spontaneous polarization without relaxation effects.
+teresting alternative to standard multiferroics such as BiFeO3
+and YMnO3, where the coercive electric fields are orders of
+magnitude larger.
+C.
+Magnetic ground state of AB3 materials
+The AB3 materials all have space group P¯3m1 correspond-
+ing to monolayers of the BI3 (or AlCl3) prototype. The mag-
+netic lattice is the honeycomb motif, thus hosting two mag-
+netic ions in the primitive cell. Several materials of this pro-
+totype have been characterized experimentally, but here we
+only present results for the Cr compounds. This is due to the
+fact that experimental data of in-plane order is missing for all
+but CrX3, FeCl3 and RuCl3. Moreover, all magnetic com-
+pounds were found to have a simple ferromagnetic ground
+state. RuCl3 is a well known insulator with stripy antiferro-
+magnetic in-plane order. However, bare LDA finds a metallic
+state and both Hubbard corrections and self-consistent spinor-
+bit coupling are required to obtain the correct insulating state
+[64]. The latter is incompatible with the GBT approach and
+we have not pursued this further here. Bulk FeCl3 is known to
+be an insulating helimagnet with Q = ( 4
+15, 1
+15, 3
+2) [65], while
+we find the monolayer to be a metallic ferromagnet.
+For CrI3 we compare the spin spiral dispersion to the spiral
+energy determined by a third nearest neighbour energy map-
+ping procedure. The prototype thus serves as a testing ground
+for applying unconstrained GBT to materials with multiple
+magnetic atoms in the unit cell. We analyse the intracell an-
+gle between the Cr atoms of CrI3 and provide an expression
+for generating good initial magnetic moments for GBT calcu-
+lations. We finally discuss the observed deviations from the
+classical Heisenberg model and to what extend the flat spiral
+spectrum can be used to obtain the magnon excitation spec-
+trum.
+CrX3
+The chromium trihalides are of considerable interest due
+to the versatile properties that arise across the three different
+halides. Monolayer CrI3 was the first 2D monolayer that were
+demonstrated to host ferromagnetic order below 45 K [1] and
+has spurred intensive scrutiny in the physics of 2D magnetism.
+The magnetic order is governed by strong magnetic easy-axis
+anisotropy, which is accurately reproduced by first principles
+simulations [66, 67]. In contrast, monolayers of CrCl3 exhibit
+ferromagnetic interactions as well, but no proper long range
+order due easy-plane anisotropy. Instead, these monolayers
+exhibit Kosterlitz-Thouless physics, which give rise to quasi
+long range order below 13 K [9].
+The GBT is not really necessary to find the ground state of
+the monolayer chromium halides. They are all ferromagnetic
+and insulating and only involve short range exchange interac-
+tions that are readily obtained from collinear energy mapping
+methods [66–68]. Nevertheless, the gap between the acoustic
+and optical magnons in bulk CrI3 has been proposed to arise
+from either (second neighbor) Dzyalosinskii-Moriya interac-
+tions [69] or Kitaev interactions [70, 71]. The former could in
+principle be extracted directly from planar spin-spiral calcu-
+lations [40], while the latter requires conical spin spirals. The
+origin of this gap is, however, still subject to debate [72] and
+here we will mainly focus on the magnetic interactions that do
+not rely on spinorbit coupling. In the following we will focus
+on CrI3 as a representative member of the family.
+The honeycomb lattice contains two magnetic atoms per
+unit cell and the magnetic moments at the two sites will in
+general differ by an angle ξ. Since we do not impose any con-
+straints except for the boundary conditions specified by q, the
+angle will be relaxed to its optimal value when the Kohn-Sham
+equations are solved self-consistently. The convergence of ξ,
+may be a tedious process since the total energy has a rather
+weak dependence on ξ. For a given q the classical energy of
+the model (1) is minimized by the angle ξ 0 given by
+tanξ 0 = −ImJ12(q)
+ReJ12(q),
+(8)
+where
+J12(q) = ∑
+i
+J12
+0i e−iq·Ri
+(9)
+is the Fourier transform of the inter-sublattice exchange cou-
+pling. If one assumes nearest neighbour interactions only, ξ 0
+becomes independent of exchange parameters and the result-
+ing expression thus comprises a suitable initial guess for the
+inter-sublattice angle. We note that the classical spiral en-
+ergy is independent of ξ (in the absence of spinorbit coupling)
+when J12(q) = 0 and the angle may be discontinuous at such
+q-points. This occurs for example in the magnetic honeycomb
+lattice at the K-point (q = (1/3,1/3)). In general, Eq. (8) has
+two solutions that differ by π and only one of these minimzes
+the energy while the other maximizes it. The maximum en-
+ergy constitutes an ”optical” spin spiral branch, which is if
+interest if one wishes to extract the exchange coupling con-
+stants.
+The spiral energies of CrI3 (with optimized intracell angles)
+are shown in figure 4, where we show both the ferromagnetic
+(ξ = 0) and the antiferromagnetic (ξ = π) results on the ΓK
+
+9
+Γ
+M
+K
+Γ
+0
+10
+20
+30
+E(q) [meV]
+Mapping
+FM
+AFM
+Γ
+M
+K
+Γ
+2.92
+2.94
+m [µB]
+Γ
+M
+K
+Γ
+0
+90
+180
+ξ [o]
+FIG. 4. (Left: spin spiral energies of CrI3 compared to third nearest neighbour energy mapping. Right: angles beteen the two magnetic
+moments. The spin spirals are initialised with angles determined by Eq. (8) which are shown in black. The moments are collinear on the ΓK
+path and so the AFM solution is also quasi-stable in DFT. Center: the magnitude of local magnetic moments along the spiral path.
+path. We also show the spiral energy obtained from the model
+(1) with exchange parameters calculated from a collinear en-
+ergy mapping using four differnet spin configurations. We get
+J1 = 2.47 meV, J2 = 0.682 meV and J3 = −0.247 meV for the
+first, second and nearest neighbour interactions respectively,
+which is in good agreement with previous LDA calculations
+[73]. The model spiral energy is seen to agree very well with
+that obtained from the GBT, which largely validates such a
+three-parameter model (when spinorbit is neglected). We do,
+however, find a small deviation in the regions between high-
+symmetry points. This is likely due to higher order exchange
+interaction, which will deviate in the two approaches. For
+example, a biquadratic exchange term [38], will cancel out
+in any collinear mapping, but will influence the energies ob-
+tained from the GBT. Biquadratic exchange parameters could
+thus be extracted from the deviation between the two calcula-
+tions.
+In figure 4 we also show the calculated values of ξ and
+the magnitude of the local magnetic moment at the Cr sites
+along the path. The self-consistent intracell angles are found
+to match very well with the initial guess, except for a slight
+deviation on the Brillouin zone boundary. This corroborates
+the fact that exchange couplings beyond second neighbours
+are insignificant (the second nearest neighbor coupling is an
+intra-sublattice interaction and does not influence the angle).
+It is also rather instructive to analyze the variation in the
+magnitude of local magnetic moments. In general, the map-
+ping of electronic structure problems to Heisenberg types of
+models like (1) rests on an adiabatic assumption where it is
+assumed that the magnitude of the moments are fixed. How-
+ever, the present variation in the magnitude of moments does
+not imply a breakdown of the adiabatic assumption, but re-
+flects that DFT should be mapped to a quantum mechanical
+Heisenberg model rather than a classical model. In particu-
+lar, the ratio of spin expectation values between the ferromag-
+netic ground state and the (anti-ferromagnetic) state of highest
+energy is approximately ⟨Si⟩AFM/⟨Si⟩FM = 0.83 in the quan-
+tized model [74]. While this ratio is somewhat smaller than
+the difference between ferromagnetic and anti-ferromagnetic
+moments found here, the result does imply that the magni-
+tude of moments should depend on q. And the fact that the
+q = 0 anti-ferromagnetic moments are smaller than the ferro-
+magnetic ones in a self-consistent treatments reflects that DFT
+captures part of the quantum fluctuations inherent to the model
+(1).
+We note that the spin spiral energy Eq calculated from the
+isotropic Heisenberg model using the optimal angle given by
+Eq. (8) is related to the dynamical excitations (magnon ener-
+gies) by ω±
+q = E±
+q /S and the spiral energies thus comprise a
+simple method to get the magnetic excitation spectrum. How-
+ever, even if a model like (1) fully describes a magnetic mate-
+rial (no anisotropy or higher order terms) there will be a sys-
+tematic error in the extracted exchange parameters (and re-
+sulting magnon spectrum) if the parameters are extracted by
+mapping to the classical model. The reason is, that the clas-
+sical energies correspond to expectation values of spin con-
+figurations with fixed magnitude of the spin, which is not ac-
+commodated in a self-consistent approach. This error is di-
+rectly reflected by the variation of the magnitude of moments
+in figure 4. The true exchange parameters can only be ob-
+tained either by mapping to eigenstates of the model [74] or
+by considering infinitesimal rotations of the spin, which may
+be handled non-selfconsistently using the magnetic force the-
+orem [37, 75–78]. Nevertheless, the deviations between ex-
+change parameters obtained from classical and quantum me-
+chanical energy mapping typically deviates by less than 5 %
+[74] and for insulators it is a good approximation to extract
+the magnon energies from planar spiral calculations although
+the mapping is only strictly valid in the limit of small q.
+IV.
+CONCLUSION AND OUTLOOK
+In conclusion, we have demonstrated the abundance of spi-
+ral magnetic order in 2D transition metal dichalcogenides
+from first principles calculations. The calculations imply that
+type II multiferroic order is rather common in these materials
+and we have calculated the spontaneous polarization in a se-
+lected subset of these using fully relaxed structures in super
+cells. While the super cell calculations does not correspond to
+the exact spirals found from the GBT, the calculations show
+that relaxation effects plays a crucial role for the induced po-
+larization and should be taken into account in any quantitative
+analysis. The spontaneous polarization in type II multifer-
+roics is in general rather small compared to what is found in
+ordinary 2D ferroelectrics and could imply that the chirality
+
+10
+of spirals are switchable by small electric fields. It would be
+highly interesting to calculate the coercive field for switch-
+ing in these materials, but due to the importance of relaxation
+effects and spin-orbit coupling this is a non-trivial computa-
+tion that cannot simply be obtained from the Born effective
+charges and force constant matrix.
+The GBT comprises a powerful framework for extract-
+ing the magnetic properties of materials from first princi-
+ples. In addition to the single-q states considered here, one
+may use super cells to extract the importance of higher order
+exchange interactions and unravel the possibility of having
+multi-q ground states. In addition, for non-centrosymmetric
+materials, the PSO approach may be readily applied to ob-
+tain the Dzyaloshinskii-Moriya interactions, which may lead
+to Skyrmion lattice ground states or stabilize other multi-q
+states.
+V.
+APPENDIX
+A.
+Implementation
+In the PAW formalism we expand the spiral spinors using the standard PAW transformation [79]
+ψq,k(r) = ˆ
+T ˜ψq,k(r) = ˜ψq,k(r)+∑
+a ∑
+i
+(φ a
+i (r)− ˜φ a
+i (r))
+�
+dr[ ˜pa
+i (r)]∗ ˜ψq,k(r),
+(10)
+where ˜ψq,k(r) is a smooth (spinor) pseudo-wavefunction that coincides with ψq,k(r) outside the augmentation spheres and devi-
+ates from ψq,k(r) by the second term inside the augmentation spheres. The all-electron wavefunction ψq,k(r) is thus expanded
+in terms of (spinor) atomic orbitals φ a
+i inside the PAW spheres and the expansion coefficients are given by the overlap between
+the pseudowavefunction and atom-centered spinor projector functions ˜pa
+i . Using Eq. (3) we may write this as
+ψq,k(r) = eik·rU†
+q(r) ˜uq,k(r)+∑
+a ∑
+i
+(φ a
+i (r)− ˜φ a
+i (r))
+�
+dr[ ˜pa
+i (r)]∗eik·rU†
+q(r) ˜uq,k(r)
+= eik·rU†
+q(r) ˜uq,k(r)+∑
+a ∑
+i
+(φ a
+i (r)− ˜φ a
+i (r))
+�
+dr[e−ik·rUq(r) ˜pa
+i (r)]∗ ˜uq,k(r)
+= eik·rU†
+q(r) ˜uq,k(r)+∑
+a ∑
+i
+(φ a
+i (r)− ˜φ a
+i (r))
+�
+dr[ ˜pa
+i,q,k(r)]∗ ˜uq,k(r)
+≡ Tq,k ˜uq,k(r),
+(11)
+where Uq(r) was given in Eq. (4) and we defined
+˜pa
+i,q,k(r) = e−ik·rUq(r) ˜pa
+i (r).
+(12)
+The PAW transformed Kohn-Sham equations then read
+˜Hq,k ˜uq,k(r) = εq,kSq,k ˜uq,k(r),
+(13)
+with
+˜Hq,k = T †
+q,kHTq,k,
+Sq,k = T †
+q,kTq,k.
+(14)
+Calculations in the framework of the GBT thus requires two modifications compared to the approach for solving the ordinary
+Kohn-Sham equations in the PAW formalism. 1) The k-dependence of the standard Bloch Hamiltonian is replaced by k →
+k ∓ q/2 for spin-up and spin down components respectively. 2) Different spin dependent projector functions has to be applied
+when calculating the projector overlaps with the spin-up and spin-down components of the psudowavefunctions (see Eq. (12)).
+B.
+Benchmark
+The LDA implementation of the GBT have been tested by
+checking that our results agree with similar calculations from
+the literature and by verifying internal consistency by compar-
+ing with super cell calculations. The case of fcc Fe has been
+found to have a spin spiral ground state [81] and the calcu-
+lation of the ordering vector Q has been become a standard
+benchmark for spin spiral implementations [82]. In previous
+simulations the ordering vector was found to be rather sensi-
+tive to the lattice constant and in figure 6 we show the spin spi-
+ral energies along the ΓXW path using the experimental lattice
+constant as well as the lattice constant which has been found
+to reproduce the experimental ordering vector [80]. The cal-
+culated value of Q is in good agreement with previous reports
+in both cases [83]. We also confirm a similar low energy bar-
+
+11
+−0.4
+−0.2
+0.0
+0.2
+0.4
+qx
+0.0
+0.1
+0.2
+E − E0 [eV]
+Spiral
+Supercell
+FIG. 5. Comparison between GBT spin spiral calculations and su-
+percell calculations without spinorbit coupling in monolayer CoPt.
+100
+120
+140
+3.58 ˚A (exp)
+3.50 ˚A
+Γ
+X
+W
+−20
+−10
+0
+E(q) [meV]
+FIG. 6. Spin spiral energies of fcc Fe for the experimental lattice
+constant (red) and a strained latice constant, which is known to re-
+produce the experimental spin spiral order in (blue). The dashed
+vertical lines indicate the minima found in Ref. [80].
+rier between the two local minima, as is expected from LDA
+[84].
+In order to check internal consistency we have investigated
+the case of monolayer CoPt [40] where we compare spin spi-
+ral energies calculated using the GBT with energies calculated
+from super cells. We thus construct a 16x1 super cell of the
+CoPt monolayer and consider spirals with qc = ( n
+16) in units of
+reciprocal lattice vectors. This allows us to extract 16 different
+spiral energies in the supercell using standard non-collinear
+DFT. In order to compare the two methods we have used a
+k-point grid of 16×16×1 for the GBT and 1×16×1 for the
+supercell and a plane wave cutoff of 700 eV for both calcu-
+lations. In Fig. 5 we compare the results without spinorbit
+coupling and find excellent agreement between supercell and
+GBT calculations. We note that when spinorbit coupling is
+neglected one has Eq = E−q.
+Since spinorbit coupling is incompatible with the GBT one
+has to resort to approximate schemes to include it in the cal-
+culations. In the present work we have used the PSO method
+proposed by Sandratskii [40]. In Fig. 7 we compare spin spi-
+ral calculations with supercell calculations where the spinorbit
+coupling has been included either fully or by the PSO method.
+The PSO method is fully compatible with the GBT and we
+−0.5
+−0.25
+0
+0.25
+0.5
+qx
+0.00
+0.05
+0.10
+0.15
+0.20
+E − E0 [eV]
+Spiral Proj. SOC
+Spiral Full SOC
+SC Full SOC
+SC Projected SOC
+FIG. 7. Comparison between GBT spin spiral calculations and super-
+cell calculations with projected and full spinorbit coupling in mono-
+layer CoPt.
+find excellent agreement between the spin spiral energies cal-
+culated with the GBT and with supercells. The PSO approach
+is, however an approximation and the correct result can only
+be obtained from the supercell using the full spinorbit cou-
+pling. We see that the PSO calculations are in good agreement
+with those obtained from full spinorbit coupling but overesti-
+mates the energies at the Brillouin zone boundary by a few
+percent. In contrast, if one tries to include the full spinor-
+bit operator in the GBT calculations (by diagonalizing HKS
+including spinorbit coupling on a basis of GBT eigenstates
+without spinorbit coupling) the energies are severely under-
+estimated with respect to the exact result (from the supercell
+calculation). We note that the spiral energies including spinor-
+bit coupling shows a slight asymmetry between points at q and
+−q, which can be related to the Dzyaloshinskii-Moriya inter-
+actions in the system [40].
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+SUPPLEMENTARY INFORMATION - TYPE II MULTIFERROIC ORDER IN TWO-DIMENSIONAL TRANSITION METAL
+HALIDES FROM FIRST PRINCIPLES SPIN-SPIRAL CALCULATIONS
+Joachim Sødequist1 and Thomas Olsen1,∗
+1CAMD, Department of Physics, Technical University of Denmark, 2820 Kgs. Lyngby Denmark
+I.
+SPIN SPIRAL DISPERSIONS
+The entire spin spiral dispersion carry more information than just the energy minima was reported we reported in the main text,
+these are shown here for in figures 2 and 4. One can find not only the stability with respect to the ferromagnetic configuration,
+but also compare to any other configuration in the energy landscape. Additionally, we can observe whether the remain magnetic
+moments are unchanged during self-consistent field cycle, and we find this is generally true except for CoI2 and perhaps the
+titanium compounds. We note that the local magnetic moments found here are the integrated inside the PAW spheres of the
+respective atoms, thus these local moments does not integrate to the total moments reported in the main text since interstitial
+magnetization density is neglected. We also provide the projected spin orbit energies in figure 3 for the lowest energy state,
+naturally the shape will depend very much on the specific spiral, in some cases such as the q = K we find quite similar energies in
+the out-of-plane orientations, whereas incommensurate spirals tend to have more well defined minima. The in-plane orientations
+reported here are related by a 90◦ phase shift, but the dashed line highlight that they are indeed degenerate as expected.
+II.
+SPIN SPIRAL CONVERGENCE
+An example of convergence of a intracell angle in a rectangular supercell of the hexagonal VI2 system is represented in 1. We
+find that for all calculations which reach the convergence criteria on particularly the density, all converge the angle within some
+narrow region of the true angle. We observe that the number of iterations required increase dramatically, when the initial guess
+is further away from the true angle, hence highlighting the importance of choosing initial conditions according to Eq. (8) in the
+main text.
+FIG. 1. Convergence of the intracell angle ξ in spin spiral ground state calculation of VI2 at the spiral vector q = (1/4,0,0) at varying different
+initial conditions. The calculations shown in red, did not reach the convergence criteria on the density within the time-wall of the calculation,
+while the blue were considered converged. The black horizontal line is the expected angle for a smooth spin spiral as it if it was an equivalent
+spin spiral in the primitive unit cell.
+
+Vl2convergence,g=(1/4,0,0),rect
+175
+150
+125
+5-angle
+100
+75
+50
+25
+0
+0
+200
+400
+600
+800
+1000
+SCF-count16
+FIG. 2. Spin spiral energies for AB2 magnets and the local magnetic moments of the magnetic atoms and ligands. For ferromagnetic refer to
+Fig. 4.
+
+TiBr2
+Wave length ^ [A]
+inf
+11.2
+5.6
+6.5
+12.9
+inf
+0
+[]
+Energy
+-10
+Ti magmom
+1.2
+[meV]
+I norm magnetic moment 
+-20
+Br magmom
+1.0
+Spin spiral energy [
+-30
+0.8
+40
+0.6
+50
+0.4
+-60
+S
+-70
+0.2
+ocal
+-80
+Lo
+0.0
+K
+M
+q vector [A-1]Til2
+Wave length ^ [A]
+inf
+12.3
+6.1
+7.1
+14.1
+inf
+0
+n magnetic moment [lμsl]
+Energy
+1.4
+Ti magmom
+M
+-10
+[me]
+I magmom
+1.2
+ Spin spiral energy
+-20
+0.8
+0.6
+-30
+ocal norm
+0.4
+-40
+0.2
+Lo
+0.0
+K
+M
+q vector [A-1]VCI2
+Wave length ^ [A]
+inf
+11.0
+5.5
+6.3
+12.7
+inf
+0
+一
+门
+Energy
+2.5
+-10
+V magmom
+[meV]
+moment
+Cl magmom
+2.0
+-20
+ Spin spiral energy 
+1.5
+5
+-30
+40
+1.0
+-50
+0.5
+-60
+0.0
+K
+M
+q vector [A-1]VBr2
+Wave length ^ [A]
+inf
+11.5
+5.8
+6.7
+13.3
+inf
+0
+门
+: moment [lμl]
+Energy
+2.5
+V magmom
+[meV]
+Br magmom
+-10
+2.0
+ Spin spiral energy
+-15
+1.5
+-20
+1.0
+25
+30
+0.5
+-35
+0.0
+K
+M
+q vector [A-1]V12
+Wave length 入 [A]
+inf
+12.4
+6.2
+7.1
+14.3
+inf
+６
+ Energy
+2.5
+V magmom
+[meV]
+4 
+I magmom
+2.0
+I energy
+2
+Local norm magnetic
+1.5
+Spin spiral 
+0
+1.0
+0.5
+0.0
+K
+M
+q vector [A-1]MnCI2
+Wave length ^ [A]
+inf
+11.2
+5.6
+6.4
+12.9
+inf
+0
+Energy
+Mn magmom
+Local norm magnetic moment [
+Spin spiral energy [meV]
+Cl magmom
+-10
+2
+15
+-20
+K
+M
+q vector [A-1]MnBr2
+Wave length ^ [A]
+inf
+11.7
+5.8
+6.7
+13.5
+inf
+0.0
+[|μB|] 
+Energy
+-2.5
+Mn magmom
+[meV]
+-5.0
+Br magmom
+7.5
+I energy
+ Spin spiral
+2
+-12.5
+-20.0
+0
+K
+M
+q vector [A-1]Mn12
+Wave length ^ [A]
+inf
+12.5
+6.2
+7.2
+14.4
+inf
+0
+Energy
+Mn magmom
+ Spin spiral energy [meV]
+I magmom
+10
+15
+-20
+K
+M
+q vector [A-1]CoBr2
+Wave length 入 [A]
+inf
+11.2
+5.6
+6.5
+12.9
+inf
+Energy
+Co magmom
+[meV]
+20
+2.0
+Br magmom
+Spin spiral energy
+15
+1.5
+Local norm magnetic i
+10
+1.0
+5
+0.0
+0
+K
+M
+q vector [A-1]Col2
+Wave length ^ [A]
+inf
+11.7
+5.8
+6.7
+13.5
+inf
+Energy
+20 
+Co magmom
+1.2
+Spin spiral energy [meV]
+I magmom
+10
+1.0
+0.8
+0.6
+0.4
+-10
+0.2
+-20
+0.0
+K
+M
+q vector [A-1]NiCI2
+Wave length 入 [A]
+inf
+10.5
+5.3
+6.1
+12.1
+inf
+50 
+Energy
+1.4
+Ni magmom
+Spin spiral energy [meV]
+1.2
+40
+Cl magmom
+1.0
+30
+0.8
+Local norm magnetic
+20
+0.6
+0.4
+10
+0.2
+0
+0.0
+K
+M
+q vector [A-1]NiBr2
+Wave length 入 [A]
+inf
+11.1
+5.5
+6.4
+12.8
+inf
+1.4
+Energy
+n magnetic moment [lμbl
+40
+Ni magmom
+1.2
+[meV]
+Br magmom
+1.0
+30
+Spin spiral energy
+0.8
+20
+0.6
+10
+0.4
+Local norm
+0.2
+0
+0.0
+K
+M
+q vector [A-1]Nil2
+Wave length ^ [A]
+inf
+11.9
+6.0
+6.9
+13.8
+inf
+1.2
+40
+Energy
+magnetic moment [lμ]
+Ni magmom
+[meV]
+1.0
+30 
+I magmom
+20 -
+0.8
+T energy
+10 -
+0.6
+Spin spiral
+0
+0.4
+ norm
+-10
+0.2
+-20
+ocal
+0.0
+-30 -
+K
+M
+q vector [A-1]17
+FIG. 3. Projected spin orbit energies of the ground state found in Fig. 2
+
+TiBr2
+-5.1
+-
+-
+-
+-
+-5.2 -
+-
+-
+-
+-
+-
+-
+-
+-
+[meV]
+-5.3
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-5.4
+-
+-
+E
+-
+-
+-
+-5.5
+-
+-
+-5.6 -
+-
+-
+-
+IP
+Screw
+OoP
+IP
+theta, phiTil2
+-
+-25.2
+-
+-
+-25.4
+-
+-
+-
+[meV]
+25.6
+-25.8
+-26.0
+-
+-
+-
+-
+IP
+Screw
+OoP
+IP
+theta, phiVCI2
+-0.68
+-0.69
+-0.70
+[meV]
+-0.71
+-0.72
+-0.73
+-0.74
+IP
+Screw
+OoP
+IP
+theta, phiVBr2
+-5.28
+-
+-
+-5.29
+-
+-5.30
+-
+-
+-
+-5.31
+[meV]
+-5.32
+Jos:
+-5.33
+E
+-5.34
+-5.35
+-
+-
+-
+-5.36
+IP
+Screw
+OoP
+IP
+theta, phiVI2
+-25.2
+-25.3
+-25.4
+25.5
+25.6
+E
+-25.7
+-25.8
+-25.9
+IP
+Screw
+OoP
+IP
+theta, phiMnCI2
+-1.780
+-
+-1.785
+-
+-
+-
+-
+-
+-
+[meV]
+790
+-
+-
+-
+Jos:
+-
+-
+E
+-1.795
+-
+-
+-1.800
+-
+-
+-1.805
+IP
+Screw
+OoP
+IP
+theta, phiMnBr2
+-
+-6.17
+-6.18
+6.19
+-6.20
+-6.21
+-
+-
+-
+-
+-
+-6.22
+IP
+Screw
+OoP
+IP
+theta, phiMn12
+-27.4
+-
+-
+-27.6
+-
+-
+-
+-27.8
+[meV]
+-28.0
+-28.2
+-28.4
+IP
+Screw
+OoP
+IP
+theta, phiCoBr2
+-8.38
+-
+-8.40
+[meV]
+-8.42
+soc
+E
+-8.44
+-8.46
+-8.48
+IP
+Screw
+OoP
+IP
+theta, phiCol2
+-40
+-41 
+[meV]
+-42
+soc
+E
+-43
+-44 -
+-45 -
+-
+IP
+Screw
+OoP
+IP
+theta, phiNiCI2
+-3.050
+-3.055
+[meV]
+-3.060
+-3.065
+-3.070
+IP
+Screw
+OoP
+IP
+theta, phiNiBr2
+-
+-8.65
+-8.70
+ [meV]
+-8.75
+-8.80
+E
+-8.85
+-8.90
+-
+IP
+Screw
+OoP
+IP
+theta, phiNil2
+-
+-40
+-
+-41 -
+[meV]
+-42
+-
+-43
+-44 :
+-
+-
+IP
+Screw
+OoP
+IP
+theta, phi18
+FIG. 4. Spin spiral energies for ferromagnetic AB2 magnets and the local magnetic moments on the atoms
+
+FeCI2
+Wave length 入 [A]
+inf
+10.2
+6.0
+6.2
+12.4
+inf
+3.5
+Energy
+: moment [lμb]
+120
+Fe magmom
+[meV]
+3.0
+Cl magmom
+100
+2.5
+Spin spiral energy
+80
+2.0
+Local norm magnetic
+60
+1.5
+40
+1.0
+0.5
+20
+0.0
+0
+M
+K
+q vector [A-1]FeBr2
+Wave length 入 [A]
+inf
+10.8
+6.4
+6.5
+13.1
+inf
+3.5
+Energy
+ moment [lμBl]
+80
+Fe magmom
+3.0
+Spin spiral energy [meV]
+Br magmom
+2.5
+60
+2.0
+Local norm magnetic
+40
+1.5
+1.0
+0>
+0.5
+0.0
+M
+K
+q vector [A-1]Fe12
+Wave length 入 [A]
+inf
+11.6
+6.8
+7.0
+14.1
+inf
+3.5
+40
+Energy
+[8l]
+35
+Fe magmom
+3.0
+ Spin spiral energy [meV]
+Local norm magnetic moment [
+I magmom
+30
+2.5
+25
+2.0
+20
+1.5
+15
+1.0
+10
+0.5
+5
+0.0
+0
+M
+K
+q vector [A-1]CoCI2
+Wave length 入 [A]
+inf
+10.6
+5.3
+6.1
+12.3
+inf
+2.5
+50
+Energy
+Local norm magnetic moment [lμbl
+Co magmom
+Spin spiral energy [meV]
+Cl magmom
+2.0
+40
+1.5
+30
+1.0
+20
+0.5
+10
+0.0
+K
+M
+q vector [A-1]

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+Primordial black holes in loop
+quantum gravity: The effect on the
+threshold
+Theodoros Papanikolaoua
+aNational Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece
+E-mail: papaniko@noa.gr
+Abstract. Primordial black holes form in the early Universe and constitute one of the
+most viable candidates for dark matter. The study of their formation process requires
+the determination of a critical energy density perturbation threshold δc, which in general
+depends on the underlying gravity theory.
+Up to now, the majority of analytic and
+numerical techniques calculate δc within the framework of general relativity.
+In this
+work, using simple physical arguments we estimate semi-analytically the PBH formation
+threshold within the framework of quantum gravity, working for concreteness within
+loop quantum gravity (LQG), which constitutes a non-perturbative and background-
+independent quantization of general relativity. In particular, for low mass PBHs formed
+close to the quantum bounce, we find a reduction in the value of δc up to 50% compared to
+the general relativistic regime quantifying for the first time to the best of our knowledge
+how quantum effects can influence PBH formation within a quantum gravity framework.
+Finally, by varying the Barbero-Immirzi parameter γ of LQG we show its effect on
+the value of δc while using the observational/phenomenological signatures associated to
+ultra-light PBHs, namely the ones affected by LQG effects, we propose the PBH portal
+as a novel probe to constrain the potential quantum nature of gravity.
+Keywords: primordial black holes, quantum gravity, loop quantum gravity
+arXiv:2301.11439v1  [gr-qc]  26 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+The fundamentals of loop quantum gravity
+2
+2.1
+The classical dynamics
+2
+2.2
+The quantum dynamics
+4
+3
+The threshold of primordial black hole formation in loop quantum
+gravity
+5
+4
+Results
+8
+5
+Conclusions
+10
+1
+Introduction
+PBHs, firstly proposed in early ’70s [1–3], form in the early universe, typically during
+the Hot Big Bang (HBB) radiation-dominated (RD) era out of the gravitational collapse
+of enhanced cosmological perturbations.
+According to recent arguments, PBHs can
+naturally account for a part or the totality of dark matter [4, 5], seed the large-scale
+structures through Poisson fluctuations [6–9] as well as the primordial magnetic fields
+through the presence of disks around them [10, 11]. At the same time, they are associated
+with a plethora of gravitational-wave (GW) signals from black-hole merging events [12–
+16] up to primordial scalar induced GWs [17–22] (for a recent review see [23]).
+In
+particular, through the aforementioned GW portal, PBHs can act as well as a novel
+probe shedding light on the underlying gravity theory [24, 25]. Other hints in favor of
+PBHs can be found here [26].
+In the standard PBH formation scenario, where PBHs form from the collapse of lo-
+cal overdensity regions, the PBH formation threshold δc depends in general on the shape
+of the energy density perturbation profile of the collapsing overdensity [27–30] as well
+as on the equation-of-state parameter at the time of PBH gravitational collapse [30–33].
+This critical threshold value is very important since it can affect significantly the abun-
+dance of PBHs, a quantity which is constrained by numerous observational probes [34].
+From a historic perspective, after a first analytic calculation of δc by B. Carr and
+S. Hawking in 1975 [31, 35], δc was studied mostly through numerical hydrodynamic
+simulations by [36–40]. Within the last decade, there has been witnessed a remarkable
+progress regarding the determination δc both at the analytic as well as at the numerical
+level.
+In particular, at the analytic level, T.Harada, C-M. Yoo & K. Kohri (HYK)
+in 2013 [32] refined the PBH formation threshold value obtained by Carr in 1975 by
+comparing the time at which the pressure sound wave crosses the overdensity collapsing
+– 1 –
+
+to a PBH with the onset time of the gravitational collapse. Their expression for δc in
+the uniform Hubble gauge reads as:
+δc = sin2
+� π√w
+1 + 3w
+�
+.
+(1.1)
+At this point, it is very important to stress that very recently there was exhibited a
+rekindled interest in the scientific community regarding the effect of non-linearities [41–
+45] and non-Gaussianities [46–50] on the value of δc. In addition, some first research
+works were also performed regarding the dependence of the PBH formation threshold on
+non sphericities [51, 52], on anisotropies [53], on the velocity dispersion of the collapsing
+matter [54] as well as within the context of modified theories of gravity [55].
+In this work, we study semi-analytically the effect of the potential quantumness of
+spacetime on the determination of the PBH formation threshold by using simple physical
+arguments studying whether the PBH portal can act as a novel way to probe the quantum
+nature of gravity. For concreteness, we work within the framework of loop quantum
+gravity (LQG) [56, 57] which constitutes a nonperturbative and background-independent
+quantization of general relativity. Very interestingly, LQG is able to solve the problem of
+past and future singularities [58] and provide the initial conditions for inflation, solving
+in this way naturally the flatness and the horizon cosmological problems [59]. It can also
+account for the large scale structure formation [60] as well as for the currently observed
+cosmic acceleration [61–63]. PBHs were studied firstly within the context of LQG in [64]
+where the PBH evolution was explored accounting for the effects of Hawking radiation
+and accretion in a LQG background. In the present work, we investigate the effect of
+LQG on the PBH formation process and in particular at the level of the determination
+of the PBH formation threshold.
+The paper is organized as follows: In Sec. 2 we revise the basics of loop quantum
+gravity. Then, in Sec. 3 we determine semi-analytically the PBH formation threshold δc
+by comparing the gravity and the sound wave pressure forces. Followingly, in Sec. 4 we
+present our results while Sec. 5 is devoted to conclusions.
+2
+The fundamentals of loop quantum gravity
+Loop quantum gravity brings conceptually together the two fundamental pillars of mod-
+ern physics, namely General Relativity (GR) and Quantum Mechanics (QM). It consti-
+tutes actually a non-perturbative and background-independent quantization of general
+relativity [65, 66]. In particular, it is based on a connection-dynamical formulation of
+GR defined on a spacetime manifold M = R × Σ, where Σ stands for the 3D spatial
+manifold.
+2.1
+The classical dynamics
+Working within the Hamiltonian framework, the classical phase space consists of the
+Ashtekar-Barbero variables which are actually the two canonically conjugate variables
+– 2 –
+
+of the theory. These variables are the densitized triad Ea
+i and the Ashtekar connection
+Ai
+a defined as follows [65–68]:
+Ea
+i = |det(eb
+j)|−1ea
+i ,
+(2.1)
+Ai
+a
+= Γi
+a + γKi
+a,
+(2.2)
+where ea
+i is the triad field, Γi
+a is the spin connection, Ki
+a is the extrinsic curvature and
+γ is the so-called Barbero-Immirzi parameter which allows the quantisation procedure
+to be performed on a compact group. Such a setup is based on a 3+1 decomposition of
+the metric written in the following form:
+ds2 = N2dt2 − qab(dxa + Nadt)(dxb + Nbdt),
+(2.3)
+where qab = ei
+aeib is the spatial metric, N is the lapse function and Ni is the shift
+vector. This metric choice simplifies the quantisation process and is chosen for conve-
+nience. However, as said before, the LQG background equations will not depend on
+the choice of the spacetime metric. This independence of the background on the choice
+of the spacetime foliation is associated to some constraints. Firstly, one should require
+the diffeomorphism constraint which renders the theory independent of the choice of
+the spatial geometry, i.e. shift vector, and secondly the Hamiltonian constraint which
+ensures the theory to be invariant under the choice of temporal coordinates1, i.e. lapse
+function. These two constraints conserve the general spacetime covariance of the theory.
+Thirdly, one imposes the Gaussian constraint which makes the theory invariant under
+any rotations of the triad fields.
+At the classical level, the two canonically conjugate variables Ea
+i and Ai
+a will be
+related with the following non-vanishing Poisson bracket:
+{Ai
+a(x), Ea
+i (y)} = 8πGγδb
+aδi
+jδ(3)(x − y),
+(2.4)
+while the dynamics of the theory will be governed by the following Hamiltonian acting
+on the canonical variables [69, 70]:
+H[N] =
+1
+8πG
+�
+Σ
+d3xN
+�
+F j
+ab − (1 + γ2)ϵjmnKm
+a Kn
+b
+� ϵjklEa
+kEb
+l
+√q
+,
+(2.5)
+where F j
+ab is the curvature of the Ashtekar connection defined as F j
+ab ≡ ∂aAj
+b − ∂bAj
+a +
+ϵijkAj
+aAk
+b.
+Working now within the spatially flat Friedman-Lemaˆıtre-Robertson-Walker (FLRW)
+model, one introduces a fiducial cell V connected to a fiducial metric oqab and a fiducial
+orthonormal triad and co-triad (oea
+i ,o ωi
+a) such as oqab =o ωi
+a
+oωi
+b. At the end, the reduced
+Ashtekar connection and densitized triad read as [67]
+Ai
+a = cV −1/3
+0
+oωi
+a,
+Eb
+i = pV −2/3
+0
+�
+det(oq) oeb
+i,
+(2.6)
+1Here, we conventionally denote as temporal coordinates the ones perpendicular to the 3D spatial
+slices. This notation is convenient but does not preassume a preferred time. As a consequence, general
+covariance is conserved.
+– 3 –
+
+where V0 is the fiducial volume as measured by the fiducial metric oqab and c, p are
+functions of the cosmic time t.
+In order now, to identify an internal clock of our theory, we introduce a dynamical
+massless scalar field described by the Hamiltonian:
+Hφ =
+p2
+φ
+2|p|3/2 .
+(2.7)
+At the end, the cosmological classical phase space is composed of two congugate pairs
+(c, p) and (φ, pφ) which obey the following Poisson brackets:
+{c, p} = 8πG
+3
+γ,
+{φ, pφ} = 1.
+(2.8)
+with |p| = a2V 2/3
+0
+and c = γ ˙aV 1/3
+0
+. Finally, using he Hamiltonian constraint one obtains
+the usual Friedmann equation within GR for a flat FRLW model,
+H2 = 8πG
+3
+ρ.
+(2.9)
+2.2
+The quantum dynamics
+Working now at the quantum level, the classical phase space variables and the clas-
+sical Hamiltonian will be promoted to quantum operators while the Poisson brackets
+will be replaced by commutation relations. However, within quantum field theory, the
+commutation algebra of quantum operators requires integration over the 3D space, thus
+assuming a well pre-defined background.
+Nevertheless, this setup cannot be applied
+within the framework of LQG since we want a background independent theory. For this
+reason, the quantisation process is performed at the level of two new canonical variables,
+namely the holonomy of the Ashtekar connection he(A) along a curve e ⊂ Σ and the
+flux of the densitized triad FS(E) along a 2-surface S defined as [67]
+he(A) ≡ Pexp
+��
+e
+τiAi
+adxa
+�
+,
+FS(E) ≡
+�
+S
+τiEi
+anad2y,
+(2.10)
+where τ = −iσi/2 (σi are the Pauli matrices) with [τi, τj] = ϵijkτ k, na is the unit vector
+vertical to the surface S and P is a path-ordering operator. These functions constitute
+non-trivial SU(2) variables satisfying a unique holonomy-flux Poisson algebra [71–74].
+Working within this representation one can then construct a kinematical Hilbert
+space for the gravity sector which is actually the space of the square integrable func-
+tions on the Bohr compactification of the real line, i.e. Hgrav
+kin ≡ L2(RBohr, dµBohr) [67].
+Regarding the matter sector, the respective kinematical Hilbert space is defined like in
+the standard Shrondigner picture as Hmatter
+kin
+≡ L2(R, dµ). Thus, the whole kinematical
+Hilbert space of the theory is defined as Hkin ≡ Hgrav
+kin ⊗ Hmatter
+kin
+.
+Focusing now on the homogeneous and isotropic FLRW model, usually called
+as Loop Quantum Cosmology (LQC) and following the conventional quantisation ¯µ
+scheme [75] one introduces two new conjugate variables defined as follows:
+u ≡ 2
+√
+3 sgn(p)/¯µ3,
+b ≡ ¯µc,
+(2.11)
+– 4 –
+
+where ¯µ =
+�
+∆/|p| and ∆ = 4
+√
+3 πγGℏ being the minimum nonzero eigenvalue of the
+area operator [76].
+Finally, one can show that the new variables obey the following
+Poisson bracket:
+{b, u} = 2
+ℏ,
+(2.12)
+and that in Hgrav there are two elementary operators, namely �
+eib/2 and ˆu related to
+the holonomy and the flux conjugate variables.
+In particular, it turns out that the
+eigenstates |u⟩ of ˆu form an orthonormal basis in Hgrav
+kin
+and the actions of these two
+operators in this basis can read as
+�
+eib/2 |u⟩ = |u + 1⟩ ,
+ˆu |u⟩ = u |u⟩ .
+(2.13)
+Letting now |φ⟩ being the orthonormal bases in Hmatter
+kin
+one can define |u, φ⟩ ≡ |u⟩ ⊗ |φ⟩
+as the generalized basis of the whole kinematic Hilbert space Hkin. Thus, after defining
+the relevant Hilbert space and the associated to it orthonormal basis, one can promote
+the Hamiltonian to a quantum operator. In particular, it is possible to define a quasi-
+classical sharped initial state living in Hkin, which can be viewed as wavepacket around
+a classical trajectory. Consequently, expressing the Hamiltonian (2.5) in terms of fluxes
+and holonomies one can derive the expectation value of the Hamiltonian operator over
+the initial semi-classical sharped state which at the end will contain first order quantum
+corrections. Finally, accounting only for the holonomy corrections (since flux or inverse
+volume corrections face the issue of a fiducial cell dependence [75]) one obtains the
+following modified Friedmann equation [75, 77, 78]:
+H2 = 8πG
+3
+ρ
+�
+1 − ρ
+ρc
+�
+,
+(2.14)
+where ρc = 2
+√
+3 M4
+Pl
+γ3
+. As it can be seen from Eq. (2.14) for ρ > ρc there is no physical evo-
+lution since H2 < 0. One then finds that the effect of holonomies leads to a non-singular
+evolution where the classical Big Bang singularity is replaced by a non-singular quantum
+bounce where ρ = ρc and H = 0. This bouncing point constitutes a transitioning point
+between a contracting (H < 0) and an expanding phase (H > 0).
+3
+The threshold of primordial black hole formation in loop quantum
+gravity
+Having introduced before the fundamentals of LQG, we estimate in this section the PBH
+formation threshold δc accounting for effects of loop quantum gravity at the level of the
+background cosmic evolution.
+To do so, we assume that the collapsing overdensity region is described by a homo-
+geneous core (closed Universe) described by the following fiducial metric:
+ds2 = −dt2 + a2(t)
+�
+dχ2 + sin2 χdΩ2�
+,
+(3.1)
+– 5 –
+
+where dΩ2 is the line element of a unit two-sphere and a(t) is the scale factor of the
+perturbed overdensity region.
+For this type of close homogeneous and isotropic spacetime foliations one can show
+that following the procedure as described in Sec. 2 the modified Friedmann equation in
+k = 1 LQC accounting only for the holonomy corrections will read as [79, 80]:
+H2 =
+� ˙a
+a
+�2
+=
+1
+3M2
+Pl
+(ρ − ρ∗)
+�
+1 − ρ − ρ∗
+ρc
+�
+,
+(3.2)
+where ρ is the energy density of the overdense region and ρ∗ = ρc
+�
+(1 + γ2)D2 + sin2 D
+�
+with D ≡ λ(2π2)1/3/v1/3, λ2 = 4
+√
+3 πγℓ2
+Pl [81], ℓPl being the Planck length and v =
+2π2a3 being the physical volume of the unit sphere spatial manifold [81]. Since v increases
+with time one can expand Eq. (3.2) in the limit u ≫ 1 [79]. At the end, keeping terms
+up to O(1/v2/3) one can show that Eq. (3.2) takes the following form:
+H2 =
+� ˙a
+a
+�2
+=
+1
+3M2
+Pl
+�
+ρ
+�
+1 − ρ
+ρc
+�
+− 3M2
+Pl
+a2
+�
+1 − 2 ρ
+ρc
+��
+= F(ρ)
+3M2
+Pl
+− G(ρ)
+a2 ,
+(3.3)
+where F(ρ) = ρ(1 − ρ/ρc) and G(ρ) = 1 − 2ρ/ρc. In the limit where γ = 0, ρc → ∞ and
+one recovers the standard GR k = 1 Friedmann equation H2 =
+ρ
+3M2
+Pl − 1
+a2 .
+As regards now the background, the latter it will behave as the standard homoge-
+neous and isotropic FLRW background whose fiducial metric reads as
+ds2 = −dt2 + a2
+b(t)
+�
+dr2 + r2dΩ2�
+(3.4)
+and whose modified Friedmann equation within LQC will read as
+H2
+b =
+� ˙ab
+ab
+�2
+=
+ρb
+3M2
+Pl
+�
+1 − ρ
+ρc
+�
+.
+(3.5)
+In this setup, the collapsing overdense region corresponds to the region where 0 ≤
+χ ≤ χa and the areal radius at the edge of the ovedensity will read as
+Ra = a sin χa.
+(3.6)
+At this point, we need to stress that the characteristic size of the overdensity is
+initially super-horizon and will reenter the cosmological horizon when the areal radius
+of the overdensity becomes equal to the cosmological horizon H−1, i.e.
+1
+Hhc
+= ahc sin χa,
+(3.7)
+where the index “hc” denotes quantities at the horizon crossing time. Writing now the
+energy density of the overdensity as ρ = ρb(1 + δ), where δ ≡ δρ
+ρb , one can plug ρ into
+Eq. (3.3) and working within the uniform Hubble gauge where H = Hb they can recast
+Eq. (3.7) as
+sin2 χa =
+1
+Gδ
+hc
+�F δ
+hc
+Fhc
+− 1
+�
+,
+(3.8)
+– 6 –
+
+where F δ
+hc = F [ρb,hc(1 + δ)] and Gδ
+hc = G [ρb,hc(1 + δ)].
+Once then the overdensity region crosses the cosmological horizon will initially
+follow the cosmic expansion and at some point it will detach from it starting to collapse
+to form a black hole horizon. This basically happens at the time of maximum expansion
+of the overdensity, when the Hubble parameter in Eq. (3.5) becomes zero, i.e. Hm = 0,
+or equivalently when
+am = 3M2
+PlGm
+Fm
+,
+(3.9)
+with the subscript “m” denoting quantities at the maximum expansion time.
+Having derived above the horizon crossing time and the time at maximum expan-
+sion we establish below a criterion for PBH formation by investigating the necessary
+conditions for the triggering of the gravitational collapse process. Doing so, we confront
+the gravitational force which pushes matter inwards and enhances in this way the black
+hole gravitational collapse with the sound wave pressure force which pushes matter out-
+wards, thus disfavoring the collapse of the overdensity. In particular, the criterion which
+we adopt is the requirement that the time at which the pressure sound wave crosses
+the radius of the overdensity region should be larger than the time at the maximum
+expansion, which is actually the time of the onset of the gravitational collapse. Thus,
+the sound pressure force will not have time to disperse the collapsing fluid matter to
+the background medium and prevent in this way the collapsing process. Equivalently,
+we require that the proper size of the overdensity χa is larger than the sound crossing
+distance by the time of maximum expansion χs, i.e.
+χa > χs.
+(3.10)
+To compute now the sound crossing distance by the time of maximum expansion we
+assume matter in terms of a perfect fluid characterized by a constant equation-of-state
+(EoS) parameter w, defined as the ratio between the pressure p and the energy density
+ρ of the fluid, w ≡ p/ρ. Within this framework, the sound wave propagation equation
+reads as
+adχ
+dt = √w ,
+(3.11)
+where we used the fact that for a perfect fluid with a constant EoS parameter the square
+sound wave c2
+s is equal to w, i.e. c2
+s = w. At the end, χs can be recast in following form:
+χs = √w
+� tm
+tini
+dt
+a =
+√w
+3
+� ρini
+ρb,m
+dρb
+(1 + w)ρb
+�� ρb,m
+ρb
+�
+2
+3(1+w) GmF(ρb)
+Fm
+− G(ρb)
+,
+(3.12)
+where we have assumed that for a perfect fluid ρb ∝ a−3(1+w) and used Eq. (3.9) to
+express am in terms of Fm and Gm.
+At the end, using Eq. (3.8) the criterion for PBH formation reads as
+1
+Gδ
+hc
+�F δ
+hc
+Fhc
+− 1
+�
+> sin2 χs.
+(3.13)
+– 7 –
+
+To determine therefore the PBH formation threshold, one can follow the following
+procedure: From Eq. (3.8), one should firstly determine the the ratio ρhc/ρm for a given
+value of χa and then solve numerically the inequality Eq. (3.13) in order to extract the
+value of the critical energy density contrast δc required for the overdensity region to
+collapse and form a PBH.
+Practically, one should compute χs from Eq. (3.12) for a given value of ρm and
+equate χs with χa. Then, solving Eq. (3.8) they will extract the ratio ρhc/ρm and then
+plugging it into Eq. (3.13) they can extract numerically δc. At the end, given the fact
+that ρb,hc < ρc, i.e. PBHs form after the quantum bounce, and that δ < 1 since we want
+to be within the perturbative regime, one can show from Eq. (3.8) that
+δc ≃ sin2 χs,
+(3.14)
+with χs given by Eq. (3.12).
+At this point, we should stress that the above expression for the value of δc is a
+lower bound estimate of its true value since it assumes the homogeneity of the collapsing
+overdensity region which in general is not the case when one is met with strong pressure
+gradients. Thus, it is strictly valid for regimes where w ≪ 1. However, PBH formation
+was never studied before in a rigorous way within the context of LQG through numerical
+simulations. To that end, Eq. (3.14) provides a reliable estimate for the value of δc.
+4
+Results
+Following the procedure described above, we calculate here the PBH formation threshold
+δc within the framework of LQG and we compare it with its value in GR. In particular,
+in the left panel of Fig. 1 we show the PBH formation threshold as a function of the
+energy density at the time of maximum expansion ρm by fixing the EoS parameter
+w = 1/3 since we study PBH formation during the RD era and the value of the Barbero-
+Immirzi parameter γ = 0.2375 obtained from the computation of the entropy of black
+holes [82]. Interestingly, we see a deviation from GR for high energies at the time of
+maximum expansion which correspond to very small mass PBHs forming close to the
+quantum bounce. This behavior is somehow expected since in this high energy regime,
+one expects to see a quantifiable effect of the quantum nature of gravity. In particular,
+we observe a drastic reduction of the value of δc in this region of high values of ρm up
+to 50% compared the GR case.
+This reduction of δc should be related with a smaller cosmological/sound horizon
+in LQC compared to GR as it can be speculated from Eq. (3.14). To see this, let us find
+the necessary conditions to get a cosmological horizon in LQC smaller than that in GR.
+Doing so, one should require that
+H2
+LQC > H2
+GR ⇔
+ρ
+3M2
+Pl
+�
+ρ − ρ
+ρc
+�
+− 1
+a2
+�
+1 − 2 ρ
+ρc
+�
+>
+ρ
+3M2
+Pl
+− 1
+a2 ⇔ ρ < 6M2
+Pl
+a2 .
+(4.1)
+For ρ = ρm and a = am as given by Eq. (3.9) one can verify that the inequality 4.1 is
+identically satisfied. Thus, indeed the cosmological horizon in LQC is smaller than in
+that GR leading to a reduction of δc compared to its GR value.
+– 8 –
+
+10−13
+10−11
+10−9
+10−7
+10−5
+10−3
+10−1
+101
+ρm/M 4
+Pl
+0.30
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+δc
+w = 1/3
+LQG
+GR
+10−5
+10−3
+10−1
+101
+103
+105
+107
+MPBH in grams
+0.30
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+δc
+w = 1/3
+γ=10−1
+γ=1
+γ=101
+γ=102
+γ=103
+Figure 1: Left Panel:The PBH formation threshold in the uniform Hubble gauge in the
+radiation-dominated era (w = 1/3) as a function of the energy density at the onset of the
+PBH gravitational collapse in LQG (green curve) and in GR (Eq. (1.1)) (black dashed
+curve).
+Right Panel: The PBH formation threshold in the uniform Hubble gauge in
+the radiation-dominated era (w = 1/3) as a function of the primordial black hole mass
+MPBH in LQG for different values of the Barbero-Immirzi parameter γ. The vertical
+black dashed line corresponds to MPBH = MPl.
+Then, we consider the Barbero-Immirzi parameter γ as a free parameter of the
+underlying quantum theory in the context of LQG. In particular, despite the fact that
+the Bekenstein-Hawking entropy has been standardly used as a way to fix the value of γ,
+the dependence of the entropy calculation on γ is controversial, and the value γ ≃ 0.2375,
+calculated using thermodynamical arguments, is not broadly accepted [83–85]. In fact,
+the choice to vary this parameter is motivated by the fact that γ is actually a coupling
+constant with a topological term in the gravitational action, with no consequence at the
+level of the classical equations of motion [86–91]. Thus, we vary the Barbero-Immirzi
+parameter within the range of 0.1 < γ < 1000 accounting for observational constraints
+for the duration of inflation after a quantum bounce [92, 93]. At the end, we plot in
+the right panel of Fig. 1 the PBH formation threshold δc as a function of the PBH
+mass for different values of the parameter γ within the observationally allowed range
+γ ∈ [0.1, 1000]. We set the lower bound on the PBH mass equal to the Planck mass as
+predicted within the quantum gravity approach [94] (See vertical black dashed line in
+the right panel of Fig. 1).
+In order to get the PBH mass, we account for the fact that the PBH mass is of
+the order of the cosmological horizon mass at horizon crossing time. Solving at the end
+numerically Eq. (3.8) we found that ρhc/ρm ∼ 10. This corresponds to
+N = ln
+� am
+ahc
+�
+= 1
+4 ln
+�ρhc
+ρm
+�
+∼ 0.6 e − folds
+(4.2)
+passing from horizon crossing time up to the onset of the gravitational collapse process at
+– 9 –
+
+ρ = ρm, thus being in agreement with the results from PBH numerical simulations [95].
+As expected, when we increase the value of γ the overall mass range moves to higher
+masses given the fact that higher values of γ are equivalent with lower values of ρc, thus
+the quantum bounce happens at later times. Consequently, PBHs if formed will form at
+later times, thus will acquire larger masses.
+Interestingly, independently on the value of the Barbero-Immirzi variable δc is re-
+duced on the low mass region, which for γ < 1000 corresponds to masses MPBH < 103g.
+In particular, this reduction in δc in this very small PBH mass range will entail an
+enhancement in their abundances with tremendous consequences on the associated to
+them phenomenology. Indicatively, we mention here that these ultra-light PBHs can
+trigger early PBH-matter dominated eras [20, 21, 96] before BBN and reheat the Uni-
+verse through their evaporation [97] while at the same time they can account for the
+Hubble tension through the injection to the primordial plasma of light dark radiation
+degrees of freedom [98, 99] while at the same time they can produce naturally the baryon
+assymetry through CP violating out-of-equilibrium decays of their Hawking evaporation
+products [100, 101].
+Consequently, one can constrain the above mentioned observa-
+tional/phenomenological signatures by studying PBH formation within the context of
+LQG while vice-versa given the above mentioned phenomenology one can constrain the
+Barbero-Immirzi parameter γ which is the fundamental parameter within LQG. In this
+way, PBHs are promoted as a novel probe to constrain the potential quantum nature of
+gravity.
+5
+Conclusions
+PBHs firstly introduced in ’70s are of great significance, since they can naturally account
+for a part or all of the dark matter sector, while at the same time they might seed the
+formation of large-scale structures through Poisson fluctuations. Moreover, they can also
+offer the seeds for the progenitors of the black-hole merging events recently detected by
+LIGO/VIRGO as well as for the supermassive black holes present in the galactic centers.
+Their formation was mainly studied within the context of general relativity using both
+analytic and numerical techniques.
+In this work, we studied PBH formation within the context of LQG by investigating
+the impact of the potential quantum character of spacetime on the critical PBH forma-
+tion threshold δc, whose value can crucially affect the abundance of PBHs, a quantity
+which is constrained by numerous observational probes. In particular, by comparing
+the gravitational force with the sound wave pressure force during the process of the
+gravitational collapse we obtained a reliable estimate on the value of δc.
+Interestingly, we found that for low mass PBHs formed close to the quantum
+bounce, the value of δc is drastically reduced up to 50% compared to the general rel-
+ativistic regime with tremendous consequences for the observational/phenomenological
+footprints of such small PBH masses. In this way, we quantified for the first time to the
+best of our knowledge how quantum effects can influence PBH formation in the early
+Universe within a quantum gravity framework.
+– 10 –
+
+Finally, by treating the Barbero-Immirzi parameter γ as the free parameter of LQG
+we varied its value by studying its effect on the value of the PBH formation threshold.
+As expected, we found an overall shift of the PBH masses affected by the choice of γ.
+Very interestingly, we showed as well that using the observational and phenomenological
+signatures associated to ultra-light PBHs, namely the ones affected by LQG effects, one
+can constrain the quantum parameter γ. At this point, we should highlight the fact that
+our formalism can be applied to any quantum theory of gravity giving an explicit form
+for the equations of the background cosmic evolution establishing in this way the PBH
+portal as a novel probe to constrain the potential quantum nature of gravity.
+Acknowledgments
+The author acknowledges financial support from the Foundation for Education and
+European Culture in Greece as well the contribution of the COST Action CA18108
+“Quantum Gravity Phenomenology in the multi-messenger approach”.
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