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@@ -0,0 +1,994 @@
+arXiv:2301.05056v1  [gr-qc]  12 Jan 2023
+Tunneling probability for the birth of universes
+with radiation, cosmological constant and an
+ad hoc potential
+G. Oliveira-Neto and D. L. Canedo
+Departamento de F´ısica,
+Instituto de Ciˆencias Exatas,
+Universidade Federal de Juiz de Fora,
+CEP 36036-330 - Juiz de Fora, MG, Brazil.
+gilneto@fisica.ufjf.br, danielcanedo.tr@hotmail.com
+G. A. Monerat
+Departamento de Modelagem Computacional,
+Instituto Polit´ecnico,
+Universidade do Estado do Rio de Janeiro,
+CEP 28.625-570, Nova Friburgo - RJ - Brazil.
+monerat@uerj.br
+January 13, 2023
+Abstract
+In this work we study the birth of Friedmann-Lemaˆıtre-Robertson-
+Walker (FLRW) models with zero (k = 0) and negative (k = −1) cur-
+vatures of the spatial sections. The material content of the models is
+composed of a radiation perfect fluid and a positive cosmological con-
+stant. The models also have the presence of an ad hoc potential which
+origin is believed to be of geometrical nature. In order to describe the
+birth of these universes, we quantize them using quantum cosmology.
+Initially, we obtain the Wheeler-DeWitt equations and solve them us-
+ing the WKB approximation. We notice that the presence of the ad
+1
+
+hoc potential produces a barrier for any value of k. It means that
+we may describe the birth of the universe through a tunneling mecha-
+nism, for any curvature of the spatial sections, not only for the usual
+case k = 1. We, explicitly, compute the tunneling probabilities for
+the birth of the different models of the universe and compare these
+tunneling probabilities.
+Keywords: Quantum cosmology, Wheeler-DeWitt equation, Cosmolog-
+ical constant, Radiation perfect fluid, Ad hoc potential
+PACS: 04.60.Ds, 98.80.Bp, 98.80.Qc
+1
+Introduction
+Quantum cosmology (QC) was the first attempt to describe the Universe as
+a quantum mechanical system. It uses general relativity (GR) in order to
+describe the gravitational interaction between the material constituents of
+the Universe. The canonical quantization was the first method used by the
+physicists working in QC. Several physicists contributed to the development
+of that research area, culminating in the introduction of the Wheeler-DeWitt
+equation [1], [2]. Another way to quantize a theory is using the path integral
+method [3, 4]. That method was first discussed in connection to the quanti-
+zation of GR by C. Misner [5]. After that, many physicists contributed to the
+development of that method of quantization in QC. Another fundamental line
+of research in QC is the problem of interpretation. Since one cannot use the
+Copenhagen interpretation of quantum mechanics to the system composed of
+the entire Universe, several new interpretations of quantum mechanics have
+been introduced. The first one was De Broglie-Bohm or Causal Interpreta-
+tion, first suggested by L. de Broglie [6, 7, 8, 9, 10] and later developed by
+D. Bohm [11, 12]. Another important interpretation was formulated by H.
+Everett, III and is known as the Many Worlds Interpretation [13]. A more
+recent interpretation of quantum mechanics that may be used in QC is the
+Consistent Histories or Decoherent Histories [14, 15, 16, 17, 18, 19, 20]. For
+a more complete introduction of the basic concepts of QC see [21, 22, 23, 24].
+One of the most interesting explanations for the regular birth of the Uni-
+verse, coming from QC, is the spontaneous creation from nothing [25, 26, 27,
+28, 29, 30, 31, 32]. In that explanation, one has to consider the Universe as
+a quantum mechanical system, initially with zero size. It is subjected to a
+potential barrier which confines it. In a FLRW quantum cosmological model,
+2
+
+that potential barrier is formed, most generally, due to the positive curvature
+of the spatial sections of the model and also due to the presence of a posi-
+tive cosmological constant or a matter content that produces an accelerated
+expansion of the universe. Since, in that explanation, the Universe should
+satisfy the quantum mechanical laws, it may tunnel through the barrier and
+emerges to the right of it with a finite size. That moment is considered the
+beginning of the Universe. Therefore, the Universe starts in a regular way
+due to its finite size. Several works, in the literature, have already considered
+cosmological models where one can compute, quantitatively, the tunneling
+probability for the birth of different universes [33, 34, 35, 36, 37, 38].
+Since there are some theoretical [39, 40] as well as observational [41, 42]
+evidences that our Universe has a flat spatial geometry, it would be inter-
+esting if we could produce a spatially flat cosmological model which birth is
+described by a spontaneous creation from nothing. As mentioned, above, the
+usual way to construct the potential barrier uses as one of the fundamen-
+tal ingredients the positive curvature of the spatial sections of the universe.
+Therefore, one has to find a different way to produce the barrier that the Uni-
+verse has to tunnel through in order to be born. In a recent paper some of us
+have introduced an ad hoc potential (Vah), that has all necessary properties
+in order to describe the regular birth of the Universe by the spontaneous cre-
+ation from nothing [37]. In addition to those properties, the universe could
+have positive, negative or nil curvature of the spatial sections. It is believed
+that such ad hoc potential may appear as a purely geometrical contribution
+coming from a more fundamental, geometrical, gravitational theory than
+general relativity [37]. As mentioned in Ref. [37], Vah has another interest-
+ing property at the classical level. It produces a large class of non-singular,
+bounce-type solutions.
+In this work we study the birth of FLRW models with zero (k = 0) and
+negative (k = −1) curvatures of the spatial sections. The model with k = 1
+was studied in Ref. [37]. Here, we are going to consider few results obtained
+in Ref. [37] in order to compare them with the new results obtained for
+the models with k = 0 and k = −1. The material content of the models is
+composed of a radiation perfect fluid and a positive cosmological constant.
+The models also have the presence of an ad hoc potential which origin is
+believed to be of geometrical nature. In order to describe the birth of these
+universes, we quantize them using quantum cosmology. Initially, we obtain
+the Wheeler-DeWitt equations and solve them using the WKB approxima-
+tion. We notice that the presence of Vah produces a barrier for any value
+3
+
+of k. It means that we may describe the birth of the universe through a
+tunneling mechanism, for any curvature of the spatial sections, not only for
+the usual case k = 1. We, explicitly, compute the tunneling probabilities for
+the birth of the different models of the universe and compare these tunneling
+probabilities.
+In Section 2, we obtain the Hamiltonians of the models and investigate the
+possible classical solutions using phase portraits. In Section 3, we canonically
+quantize the models and write the appropriate Wheeler-DeWitt equations.
+Then, we find the approximated WKB solutions to those equation. In Section
+4, we compute the quantum WKB tunneling probabilities as functions of the
+parameters: (i) the ad hoc potential parameter (σ), (ii) the cosmological
+constant (Λ), (iii) the radiation energy (E) and (iv) the curvature parameter
+(k). As the final result of that section, we compare the TPW KB’s for models
+with different values of k. The conclusions are presented in Section 5. In
+Appendix A, we give a detailed calculation of the fluid total hamiltonian
+used in this work.
+2
+The Classical Model
+In the present work, we want to study homogeneous and isotropic universes
+with constant negative and nil curvatures of the spatial sections. Therefore,
+we start introducing the FLRW metric, which is the appropriate one to treat
+those universes,
+ds2 = −N2(t)dt2 + a2(t)
+�
+dr2
+1 − kr2 + r2dΩ2
+�
+,
+(1)
+where a(t) is the scale factor, k gives the type of constant curvature of the
+spatial section, dΩ is the angular line element of a 2D sphere and N(t) is
+the lapse function introduced in the ADM formalism [2]. The action of the
+geometrical sector of the model is given by,
+S = 1
+2
+�
+M d4x√−g(R − 2Λ) +
+�
+∂M d3x
+√
+hhabKab
+(2)
+where R is the Ricci scalar, Λ is the cosmological constant, hab is the 3-metric
+induced on the boundary ∂M of the four-dimensional space-time M and Kab
+is the extrinsic curvature tensor of the boundary. We use the natural unit
+system where ¯h = 8πG = c = kB = 1. After some calculations we obtain
+4
+
+from the action Eq. (2), with the aid of the metric coming from Eq. (1), the
+following hamiltonian for the gravitational sector,
+NH = −p2
+a
+12 − 3ka2 + Λa4,
+(3)
+where pa is the canonically conjugated momentum to a. Here, we are working
+in the conformal gauge N = a.
+The matter content of the models is a
+radiation perfect fluid, which is believed to have been very important in the
+beginning of our universe. That perfect fluid has the following equation of
+state,
+prad = 1
+3ρrad,
+(4)
+where prad is the radiation fluid pressure and ρrad is its energy density. In
+order to obtain the hamiltonian associated to that fluid, we use the Schutz
+variational formalism [43, 44]. The starting point for that task is the following
+perfect fluid action [45],
+�
+M d4x√−gprad
+(5)
+The necessary calculations in order to obtain the hamiltonian from that ac-
+tion Eq. (5), using the Schutz variational formalism, are presented in Ap-
+pendix A.
+Using Eq. (3) and Eq. (39) from Appendix A, we may write the total
+hamiltonian of the model, in the conformal gauge N = a, as,
+NH = −p2
+a
+12 + pT − 3ka2 + Λa4 + Vah,
+(6)
+where pa and pT are the canonically conjugated momenta to a and T, re-
+spectively. The variable T is associated to the radiation fluid, as discussed
+in Appendix A. Vah is the ad hoc potential, which is defined as,
+Vah = −
+σ2a4
+(a3 + 1)2,
+(7)
+where σ is a dimensionless parameter associated to the magnitude of that
+potential. As discussed in Ref. [37], if one observes the limits of the ad hoc
+potential Eq.(7), when a assumes small as well as large values, one notices
+that it produces a barrier. In FLRW cosmological models constructed using
+the Hoˇrava-Lifshitz gravitational theory [46, 47, 48, 49, 50], one may have,
+5
+
+in the hamiltonian, terms similar to the asymptotic limits of Vah, which
+have purely geometrical origin. Then, it is not difficult to imagine that Vah
+should come from a purely geometrical contribution of a more fundamental
+gravitational theory.
+From the total hamiltonian Eq. (6) it is possible to identify an effective
+potential (Veff(a)) that comprises the terms related to the curvature of the
+spatial sections, cosmological constant and ad hoc potential. With the aid
+of Eq. (7), Veff(a) is given by,
+Veff(a) = 3ka2 − Λa4 +
+σ2a4
+(a3 + 1)2.
+(8)
+Observing Veff(a) Eq.
+(8), it is possible to see that for all values of the
+parameters Λ, σ and k = −1 or k = 0, that potential is well defined at a = 0.
+In fact, it goes to zero when a → 0. It is, also, possible to see that when
+a → ∞ the potential Veff → −∞. Another important property of Veff(a)
+Eq. (8), is that for all values of the parameters Λ, σ and k = −1 or k = 0,
+it has only one barrier. That situation is different from the case where the
+curvature of the spatial section is positive (k = 1), which was studied in Ref.
+[37]. There, depending on the values of Λ and σ, Veff(a) could have one or
+two barriers. Examples of all those properties can be seen in Figures (1-4).
+Figure 1: Veff(a) for k = −1 with
+Λ = 0.01 and different values of σ.
+Figure 2: Veff(a) for k = −1 with
+Λ = 1.5 and different values of σ.
+6
+
+Figure 3: Veff(a) for k = 0 with
+Λ = 0.01 and different values of σ.
+Figure 4: Veff(a) for k = 0 with
+Λ = 1.5 and different values of σ.
+Now, we can study the classical dynamical behavior of the model with
+the aid of the hamilton’s equation. We may compute them from the total
+hamiltonian Eq. (6), to obtain,
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+˙a =
+∂NH
+∂pa = −1
+6pa,
+˙pa =
+−∂NH
+∂a
+= ∂Veff
+∂a ,
+˙T =
+∂NH
+∂pT = 1,
+˙pT =
+−∂NH
+∂T
+= 0,
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(9)
+where the dot means derivative with respect to the conformal time η.
+One may have the general idea on how the scale factor behaves by study-
+ing the phase portraits of the models in the plane (a, pa). Due to the fact
+that, as mentioned above, Veff(a) Eq. (8) for the present models have only
+one barrier, the phase portraits are simpler than the ones for the models with
+k = +1 [37].
+7
+
+Figure 5: Phase portraits in the
+plane (a, pa) for the model with
+k = −1, Λ = 0.01, σ = −50 and
+different values of pT.
+Figure 6: Phase portraits in the
+plane (a, pa) for the model with
+k = 0, Λ = 0.01, σ = −50 and
+different values of pT.
+The dashed curves in Figures 5 and 6 are called separatrixes. They sepa-
+rate different classes of solutions for a given energy pT. Those phase portraits
+Figures 5 and 6 have, also, two fixed points, which represent stationary so-
+lutions of the model. Let us call those points A1 e A2. In particular, A2 is
+called Einstein’s Universe, there the gravitational attraction and the cosmo-
+logical expansion balance each other. A1 is located, on the plane (a, pa), by
+(a = 0, pa = 0) and energy pT = 0. It is the same point for Figures 5 and
+6. For A2, the points on the plane (a, pa) and the values of pT are given in
+Table 1.
+Table 1: Location of A2 for Figures 5 and 6
+A2
+pT
+Figure 5
+(a = 1.255633946, pa = 0)
+695.1851495
+Figure 6
+(a = 1.259875701, pa = 0)
+699.9309422
+Observing Figures 5 and 6, we may identify a first class of solutions
+present in the model. For a and pT smaller than the ones for the fixed point
+A2 and for pa greater than the ones for the fixed point A2, we have a class
+of solutions where the universe starts expanding from an initial Big Bang
+8
+
+singularity, reaches a maximum size and then contracts to a final Big Crunch
+singularity. These solutions are located in Region I of Figures 5 and 6.
+Now, for pa and pT greater than the ones for the fixed point A2, we have a
+second class of solutions where the universe starts expanding from an initial
+Big Bang singularity (a = 0) and continues expanding to infinity values of
+a. It tends asymptotically to a De Sitter type solution. These solutions are
+located in Region II of Figures 5 and 6.
+Now, for pa < 0 (initially), pT smaller than the ones for the fixed point
+A2 and a greater than the ones for the fixed point A2, we have a third class
+of solutions where the universe starts contracting from an initial scale factor
+value, reaches a minimum size for pa = 0 and then expands to infinity values
+of a, for pa > 0. It tends asymptotically to a De Sitter type solution. These
+are the bouncing solutions for the present models. These solutions are located
+in Region III of Figures 5 and 6.
+Finally, a fourth class of solutions appears if we choose pa < 0 and pT
+greater than the ones for the fixed point A2. In that class of solutions the
+universe starts contracting from a large finite value of a and continues con-
+tracting until it reaches a final Big Crunch singularity. These solutions are
+located in Region IV of Figures 5 and 6.
+The classical scale factor behavior may be computed by solving a system
+of ordinary differential equations. The first equation is obtained by imposing
+the hamiltonian constraint H = 0 Eq. (6) and substituting, in the resulting
+equation, the value of pa in terms of ˙a, with the aid of Eqs.
+(9).
+That
+equation is the Friedmann equation for the present model and is given by,
+˙a(0) = ±1
+6
+�
+12(pT − Veff(a0)),
+(10)
+where a0 = a(η = 0) is the scale factor initial condition. The second equa-
+tion is obtained by combining the hamilton’s equations (9), resulting in the
+following second order, ordinary, differential equation for a(η),
+∂2a(η)
+∂η2
++ ka(η) − 2Λ
+3 a(η)3 + 2σ2
+3
+a(η)3
+(a(η)3 + 1)2 −
+σ2a(η)6
+(a(η)3 + 1)3 = 0.
+(11)
+We solve that system of equations (10), (11), in the following way. Initially,
+we choose a value for a0 and substitute it in the Friedmann equation (10), in
+order to find the initial value for ˙a (˙a0). Then, we use these initial conditions
+9
+
+in order to solve equation (11). Due to the complexity of both equations
+(10), (11), we solve the system numerically. As we mentioned above, the
+Veff(a) (8) for both values of k (-1 or 0) have just one barrier. Therefore,
+the results for a(η) for both cases are very similar. Next, we solve the system
+of equations (10), (11), for Λ = 0.01, σ = −50 and k = 0 or k = −1, which
+correspond to the phase portraits shown in Figures 5 and 6. For those models
+we find the four classes of solutions described, qualitatively, above.
+In Figure 7, we see examples of the first class of solutions described above,
+for k = −1 and k = 0. In order to obtain them, we set a(0) = 0, pT = 164,
+˙a0 = −7.393691003, which gives pa > 0 from Eq. (9).
+In Figure 8, we see examples of the second class of solutions described
+above, for k = −1 and k = 0. In order to obtain them, we set a(0) = 0,
+pT = 800, ˙a(0) = −16.32993162, which gives pa > 0 from Eq. (9).
+In Figure 9, we see examples of the third class of solutions described
+above, for k = −1 and k = 0. In order to obtain the solution for k = −1, we
+set a(0) = 1000, pT = 500 and ˙a(0) = −57743.68797. In order to obtain the
+solution for k = 0, we set a(0) = 1000, pT = 500 and ˙a(0) = −57735.02837.
+We can, clearly, see from Figure 9 two examples of bouncing solutions for
+the present models.
+Finally, in Figure 10, we see examples of the fourth class of solutions
+described above, for k = −1 and k = 0. In order to obtain the solution for
+k = −1, we set a(0) = 10, pT = 800, ˙a(0) = 19.79099058, which gives pa < 0
+from Eq. (9). In order to obtain the solution for k = 0, we set a(0) = 10,
+pT = 800, ˙a(0) = 17.07873849, which gives pa < 0 from Eq. (9).
+3
+Canonical Quantization, WKB Solution and
+WKB Tunneling Probability
+3.1
+Canonical Quantization
+In order to study the birth of the universes described by the cosmological
+models introduced in the present paper, we must quantize these models.
+We do that by using the Dirac’s formalism for quantization of constrained
+systems [51, 52, 53, 54]. The first step consists in introducing a wave-function
+(Ψ) which is a function of the canonical variables. In the present model these
+10
+
+Figure 7: Classical scale factor be-
+havior for universes with k = −1
+and k = 0, Λ = 0.01 and σ = −50.
+Figure 8: Classical scale factor be-
+havior for universes with k = −1
+and k = 0, Λ = 0.01 and σ = −50.
+Figure 9: Classical scale factor be-
+havior for universes with k = −1
+and k = 0, Λ = 0.01 and σ = −50.
+Figure 10:
+Classical scale factor
+behavior for universes with k = −1
+and k = 0, Λ = 0.01 and σ = −50.
+variables are ˆa and ˆT, then,
+Ψ = Ψ(ˆa, ˆT) .
+(12)
+11
+
+In the second step, we demand that the operators ˆa and ˆT and their con-
+jugated momenta ˆPa and ˆPT, satisfy suitable commutation relations. In the
+Schr¨odinger picture ˆa and ˆT become multiplication operators, while their
+conjugated momenta become the following differential operators,
+pa → −i ∂
+∂a ,
+pT → −i ∂
+∂T .
+(13)
+In the third and final step, we impose that the operator associated to NH (6)
+annihilates the wave-function Ψ (12). The resulting equation is the Wheeler-
+DeWitt equation for the present models.
+It resembles a time dependent,
+one-dimensional, Schr¨odinger equation,
+� 1
+12
+∂2
+∂a2 − 3ka2 + Λa4 −
+σ2a4
+(a3 + 1)2
+�
+Ψ(a, τ) = −i ∂
+∂τ Ψ(a, τ)
+(14)
+where the new variable τ = −T has been introduced.
+3.2
+WKB Solution
+Now, we want to determine the WKB approximated solution to the Wheeler-
+DeWitt equation (14). We start imposing that the solution to equation (14)
+may be written as [55, 56],
+Ψ(a, τ) = ψ(a)e−Eτ
+(15)
+where E is the energy associated to the radiation fluid. Introducing Ψ(a, τ)
+Eq. (15) in the Wheeler-DeWitt equation (14), we obtain,
+∂2ψ(a)
+∂a2
++ 12(E − Veff(a))ψ(a) = 0,
+(16)
+where Veff(a) is given in Eq. (8). Next, in Eq. (15), we consider that ψ(a)
+is given by,
+ψ(a) = A(a)eiφ(a),
+(17)
+where A(a) is the amplitude and φ(a) is the phase. Introducing ψ(a) Eq.
+(17) in Eq. (16) and supposing that the amplitude A(a) varies slowly as a
+function of a, we find the following general solutions for Eq. (16):
+(i) For regions where E > Veff(a),
+ψ(a) =
+C
+�
+K(a)
+e± i
+¯h
+�
+K(a)da,
+(18)
+12
+
+where C is a constant and
+K(a) =
+�
+12(E − Veff(a)).
+(19)
+(ii) For regions where E < Veff(a),
+ψ(a) =
+C1
+�
+k(a)
+e± 1
+¯h
+�
+k(a)da,
+(20)
+where C1 is a constant and
+k(a) =
+�
+12(Veff(a) − E).
+(21)
+3.3
+WKB Tunneling Probability
+Finally, using those WKB solutions, we want to determine the quantum me-
+chanical tunneling probabilities for the birth of the present universes. More
+precisely, the probabilities that the present universes will tunnel through
+Veff. An important condition for the tunneling process is that the energy E,
+of the wavefunction, be smaller than the maximum value of Veff(a). If we
+impose that condition, we may divide the a axis in three distinct regions with
+respect to the points where E intercepts Veff(a) (8), which are: (1) Region
+I - It extends from the origin until the point where E intercepts Veff(a) at
+the left (al), 0 < a < al; (2) Region II - It extends from the point where E
+intercepts Veff(a) at the left until the point where E intercepts Veff(a) at
+the right (ar), al < a < ar. That region is entirely inside Veff(a); (3) Region
+III - It extends from the point where E intercepts Veff(a) at the right until
+the infinity, ar < a < ∞. Now, we may write the WKB solutions Eqs. (18)
+and (20) for each one of these three regions,
+ψ(a)
+=
+A
+�
+K(a)
+ei� al
+a
+K(a)da +
+B
+�
+K(a)
+e−i� al
+a
+K(a)da
+I
+(0 < a < al)
+ψ(a)
+=
+C
+�
+k(a)
+e
+−� ar
+al k(a)da +
+D
+�
+k(a)
+e
+� ar
+al k(a)da
+II
+(al < a < ar)
+ψ(a)
+=
+F
+�
+K(a)
+ei� a
+ar K(a)da +
+G
+�
+K(a)
+e−i� a
+ar K(a)da
+III
+(ar < a < ∞)
+(22)
+13
+
+where A, B, C, D, E, F, G are constant coefficients to be determined. One
+may establish a relationship between all these coefficients A, B, C, D, E, F, G
+with the aid of the connections formulas, which are important formulas of
+the WKB approximation [55, 56]. The relationship is given by the following
+equation,
+�
+A
+B
+�
+= 1
+2
+�
+2θ + 1
+2θ
+i(2θ − 1
+2θ)
+−i(2θ − 1
+2θ)
+2θ + 1
+2θ
+� �
+F
+G
+�
+,
+(23)
+where θ is given by,
+θ = e
+� ar
+al k(a)da = e
+� ad
+ae da
+�
+12(3ka2−Λa4+
+σ2a4
+(a3+1)2 −E).
+(24)
+Let us consider, now, that the incident wavefunction (ψinc) with energy
+E propagates from the origin to the left of Veff(a) in Region I. When the
+wavefunction reaches Veff(a) at al, part of the incident wavefunction is re-
+flected back to Region I and part tunnels through Veff(a) in Region II. When
+the wavefunction emerges from Veff(a) at ar, it produces a transmitted com-
+ponent (ψtrans) which propagates to infinity in Region III. By definition the
+tunneling probability (TPW KB) is given by,
+TPW KB = |ψtrans
+√ktrans|2
+|ψinc
+√kinc|2
+= |F|2
+|A|2 ,
+(25)
+where we are assuming that there is no incident wavefunction from the right,
+it means that G = 0 in Eq. (23). With the aid of Eq. (23), TPW KB becomes,
+TPW KB =
+4
+(2θ + 1
+2θ)2.
+(26)
+4
+Results
+Now, we want to quantitatively compute the tunneling probabilities for the
+birth of the universes described by the present models.
+These tunneling
+probabilities are measured by TPW KB (26). They depend on: (i) the radia-
+tion energy E, (ii) the cosmological constant Λ and (iii) the ad hoc potential
+parameter σ.
+14
+
+4.0.1
+TPW KB as a function of E
+If we fix the values of Λ, σ and k, TPW KB Eq. (26) becomes a function of the
+energy E. In order to determine how that tunneling probability depends on
+E, we compute TPW KB Eq. (26) for 70 different values of E with σ = −50
+and Λ = 1.5. As a matter of completeness and in order to facilitate the
+comparison, we shall compute the values for the models with k = 1 besides
+the ones for models with k = −1, 0. Therefore, we repeat those calculations
+three times, one for each value of k.
+We choose values of E, such that,
+they are smaller than the maximum barrier value (Veffmax). For k = −1
+Veffmax = 691.5188154, for k = 0 Veffmax = 696.2063154 and for k = 1
+Veffmax = 700.8938154. The energies are given by: E = {E1 = 5, E2 =
+10, E3 = 20, ..., E68 = 670, E69 = 680, E70 = 690}. The curves ln(TPW KB)
+versus E, for each k, are given in Figure 11. We use the natural logarithm
+of TPW KB because some values of that tunneling probability are very small.
+Observing Figure 11, we notice that TPW KB increases for greater values of
+E. Thus, it is more likely that the universe is born with the greatest value
+of the radiation energy E. From Figure 11, we also notice that TPW KB is
+greatest for k = −1, decreases for k = 0 and decreases even further for k = 1.
+So, it is more likely that the universe is born with negatively curved spatial
+sections.
+4.0.2
+TPW KB as a function of Λ
+If we fix the values of E, σ and k, TPW KB Eq. (26) becomes a function
+of the cosmological constant Λ. In order to determine how that tunneling
+probability depends on Λ, we compute TPW KB Eq.
+(26) for 21 different
+values of Λ with σ = −50 and E = 690. We repeat those calculations three
+times, one for each value of k. We choose values of Λ, such that, E = 690
+is always smaller than Veffmax. The cosmological constant values are given
+by: Λ = {Λ1 = 0.6, Λ2 = 0.65, Λ3 = 0.7, ..., Λ19 = 1.5, Λ20 = 1.55, Λ21 = 1.6}.
+The curves ln(TPW KB) versus Λ, for each k, are given in Figure 12. We
+use the natural logarithm of TPW KB because some values of that tunneling
+probability are very small.
+Observing Figure 12, we notice that TPW KB
+increases for greater values of Λ. Therefore, it is more likely that the universe
+is born with the greatest value of Λ. From Figure 12, we also notice that
+TPW KB is greatest for k = −1, decreases for k = 0 and decreases even further
+for k = 1. Thus, it is more likely that the universe is born with negatively
+15
+
+Figure 11: WKB Tunneling Probabilities as functions of the energy E, for
+σ = −50 and Λ = 1.5. Each curve corresponds to a different value of the
+spatial curvature k.
+curved spatial sections.
+4.0.3
+TPW KB as a function of σ
+If we fix the values of E, Λ and k, TPW KB Eq. (26) becomes a function of
+the ad hoc potential parameter σ. In order to determine how that tunneling
+probability depends on σ, we compute TPW KB Eq.
+(26) for 29 different
+values of σ with Λ = 1.5 and E = 680. We repeat those calculations three
+times, one for each value of k. We choose values of σ, such that, E = 680
+is always smaller than Veffmax. The ad hoc potential parameter values are
+given by: σ = {σ1 = −50, σ2 = −50.5, σ3 = −51, ..., σ27 = −63, σ28 =
+−63.5, σ29 = −64}. The curves ln(TPW KB) versus σ, for each k, are given
+in Figure 13. We use the natural logarithm of TPW KB because some values
+of that tunneling probability are very small. Observing Figure 13, we notice
+that TPW KB decreases for greater absolute values of σ. Therefore, it is more
+likely that the universe is born with the smallest possible absolute value of σ.
+16
+
+Figure 12: WKB Tunneling Probabilities as functions of the cosmological
+constant Λ, for σ = −50 and E = 690. Each curve corresponds to a different
+value of the spatial curvature k.
+From Figure 13, we also notice that TPW KB is greatest for k = −1, decreases
+for k = 0 and decreases even further for k = 1. So, it is more likely that the
+universe is born with negatively curved spatial sections.
+5
+Conclusions
+In this work we studied the birth of FLRW models with zero (k = 0) and
+negative (k = −1) curvatures of the spatial sections. The model with k = 1
+was studied in Ref. [37]. Here, we considered few results obtained in Ref.
+[37] in order to compare them with the new results obtained for the models
+with k = 0 and k = −1. The material content of the models is composed of
+a radiation perfect fluid and a positive cosmological constant. The models
+also have the presence of an ad hoc potential which origin is believed to
+be of geometrical nature. At the classical level, we studied the models by
+drawing phase portraits in the plane (a, pa). We identified all possible types
+17
+
+Figure 13: WKB Tunneling Probabilities as functions of the ad hoc potential
+parameter σ, for Λ = 1.5 and E = 680. Each curve corresponds to a different
+value of the spatial curvature k.
+of solutions, including some new bouncing solutions. We explicitly, solved
+the Einstein’s equations and gave examples of all possible types of classical
+solutions.
+In order to describe the birth of these universes, we quantized them using
+quantum cosmology. Initially, we obtained the Wheeler-DeWitt equations
+and solved them using the WKB approximation. We notice that the presence
+of Vah produces a barrier for any value of k. It means that we may describe
+the birth of the universe through a tunneling mechanism, for any curvature
+of the spatial sections, not only for the usual case k = 1. We, explicitly,
+computed the tunneling probabilities for the birth of the different models
+of the universe, as functions of the radiation energy E, the cosmological
+constant Λ and the ad hoc potential parameter σ. We compared the WKB
+tunneling probability behavior for different values of k.
+From our results, we noticed that TPW KB increases for greater values of
+E. Therefore, it is more likely that the universe is born with the greatest
+18
+
+value of the radiation energy E. We, also, noticed that TPW KB increases for
+greater values of Λ. Thus, it is more likely that the universe is born with
+the greatest value of Λ. We, also, noticed that TPW KB decreases for greater
+absolute values of σ. Hence, it is more likely that the universe is born with
+the smallest possible absolute value of σ. In all models we have studied, we
+noticed that TPW KB is greatest for k = −1, decreases for k = 0 and decreases
+even further for k = 1. So, it is more likely that the universe is born with
+negatively curved spatial sections.
+Acknowledgments. D. L. Canedo thanks Coordena¸c˜ao de
+Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES) and Universidade
+Federal de Juiz de Fora (UFJF) for his scholarships. G. A. Monerat thanks
+FAPERJ for financial support and Universidade do Estado do Rio de Janeiro
+(UERJ) for the Prociˆencia grant.
+A
+Radiation fluid hamiltonian
+In the present model, the starting point of the Schutz formalism is the de-
+scription of the fluid four-velocity Uν in terms of the potentials µ, φ, θ and
+S,
+Uν = 1
+µ(φ,ν + θS,ν) ,
+(27)
+where µ is the specific enthalpy, S is the specific entropy and the potentials
+φ and θ have no clear physical meaning. The four-velocity is subjected to
+the normalization condition,
+UνUν = −1
+(28)
+In what follows, we will use the following thermodynamic equations,
+ρ = ρ0(1 + Π),
+µ = (1 + Π) + p
+ρ0
+,
+TdS = dΠ + pd
+� 1
+ρ0
+�
+,
+(29)
+where Π is the specific internal energy, T is the absolute temperature and ρ0
+is the rest mass density. Combining those equations, we may write,
+T = 1 + Π,
+S = ln(1 + Π) 1
+ρ
+1
+3
+0
+(30)
+19
+
+Now, we can write the specific enthalpy µ in terms of the other thermo-
+dynamic potentials presents in Eq.(27), with the aid of the normalization
+condition Eq.(28),
+µ = 1
+N ( ˙φ + θ ˙S).
+(31)
+If we combine the Eqs. (29), (30) and (31), we may write the radiation energy
+density as,
+ρ =
+� 1
+N ( ˙φ + θ ˙S)
+4
+3
+�4
+e−3S
+(32)
+Introducing the above expression of ρ Eq.(32) in the radiation fluid action
+Eq.(5), we find with the aid of Eq.(4),
+�
+M d4x√−g1
+3ρrad =
+�
+M d4x√−g1
+3
+� 1
+N ( ˙φ + θ ˙S)
+4
+3
+�4
+e−3S,
+(33)
+Next, we identify from the radiation fluid action Eq.(33) its lagrangian Lf,
+Lf = 27
+256
+a3
+N3( ˙φ + θ ˙S)
+4e−3S
+(34)
+From that lagrangian, we compute the canonically conjugated momenta to
+the canonical variables φ (pφ) and S (pS), in the usual way,
+pφ = ∂Lf
+∂ ˙φ = 27
+64
+a3
+N3( ˙φ + θ ˙S)
+3e−3S,
+pS = ∂Lf
+∂ ˙S = θpφ
+(35)
+The general expression for the fluid total hamiltonian NHf, in the present
+model, is given by,
+NHf = ˙φpφ + ˙SpS − NLf,
+(36)
+Introducing the fluid lagrangian Eq.(34) and the canonically conjugated mo-
+menta Eq.(35) in the fluid total hamiltonian expression Eq.(36), we find,
+NHf = pφ
+4
+3
+a eS.
+(37)
+We may greatly simplify the fluid total hamiltonian expression Eq.(37) by
+performing the following canonical transformations [31],
+T = pse−Spφ
+− 4
+3,
+pT = pφ
+4
+3eS,
+¯φ = φ − 4
+3
+pS
+pφ
+,
+¯pφ = pφ.
+(38)
+20
+
+If we rewrite the fluid total hamiltonian Eq.(37) in terms of the new canonical
+variables and their conjugated momenta Eqs.(38), we obtain,
+NHf = PT
+a .
+(39)
+Observing that last equation, we notice that the canonical variable T, asso-
+ciated to the radiation fluid, will play the role of time in the quantum version
+of those models.
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+24
+

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+Draft version January 3, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Bubble in the Whale: Identifying the Optical Counterparts and Extended Nebula for the
+Ultraluminous X-ray Sources in NGC 4631
+Jing Guo (郭静)
+,1 Jianfeng Wu
+,1 Hua Feng
+,2, 3 Zheng Cai
+,2 Ping Zhou
+,4, 5 Changxing Zhou
+,3
+Shiwu Zhang
+,2 Junfeng Wang
+,1 Mouyuan Sun
+,1 Wei-Min Gu
+,1 Shan-Shan Weng
+,6 and
+Jifeng Liu
+7, 8, 9
+1Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, China
+2Department of Astronomy, Tsinghua University, Beijing 100084, China
+3Department of Engineering Physics, Tsinghua University, Beijing 100084, China
+4School of Astronomy & Space Science, Nanjing University, 163 Xianlin Avenue, Nanjing 210023, China
+5Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Ministry of Education, Nanjing 210023, China
+6Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing 210023, China
+7Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
+8School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
+9WHU-NAOC Joint Center for Astronomy, Wuhan University, Wuhan, Hubei 430072, China
+ABSTRACT
+We
+present
+a
+deep
+optical
+imaging
+campaign
+on
+the
+starburst
+galaxy
+NGC
+4631
+with
+CFHT/MegaCam.
+By supplementing the HST/ACS and Chandra/ACIS archival data, we search
+for the optical counterpart candidates of the five brightest X-ray sources in this galaxy, four of which
+are identified as ultraluminous X-ray sources (ULXs). The stellar environments of the X-ray sources
+are analyzed using the extinction-corrected color-magnitude diagrams and the isochrone models. We
+discover a highly asymmetric bubble nebula around X4 which exhibits different morphology in the
+Hα and [O iii] images. The [O iii]/Hα ratio map shows that the Hα-bright bubble may be formed
+mainly via the shock ionization by the one-sided jet/outflow, while the more compact [O iii] structure
+is photoionized by the ULX. We constrain the bubble expansion velocity and interstellar medium den-
+sity with the MAPPINGS V code, and hence estimate the mechanical power injected to the bubble as
+Pw ∼ 5 × 1040 erg s−1 and the corresponding bubble age of ∼ 7 × 105 yr. Relativistic jets are needed
+to provide such level of mechanical power with a mass-loss rate of ∼ 10−7 M⊙ yr−1. Besides the
+accretion, the black hole spin is likely an additional energy source for the super-Eddington jet power.
+1. INTRODUCTION
+Ultraluminous X-ray sources (ULXs) are non-nuclear
+point-like X-ray sources with isotropic luminosity LX ≳
+1039 erg s−1, which corresponds to the Eddington limit
+for a ∼ 10 M⊙ black hole (Feng & Soria 2011; Kaaret
+et al. 2017).
+Two mechanisms are likely to explain
+the high luminosity: the sub-Eddington accretion onto
+intermediate-mass black holes (IMBHs) and stellar-mass
+compact objects undergoing super-Eddington accretion.
+The minority of ULXs at the higher end of the luminos-
+ity range can be explained by the first mechanism, such
+as ESO 243−49 HLX-1 with LX ∼ 1042 erg s−1 (Farrell
+et al. 2009; Webb et al. 2012). Meanwhile, the X-ray
+Corresponding author: Jianfeng Wu
+wujianfeng@xmu.edu.cn
+spectral properties of most ULXs are consistent with
+the super-Eddington accretion scenario (e.g., Gladstone
+et al. 2009; Walton et al. 2014; Salvaggio et al. 2022).
+Recent studies further identified several ULXs powered
+by neutron stars from the detections of pulsating radia-
+tions (Bachetti et al. 2014; F¨urst et al. 2016; Israel et al.
+2017a,b; Weng et al. 2017; Carpano et al. 2018; Wilson-
+Hodge et al. 2018; Sathyaprakash et al. 2019; Rodr´ıguez
+Castillo et al. 2020; Quintin et al. 2021).
+The definitive approach to decipher the nature of non-
+pulsating ULXs is the dynamical mass measurement of
+the accretors, which relies on the optical spectroscopy of
+the donor stars. However, the archived optical data on
+ULXs are far less abundant than X-ray data because
+most of the ULX optical counterparts are very faint
+(mV > 21 mag) and located in fairly crowded regions.
+Previous studies found that most of the ULXs are asso-
+arXiv:2301.00022v1  [astro-ph.HE]  30 Dec 2022
+
+ID2
+ciated with young star clusters, showing the donor stars
+might be the OB type (Roberts et al. 2008; Poutanen
+et al. 2013). For a limited number of ULXs, the nature
+of the donor stars are unambiguously identified (e.g.,
+M101 ULX-1 and NGC 7793 P13), while the dynami-
+cal studies on these systems supported the stellar-mass
+accretor scenario (Liu et al. 2013; Motch et al. 2014).
+A number of ULXs have surrounding bubble nebulae
+detected from deep optical imaging observations (e.g.,
+Pakull & Mirioni 2002; Ramsey et al. 2006; Soria et al.
+2010, 2021), the majority of which are considered to
+be formed via shock ionizations driven by the interac-
+tions of strong jets/outflows and the ambient interstellar
+medium (ISM). Strong outflows may be ubiquitous for
+ULXs under supercritical accretion (e.g., Narayan et al.
+2017; Weng & Feng 2018; Zhou et al. 2019; Qiu & Feng
+2021; Kosec et al. 2021). The kinetic power and age of
+the bubble can be inferred from its size and expanding
+velocity (Weaver et al. 1977), and hence may reveal the
+kinematics of jets/outflows and the accretion physics of
+ULXs (Pakull et al. 2010; Cseh et al. 2012; Soria et al.
+2021). For the other few cases, the high-ionization fea-
+tures (e.g., He ii λ4686) in the spectra of the optical
+nebulae imply that the photoionization could be the ma-
+jor origin of the extended structure (Pakull & Mirioni
+2002). Both shock ionization and photoionization may
+certainly be working at the same time while dominating
+different parts of the same optical nebula (G´urpide et al.
+2022; Zhou et al. 2022).
+In this work, we report on an optical broad-band and
+narrow-band imaging campaign for the Whale Galaxy
+NGC 4631 to identify the optical counterparts and sur-
+rounding extended nebulae of the ULXs, for which Soria
+& Ghosh (2009) presented a detailed study of their X-
+ray properties. As a late-type starburst galaxy 7.35 Mpc
+away, NGC 4631 (Figure 1) has been extensively studied
+in multiwavelengths.
+The existence of molecular out-
+flows, abundant gas and the X-ray halo reveals the di-
+versity of objects and astrophysical processes (e.g., Ya-
+masaki et al. 2009; Irwin et al. 2011; Mel´endez et al.
+2015). From the archival XMM-Newton data, Soria &
+Ghosh (2009) identified five brightest X-ray sources scat-
+tered in NGC 4631 and found that four of them (X1,
+X2, X4, X5) can be classified as ULXs.1 For the pur-
+pose of studying their physical nature and stellar envi-
+ronments, we analyze the optical images of all five X-
+ray sources in this paper combining the Canada-France-
+1 While Mineo et al. (2012) classified X4 as a high mass X-
+ray binaries in the sub-Eddington state based on the Chandra-
+measured luminosity, we adopt Soria & Ghosh (2009)’s classifi-
+cation throughout this work.
+Hawaii Telescope (CFHT) and Hubble Space Telescope
+(HST) observations, supplemented with Chandra data
+to determine the precise astrometry. The details of the
+five X-ray sources can be found in Table 1.
+This paper is organized as follows. In Section 2 we
+present the optical and X-ray observations and data re-
+duction. In Section 3, we improve the relative astrom-
+etry and identify optical counterpart candidates for the
+X-ray sources, which are investigated in Section 4 based
+on their locations on the isochrone diagrams. In Sec-
+tion 5, we present a newly discovered bubble nebula
+around X4 and the analyses on its morphology and ki-
+netic power. Section 6 summarizes our conclusions.
+2. OBSERVATIONS AND DATA REDUCTION
+2.1. CFHT
+We obtained optical broad-band and narrow-band
+imaging of NGC 4631 with the 3.6-m CFHT located
+on Mauna Kea, Hawaii.
+The MegaCam instrument
+mounted on CFHT has a wide field of view (1 deg2)
+which can fully cover all the 5 luminous X-ray sources
+in NGC 4631. The detector consists of 40 CCDs, each
+of which has 2048 × 4612 pixels with 0.′′187 × 0.′′187 per
+pixel. We are awarded a total of 4.5-hour exposure time
+(PI: Jing Guo, ObsId: 20AS01) executed in 2020 March
+and June. The images are taken with three broad bands
+(u, g, and r) and two narrow bands (Hα and [O iii]).
+The Hα and [O iii] filters have a width of ∼ 100 ˚A, cen-
+tered at 6590 ˚A and 5006 ˚A, respectively. A dithering
+pattern was applied during the observations to cover the
+CCD gaps, which requires at least five exposures for each
+band. The detailed observation log is listed in Table 2.
+The data products we received have been prepro-
+cessed with the Elixir pipeline, which includes bias-
+subtraction, flat-fielding, etc., for each individual frame
+(Magnier & Cuillandre 2004). The first step is to per-
+form precise astrometric calibration and to stack im-
+ages from individual exposures in the same band for
+the purposes of eliminating CCD gaps and reaching
+the desired sensitivity level. In the stacking procedure,
+SExtractor (Bertin & Arnouts 1996) was applied on
+each image of single exposures to generate the catalog of
+all point sources with coordinates. The astrometric solu-
+tions were then computed with the SCAMP (Bertin 2006)
+software by referencing the catalog from Gaia Data Re-
+lease 1 (DR1). We utilized the SWarp (Bertin 2010) task
+
+3
+Table 1. List of the Five Brightest X-ray Sources in NGC 4631
+Source ID
+R.A.
+Dec.
+Chandra Net Counts
+Off-axis
+Opt-X Error Circle
+NH
+E(F606W-F814W)
+(J2000)
+(J2000)
+(0.5–8.0 keV)
+(arcmin)
+(arcsecond)
+(1021 cm−2)
+X1
+12 42 15.99
++32 32 49.47
+6.7±2.8
+4.10
+0.673
+2.4+0.3
+−0.3
+0.35+0.15
+−0.15
+X2
+12 42 11.12
++32 32 35.63
+981.2±32.9
+3.05
+0.275
+28.3+3.6
+−3.2
+3.80+1.74
+−1.55
+X3
+12 42 06.13
++32 32 46.43
+357.6±19.6
+2.11
+0.269
+2.0+1.0
+−0.9
+0.29+0.48
+−0.43
+X4
+12 41 57.42
++32 32 02.79
+77.7±9.2
+0.19
+0.280
+0.32+1.02
+−0.32
+0.05+0.49
+−0.16
+X5
+12 41 55.57
++32 32 16.77
+2977.8±55.8
+0.51
+0.268
+2.0+0.2
+−0.2
+0.29+0.10
+−0.10
+Note—The NH values are retrieved from Soria & Ghosh (2009) which were obtained from the Chandra spectral analyses.
+Table 2. Observation Log of NGC 4631
+Instrument
+Source ID
+ObsID
+Filter
+Observation Date (UT)
+Exposure Time
+CFHT/MegaCam
+X1-X5
+20AS01
+Hα
+2020-03-23
+12×900 sec
+(PI: Jing Guo)
+[O iii]
+2020-05-19
+5×750 sec
+u
+2020-03-23
+5×126 sec
+g
+2020-03-23
+5×126 sec
+r
+2020-03-23
+5×126 sec
+HST/ACS
+X1,X2,X3
+j8r331010
+F606W
+2003-08-03
+676 sec
+X1,X2,X3
+j8r331020
+F814W
+2003-08-03
+700 sec
+X4, X5
+j8r332010
+F606W
+2004-06-09
+676 sec
+X4, X5
+j8r332020
+F814W
+2004-06-09
+700 sec
+Chandra/ACIS
+X1-X5
+797
+2000-04-16
+60 ksec
+Table 3. List of the reference stars
+Reference ID
+X-ray Coordinates
+Optical Coordinates
+Off-axis
+Net Counts
+X-ray positional error
+Chandra
+HST
+(arcmin)
+Chandra
+(arcsecond)
+Ref.1
+12 42 25.78 +32 33 21.40
+12 42 25.79 +32 33 21.23
+6.2
+120 ± 11
+0.32
+Ref.2
+12 42 04.03 +32 34 08.60
+12 42 04.03 +32 34 08.41
+2.7
+107 ± 10
+0.19
+Note—The HST coordinates are given by the Dolphot package.
+to perform the image stacking. The astrometric error
+during these processes are < 0.′′03. A multi-color im-
+age of NGC 4631 is shown in Figure 1. This RGB-like
+image combines the three broad bands and two narrow
+bands.
+The five red circles label the positions of the
+five luminous X-ray sources analyzed in Soria & Ghosh
+(2009). X3 is more likely a black hole X-ray binary in its
+high/soft state, while the remaining four X-ray sources
+are classified as ULXs, among which X1 is a supersoft
+ULX.
+We select 40 point sources from the Pan-STARRS1
+DR2 catalog (Flewelling 2018; Flewelling et al. 2020)
+that are isolated and have an appropriate magnitude
+(17–19 mag) to serve as photometry references.
+The
+Pan-STARRS1 DR2 catalog does not flag the source
+whether it is a star. Thus we select such a relatively
+large set of referencing sources aiming to obtain a more
+
+4
+12H42'30"
+15"
+00"
+41'45"
+30"
+32°36'00"
+34'00"
+32'00"
+30'00"
+R.A.
+Dec.
+X1
+X2
+X2
+X3
+X4
+X5
+Figure 1. The RGB-like image combines five CFHT/MegaCam filters, including the three broad bands (u, g, r) and two narrow
+bands (Hα, [O iii]). The u, g, and r bands are shown in blue, green, and red colors, respectively, while the Hα and [O iii] filters
+are represented by crimson and teal colors, respectively. The red circles label the positions of the five X-ray sources.
+12h42m17.0s
+16.5s
+16.0s
+15.5s
+15.0s
+32°33'00"
+32'55"
+50"
+45"
+40"
+R.A.
+Dec.
+X 1 
+12h42m12.0s
+11.5s
+11.0s
+10.5s
+10.0s
+32°32'45"
+40"
+35"
+30"
+25"
+R.A.
+Dec.
+X 2 
+12h42m07.0s
+06.5s
+06.0s
+05.5s
+05.0s
+32°32'55"
+50"
+45"
+40"
+35"
+R.A.
+Dec.
+X 3 
+12h41m58.5s
+58.0s
+57.5s
+57.0s
+56.5s
+32°32'15"
+10"
+05"
+00"
+31'55"
+R.A.
+Dec.
+X 4 
+12h41m56.5s
+56.0s
+55.5s
+55.0s
+54.5s
+32°32'25"
+20"
+15"
+10"
+05"
+R.A.
+Dec.
+X 5 
+Figure 2. The CFHT/MegaCam g-band images of X1-X5 and their vicinity, respectively. The green circles are centered at the
+X-ray location of each source with a radius of 3′′. Optical counterparts are difficult to identify in these broad-band images due
+to the seeing limit (0.′′45–0.′′75) for the ground-based CFHT.
+
+5
+statistically reliable photometry calibration. We adopt
+the conversion equations from Pan-STARRS filters to
+MegaCam filters provided by the Canadian Astronomy
+Data Centre (CADC),2 except for Hα we use the for-
+mula provided by Boselli et al. (2018). Finally, for each
+given source, we can derive an array of magnitude val-
+ues calibrated from the 40 reference stars.
+The peak
+value of the best-fit Gaussian profile to the histogram
+of the magnitude values was adopted as the measured
+magnitude for this source.
+The zoom-in CFHT/MegaCam g-band images of the
+five luminous X-ray sources are shown in Figure 2. For
+the stacked image in each band, we estimate the expo-
+sure depth reaching 25–26 mag arcsec−2 at 3σ level. Due
+to the seeing limit of ground-based imaging, it is difficult
+to identify the exact optical counterparts to the X-ray
+sources at their crowded locations in the edge-on galaxy
+NGC 4631. However, in the Hα narrow-band images we
+discover a bubble-like extended nebula surrounding X4,
+which may be inflated by the jet or wind launched from
+the ULX accretion disk (Figure 3). The projected size
+of this bubble structure is ∼ 130 pc × 100 pc.
+To obtain a more precise profile of the extended bub-
+ble structure, we need to subtract the continuum con-
+tribution from the Hα image. Boselli et al. (2018) uti-
+lized a large set of unsaturated stars and derived an
+empirical equation (see their Eqn. 4) to relate the g − r
+color and the Hα magnitude. Following Boselli et al.
+(2018), we use the data products which were processed
+by CADC with MegaPipe upon our request. MegaPipe
+will subtract the sky background, normalize the flux in
+the whole stacked image, and provide a catalog of de-
+tected point sources (Gwyn 2008). We filtered out the
+pixels with low signal-to-noise ratio (S/N ⩽ 5), and
+then applied the equation in Boselli et al. (2018) pixel by
+pixel. The generated image is shown in the upper right
+panel of Figure 3. Most of the point sources around X4
+have been removed from the Hα image. The morphology
+of the extended structure are clearly revealed.
+For the [O iii] narrow-band image, we derived a sim-
+ilar equation connecting the g-r color and the [O iii]
+magnitude by roughly assuming the continuum magni-
+tude is linearly related to wavelength in the given range
+(i.e., the continuum follows a power-law spectral profile;
+see details in Appendix A):
+mg/[O III] ≈ mg − 0.155 × (mg − mr),
+(1)
+where mg/[O III] is the magnitude of the continuum that
+falls within the [O iii] narrow-band filter. After applying
+2 https://www.cadc-ccda.hia-iha.nrc-
+cnrc.gc.ca/en/megapipe/docs/filt.html
+the equation pixel by pixel, we obtain an [O iii] narrow-
+band image for which most of the continuum contribu-
+tion has been eliminated (see the lower right panel of
+Figure 3). As for the continuum-subtracted Hα image,
+the point sources have been mostly removed from the
+[O iii] image, proving the efficacy of our continuum sub-
+traction method. We will discuss this bubble structure
+in details in Section 5.
+2.2. HST
+NGC 4631 has been observed with the Advanced
+Camera for Surveys (ACS; Ford et al. 1998) onboard
+HST (see Table 2).
+The five luminous X-ray sources
+were completely covered by the observations in Proposal
+9765. The field containing X1, X2, and X3 was observed
+in 2003 August (ObsID j8r331010 for the F606W filter
+and j8r331020 for F814W), while X4 and X5 were cov-
+ered by the observations in 2004 June (ObsID j8r332010
+for F606W and j8r332020 for F814W). Each observa-
+tion has a total exposure time of 1376 sec. The images
+of the five X-ray sources in the F606W band are shown
+in Figure 4.
+We aim to identify the optical counterparts of X-ray
+sources with the HST imaging and derive their mag-
+nitudes. Astrometric calibration is also needed for the
+HST images. As the lack of coverage upon the galaxy
+disk of NGC 4631 in Gaia, we are not able to directly
+align HST images with the Gaia references. CFHT im-
+ages with a large field of view are reused as the reference
+images to align the HST data. We selected seven refer-
+ence sources in each HST observation to perform astro-
+metric calibration, for which we obtained the RMS resid-
+ual of 0.′′03. Then we employed the Dolphot package
+to perform Point Spread Function (PSF) photometry.
+Dolphot can identify point sources in heavily crowded
+areas and return their Vega magnitudes (Dolphin 2000).
+The acsmask task was used to flag bad pixels, and the
+calcsky task can calculate the sky background.
+After
+these preprocessing, the PSF photometry was accom-
+plished by the dolphot task. The parameters in dolphot
+are configured referring to Williams et al. (2014) where
+they made a series of artificial stars to test a mesh grid
+parameters and found out the most suitable parameter
+set for crowded fields.
+2.3. Chandra
+In the X-ray band, NGC 4631 has been observed by
+Einstein, ROSAT, Chandra, and XMM-Newton. To ob-
+tain precise locations of the X-ray sources, we repro-
+cessed the Chandra/ACIS data which have a sub-arcsec
+angular resolution.
+The Chandra observation (ObsID
+797) was carried out on 2000 April 16 for a total of 60
+
+6
+12h41m57.8s
+57.6s
+57.4s
+57.2s
+57.0s
+32°32'08"
+06"
+04"
+02"
+00"
+R.A.
+Dec.
+12h41m57.8s
+57.6s
+57.4s
+57.2s
+57.0s
+32°32'08"
+06"
+04"
+02"
+00"
+R.A.
+Dec.
+B
+C
+A
+12h41m57.8s
+57.6s
+57.4s
+57.2s
+57.0s
+32°32'08"
+06"
+04"
+02"
+00"
+R.A.
+Dec.
+12h41m57.8s
+57.6s
+57.4s
+57.2s
+57.0s
+32°32'08"
+06"
+04"
+02"
+00"
+R.A.
+Dec.
+12h41m57.8s
+57.6s
+57.4s
+57.2s
+57.0s
+32°32'08"
+06"
+04"
+02"
+00"
+R.A.
+Dec.
+B
+C
+A
+12h41m57.8s
+57.6s
+57.4s
+57.2s
+57.0s
+32°32'08"
+06"
+04"
+02"
+00"
+R.A.
+Dec.
+Figure 3. From left to right in the first row, the first panel is the CFHT/MegaCam r-band image, where the cyan circle is
+centered at the X-ray location (the red cross symbol) of X4 with a 3′′ radius. The half length of the red cross represents the
+X-ray positional error of X4. The second is the Hα image, residing with a bubble-like structure around X4. The brightest region
+is marked as A region in the white circle. When performing the photometry for the whole bubble, the shape is adopted as the
+region between the two red ellipses, marked as B (which includes the A region). The cavity in the center is marked as C. The
+third panel is the result of subtracting the underlying continuum component from the Hα image. Most of the stellar sources
+have been removed here. The blank regions represent the dropped pixels that do not have adequate S/N. In the second row,
+the images of g, [O iii] and [O iii] with continuum removed are shown in turn.
+ksec exposure time. We perform X-ray astrometry and
+photometry in this work. The spectral and timing prop-
+erties of these X-ray sources were presented in details in
+Soria & Ghosh (2009).
+The data were reprocessed with the CIAO (v4.13)
+package. The chandra_repro task was applied to cre-
+ate a new level = 2 event file calling the latest calibra-
+tion products (CALDB v4.9.4) and more advanced al-
+gorithms.
+For astrometric calibration, we aligned the
+Chandra/ACIS images to the HST/ACS images (see
+Section 3 for details).
+The CIAO script deflare was used to remove the
+background flares (> 3σ) which only accounts for ≈ 4%
+of the total exposure time. The full-band (0.5–8.0 keV)
+X-ray image was then generated using the ASCA grade
+0,2,3,4,6 events. The PSF and exposure maps were pro-
+duced accordingly. The final X-ray point source detec-
+tion was carried out using wavdetect. The detection
+threshold is set to be 10−6, while the wavelet scales are
+1,
+√
+2, 2, 2
+√
+2, and 4 pixels. The coordinates return by
+wavdetect are adopted as the X-ray positions for the
+five luminous X-ray sources.
+3. IDENTIFYING THE OPTICAL
+COUNTERPARTS
+To identify the optical counterparts of the X-ray
+sources, we improve the astrometry of Chandra/ACIS
+images relative to the HST/ACS images following the
+methodology laid out in Yang et al. (2011).
+Because
+of the small field of view of HST/ACS, only one com-
+mon source can be registered from the Chandra to the
+HST images. Therefore, we supplement the HST/ACS
+observation on an adjacent and partly overlapping field
+(ObsID jc9l04010) to taking a mosaic image using the
+AstroDrizzle package. The second common source is
+therefore added.
+These two objects are identified as
+point X-ray sources (Wang et al. 2016; Evans et al.
+2010), and their coordinates and other information are
+listed in Table 3.
+We use the CIAO task wcs_match
+to register the Chandra image to the HST image. The
+
+7
+Figure 4. The HST/ACS F606W images of each X-ray source, with the overlaid white contours representing the X-ray flux
+level from the Chandra/ACIS data (contours not in uniform scales among the five panels). In each panel, the green circle is
+centered at the X-ray location with the radius represents the respective error circle. The numbered red circles are the optical
+counterpart candidates of the X-ray source. The white dashed circle has a radius of 1′′. The cyan circle in the middle right
+panel marks the young star associations northeast to X4, while that in the bottom left panel labels the compact young star
+group associated with X5.
+
+32°32'52"
+32°32'38"
+×2
+X1
+37"
+50"
+36"
+Dec.
+Dec.
+5
+12h42m16.1s
+16.0s
+15.9s
+15.8s
+12h42m11.2s
+11.1s
+11.0s
+10.9s
+R.A.
+R.A.
+32°32'05"
+×3
+X 4
+04"
+47"
++
+O
+4
+01"
+06.2s
+57.3s
+12h42m06.3s
+06.15
+06.0s
+12h41m57.5s
+57.45
+57.2s
+R.A.
+R.A.
+32°32'19"
+X 5
+18"
+5.
+12h41m55.7s
+55.6s
+55.5s
+55.4s
+R.A.8
+RMS residual is 0.′′02.
+The updated positions of five
+X-ray sources are listed in Table 1.
+We calculated the 95% Chandra positional error ra-
+dius for each source using Equation 5 in Hong et al.
+(2005) which has considered the PSF variations across
+the field of view. We then converted it to the 1σ error
+radius by applying the relation rX = rX(95%)/1.95996
+in Zhao et al. (2005). The size of the error circle is pri-
+marily related to the number of counts and the off-axis
+angle of the source.
+Finally, we adopt the positional
+uncertainty of each source as the quadratical combina-
+tion of all kinds of errors: the average X-ray positional
+error of the two reference objects (0.′′19), the X-ray po-
+sitional error of each X-ray source (0.′′15-0.′′63), the error
+caused by the alignment between the HST and Chandra
+images, and the error of optical coordinates which were
+generated during the astrometric calibration of HST and
+CFHT images. The latter two kinds of errors are both
+ignorable compared to the X-ray positional errors. The
+final positional uncertainties of the five X-ray sources
+are listed in Table 1.
+In Figure 4, we overlay the X-ray flux contours (solid
+white lines) onto the HST/ACS/F606W images for each
+X-ray source.
+The green circles represent the uncer-
+tainties of X-ray positions.
+X1 has the largest error
+circle (0.′′66) because of the small number of Chandra
+net counts (≈ 7). There are multiple candidate opti-
+cal counterparts for X1 detected by Dolphot, which are
+labeled by small red circles in the upper left panel of
+Figure 4. The error radii of X2–X5 are similar (∼ 0.′′3).
+X2 has one candidate optical counterpart in its X-ray
+positional error circle, while both X3 and X4 have a few
+candidates in their respective error circles. X5 is located
+within a crowded region while two individual sources are
+detected in the error circle. We will analyze these can-
+didate optical counterparts and the surrounding stellar
+environments in the next section.
+4. COLOR-MAGNITUDE DIAGRAM
+For most ULXs, the optical emission is dominated by
+the X-ray reprocessing on the accretion disk (Tao et al.
+2011). Nevertheless, the optical Color-Magnitude Dia-
+grams (CMD) can be used to infer the age of the stel-
+lar environments around ULXs, which could potentially
+suggest the nature of the ULX donor stars. The Padova
+Stellar Evolution Code (PARSEC; Bressan et al. 2012)
+provides the isochrone databases for almost all the main-
+stream telescope filters.3 Here we utilize the isochrones
+based on the HST/ACS filters system.
+3 http://stev.oapd.inaf.it/cgi-bin/cmd
+We derive the extinction AV from the hydrogen col-
+umn density obtained by Soria & Ghosh (2009) via X-
+ray spectral analyses of the Chandra observations based
+on the relation of NH (cm−2) = (2.21 ± 0.09) × 1021
+AV (mag) presented in G¨uver & ¨Ozel (2009). To con-
+vert AV to the extinction in the HST/ACS filter sys-
+tem E(F606W − F814W), we then interpolate the cen-
+tral wavelengths of these filters to the extinction law
+derived in Cardelli et al. (1989). The extinction value
+for each X-ray source is listed in Table 1.
+Closely aligned with a young stellar cluster, X2 has
+large extinction E(F606W − F814W) = 3.8 mag, which
+may introduce significant uncertainties when applying
+the CMD to derive the ages of its surrounding stars.
+Therefore, we only plot the isochrones for the other four
+X-ray sources (Figure 5). For each panel, the solid blue
+dots stand for the optical counterpart candidates of the
+X-ray source. The light blue dots represent the stars
+within 1′′ (36 pc; see the white dashed circles in Fig-
+ure 4) from the X-ray position but outside the optical-
+to-X-ray error circle.
+The immediate surrounding stars of X1 do not appear
+to be closely associated like in a star group or cluster.
+Its optical counterpart candidates, as well as the nearby
+stars within 1′′, span a wide range of age from 5 Myr
+to 80 Myr, which indicates that they are unlikely born
+at the same time or in the same environment. As dis-
+cussed in Section 3, the positional error circle of X1 is
+also much larger (0.′′673), corresponding to ≈ 25 pc. For
+X2, the sole optical counterpart shown in Fig 4 is likely
+to be not reliable because of the large extinction. For
+X3, the three optical counterpart candidates have ages
+of ∼50–80 Myr which are consistent with the ages of
+the environmental sources. For X4, the ages of the sur-
+rounding stars range mostly in ∼ 20–80 Myr. The six
+candidate optical counterparts also show similar ages.
+There appears to be a star association northeast of X4
+with the size of ≈ 2′′ across (71 pc). The CMD shows
+that most of its member stars are very young with ages
+of 5–20 Myr (the green tri up symbols in the lower left
+panel of Figure 5). It is worth noting that the NH value
+of X4 derived from the XMM-Newton spectral model-
+ing is one order of magnitude higher than that with the
+Chandra data (Soria & Ghosh 2009). The optical coun-
+terpart candidates of X4 would be younger, < 20 Myr
+(see fig 5), if the XMM-Newton extinction value were
+adopted. X5 appears to locate within a compact star
+group (≈ 0.′′8 across; corresponding to 28 pc). Two point
+sources are identified within the error circle with ages of
+∼ 5 Myr, while three more individual sources in this star
+group are resolved by Dolphot, which have ages ∼ 5–
+10 Myr (the green tri up symbols in Figure 5 lower right
+
+9
+2
+1
+0
+1
+2
+F606W-F814W
+10
+9
+8
+7
+6
+5
+4
+3
+2
+F814W
+X 1 
+5 Myr
+10 Myr
+20 Myr
+50 Myr
+80 Myr
+2
+1
+0
+1
+2
+F606W-F814W
+10
+9
+8
+7
+6
+5
+4
+3
+2
+F814W
+X 3 
+5 Myr
+10 Myr
+20 Myr
+50 Myr
+80 Myr
+2
+1
+0
+1
+2
+F606W-F814W
+10
+9
+8
+7
+6
+5
+4
+3
+2
+F814W
+X 4 
+5 Myr
+10 Myr
+20 Myr
+50 Myr
+80 Myr
+2
+1
+0
+1
+2
+F606W-F814W
+10
+9
+8
+7
+6
+5
+4
+3
+2
+F814W
+X 5 
+5 Myr
+10 Myr
+20 Myr
+50 Myr
+80 Myr
+Figure 5. The color-magnitude diagrams (CMDs) for optical point sources around X1, X3, X4, and X5, respectively. The solid
+blue dots are the optical counterpart candidates within the error circle. The light blue dots are the point sources within 1′′ but
+outside the error circle which could be born in the same environment. The green tri up symbols in left-lower panel represent
+the sources in the star group northeast of X4. Extinction correction has been applied based on the X-ray hydrogen column
+density. The open blue circles labels the loci of the optical counterpart candidates of X4 when the extinction value is adopted
+from the XMM-Newton spectral modeling.
+
+10
+12h41m57.8s 57.6s
+57.4s
+57.2s
+57.0s
+32°32'06"
+04"
+02"
+00"
+R.A.
+Dec.
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Figure 6. The [O iii]/Hα flux ratio map for the extended
+structure around X4, where the vertical color bar shows the
+line ratio values. The red ellipse represents the profile of the
+Hα bubble, while the red cross marks the position of X4.
+The white horizontal and vertical bars illustrate the major
+and minor axes of the extended structure.
+panel). Therefore, X5 is likely associated with a young
+star cluster.
+It is worth noting that the extinction derived from
+the hydrogen column density obtained with X-ray spec-
+tra represent an upper limit for the candidate optical
+counterparts and their surrounding stars.
+If signifi-
+cant intrinsic absorption exists for the X-ray source,
+the extinction would be much smaller. A conservative
+lower limit would be the Galactic extinction along the
+light of sight of NGC 4631, which is AV = 0.015 mag
+(Schlafly & Finkbeiner 2011), corresponding to NH =
+3.3 × 1019 cm−2.
+This is ignorable compared to that
+from the X-ray spectroscopy.
+The true values of ex-
+tinction should be in between the above lower and up-
+per limits. Overestimation of the extinction would place
+the stars at younger age regions on the CMD. However,
+it is probably reasonable to assume that the candidate
+optical counterparts and surrounding stars would suf-
+fer from high extinction (i.e., close to the upper limits),
+since NGC 4631 appears as an edge-on disk galaxy.
+5. A NEWLY DISCOVERED BUBBLE
+STRUCTURE AROUND X4
+5.1. Morphology Analysis
+Both of the continuum-subtracted Hα and [O iii] im-
+ages display a clear extended structure around X4 (see
+the right two panels in Figure 3), while exhibiting differ-
+ent morphology in the two bands. In the Hα image, the
+structure appears more like an inflated bubble with the
+size of ∼ 130 pc × 100 pc. The X-ray source X4 is not
+located in the center. Instead, this Hα bubble structure
+appears to be sourced from the location of the ULX
+(see the red cross in Figure 3) and is oriented toward
+the southwest direction, reaching maximum luminosity
+in the outermost region, ∼ 100 pc away from X4. The
+extended nebula in the [O iii] image has a smaller size.
+In contrast to the Hα bubble, the brightest region of the
+[O iii] structure is to the east of X4, and is substantially
+closer to the X-ray source ( <
+∼ 25 pc).
+The extended structures around ULXs may originate
+from photoionization or shock ionization, both of which
+could coexist while playing major roles in different parts
+of the structure (Moon et al. 2011; G´urpide et al. 2022;
+Zhou et al. 2022). Generally, in the photoionization pro-
+cess, the line flux ratio [O iii]/Hβ tends to peak at or
+near the ionizing source and declines outwards. For the
+shock-ionized bubble, the edge region has higher excita-
+tion level and exhibits the higher [O iii]/Hβ ratio than
+in the central area. Here we use the [O iii]/Hα ratio
+as a proxy, since it is reasonable to assume that the
+line ratio Hα/Hβ ≡ τ remains constant in the bub-
+ble area.
+The typical τ value is ∼ 3 for ULX bub-
+bles (Allen et al. 2008).
+We will calculate the exact
+value for our case in the next subsection. Derived from
+the continuum-subtracted [O iii] and Hα images, the
+[O iii]/Hα ratio map is shown in Figure 6, in which
+the red ellipse marks the bubble shape in the Hα band
+(same as that in the upper middle panel of Figure 3).
+We extract the [O iii]/Hα line ratio roughly along the
+major and minor axes of the bubble (the white horizon-
+tal and vertical bars in Figure 6 respectively) and obtain
+a clearer spatial profile, which is illustrated in Figure 7.
+The [O iii]/Hα ratio reaches its minimum in the bubble
+center and increases outwards along both axes, which
+suggests that the Hα bubble is mostly dominated by
+the shock ionization. There is a bump of the [O iii]/Hα
+ratio in the east edge of major axis, coinciding with the
+brightest [O iii] region. This area is likely formed pre-
+dominantly by photoionization. It is indeed close to the
+ULX which is presumably the source of ionizing photons.
+The peak [O iii]/Hα value in this area is >
+∼ 1. Combined
+with the calculated Hα/Hβ line ratio of τ ∼ 3.75 (see
+Section 5.2), the peak [O iii]/Hβ would be ∼ 4, which
+is similar to that of photoionization-dominated nebulae
+found in previous ULX bubble studies (e.g., Soria et al.
+2021).
+The shock-ionized bubbles around ULXs can be
+formed via two mechanisms: through explosive events
+like supernovae (i.e., supernova remnants) or being in-
+flated by continuous jet/outflow from ULXs (Pakull
+et al. 2006).
+However, ULX bubbles often have sizes
+of a few hundred pc (e.g., Ramsey et al. 2006; Gris´e
+
+11
+2
+1
+0
+1
+2
+arcsecond distance from the bubble center
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+[OIII]/H
+Major axis
+Minor axis
+Figure 7. The spatial profile of the [O iii]/Hα flux ratio
+along the major and minor axes of the extended nebula (see
+the white bars in Figure 6).
+et al. 2011), which are one order of magnitude larger
+than normal supernova remnants. Although there exists
+the possibility of very energetic hypernova explosions, it
+is unlikely considering the stellar environments and the
+survival of ULXs as binary systems during the events
+(Feng & Soria 2011). Therefore, we suggest the Hα bub-
+ble structure around X4 is more likely to be inflated by
+the ULX jet/outflow.
+It is worth noting that, unlike many other ULX bub-
+bles, this extended structure around X4 only has a one-
+sided lobe to the southwest direction, while X4 itself is
+close to the east edge of the bubble. The missing of the
+lobe in the other direction is not caused by the relativis-
+tic beaming because the bubble expansion velocity vs
+is only at the order of hundred km s−1, far below the
+speed of light. For example, the bubble around the ULX
+in NGC 5585 has an expansion velocity of 125 km s−1
+(Soria et al. 2021). For this bubble around X4, the ex-
+pansion velocity is estimated to be vs ∼ 110 km s−1 (see
+Section 5.2).
+The asymmetric profile of an extended nebula could
+imply the density gradient of ISM or outflows, as sug-
+gested for IC 342 X-1 (Cseh et al. 2012). Here in our
+case, one viable scenario is that the ISM is much denser
+to the east of X4, resulting in an outflow blocked by
+the dense medium. Hence, the east area is mainly pho-
+toionized by the ULX itself (as shown by the [O iii]/Hβ
+ratio profile), while most other regions are dominated by
+shock ionization through the outflow. Similar situations
+can be found in the simulations of supernova feedback
+(Creasey et al. 2011; Pardi 2017). In their simulations, if
+the ISM density reaches 102–104 cm−3 and the ejection
+temperature is lower than 106 K, the injected energy of
+the outflow will be immediately lost due to the strong
+radiative cooling in the high density regions, dubbed the
+overcooling problem.
+An alternative interpretation of this unusual morphol-
+ogy is that the accretion disk of X4 has launched an
+asymmetric outflow, i.e., the outflow to the east direc-
+tion is much weaker or nonexistent. There have been
+numerical simulations showing that an asymmetric or
+even one-sided outflow can be formed from the accretion
+disk if the accretor is rotating and is accompanied with
+a complex magnetic field (Lovelace et al. 2010; Dyda
+et al. 2015).
+It is also possible that both of the two mechanisms
+are responsible for this asymmetric morphology.
+The
+side with the weaker/absent outflow has more dense am-
+bient ISM, leading to photoionization dominating the
+compact east area close to the ULX, while the shock-
+ionized bubble is only formed to the opposite direction.
+5.2. Mechanical Power Estimation
+To estimate the mechanical power needed to inflate
+the bubble, we first calculate the Hα luminosity from
+the surface brightness of the structure measured with
+Python/Photutils.
+The brightest region in the Hα
+band (marked with “A” in Figure 3) has a surface bright-
+ness of 19.34 ± 0.01 mag arcsec−2. The whole bubble
+structure, which is confined in a donut shape (region B
+in Figure 3, subtracting region C while including region
+A), has an average surface brightness of 19.64 ± 0.01
+mag arcsec−2. Using the surface brightness and bubble
+size R, We can estimate the injected mechanical power
+Pw.
+Based on the standard bubble theory (Weaver et al.
+1977; Pakull et al. 2006), Pw can be calculated as
+P39 ≈ 3.8R2
+2v2
+3n erg s−1,
+(2)
+where P39 ≡ Pw/(1039 erg s−1); R2 ≡ R/(100 pc);
+v2 ≡ vs/(100 km s−1); n is the ISM number density
+in unit of cm−3. As the ULX is located at the edge of
+the Hα bubble, we conceive that the one-sided jet/wind
+formed this one-lobe bubble. Therefore, we adopt the
+scale of the whole bubble to substitute the radius, i.e.,
+R = 130 pc and R2 = 1.3. The number density n can be
+derived from the Hβ luminosity LHβ, the expanding ve-
+locity vs, and the area of the spherical bubble A (Dopita
+& Sutherland 1996),
+n = 1.3 × 105LHβA−1v−2.41
+2
+cm−3.
+(3)
+The surface area A of the spherical bubble with a di-
+ameter of 130 pc is calculated as 5 × 1041 cm2. Without
+available Hβ imaging, the Hβ luminosity can be derived
+
+12
+60
+70
+80
+90
+100
+110
+120
+velocity  km s
+1
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+[O III]/H
+(108, 0.3)
+(145, 0.3)
+Z = 0.5 Z , n = 6 cm
+3, B = 3.0 G, T = 20000 K
+Figure 8. The adopted solution from MAPPING V code.
+The ISM number density is 6 cm−3. When the shock velocity
+vs ≈ 108 km s−1, the [O iii]/Hα flux ratio is consistent with
+the observed value of ≈ 0.3.
+from Hα luminosity using the Balmer line ratio τ. The
+intrinsic Hα luminosity is LHα = 1.2×1038 erg s−1, cal-
+culated from the bubble surface brightness. With the
+lack of optical spectroscopy on the bubble, the precise
+expanding velocity vs is not available.
+We employed
+the widely used shock-ionization model MAPPINGS V
+(Allen et al. 2008) to estimate τ and vs.
+We carried
+out a series of calculations with MAPPINGS V which
+returned values for a variety of line ratios to compare
+with the observation.
+We fixed the metallicity to 0.5
+solar abundance (Pilyugin et al. 2014), magnetic field
+to a typical value of 0.3 µG, and arranged a large input
+grid of the shock velocity vs and hydrogen number den-
+sity n, finding a set of solution matching the observed
+[O iii]/Hα line ratio, which is ≈ 0.3 along the most parts
+of the Hα bubble (see Figure 7). The result is shown in
+Figure 8. We obtained the following parameter values
+in this solution: n ∼ 6 cm−3, vs ∼ 110 km s−1, and the
+Balmer line ratio τ ∼ 3.75.
+Combining Eqns. (2) and (3), we can derive a relation
+where the mechanical power Pw is determined by the Hα
+luminosity LHα, the line ratio τ and expanding velocity
+vs:
+P39 ≈ 5.0 × 105R2
+2(LHα/τ)A−1v0.59
+2
+.
+(4)
+By substituting the MAPPING V solution, we calcu-
+lated the value of P39 ≈ 51, i.e., the injected mechanical
+power Pw ∼ 5 × 1040 erg s−1.
+The lifetime t of the
+bubble is estimated to be t = 3
+5R/vs ∼ 7 × 105 yr.
+We can now calculate the mass-loss rate
+˙M of the
+ULX jet/wind. In the non- and mildly-relativistic sce-
+nario, the injected power can also be expressed as Pw =
+1
+2 ˙Mv2
+w (Weaver et al. 1977), where vw is the velocity of
+0.0
+0.1
+0.2
+0.3
+0.4
+c
+5.0
+4.5
+4.0
+3.5
+3.0
+2.5
+2.0
+1.5
+log(M)
+0
+2
+4
+6
+8
+10
+12
+14
+7.5
+7.0
+6.5
+6.0
+5.5
+5.0
+log(M)
+=2
+=10
+Figure 9.
+The estimated mass-loss rate
+˙M of non- and
+mildly-relativistic wind (left panel) and the relativistic jet
+(right panel). The x-axis is the wind velocity normalized by
+the speed of light vc in the left panel and the bulk Lorentz
+factor Γ in the right panel.
+jet/wind. This equation can be transformed to
+˙M ≈ 3.3 × 10−8P39/v2
+c M⊙ yr−1,
+(5)
+where vc(≡ vw/c) is the wind velocity in unit of the
+speed of light c. The left panel of Figure 9 shows the
+range of the mass-loss rate
+˙M and its dependence on
+wind velocity vc. With a typical value of vc ∼ 0.2 (Pinto
+et al. 2016, 2021; Kosec et al. 2018), the mass-loss rate is
+calculated as ∼ 10−5M⊙ yr−1 (the green vertical line in
+the left panel of Figure 9), which means that ∼ 10 M⊙
+will be lost through ULX wind in the bubble lifetime.
+Such a high mass loss rate will make it difficult to sustain
+the long-term stable accretion activity.
+Instead, we consider the jet-powered bubble scenario
+with highly relativistic ejection velocity.
+The mass-
+loss rate
+˙M can be inferred with the following equation
+(Kaiser & Alexander 1997; Cseh et al. 2012),
+˙M =
+Pw
+(Γ − 1)c2 .
+(6)
+Adopting a minimum bulk Lorentz factor Γ = 2, we
+can derive the
+˙M ranges as ∼ 10−6 M⊙ yr−1, while for
+a higher bulk Lorentz factor, like Γ = 10, the mass-
+loss rate would decrease to ∼ 10−7 M⊙ yr−1 (Figure 9
+right panel), which is clearly more realistic for sustain-
+ing the accretion of ULXs. This would suggest that rel-
+ativistic jets are necessary to generate the shock-ionized
+ULX bubbles like the one we found around X4. Mildly-
+relativistic winds with typical velocity of ∼ 0.2c alone
+would not provide adequate mechanical power. Steady
+jets at the distance of NGC 4631 would have a radio flux
+level of ∼ 1 µJy, which is difficult to detect with current
+facilities, while flaring jets that are 1–2 orders of magni-
+tude brighter could be detected with, e.g., the Karl G.
+Jansky Very Large Array (VLA), like the case of Holm-
+berg II X-1 (Cseh et al. 2015). We have searched the
+
+13
+VLA database; sensitive radio imaging data with sub-
+arcsec resolution on NGC 4631 will be publicly available
+in the near future.
+The estimated jet mechanical power of NGC 4631 X4
+is greater than its radiative luminosity. It would also be
+above the Eddington limit if the accretor mass is less
+than ∼ 100 M⊙, as for most ULXs. This is similar to
+the cases of several microquasars found in nearby galax-
+ies, e.g., NGC 7793 S26 (Pakull et al. 2010) and M83
+MQ1 (Soria et al. 2014), both of which have the same
+level of jet power at ∼ 1040 erg s−1. The Galactic micro-
+quasar SS 433 also has the jet power far exceeding its
+X-ray luminosity (Fabrika 2004). These microquasars
+also have surrounding shock-ionized bubble structures
+detected with optical/infrared emission lines. Their X-
+ray luminosity are admittedly orders of magnitude be-
+low the canonical definition of ULXs.
+However, they
+could have had episodes of super-Eddington radiative
+luminosity in the past, while NGC 4631 X4 itself also
+had sub-Eddington X-ray luminosity (∼ 1037 erg s−1)
+during its Chandra observation. Furthermore, the low
+X-ray luminosity of SS 433 is also due to the heavy ob-
+scuration along the line of sight; only reflected X-ray
+flux is detectable (e.g., Begelman et al. 2006; Middle-
+ton et al. 2021). On a much larger scale, some powerful
+Fanaroff-Riley II radio galaxies and blazars have been
+found to have jet power much greater than the radiative
+luminosity (Ito et al. 2008; Ghisellini et al. 2014). NGC
+4631 X4 and the aforementioned microquasars appears
+to be analogs of these active galaxies at stellar scales.
+From another perspective, we consider the energy
+sources of the injected mechanical power. In case of all
+the jet mechanical power originates from the accretion,
+i.e., the release of gravitational potential energy of the
+accreted material, the needed accretion rate ˙m can be
+calculated from Pw = ϵ ˙mc2., where ϵ is the fraction of
+accretion power converted into mechanical energy. Un-
+der the assumption of ϵ = 0.1, which is already consid-
+ered as exceptionally high, the needed accretion rate is
+˙m ∼ 10−5 M⊙ yr−1. For more realistic ϵ values, the
+needed accretion rate would be even higher. This would
+suggest that there should be additional source(s) of the
+jet mechanical power. For the cases of black hole ac-
+cretion, a promising energy source would be the black
+hole spin, i.e., the Blandford-Znejak (BZ) mechanism
+(Blandford & Znajek 1977). There have been evidences
+supporting this jet power origin for Galactic black hole
+binaries (e.g., Narayan & McClintock 2012; but also
+see, e.g., Russell et al. 2013).
+From our analyses for
+NGC 4631 X4, the presumable jet requires additional
+energy source besides the accretion to provide sufficient
+mechanical power to inflate the bubble structure. Nu-
+merical simulations on super-Eddington accretions by
+Narayan et al. (2017) demonstrate that the total energy
+conversion efficiency (including both radiative and me-
+chanical power) of ULXs can be as high as ∼ 0.7 when
+introducing the high black hole spin (a∗ = 0.9) and the
+“magnetic arrested disk” (MAD; e.g., Bisnovatyi-Kogan
+& Ruzmaikin 1976; Narayan et al. 2003) models, where
+the majority of energy is carried out in the form of me-
+chanical power. The black hole spin energy is extracted
+into the jets via the BZ mechanism.
+6. CONCLUSIONS
+We present an optical imaging study of the five bright-
+est X-ray sources in NGC 4631, among which Soria &
+Ghosh (2009) identified four ULXs (X1, X2, X4, X5).
+Chandra/ACIS data are utilized to obtain precise as-
+trometry and to identify possible optical counterparts
+from the HST/ACS images. A broad-band and narrow-
+band imaging campaign with CFHT/MegaCam is car-
+ried out to search for the bubble structures around the
+X-ray sources and to investigate their accretion states.
+The supersoft X1 has a large optical-to-X-ray posi-
+tional error (≈ 0.′′5) due to its low counts during the
+Chandra observation.
+The candidate optical counter-
+parts and the surrounding stars of X1 span a wide range
+of ages from 5 Myr to 80 Myr in the CMD, suggesting
+that they are likely not physically associated. X3 resides
+in a stellar environment with the age range of ∼ 50–80
+Myr, while its three candidate counterparts show sim-
+ilar ages.
+X4 has six optical counterpart candidates,
+all of which show the age range consistent with that of
+the surrounding stars at ∼ 20–80 Myr. X5 appears to
+be associated with a star group with the age of ∼ 5–
+10 Myr, which is typical for the star clusters related to
+ULXs (Poutanen et al. 2013). This young star group
+is a manifestation of the strong star forming activity in
+the starburst galaxy NGC 4631. We do not provide the
+CMD for X2 due to its high extinction.
+A bubble nebula with a size of ∼ 130 pc × 100 pc
+around X4 is firstly detected in our CFHT/MegaCam
+Hα narrow-band image. Unlike many other ULXs re-
+siding in the interior of their respective bubbles, this
+ULX is located at the east edge. It appears the Hα bub-
+ble originates from X4 and expands one-sided towards
+the west direction, reaching maximum luminosity in the
+outermost region. In contrast, the extended structure
+appears smaller in the [O iii] image, while its bright-
+est section is much closer to the ULX and located to
+the east. The [O iii]/Hα line ratio map suggests that
+the Hα bubble is generated mainly by shock ionization,
+while the [O iii] structure is illuminated by the ULX via
+photoionization.
+
+14
+3000
+4000
+5000
+6000
+7000
+8000
+Wavelength
+Magnitude
+g
+r
+OIII
+mg
+mr
+Figure 10. The supposed linear relation between continuum
+magnitude and wavelength. The two pairs of black dots mark
+the boarders of the g and r bands of CFHT/MegaCam. The
+continuum in the [O iii] band is illustrated by the red shaded
+area.
+The X4 bubble has an average surface brightness of
+19.64 ± 0.01 mag arcsec−2 in the Hα band. By match-
+ing the observed [O iii]/Hα line ratio, we estimate the
+bubble expansion velocity vs ∼ 110 km s−1 and the am-
+bient ISM density n ∼ 6 cm−3 using the MAPPINGS
+V code. With these parameters, we constrain the me-
+chanical power to inflate the bubble being ∼ 5×1040 erg
+s−1 and the bubble age of ∼ 7 × 105 yr. Furthermore,
+we demonstrate that for non- or mildly- relativistic wind
+alone to generate the observed bubble, the needed mass-
+loss rate would be too high to sustain the long-term ac-
+cretion. Instead, in the case of a relativistic jet (with a
+bulk Lorentz factor Γ ∼ 10) to inflate the bubble, the
+mass-loss rate would decrease to a more realistic level
+of ∼ 10−7 M⊙ yr−1. Similar to the cases of a few mi-
+croquasars found in the Milky Way and nearby galaxies
+(e.g., SS 433, NGC 7793 S26, and M83 MQ1), the esti-
+mated mechanical jet power of NGC 4631 X4 is above
+the Eddington limit for a stellar-mass black hole. The
+black hole spin is likely to contribute to the jet power
+via the BZ mechanism.
+For future perspectives, optical spectroscopy, espe-
+cially those with the integral-field instruments, will pro-
+vide the bubble expansion velocity field and flux ratio
+map for a variety of emission lines, from which a more
+precise estimate of the mechanical power can be ob-
+tained. High-resolution X-ray spectroscopy will enable
+the measurement of outflow velocity, while deeper ra-
+dio imaging with high angular resolution could reveal
+the ULX jet. With all these combined, we can derive a
+more reliable mass-loss rate of the outflow and further
+constrain the accretion models of ULXs.
+We thank S. Gwyn for processing the CFHT/MegaCam
+data with MegaPipe and S. Prunet for the help on
+imaging stacking.
+We thank A. Boselli and M. Fos-
+sati for helpful discussions on the continuum subtrac-
+tion of Hα images. We also thank S. Feng and Z. Li
+for archival VLA data enquiry. J.G. thank the CFHT
+staff for their hospitality during her visit to CFHT.
+This work is supported by the National Natural Science
+Foundation of China (grant No. U1938105, 12033004,
+U2038103) and the science research grants from the
+China Manned Space Project with NO. CMS-CSST-
+2021-A05 and CMS-CSST-2021-A06.
+This research uses data obtained through the Tele-
+scope Access Program (TAP), which has been funded
+by the TAP member institutes. Based on observations
+obtained with MegaPrime/MegaCam, a joint project
+of CFHT and CEA/DAPNIA, at the Canada-France-
+Hawaii Telescope (CFHT) which is operated by the Na-
+tional Research Council (NRC) of Canada, the Institut
+National des Sciences de l’Univers of the Centre Na-
+tional de la Recherche Scientifique of France, and the
+University of Hawaii. The observations at the Canada-
+France-Hawaii Telescope were performed with care and
+respect from the summit of Maunakea which is a signifi-
+cant cultural and historic site. Based on observations
+made with the NASA/ESA Hubble Space Telescope,
+and obtained from the Hubble Legacy Archive, which is
+a collaboration between the Space Telescope Science In-
+stitute (STScI/NASA), the Space Telescope European
+Coordinating Facility (ST-ECF/ESAC/ESA) and the
+Canadian Astronomy Data Centre (CADC/NRC/CSA).
+The data described here may be obtained from the
+MAST archive at doi:10.17909/T9RP4V. This research
+has made use of data obtained from the Chandra Data
+Archive and the Chandra Source Catalog, and software
+provided by the Chandra X-ray Center (CXC) in the
+application packages CIAO and Sherpa.
+Facilities:
+CFHT/MegaCam,
+HST/ACS, Chan-
+dra/ACIS
+Software:
+Astropy (Astropy Collaboration et al.
+2013, 2018), CIAO (Fruscione et al. 2006), Dolphot (Dol-
+phin 2000), Matplotlib (Hunter 2007), NumPy (Harris
+et al. 2020), Pandas (Wes McKinney 2010), PyRAF (Sci-
+ence Software Branch at STScI 2012), SCAMP (Bertin
+2006), SciPy (Virtanen et al. 2020), SExtractor (Bertin
+& Arnouts 1996), SWarp (Bertin 2010).
+
+15
+APPENDIX
+A. SUBTRACTING THE CONTINUUM FROM THE [O iii] BAND IMAGES
+To remove the g-band contribution from the [O iii] images, we assume the magnitude of continuum at a given
+wavelength is linearly correlated to this wavelength λ in the range of the g and r bands (Figure 10), i.e., the continuum
+follows a power-law spectral model (f ∝ λ−α). Then the g-band part in [O iii] can be described in the equation,
+mr − mg
+λr − λg
+= mg/OIII − mg
+λOIII − λg
+.
+(A1)
+With replacing the corresponding central wavelength in the filters of Megacam (λg = 4750 ˚A, λOIII = 5006 ˚A, λr =
+6400 ˚A), the equation can be transformed to,
+mg/OIII ≈ mg − 0.155 × (mg − mr).
+(A2)
+It is worth noting that this relation is only valid to the images generated by MegaPipe for which the counts have been
+normalized for each band. The continuum component for each pixel can be derived after the equation is applied pixel
+by pixel.
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+

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1tFRT4oBgHgl3EQfmjcL/content/tmp_files/2301.13601v1.pdf.txt

@@ -0,0 +1,1069 @@
+Qubit Lattice Algorithm Simulations of Maxwell’s
+Equations for Scattering from Anisotropic Dielectric
+Objects
+George Vahala 1, Min Soe 2, Linda Vahala 3, Abhay K. Ram 4, Efstratios Koukoutsis 5,
+Kyriakos Hizanidis 5
+1 Department of Physics, William & Mary, Williamsburg, VA23185
+2 Department of Mathematics and Physical Sciences, Rogers State University,
+Claremore,OK 74017
+3 Department of Electrical & Computer Engineering, Old Dominion University, Norfolk,
+VA 23529
+4 Plasma Science and Fusion Center, MIT, Cambridge, MA 02139
+5 School of Electrical and Computer Engineering, National Technical University of
+Athens,Zographou 15780, Greece
+Abstract
+A Dyson map explicitly determines the appropriate basis of electromagnetic fields which
+yields a unitary representation of the Maxwell equations in an inhomogeneous medium.
+A
+qubit lattice algorithm (QLA) is then developed perturbatively to solve this representation
+of Maxwell equations. QLA consists of an interleaved unitary sequence of collision operators
+(that entangle on lattice-site qubits) and streaming operators (that move this entanglement
+throughout the lattice).
+External potential operators are introduced to handle gradients in
+the refractive indices, and these operators are typically non-unitary, but sparse matrices. By
+also interleaving the external potential operators with the unitary collide-stream operators one
+achieves a QLA which conserves energy to high accuracy. Some two dimensional simulations
+results are presented for the scattering of a one-dimensional (1D) pulse off a localized anisotropic
+dielectric object.
+1
+Introduction
+There is much interest in developing algorithms to solve specific classical problems that can be
+encoded onto a quantum computer. One class of such algorithms is the qubit lattice algorithm
+(QLA) [1-21]. After identifying an appropriate set of qubits, QLA proceeds to define a unitary set
+of interleaved non-commuting collision-streaming operators which acts on this basis set of qubits
+so as to perturbatively recover the classical physics of interest.
+The entanglement of qubits is at the essence of an efficient quantum algorithm. A maximally
+entangled 2-qubit state is known as a Bell state [22]. Now the Hilbert space of a 2-qubit basis
+consists of the states {|00⟩, |01⟩, |10⟩, |11⟩}. Consider the collision operator
+C =
+�
+cos θ
+sin θ
+− sin θ
+cos θ
+�
+(1)
+1
+arXiv:2301.13601v1  [physics.plasm-ph]  31 Jan 2023
+
+acting on the subspace {|00⟩, |11⟩}. The most general tensor product state that can be generated
+from the qubit states {a0|0⟩ + a1|1⟩} , and {b0|0⟩ + b1|1⟩} is
+a0b0|00⟩ + a0b1|01⟩ + a1b0|10⟩ + a1b1|11⟩
+(2)
+Now consider the so-called Bell state
+B+ = |00⟩ + |11⟩
+√
+2
+.
+(3)
+This state cannot be recovered from the tensor-product state of the 2 qubits, Eq. (2). Indeed, to
+eliminate the |01⟩ state from Eq. (2) one requires either a0 = 0 or b1 = 0 - and this would eliminate
+either the state |00⟩ or the state |11⟩. States that can not be recovered from tensor product states
+are called entangled states. The entangled Bell state Eq, (3) is obtained usingthe collision operator
+C, Eq. (1), with angle θ = π/4.
+It is simplest to develop a QLA for the two curl-Maxwell equations, treating the divergence
+equations as initial constraints on the electromagnetic fields E, H. We shall do this in a Hermitian
+tensor dielectric medium, and comment on the discreteness effects on the time evolution of ∇ · B.
+In Sec. 2 we shall see that in an inhomogeneous medium, the electromagnetic basis set (E, B)
+cannot lead to a unitary evolution of the two curl Maxwell equations. However, a Dyson map is
+introduced that will map the basis (E, B) into the basis (nxEx, nyEy.nzEz, B) resulting in a fully
+unitary evolution for this basis set [23]. Here we have transformed to principal axes making the
+dielectric tensor diagonal with ϵi = n2
+i , i = x, y, z. The more familiar complex Riemann-Silberstein-
+Weber basis F ±
+i
+= (niEi ±iBi) is immediately generated from the real basis (niEi, Bi) by a unitary
+transformation so that this will also lead to a unitary time evolution representation.
+In Sec. 2 we will develop a QLA for the solution of 2D Maxwell equations in a tensor Hermitian
+dielectric medium. All our previous Maxwell QLA [16-18, 21] were restricted to scalar dielectrics.
+We will present a simplified discussion of the Dyson map [23] that will permit us to transform
+from a non-unitary to unitary basis for the representation of the two curl equations of Maxwell.
+For these continuum qubit partial differential equations we will generate in Sec.
+3 a discrete
+QLA for tensor dielectric media that recovers the desired equations to second order perturbation.
+While the collide-stream operator sequence of QLA is fully unitary, the external potential operators
+required to recover the derivatives of the refractive indices in Maxwell equations are not. However
+these non-unitary matrices are very sparse and should be amenable to some unitary approximate
+representation.
+The role of the perturbation parameter δ introduced in the QLA for Maxwell
+equations is quite subtle. One important test of the QLA is the conservation of electromagnetic
+energy density. This will be seen to be very well satisfied, as δ → 0. In Sec. 4 we present some 2D
+QLA simulations for a 1D Gaussian electromagnetic pulse scattering from an anisotropic dielectric
+localized object - showing results for both polarizations. Finally, in Sec. 5 we summaries the results
+of this paper.
+2
+A Unitary Representation of the two curl Maxwell Equations
+2.1
+Scalar dielectric medium
+First, consider a simple dielectric non-magnetic medium with the constitutive equations
+D = ϵE,
+B = µ0H.
+(4)
+2
+
+It is convenient to define u = (E, H)T as the fundamental fields, and d = (D, B)T the derived
+fields. Eq. (4), in matrix form, is
+d = Wu.
+(5)
+W is a Hermitian 6 × 6 matrix
+W =
+� ϵI3×3
+03×3
+03×3
+µ0I3×3
+�
+.
+(6)
+I3×3 is the 3 × 3 identity matrix. and the superscript T is the transpose operator. The curl-curl
+Maxwell equations ∇ × E = −∂B/∂t, and ∇ × H = ∂D/∂t can then be written
+i∂d
+∂t = Mu
+(7)
+where, under standard boundary conditions, the curl-matrix operator M is Hermitian :
+M =
+�
+03×3
+i∇×
+−i∇×
+03×3
+�
+.
+(8)
+Now W is invertible, so that Eq. (7) can finally be written in terms of the basic electromagnetic
+fields u = (E, H)
+i∂u
+∂t = W−1Mu
+(9)
+2.1.1
+inhomogeneous scalar dielectric media
+We immediately note that for inhomogeneous dielectric media, W−1 will not commute with M.
+Thus Eq. (9) will not yield unitary evolution for the fields u = (E, H)T . However Koukoutsis et.
+al. [23] have shown how to determine a Dyson map from the fields u to a new field representation U
+such that the resultant representation in terms of the new field U will result in a unitary evolution.
+In particular, the Dyson map [23]
+U = W1/2u
+(10)
+yields a unitary evolution equation for U :
+i∂U
+∂t = W−1/2MW−1/2U
+(11)
+since now the matrix operator W−1/2MW−1/2 is indeed Hermitian.
+Explicitly, the U vector for non-magnetic materials,is just
+U =
+�
+ϵ1/2E, µ1/2
+0
+H
+�T
+(12)
+This can be rotated into the RWS unitary representation by the unitary matrix
+L =
+1
+√
+2
+� I3×3
+iI3×3
+I3×3
+−iI3×3
+�
+(13)
+yielding URSW = LU with
+URSW =
+1
+√
+2
+�
+ϵ1/2E + iµ1/2
+0
+H
+ϵ1/2E − iµ1/2
+0
+H
+�
+.
+(14)
+3
+
+2.2
+Inhomogeneous tensor dielectric media
+The theory can be immediately extended to diagonal tensor dielectric media, with (assuming non-
+magnetic materials) the 6-qubit representation Q of the field
+U =
+�
+nxEx, nyEy, nzEz, µ1/2
+0
+H
+�T
+≡ Q.
+(15)
+(nx, ny, nz) is the vector (diagonal) refractive index, with ϵx = n2
+x ... .
+The explicit unitary representation of the Maxwell equations for 2D x-y spatially dependent
+fields written in terms of the 6-Q qubit components are
+∂q0
+∂t = 1
+nx
+∂q5
+∂y ,
+∂q1
+∂t = − 1
+ny
+∂q5
+∂x ,
+∂q2
+∂t = 1
+nz
+�∂q4
+∂x − ∂q3
+∂y
+�
+∂q3
+∂t = −∂(q2/nz)
+∂y
+,
+∂q4
+∂t = ∂(q2/nz)
+∂x
+,
+∂q5
+∂t = −∂(q1/ny)
+∂x
++ ∂(q0/nx)
+∂y
+(16)
+3
+A Qubit Lattice Representation for 2D Tensor Dielectric Media
+We develop a QLA for the unitary system Eq. (16) by determining unitary collision and streaming
+operators that recover the derivatives ∂qi/∂t, ∂qj/∂x and ∂qj/∂y. (i, j = 1..6). Our finite difference
+scheme is to recover Eq. (16) to second order in a perturbation parameter δ, where the spatial
+lattice spacing is defined to be O(δ). To recover the partial derivatives on the 6-qubit Q in the
+x−direction, we consider the unitary collision entangling operator
+CX =
+�
+�������
+1
+0
+0
+0
+0
+0
+0
+cos θ1
+0
+0
+0
+−sin θ1
+0
+0
+cos θ2
+0
+−sin θ2
+0
+0
+0
+0
+1
+0
+0
+0
+0
+sin θ2
+0
+cos θ2
+0
+0
+sin θ1
+0
+0
+0
+cos θ1
+�
+�������
+(17)
+where we shall need two collision angles θ1 and θ2. The unitary streaming operators will be of the
+form S+x
+14 which shifts qubits q1 and q4 one lattice unit δ in the +x−direction, while leaving the
+other 4 qubit components invariant. The final unitary collide-stream sequence in the x-direction is
+UX = S+x
+25 .C†
+X.S−x
+25 .CX.S−x
+14 .C†
+X.S+x
+14 .CX.S−x
+25 .CX.S+x
+25 .C†
+X.S+x
+14 .CX.S−x
+14 .C†
+X
+(18)
+.
+Similarly for the y-direction, the corresponding unitary collision entangling operator is
+CY =
+�
+�������
+cos θ0
+0
+0
+0
+0
+sin θ0
+0
+1
+0
+0
+0
+0
+0
+0
+cos θ2
+0
+sin θ2
+0
+0
+0
+−sin θ2
+cos θ2
+0
+0
+0
+0
+0
+0
+1
+0
+−sin θ0
+0
+0
+0
+0
+cos θ0
+�
+�������
+,
+(19)
+and the corresponding unitary collide-stream sequence in the y-direction
+UY = S+y
+25 .C†
+Y .S−y
+25 .CY .S−y
+03 .C†
+Y .S+y
+03 .CY .S−y
+25 .CY .S+y
+25 .C†
+Y .S+y
+03 .CY .S−y
+03 .C†
+Y
+(20)
+4
+
+We will discuss the specific collision angles θ0, θ1 and θ2 after introducing the external potential
+operators.
+The terms that remain to be recovered by the QLA are the spatial derivatives on the refractive
+index components ∂ni/∂x and ∂ni/∂y. These terms will be recovered by the following (non-unitary)
+sparse external potential operators:
+VX =
+�
+�������
+1
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+−sin β2
+0
+cos β2
+0
+0
+sin β0
+0
+0
+0
+cos β0
+�
+�������
+(21)
+and
+VY =
+�
+�������
+1
+0
+0
+0
+0
+o
+0
+1
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+cos β3
+sin β3
+0
+0
+0
+0
+0
+0
+1
+0
+−sin β1
+0
+0
+0
+0
+cos β1
+�
+�������
+(22)
+for particular angles β0 .. β3.
+Thus one possible QLA algorithm that advances the 6-qubit Q from time t to time t + ∆t is
+Q(t + ∆t) = VY .VX.UY.UX.Q(t)
+(23)
+Indeed, using Mathematica, one can show that with the collision angles
+θ0 =
+δ
+4nx
+,
+θ1 =
+δ
+4ny
+,
+θ2 =
+δ
+4nz
+,
+(24)
+and
+β0 = δ2 ∂ny/∂x
+n2y
+,
+β1 = δ2 ∂nx/∂y
+n2x
+,
+β2 = δ2 ∂nz/∂x
+n2z
+,
+β3 = δ2 ∂nz/∂y
+n2z
+(25)
+we will have a second order QLA representation of the 2D Maxwell continuum equations
+∂q0
+∂t = δ2
+∆t
+1
+nx
+∂q5
+∂y + O( δ4
+∆t)
+∂q1
+∂t = − δ2
+∆t
+1
+ny
+∂q5
+∂x + O( δ4
+∆t)
+∂q2
+∂t = δ2
+∆t
+1
+nz
+�∂q4
+∂x − ∂q3
+∂y
+�
++ O( δ4
+∆t)
+∂q3
+∂t = − δ2
+∆t
+� 1
+nz
+∂q2
+∂y − ∂nz/∂y
+n2z
+q2
+�
++ O( δ4
+∆t)
+∂q4
+∂t = δ2
+∆t
+� 1
+nz
+∂q2
+∂x − ∂nz/∂x
+n2z
+q2
+�
++ O( δ4
+∆t)
+∂q5
+∂t = δ2
+∆t
+�
+− 1
+ny
+∂q1
+∂x + ∂ny/∂x
+n2y
+q1 + 1
+nx
+∂q0
+∂y − ∂nx/∂y
+n2x
+q0
+�
++ O( δ4
+∆t)
+(26)
+under diffusion ordering, ∆t ≈ δ2.
+5
+
+3.1
+Conservation of Instantaneous Total Electromagnetic Energy in QLA Sim-
+ulations
+It is important to monitor the conservation of energy in the QLA, particularly since our current
+QLA is not fully unitary. The normalized total electromagnetic energy for a square lattice domain
+of length L is E(t)
+E(t) = 1
+L2
+� L
+0
+� L
+0
+dxdy
+�
+n2
+xE2
+x + n2
+yE2
+y + n2
+zE2
+z + B2�
+= 1
+L2
+� L
+0
+� L
+0
+dxdyQ · Q
+,
+(27)
+In our QLA simulations, we will consider the scattering of a 1D Gaussian pulse propagating in the
+x−direction, and scattering from a localized tensor 2D dielectric object in the x − y plane. We
+choose L to be significantly greater than the dielectric object so that for y ≈ 0, and for y ≈ L the
+electromagnetic fields there will be that of the 1D Gaussian pulse yielding a Poynting vector E×B
+in the ˆx. Thus the contribution to the Poynting flux
+�
+C E × B · dℓ on y = 0 and on y = L is zero.
+In our time evolution QLA simulations, we integrate only to t < tmax so that there are no fields
+generated on the sides x = 0 and x = L. Thus, in our QLA simulations we have set up parameters
+such that the total electromagnetic energy E(t) = const., Eq. (27), for t < tmax.
+E(t) is nothing but the norm of Q−qubits , and will be exactly conserved in a fully unitary
+QLA. One must also be careful in the ordering of the external potential angles, Eq. (25) : they
+must be O(δ2) in order to recover Maxwell equations.
+While we will discuss in detail in Sect. 4 our numerical QLA simulation of a 1D electromagnetic
+pulse scattering from a localized dielectric object it is appropriate to discuss here some QLA
+simulation results for the total energy. Since QLA, Eq. (23) is a perturbation theory, it will recover
+the 2D Maxwell equations as δ → 0. For δ = 0.3, we find the following time variation in the total
+energy E(t) in Fig. 1a. tmax = 20, 000 lattice time steps.
+(a) E(t) , δ = 0.3 ,
+(b) E(t) , δ = 0.1
+Figure 1: The instantaneous total electromagnetic energy E(t), Eq.
+(27), for various values of
+the perturbation parameter δ : (a) δ = 0.3, (b) δ = 0.1.
+A more accurate QLA results from
+interleaving the external potentials with the unitary collide-stream operators. For δ = 0.01, E(t)
+shows no variation on this scale, with variations in the 9th significant figure. Lattice grid L = 8192.
+On lowering the perturbation parameter to δ = 0.1 there is a nice reduction in the time variation
+of E(t), Fig 1(b). To reach the same physics tmax = 60K.
+6
+
+4.032
+X 10'4
+4.03
+4.028
+ 0.3 
+Energy.
+4.026
+4.024
+4.022
+4.02
+0
+5000
+10000
+15000
+20000
+time4.022
+ × 104
+4.0215
+: 0.1
+Energy. 8= (
+4.021
+Interleavedpotential
+4.0205
+4.02
+0
+10000
+20000
+30000
+40000
+50000
+60000
+timeHowever, if we interleave the external potential operators among the unitary collide-stream
+sequence (and similarly for the y-direction) in the form
+V′
+XUX = V ′
+XS+x
+25 .C†
+X.S−x
+25 .CX.S−x
+14 .C†
+X.S+x
+14 .CX.V ′
+X.S−x
+25 .CX.S+x
+25 .C†
+X.S+x
+14 .CX.S−x
+14 .C†
+X
+(28)
+(with the corresponding potential angle reduced by a factor of 2) we find E(t) ≈ const. for all times,
+see Fig. 1(b). There is a further strong improvement in E(t) = const. for δ = 0.01.
+4
+Scattering of a Polarized Pulse from an Anisotropic Dielectric
+Object
+We first consider a 1D Gaussian pulse propagating in a vacuum in the x-direction towards a localized
+anisotropic dielectric object, with diagonal tensor components which are conical in nz(x, y), and
+cylindrical in the x and y directions with nx(x, y) = ny(x, y), Fig. 2
+(a) nz(x, y)
+,
+(b) nx(x, y) = ny(x, y)
+Figure 2: Anisotropic tensor dielectric : (a) conical in nz, and (b) cylindrical in nx = ny. Initially,
+a 1D Gaussian pulse propagates in the x-direction, with either a polarization Ez(x, t) < 0 or a
+polarization Ey(x, t) and scatters off this tensor dielectric object. In the region away from the
+tensor dielectric object, we have a vacuum with ni = 1.0. In the dielectric, ni,max = 3.0. Lattice
+domain 81922.
+4.1
+Scattering of 1D pulse with Ez polarization
+When the 1D pulse with non-zero Ez(x, t), By(x, t) fields starts to interact with the 2D tensor
+dielectric n(x, y), the scattered fields become 2D (see Fig. 3), with By(x, y, t) dependence. The
+QLA will then spontaneously generate a Bx(x, y, t) field so that ∂Bx/∂x + ∂By/∂y ≈ 0.
+Because of the relatively weak dielectric tensor gradients for a cone, there is very little reflection
+back into the vacuum of the incident Ez field (Fig. 4). There is a localized transmitted Ez within
+the dielectric.
+At t = 36k we plot both the Ez and the By , Fig.
+5-6.
+Of considerable interest is the
+spontaneously generated Bx(x, y, t) field so that ∇ · B = 0. From Fig. 7 we see that Bx has dipole
+7
+
+8000
+6000 x
+4000
+2000
+0
+2.5
+2.0
+1.5
+1.0
+0
+2000
+4000
+6000
+y
+80008000
+3.0
+6000
+2.5
+2.0
+x4000
+1.5
+2000
+1.0
+0
+2000
+4000
+0
+6000
+8000
+y(a) Ez(x, y, t0) < 0 at t0 = 18k
+,
+(b) Ez(x, y, t0) > 0 at t0 = 18k
+Figure 3: Ez after interacting with the localized tensor dielectric. Since the phase speed in the
+tensor dielectric is less than in the vacuum, the 2D structure in Ez lags the rest of the 1D pulse
+that has not interacted with the localized dielectric object (Fig. 1). The perspective (b) is obtained
+from (a) by rotating by π about the line y = L/2.
+structure, since the plane of the plot Fig. 7(b) for the field Bx < 0 is generated by rotating the
+plane through π about the axis x = L/2
+We find in our QLA simulations, that maxx,y
+�
+∇ · B/|B| < 10−3�
+4.2
+Scattering of 1D pulse with Ey polarization
+We now turn to the 1D pulse with Ey polarization, propagating in the x−direction toward the 2D
+tensor dielectric object, Fig. 1. The other non-zero vacuum electromagnetic field is Bz(x, t). On
+interacting with the tensor dielectric n(x, y), the scattered fields will develop a spatial dependence
+on (x, y). Thus ∇ · B = 0 exactly, and no new magnetic filed components need be generated, This
+is recognized by the QLA and so the only non-zero magnetic field throughout the run is Bz(x, y, t).
+In Fig. 8 we plot the Ey-field at time t = 18k, the same time snapshot as for the case of Ez
+polarization, Fig. 3. The significant differences in the scattered field arise from the differences
+between the cylinder dielectric dependence of ny(x, y) and the cone nz(x, y).
+Also, what can be seen in Fig. 8 is the outward propagating circular-like wavefront which seems
+to be reminiscent of the reflected pulse in 1D scattering. In particular, one sees elements of a π
+phase change in this reflected wavefront.
+The corresponding Ey wavefronts at t = 36k are shown in Fig. 9
+The accompanying Bz field of the initial 1D electromagnetic pulse is shown after its scattering
+from the tensor dielectric at times t = 18k, Fig 10, and at t = 36k, Fig. 11
+Finally we consider the last of the Maxwell equations to be enforced: ∇ · D = 0. The QLA
+established a qubit basis for the curl-curl subset of Maxwell equations. For the initial polarization
+Ez(x, t) and refractive indices n = n(x, y), the ∇ · D = 0 is automatically satisfied, while ∇ · B = 0
+8
+
+-0.09998 -0.07448 -0.04899 -0.02350
+DB: Ezf.118000.bov
+0.001997
+Max: 0.001997
+Min: -0.09998-0.09998 -0.07448-0.04899 -0.02350
+DB: Ezf.1 18000.b0v
+0.001997
+Max: 0.001997
+Min: -0.09998(a) Ez(x, y, t1) < 0 at t1 = 24k
+,
+(b) Ez(x, y, t1) > 0 at t1 = 24k
+Figure 4: For early times, from the perspective of the tensor dielectric object the electromagnetic
+pulse within the dielectric is the transmitted field and has a localized Ez which becomes greater than
+the original Ez in the vacuum region. There is little reflected field since Ez will be predominantly
+interacting with the nz component of the tensor dielectric. The perspective (b) is obtained from
+(a) by rotating by π about the line y = L/2.
+(a) Ez(x, y, t2) < 0 at t2 = 36k
+,
+(b) Ez(x, y, t2) > 0 at t2 = 36k
+Figure 5: The Ez field at a late stage of development. The perspective (b) is obtained from (a) by
+rotating by π about the line y = L/2.
+9
+
+DB: Ezf.136000.b0v
+Max: 0.04157
+Min: -0.09995-0.1823
+-0.1241
+-0.06594 -0.007730 0.05048
+DB: Ezf.124000.b0v
+Max: 0.05048
+Min: -0.1823-0.1823
+-0.1241
+DB: Ezf.124000.bov
+-0.06594 -0.007730 0.05048
+Max: 0.05048
+Min: -0.1823DB: Ezf.136000.b0v
+Max: 0.04157
+Min: -0.09995(a) By(x, y, t0) > 0 at t0 = 36k
+,
+(b) By(x, y, t0) < 0 at t0 = 36k
+Figure 6: The corresponding By field at time t = 36k to the Ez field in Fig. 5. The perspective
+(b) is obtained from (a) by rotating by π about the line y = L/2.
+(a) Bx(x, y, t0) > 0 at t0 = 36k
+,
+(b) Bx(x, y, t0) < 0 at t0 = 36k
+Figure 7: The spontaneously Bx field at time t = 36k that is generated by the QLA so that
+∇ · B = 0. This time corresponds to the Ez field in Fig. 5, and By field in FIg. 6. The dipole
+structure of Bx is clear on comparing (a) and (b). The perspective (b) is obtained from (a) by
+rotating by π about the line y = L/2.
+10
+
+-0.03172 0.001195 0.03411
+0.06703
+0.09995
+DB: Byf.136000.bov
+Max: 0.09995
+Min: -0.03172-0.03172 0.001195 0.03411
+0.06703
+0.09995
+DB: Byf.136000.bov
+Max: 0.09995
+Min: -0.03172
+X-0.07529 -0.03764 1.051e-05 0.03766
+0.07531
+DB: Bxf.136000.b0v
+Max: 0.0753 1
+Min: -0.07529-0.07529 -0.03764 1.051e-05 0.03766
+0.07531
+DB: Bxf.136000.b0v
+Max: 0.0753 1
+Min: -0.07529
+X
+Z(a) Ey(x, y, t0) > 0 at t0 = 18k
+,
+(b) Ey(x, y, t0) < 0 at t0 = 18k
+Figure 8: Ey after interacting with the localized tensor dielectric. Since the cylindrical ny dielectric
+has a sharper boundary layer than the conic nz dielectric, there is now a marked ”reflected”
+wavefront propagating into the vacuum region together with the ”transmitted” part of the pulse
+into the dielectric region itself. This ”reflected” wavefront is absent when the major scattering is
+off the conic dielectric component, Fig. 3. The perspective (b) is obtained from (a) by rotating by
+π about the line y = L/2.
+11
+
+DB: Eyf. 118000.bov
+-0.03580-0.001856 0.03209
+0.06603
+0.09998
+Max:0.09998
+Min: -0.03580
++DB: Eyf. 118000.bov
+-0.03580-0.0018560.03209
+0.06603
+0.09998
+Max:0.09998
+Min: -0.03580
+X(a) Ey(x, y, t2) > 0 at t2 = 36k
+,
+(b) Ey(x, y, t2) < 0 at t2 = 36k
+Figure 9: The Ey wavefronts at a late stage of development, as the ”reflected” pulse is about to
+reach the lattice boundaries. The perspective (b) is obtained from (a) by rotating by π about the
+line y = L/2.
+(a) Bz(x, y, t1) > 0 at t1 = 18k
+,
+(b) Bz(x, y, t1) < 0 at t1 = 18k
+Figure 10: The Bz wavefronts corresponding to the Ey - field in Fig. 8. The perspective (b) is
+obtained from (a) by rotating by π about the line y = L/2.
+12
+
+DB: Eyf.136000.bov
+0.08481-0.038620.0075710.05376
+0.09996
+Max:0.09996
+Min: -0.08481DB: Eyf.136000.bov
+0.08481-0.038620.0075710.05376
+0.09996
+Max:0.09996
+Min: -0.08481
+X-0.1032
+0.0007075
+0.1046
+0.2086
+0.3125
+DB: Bzf. 1 18000.bov
+Max: 0.3125
+Min: -0.1032
+XDB: Bzf.118000.bov
+-0.1032
+0.00070750.1046
+0.2086
+0.3125
+Max: 0.3125
+Min:0.1032
+X(a) Bz(x, y, t2) > 0 at t2 = 36k
+,
+(b) Bz(x, y, t2) < 0 at t2 = 36k
+Figure 11: The Bz wavefronts at a late stage of development, as the ”reflected” pulse is about to
+reach the lattice boundaries. The corresponding Ey field is shown in Fig. 9. The perspective (b) is
+obtained from (a) by rotating by π about the line y = L/2.
+was spontaneously satisfied by the self-consistent generation of a Bx field.
+Now, if the initial
+polarization was Ey(x, t), then ∇·B = 0 is automatically satisfied, while a spontaneously generated
+Ex field is generated by the QLA so that ∇ · D = 0 is satisfied. In Fig. 12 we show the wavefronts
+of the Ex field at time t = 18k
+5
+Summary
+Determining a Dyson map, we have been able to develop a required basis from which the evolution
+equations for Maxwell equations can be unitary. In particular, we have shown that for inhomoge-
+neous non-magnetic dielectric media, the field basis (E, B) will not lead to a unitary representation.
+However, a particular Dyson map shows that (n.E, B), where n is a diagonal tensor dielectric, is a
+basis for a unitary representation. Other unitary representations can be immediately determined
+from this basis by unitary transformation, in particular the Riemann-Silberstein-Weber basis.
+Here we have concentrated on the basis (n.E, B), primarily because the fields are real and so
+lead to quicker computations. Our QLA directly encodes these fields into qubit representation. A
+unitary set of interleaved collision-streaming operators are then applied to these qubits: the unitary
+collision operators entangle the qubits, while the streaming operators move this entanglement
+throughout the lattice. With our current set of unitary collision-streaming operators, we do not
+generate the effects of derivatives on the inhomogeneous medium. These effects are included by
+the introduction of external potential operators - but at the expense of loosing the unitarity of the
+complete algorithm.
+In this paper we have performed QLA simulations on 2D scattering of a 1D electromagnetic pulse
+from a localized Hermitian tensor dielectric object. Both polsrizations are considered with different
+field evolutions because of the anisotropic in the tensor dielectric. The QLA we consider here are
+13
+
+-0.2197
+-0.1345
+-0.04931
+0.03587
+DB: Bzf.136000.bov
+0.1210
+Max: 0.1210
+Min: -0.2197DB: Bzf.136000.bov
+-0.2197
+-0.1345
+-0.04931
+0.03587
+0.1210
+Max: 0.1210
+Min: -0.2197
+X(a) Ex(x, y, t1) > 0 at t1 = 18k
+,
+(b) Ex(x, y, t1) < 0 at t1 = 18k
+Figure 12: The Ex wavefronts at time 18k spontaneously generated by QLA so as to (implicitly)
+satisfy the Maxwell equation ∇ · D = 0. As the Ex < 0 plot perspective is generated from the
+Ex > 0 plot by rotating about the y = L/2 axis through an angle π, it is immediately seen the the
+Ex field strongly exhibits dipole structure.
+based on the two curl equations of Maxwell. Moreover the QLA is a perturbative representation,
+with small parameter δ representative of the spatial lattice width, with QLA → curl − curl −
+Maxwell as δ → 0. It is not at all obvious that the QLA has the right structure to recover Maxwell
+equations - but only through symbolic manipulations (Mathematica) do we determine this Maxwell
+limit. Hence it is of some interest to see how well QLA satisfies to two divergence equations of
+Maxwell that are not directly encoded in the QLA process. We find spontaneous generation in the
+QLA so that ∇ · B = 0, ∇ · D = 0.
+Finally we comment on the conservation of energy E:
+E(t) = 1
+L2
+� L
+0
+� L
+0
+dxdy
+�
+n2
+xE2
+x + n2
+yE2
+y + n2
+zE2
+z + B2�
+In QLA, E = E(t, δ). Under appropriate scaling of the QLA operator angles, one recovers perturba-
+tively the curl-curl Maxwell equations as δ → 0. Moreover, we find that EQLA → const. as δ → 0.
+The QLA simulations presented here were run on a lattice grid of 81922, with δ = 0.1.
+The next step is to determine a fully unitary QLA for the Maxwell equations in anisotropic
+media.
+The conservation of energy would be automatically satisfied as the norm of the qubit
+basis.
+This unitary would then permit the QLA to be immediately encodable on a quantum
+computer.
+In the meantime, while we await error-correcting qubits and long decoherence time
+quantum computes, our current QLA’s are ideally parallelized on classical supercomputers without
+core saturation effects.
+14
+
+DB: Exf.1 18000.bov
+-0.04137
+-0.02067
+3.775e-05
+0.02074
+0.04145
+Max: 0.04145
+Min: -0.04137DB: Exf.1 18000.bov
+0.04137
+-0.02067
+3.775e-05
+0.02074
+0.04145
+Max: 0.04145
+Min: -0.04137
+Z6
+Acknowledgments
+This research was partially supported by Department of Energy grants DE-SC0021647, DE-FG0291ER-
+54109, DE-SC0021651, DE-SC0021857, and DE-SC0021653. This work has been carried out par-
+tially within the framework of the EUROfusion Consortium. E.K has received funding from the
+Euratom research and training program WPEDU under grant agreement no. 101052200 as well
+as from the National Program for Controlled Thermonuclear Fusion, Hellenic Republic. K.H is
+supported by the National Program for Controlled Thermonuclear Fusion, Hellenic Republic. The
+views and opinions expressed herein do not necessarily reflect those of the European Commission.
+7
+References
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+classical solitons. Phys. Lett A310, 187-196
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+representations of one-dimensional magnetohydrodynamic turbulence. Phys. Lett A306, 227-234.
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+lattice representations for vector solitons in external potentials. Physica A362, 215-221.
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+of complex vortex structures. Comput. Sci. Discovery 5, 014013
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+Fluid Dynamics, ed. H. W. Oh, (InTech Publishers, Croatia)
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+the Dirac-generated unitary lattice qubit collision-stream algorithm for 1D vector Manakov soliton
+collisions. Computers Math. with Applic. 72, 386
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+Imaginary time integration method using a quantum lattice gas approach. Rad Effects Defects
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+16
+

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+Spin-lattice and magnetoelectric couplings enhanced by orbital degrees of freedom in
+polar magnets
+Vilmos Kocsis,1, 2 Yusuke Tokunaga,1, 3 Toomas R˜o˜om,4 Urmas Nagel,4
+Jun Fujioka,5 Yasujiro Taguchi,1 Yoshinori Tokura,1, 6 and S´andor Bord´acs7, 8
+1RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
+2Institut f¨ur Festk¨orperforschung, Leibniz IFW-Dresden, 01069 Dresden, Germany
+3Department of Advanced Materials Science, University of Tokyo, Kashiwa 277-8561, Japan
+4National Institute of Chemical Physics and Biophysics, 12618 Tallinn, Estonia
+5Institute of Materials Science, University of Tsukuba, Ibaraki 305-8573, Japan
+6Tokyo College and Department of Applied Physics,
+University of Tokyo, Hongo, Tokyo 113-8656, Japan
+7Department of Physics, Institute of Physics, Budapest University of
+Technology and Economics, M˝uegyetem rkp. 3., H-1111 Budapest, Hungary
+8Quantum Phase Electronics Center and Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
+Orbital degrees of freedom mediating an interaction between spin and lattice were predicted
+to raise strong magnetoelectric effect, i.e.
+realize an efficient coupling between magnetic and
+ferroelectric orders.
+However, the effect of orbital fluctuations have been considered only in a
+few magnetoelectric materials, as orbital degeneracy driven Jahn-Teller effect rarely couples to
+polarization. Here, we explore the spin-lattice coupling in multiferroic Swedenborgites with mixed
+valence and Jahn-Teller active transition metal ions on a stacked triangular/Kagome lattice using
+infrared and dielectric spectroscopy. On one hand, in CaBaM4O7 (M = Co, Fe), we observe strong
+magnetic order induced shift in the phonon frequencies and a corresponding large change in the
+dielectric response. Remarkably, as an unusual manifestation of the spin-phonon coupling, the spin-
+fluctuations reduce the phonon life-time by an order of magnitude at the magnetic phase transitions.
+On the other hand, lattice vibrations, dielectric response, and electric polarization show no variation
+at the N´eel temperature of CaBaFe2Co2O7, which is built up by orbital singlet ions. Our results
+provide a showcase for orbital degrees of freedom enhanced magnetoelectric coupling via the example
+of Swedenborgites.
+Spin-orbit coupling (SOC) is considered among the
+most essential interactions in condensed matter science,
+standing in the background of topological insulators [1]
+and superconductors [2], Dirac and Weyl semimetals [3,
+4], Kitaev physics [5] as well of multiferroics [6, 7]. In the
+latter compounds, SOC induces magnetoelectric (ME)
+coupling between electric polarization and magnetism
+making them interesting for basic research and appealing
+for applications, however, this interaction is usually weak
+due to its relativistic nature. [8–12]. While the relativistic
+spin-orbit interaction enables the ME coupling on a single
+(a pair) of magnetic ion(s), theoretical works proposed
+early that the charge and orbital degrees of freedoms
+can mediate an enhanced ME interaction via the Kugel-
+Khomski˘ı-type spin-orbital coupling [13–16].
+However,
+materials realizing this scenario are exceptional,
+as
+charge and orbital order alone rarely break the inversion
+symmetry [16–22].
+The two most studied cases are
+Fe3O4, where the ME effect is attributed to the charge
+and orbital orderings [16–19], and LuFe2O4 in which
+the ferroelectricity is debated to emerge from charge
+ordering [20].
+Recently, CaMn7O12 was also identified
+with a chiral magnetic structure stabilized by the charge
+and orbital ordering [21, 22].
+Swedenborgites
+CaBaM4O7
+(M=Co,
+Fe)
+provide
+another platform to study the interplay between spins
+and orbitals, but there, unlike the previous examples,
+the charge degree of freedom is quenched.
+The polar
+Swedenborgites are built up by alternating layers of
+triangular and kagom´e sheets of MO4 tetrahedra, all
+pointing to the c axis, as shown in Fig. 1(a). The M 2.5+
+nominal valence, suggests a 1:1 mixture of M 2+ and M 3+
+ions, subjected to geometric frustration. The buckling of
+the kagom´e lattice releases the frustration and reduces
+the symmetry to orthorhombic at TS=450 K [23, 24]
+and TS=380 K [25, 26] in CaBaCo4O7 and CaBaFe4O7,
+respectively.
+In both compounds, X-ray spectroscopy
+studies confirmed the coexistence of distinct valences,
+M 2+ and M 3+ (electron configurations sketched in
+Fig. 2), and suggested charge order with the M 3+
+ions occupying the triangular and one of the kagom´e
+sites [23, 25, 27–29]. Therefore, both CaBaCo4O7 and
+CaBaFe4O7 contain the Jahn-Teller active Co3+ and
+Fe2+ ions, respectively, though, no further information
+is available on orbital ordering.
+However, the solid-
+solution CaBaFe2Co2O7 lacks orbital degeneracy, namely
+solely the orbital singlet Fe3+ and Co2+ charge states are
+present in this compound [28–30].
+In CaBaCo4O7, spins order antiferromegnetically at
+TN=70 K [31], and then a ferrimagnetic structure emerges
+below TC=60 K [23, 32], as shown in Fig. 1(c). The latter
+phase is accompanied by one of the largest magnetic-
+order-induced polarization detected so far [33, 34] as
+well as exceptionally large magnetostriction [35].
+Its
+arXiv:2301.03292v1  [cond-mat.str-el]  9 Jan 2023
+
+2
+FIG. 1.
+(a) The polar structural unit cell of trigonal
+Swedenborgites are built up by alternating triangular and
+Kagom´e layers of co-aligned MO4
+tetrahedra.
+(b) In
+the trigonal CaBaFe2Co2O7, one Fe3+ ion occupies the
+triangular lattice, while the remaining Fe3+/Co2+/Co2+ ions
+are distributed randomly on the Kagome lattice. The
+√
+3 ×
+√
+3-type antiferromagnetic order develops below TN=152 K
+(spin S,
+green arrow,
+reproduced after Ref. 37.)
+(c)
+The orthorhombic CaBaCo4O7 has charge order and a
+ferrimagnetic order below TC=60 K, reproduced after Ref. 23
+and 27.
+sister compound, CaBaFe4O7 also show peculiar ME
+properties.
+It becomes multiferroic close to room
+temperature, TC1=275 K upon a ferrimagnetic ordering,
+which is followed by a reorientation transition below
+TC2=211 K
+[25,
+26].
+CaBaFe2Co2O7
+develops
+an
+antiferromagnetic structure at TN=152 K [30, 36, 37]
+(Fig. 1(b)),
+however,
+its ME properties have been
+unknown.
+In this Letter, we investigate the effect of magnetic
+ordering on the charge dynamics of Swedenborgites
+via infrared and dielectric spectroscopy. We compared
+members of the material family with and without orbital
+degree of freedom,
+and found a strong spin-lattice
+coupling only in CaBaM4O7 (M = Co, Fe) with Jahn-
+Teller active ions.
+In these pristine compounds, the
+phonon frequencies show a sudden shift at TC, related
+to the large magnetic-order-induced polarization and
+magnetocapacitance.
+Moreover, we observed an order
+of magnitude decrease of the phonon life-times at the
+ferrimagnetic phase transitions. In contrast, we found no
+phonon nor dielectric anomalies and negligible change in
+the pyroelectric polarization upon the magnetic ordering
+in the orbital-singlet CaBaCo2Fe2O7.
+Therefore, our
+results highlight the importance of orbital degrees
+of freedom in the enhancement of the spin-lattice
+interaction and the ME effect in multiferroics.
+Large single crystals of CaBaCo4O7, CaBaFe4O7,
+CaBaFe2Co2O7, and YBaCo3AlO7 were grown by the
+floating zone technique [26, 30, 33, 38, 39]. Polarized,
+near normal incidence reflectivity was measured on
+polished cuts.
+Temperature dependent experiments
+were carried out up to 40000 cm−1 with an FT-
+IR spectrometer (Vertex80v, Bruker) and a grating-
+monochromator
+spectrometer
+(MSV-370YK,
+Jasco).
+The
+reflectivity
+spectrum
+of
+each
+compound
+was
+measured up to 250000 cm−1 at room temperature
+with use of synchrotron radiation at UVSOR Institute
+for Molecular Science, Okazaki, Japan.
+The optical
+conductivity was calculated using the Kramers-Kronig
+transformation [24].
+The pyroelectric polarization was
+obtained by measuring and integrating the displacement
+current with an electrometer (6517A, Keithley) while
+the temperature was swept in a Physical Property
+Measurement System (PPMS, Quantum Design).
+The
+dielectric properties were also measured in a PPMS,
+using an LCR meter (E4980A, Keysight Technologies)
+while the ac magnetization was measured in a Magnetic
+Property
+Measurement
+System
+(MPMS3,
+Quantum
+Design).
+For quantitative analysis, we fitted the real part of the
+optical conductivity as a sum of Lorentz oscillators:
+σ (ω) = −iωϵ0
+�
+�ϵ∞ +
+�
+j
+Sj
+ω2
+0,j − ω2 − iγjω
+�
+� ,
+(1)
+where ω0,j, Sj, and γj are the frequency, oscillator
+strength, and damping rate of the jth mode, and ϵ∞ is
+the high-frequency dielectric constant, respectively.
+In Fig. 2,
+we show the temperature dependence
+of the reflectivity and optical conductivity spectra
+around the lowest energy phonon modes of CaBaCo4O7,
+CaBaFe2Co2O7, and CaBaFe4O7 for light polarization
+Eω ∥ z. The reflectivity spectra over the whole photon
+energy range covered by our experiment for both Eω ∥ z
+and Eω
+⊥ z are presented in the supplement [24].
+The phonon spectra of CaBaCo4O7 and CaBaFe4O7
+(see Figs. 2(a,d),
+S3 and 2(c,f),
+S4,
+respectively)
+change markedly with temperature. The resonances are
+narrow at low temperatures and get significantly broader
+above the magnetic ordering temperature.
+Contrary
+to the pristine compounds,
+the phonon modes of
+CaBaFe2Co2O7 depend weakly on the temperature and
+show no anomaly at TN, as shown in Figs. 2(b,e), and S5.
+In Fig. 3, we compare the temperature dependence of
+the phonon parameters, frequency (ω0,j) and damping
+rate (γj) in CaBaCo4O7 and CaBaFe2Co2O7 for selected,
+
+3
+50
+60
+70
+80
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+50
+60
+70
+80
+0
+20
+40
+60
+80
+50
+60
+70
+80
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+50
+60
+70
+80
+0
+10
+20
+30
+40
+50
+60
+80
+100
+120
+140
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+60
+80
+100
+120
+140
+0
+10
+20
+30
+40
+50
+e
+5E
+ Reflectivity
+CaBaCo4O7
+Eω || z
+(a)
+4A2
+t2
+Co2+ Co3+
+#1
+#2
+Conductivity (Ω-1cm-1)
+CaBaCo4O7
+Eω || z
+Wavenumber (cm-1)
+  70K
+200K
+300K
+ T = 10K
+       50K
+       55K
+ TC=60K
+       65K
+(d)
+#2
+#1
+6A1
+ Reflectivity
+CaBaFe4O7
+Eω || z
+(c)
+5E
+e
+t2
+Fe2+ Fe3+
+#2
+#1
+ T = 10K
+       50K
+     100K
+     150K
+     200K
+TC2=211K
+     250K
+TC1=275K
+     300K
+CaBaFe4O7
+Eω || z
+Conductivity (Ω-1cm-1)
+Wavenumber (cm-1)
+(f)
+#2
+#1
+ Reflectivity
+CaBaFe2Co2O7
+Eω || z
+(b)
+e
+4A2
+t2
+Co2+ Fe3+
+6A1
+#2
+#1
+ Wavenumber (cm-1)
+ T = 10K
+       50K
+     100K
+     150K
+TN=152K
+     200K
+     250K
+     300K
+ Conductivity (Ω-1cm-1)
+CaBaFe2Co2O7
+Eω || z
+(e)
+#2
+#1
+FIG. 2. The reflectivity and the optical conductivity spectra of (a,d) CaBaCo4O7, (b,e) CaBaFe2Co2O7, and (c,f) CaBaFe4O7
+at selected temperatures in the frequency range of the lowest energy phonon modes.
+well-separated phonon modes.
+In the orthorhombic
+CaBaCo4O7 and CaBaFe4O7, the phonon modes are
+non-degenerate already at room temperature, and we
+did not resolve new modes below the magnetic phase
+transition temperatures. However, in both compounds
+the phonon frequencies change abruptly at the onset
+of the ferrimagnetic phase transitions. As an example,
+the magnitude of phonon energy shift becomes as
+large as ∆ω0/ω0
+∼4 % for modes #1 and #2 in
+CaBaCo4O7, shown in Fig. 3(a).
+This is significantly
+higher than ∆ω0/ω0 ∼1 %, the highest value observed
+in other multiferroics [40–42] and in magnets with
+strong spin-phonon coupling [43, 44].
+This indicates
+an extremely strong spin-lattice coupling [45–48], which
+agrees with recent experiments demonstrating giant
+magnetostriction [35]. In CaBaFe2Co2O7, however, the
+phonon frequencies change slightly with the temperature
+and we could not resolve any splitting of the phonon
+modes (see Fig. S5 and S8).
+The most remarkable changes in the infrared spectra
+of CaBaCo4O7 and CaBaFe4O7 are the drastic increase
+in the damping rates of all phonon modes as warmed
+above the ferrimagnetic phase transitions, see Fig. 3 and
+S8, respectively. Modes #1 and #2 of CaBaCo4O7 well
+exemplify this tendency: At T=10 K the damping rates
+of these modes are as low as 0.5 cm−1.
+Such sharp
+phonons with γ/ω0 < 1 % are unusual in condensed
+matter systems, and only observed in non-magnetic
+molecular crystals [49–52]. However, in the vicinity of
+TC the phonon lifetime decreases, i.e.
+the damping
+rate grows by an order of magnitude indicating a strong
+scattering of phonons by spin-fluctuations.
+In the
+paramagnetic phase, γ keeps increasing and at room
+temperature the phonon modes are strongly damped with
+γ/ω0 ratios exceeding 10 %.
+The strong temperature
+dependence of the damping rates away from TC, besides
+the strong spin-lattice coupling, suggests strong lattice
+anharmonicity [53, 54]. The damping rates of modes #16
+and #21, and those of CaBaFe4O7 (see Fig. S8) follow
+similar temperature dependence with pronounced change
+at the ferrimagnetic phase transitions. In contrast, the
+damping rates in CaBaFe2Co2O7 show weak temperature
+dependence and no anomalies at TN.
+As demonstrated in Fig. 4 and S9, the emergence
+of magnetic order strongly influences the pyroelectric
+polarization and the low-frequency dielectric response of
+CaBaCo4O7. We observed large magnetic-order-induced
+polarization change for P ∥ z in agreement with former
+results [33, 34] and negligible for P ⊥ z [55]. The real
+part of the dielectric constants, both ϵ⊥z and ϵ∥z, exhibit
+a step-like change when crossing TC [see Fig. 4(d,f)],
+with similar magnitude to that of in DyMn2O5 showing
+colossal magnetodielectric effect [56].
+Since the step
+height is independent of frequency between 102 and
+
+川川川川4
+66
+67
+68
+69
+415
+420
+425
+CaBaCo4O7
+525
+530
+52
+53
+54
+0
+100
+200
+300
+1
+10
+0
+100
+200
+300
+1
+10
+CaBaFe2Co2O7
+555
+560
+565
+262
+264
+266
+80
+85
+90
+95
+#2
+ ω0 (cm-1)
+#16
+#21
+TC
+(a)
+#1
+�  (cm-1)
+Temperature (K)
+(c)
+Temperature (K)
+(d)
+#10
+TN
+(b)
+#5
+#2
+#1
+FIG. 3. (a,b) Temperature dependence of the fitted phonon
+frequencies (ω0) and (c,d) damping rates (γ) in CaBaCo4O7
+and CaBaFe2Co2O7,
+respectively.
+The strong coupling
+between magnetic and elastic properties in CaBaCo4O7 is
+demonstrated by the changes in ω0 and γ around the magnetic
+phase transition (TC), indicated by dashed lines.
+105 Hz, and observed for both ϵ⊥z and ϵ∥z, the drop in the
+static dielectric function is related to the sudden changes
+in the phonon resonances. In addition to the step-edge
+in the real part, both the real and the imaginary parts of
+ϵ∥z have a peak at the close vicinity of TC. The frequency
+dependence and the related finite dissipation indicate
+electric dipoles with low-frequency dynamics and strong
+scattering.
+The peak shape in the real part suggests
+that the magnetic fluctuations can couple to electric
+dipoles and contribute to the phonon scattering [57, 58].
+Toward higher temperatures, the dielectric constants
+increase, not due to the change of phonon frequency
+but due to the decrease of the resistivity caused by
+the thermally activated carriers, as shown in Fig. S2.
+Although CaBaFe2Co2O7 has a similar pyroelectric
+crystal structure and a relatively high TN, its polarization
+is not affected by the antiferromagnetic order,
+as
+displayed in Fig. 4(c). The dielectric properties of this
+compound show a smooth variation on temperature in
+accordance with the phonon spectrum.
+We now discuss the enhanced scattering of phonons
+by spin fluctuations and the origin of the strong
+anomaly in the dielectric constant observed only in the
+pristine Swedenborgites, CaBaCo4O7 and CaBaFe4O7.
+Remarkably, such a large drop of the phonon damping
+rate induced by magnetic ordering is rare. Only minor
+changes in the damping rate have been detected in
+emblematic multiferroics including manganites RMnO3
+(R = Ho, Y) [59, 60], TbMnO3 [61], RMn2O5 (R =
+Tb, Eu, Dy, Bi) [62, 63], delafossite CuFeO2 [64] or
+Ni3V2O8 [65].
+Although several different mechanisms
+are responsible for the spin-lattice coupling in these
+materials, ranging from exchange striction [11, 66],
+inverse Dzyaloshinskii-Moriya interaction [8, 9] to on-
+site anisotropy term [12], none of them results in such
+a strong magnetic-order-induced change of phonon life-
+time.
+We note that charge fluctuations are frozen
+in the studied Swedenborgites as indicated by the
+large dc resistivity and the corresponding few-100 meV
+optical charge gap (see Fig. S2 and S7), thus, these
+cannot modify the spin-lattice interaction.
+Instead,
+we argue that low-energy fluctuations of the orbital
+degrees of freedom open a new channel and mediate
+a more efficient spin-lattice interaction in CaBaCo4O7
+and CaBaFe4O7 since orbitals can strongly interact with
+both spin fluctuations and phonons.
+This may lead
+to considerable broadening of phonon modes when the
+ordered state becomes paramagnetic as demonstrated
+in LaTiO3 [44].
+It is instructive to compare the case
+of Swedenborgites to that of hexagonal manganites.
+Although both class of compounds crystallize in a polar
+structure with geometric frustration, the phonons are
+scattered strongly by spin fluctuations exclusively in the
+Swedenborgites. In hexagonal mangnites, Mn3+ ions sit
+in a trigonal bipyramid, thus, they have S = 2 spins
+just like tetrahedrally coordinated Co3+ and Fe2+ ions,
+however, they are not Jahn-Teller active and their orbital
+singlet ground state is well separated from other 3d
+states [67, 68]. This fact also suggests that presence of
+orbital degrees of freedom allows the unusually strong
+spin-lattice coupling in Swedenborgites.
+Finally, we
+mention that a recent study of infrared phonons in
+Fe2Mo3O8 shows similar enhancement of the damping
+rate across its antiferromagnetic phase transition [42].
+In
+CaBaCo4O7
+and
+CaBaFe4O7,
+both
+the
+tetrahedrally coordinated Co3+ and Fe2+ ions possess
+the orbital-degenerate
+5E ground state multiplet as
+shown in the inset of Fig. 2. The orbital degeneracy is
+released by the trigonal to orthorhombic phase transition
+at TS, as illustrated in Fig. 4(a). The symmetry of the
+surrounding oxygen ligands is reduced to monoclinic,
+the dx2−y2 and dxy orbitals are separated by a small
+energy gap, and mixed with d3z2−r2 orbitals [25, 27].
+Since these strongly fluctuating low-symmetry orbitals
+can
+efficiently
+couple
+to
+the
+lattice,
+the
+phonons
+strongly scatter on this hybridized ground state in
+the paramagnetic phase.
+As the magnetic order
+develops, the second-order spin-orbit interaction can
+further polarize the orbitals, as an example spins along
+
+5
+0
+100
+200
+0
+10
+20
+30
+0
+100
+200
+0
+10
+20
+30
+10
+20
+30
+10
+20
+30
+57 60 63
+0
+1
+2
+3
+0
+100
+200
+0.0
+0.5
+1.0
+0
+100
+200
+0.0
+0.5
+1.0
+Temperature (K)
+Eω || z
+ε||z
+Im{ε}
+  ×5
+Re{ε}
+(f)
+Temperature (K)
+Re{ε}
+Eω || z
+Im{ε}
+  ×5
+(g)
+••10kHz
+••100kHz
+••100Hz
+Eω⊥ z
+Im{ε}
+  ×5
+ε⊥z
+••1kHz
+Re{ε}
+(d)
+Eω ⊥ z
+Im{ε} ×5
+Re{ε}
+(e)
+(h)
+P (� C/cm2)
+P⊥z
+P||z
+CaBaCo4O7
+(b)
+TC
+(a)
+P||z
+CaBaFe2Co2O7
+(c)
+TN
+Co3+ in 
+CaBaCo4O7
+FIG. 4. (a) Schematics of the ground state multiplet structure
+of Jahn-Teller active Co3+ ion in CaBaCo4O7.
+The Jahn-
+Teller active Fe2+ ion in CaBaFe4O7 has the same multiplet
+structure. The magnetic ions in the tetrahedral environment
+(Td) have the orbital-degenerate 5E ground state, which is
+preserved by the spin orbit interaction. At high temperature
+(TS < T), the oxygen environment is distorted to the polar
+C3v symmetry, but the orbital degeneracy is preserved by the
+E ground states {dx2−y2, dxy}. The trigonal to orthorombic
+distortion decreases the local symmetry to monoclinic Cs
+(TC < T < TS), releases the orbital degeneracy ({d∗
+x2−y2}),
+and deforms the orbitals. The ordering to the ferrimagnetic
+magnetic state (T < TC) further distorts the orbitals and
+selects only one (d∗∗).
+Temperature dependence of the
+(b,c) pyroelectric polarization and (d-g) dielectric constant of
+CaBaCo4O7 and CaBaFe2Co2O7, respectively. The (h) inset
+shows the peak in the imaginary part of ϵ∥z at TC.
+the y axis favours the dz2−x2 orbital [69, 70].
+The
+magnetic order in CaBaCo4O7 selects the same orbital
+shape at each Co3+ site and consequently reduces the
+fluctuations. According to this scenario, the quenching
+of the orbitals at TC strongly influences the lattice
+as well [23], which explains the exceptionally large
+magnetostriction, magnetic-order-induced polarization,
+and change in the dielectric response in CaBaCo4O7
+and CaBaFe4O7.
+The on-site anisotropy as well as
+the orbital dependence of the exchange interactions
+(Kugel-Khomski˘ı-type interaction) may equally play an
+important role in the enhanced spin-phonon coupling,
+however, our experiment is sensitive only to the Γ-point
+lattice vibrations, thus it cannot distinguish between
+these mechanisms. On one hand, the orbitals may affect
+the bond orientation dependence of the exchange and
+its bond-length variation.
+On the other hand, they
+may distort the local environment and spins drive a
+distortion of the local coordination. This question may
+be addressed by studying the momentum dependence
+of the phonon dispersion and lifetime in a scattering
+experiment.
+As the magnetic ions in CaBaFe2Co2O7
+have
+exclusively
+orbital-singlet
+ground
+states,
+the
+magnetic order has no effect on the orbitals and the
+absence of orbital degrees of freedom diminishes the
+spin-lattice coupling.
+Furthermore, orbital degeneracy
+can be the driving force behind the phonon anomalies in
+Fe2Mo3O8 [42], as it contains tetrahedrally coordinated
+Fe2+
+ions with orbital degrees of freedom,
+which
+suggests that the orbitals can enhance magetoelastic and
+magnetoelectric couplings not only in Swedenborgites,
+but also in broader classes of multiferroics.
+This idea
+is further supported by the effect of Ni-doping in
+CaBaCo4O7, where the substitution of orbital singlet
+Co2+ to Ni2+ ions with orbital degeneracy leads to
+further enhancement of the ME effect [71].
+Although
+precise theoretical description of these materials is
+challenging,
+we believe these findings will motivate
+further experimental and theoretical research.
+ACKNOWLEDGMENTS
+The authors are grateful to Karlo Penc for fruitful
+discussions, and to Akiko Kikkawa and Markus Kriener
+for the technical assistance.
+V.K. was supported by
+the Alexander von Humboldt Foundation.
+This work
+was supported by the Hungarian National Research,
+Development and Innovation Office – NKFIH grants
+FK 135003 and the bilateral program of the Estonian
+and Hungarian Academies of Sciences under the contract
+NKM 2021-24, and by the Estonian Research Council
+grant PRG736, institutional research funding IUT23-3 of
+the Estonian Ministry of Education and Research and the
+European Regional Development Fund project TK134.
+
+W
+sAE+ A&
+sAS+ AS
+2
+m
+3E
+sAS+ AS
+1XX6
+Illustration of the structural unit cell was created using
+the software VESTA[72].
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+and
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+of
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+Crystallography 41, 653 (2008).
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+A. Reller, and A. Loidl, The European Physical Journal
+Special Topics 180, 61 (2009).
+
+8
+Supplementary Material
+ADDITIONAL EXPERIMENTAL DATA
+200
+300
+400
+500
+600
+700
+800
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+CaBaFe4O7
+ 
+Cp (Jg-1K-1)
+Temperature (K)
+(b) before
+(a)
+(c) after
+(d)
+TS= 455K
+CaBaCo4O7
+CaBaCo4O7
+CaBaCo4O7
+CaBaCo4O7
+TS= 380K
+TC1= 275K
+TC2= 211K
+500µm
+100µm
+FIG. S1. (Color online) (a) Specific heat of CaBaCo4O7 and
+CaBaFe4O7 measured for warming runs. Specific heat data
+of CaBaFe4O7 is reproduced after Ref. 26.
+(b-d) Optical
+microscopy images (b) before and (c) after the specific heat
+measurements on CaBaCo4O7.
+Dark and light contrasted
+regions correspond to the orthorombic domains. CaBaCo4O7
+shows strong twinning on the microscopic scale. (c) Following
+the specific heat measurements, the orthorombic domains
+rearrange in a meander-like pattern.
+Panel (d) shows a
+magnified region in panel (c).
+Figure S1(a) shows the specific heat of CaBaCo4O7
+and CaBaFe4O7 measured in the warming runs.
+The
+orthorombic to trigonal phase transition temperatures
+are
+TS=455 K
+for
+CaBaCo4O7
+and
+TS=380 K
+for
+CaBaFe4O7.
+Above TS, neither materials show any
+further phase transitions. Figures S1(b-d) show optical
+microscopy images of CaBaCo4O7 before and after a
+high-temperature heat treatment procedure. The sample
+was heated to T=600 K for 4 h in air, then quenched to
+room temperature. The microscope images were made in
+the so-called crossed Nicholson configuration; dark and
+light contrasted regions correspond to the orthorombic
+domains.
+CaBaCo4O7 shows strong twinning on the
+microscopic scale. After the heat treatment procedure
+in Fig. S1(c,d), the orthorombic domains rearrange in a
+meander-like pattern.
+Figure S2 shows the temperature dependence of the
+resistivity (ρ) in CaBaCo4O7,
+CaBaFe2Co2O7,
+and
+CaBaFe4O7. The resistivity was measured with currents
+parallel (ρ∥z) and perpendicular to the z axis (ρ⊥z). The
+resistivity data of CaBaFe4O7 shows only a very subtle
+anomaly at TS, and has semiconductor-like temperature
+dependence both below and above TS. The absence of
+strong anomalies in the specific heat and resistivity data
+at temperatures above TS in Figs. S1 and S2 implies
+that the charge ordered state is not melted up to the
+decomposition temperatures in CaBaFe4O7.
+Figures S3, S4, S5, and S6 show the reflectivity
+and
+optical
+conductivity
+spectra
+of
+CaBaCo4O7,
+CaBaFe4O7,
+CaBaFe2Co2O7,
+and
+YBaCo3AlO7
+at
+selected temperatures. In each figure, panels (a,c) and
+(b,d) correspond to measurements with Eω ⊥ z and Eω ∥
+z, respectively. YBaCo3AlO7 is a spin-glass (Tf=17 K),
+has the hexagonal P63mc structure [38], and it hosts
+exclusively orbital-singlet Co2+ ions.
+Comparison of
+YBaCo3AlO7 to CaBaFe2Co2O7 and to the ferrimagnetic
+compounds helped us to examine the role of long-range
+order.
+Similarly to the solid solution CaBaFe2Co2O7,
+the phonons of YBaCo3AlO7 in Fig. S5 show only weak
+temperature dependence.
+The
+optical
+conductivity
+spectra
+were
+calculated
+from the reflectivity data using the Kramers-Kronig
+transformation.
+The low-energy part of the measured
+reflectivity spectra was extrapolated to zero photon
+energy as a constant value.
+Figure S7 shows the UV
+and hard-UV optical reflectivity and optical conductivity
+of CaBaCo4O7, CaBaFe2Co2O7, and CaBaFe4O7, which
+was used as a high-energy extension for the Kramers-
+Kronig transformation. Spectra above k=106 cm−1 were
+assumed to follow the free electron model.
+Figure S8 summarizes the temperature dependence
+of
+the
+fitted
+phonon
+frequencies
+(ω0),
+oscillator
+strengths (S), and damping rates (γ) in CaBaCo4O7,
+CaBaFe2Co2O7,
+and CaBaFe4O7.
+Data shown in
+Fig. S8(a,b,g,h) are the same as those in Fig. 2. At the
+magnetic phase transitions, the phonon frequencies and
+damping rates show remarkable changes in the pristine
+compounds.
+In CaBaFe2Co2O7, none of the phonon
+parameters show change in the vicinity of TN.
+We
+detect the appearance of no new phonon modes, which
+is supported by the temperature dependence of S in
+S8(d,e,f), which show changes only around the magnetic
+phase transitions.
+Figures S9(a-c) show the real and imaginary parts
+of the dielectric constants (ϵ) measured in CaBaCo4O7,
+CaBaFe2Co2O7, and CaBaFe4O7, respectively for Eω ⊥
+z in the upper panels and Eω
+∥
+z in the lower
+
+9
+panels.
+The real and imaginary parts of the ac
+magnetic susceptibility (ac-χ) measured in CaBaCo4O7,
+CaBaFe2Co2O7, and CaBaFe4O7 for Hω ⊥ z and Hω ∥ z
+are shown in Figs. S9(d), S9(e), and S9(f), respectively.
+The magnitude of the oscillating magnetic field was
+δHω=5 Oe. The ac-χ measurements were performed in
+the absence of static H field, except for the lower panel
+of Fig. S9(f), where a moderate H = 3 kOe static field
+was applied. In CaBaCo4O7, the real part of ϵ⊥z has a
+step like jump, while the imaginary part rapidly increases
+above TC.
+The real part of ϵ∥z has a double peak
+structure (strongest peak at TC), and the imaginary part
+has a single peak at TC. Above TC, all components of the
+dielectric constants show strong frequency dependence
+and anisotropy, i.e.
+ϵ∥z increases more rapidly with
+temperature than ϵ⊥z. However, for low frequencies these
+features are an aggregate of the anisotropic resistivity
+and the Maxwell-Wagner relaxation caused by Schottky
+barriers forming at sample electrode interfaces [73,
+74].
+Figures S9(d) and S9(f) also show the frequency
+dependence of the imaginary part of the ac-χ in
+CaBaCo4O7 and CaBaFe4O7, respectively. The inset of
+panel (d), Fig. S9(g), shows the imaginary part of the ac-
+χ measured in CaBaCo4O7 for a magnified region. The
+imaginary parts of the ac-χ both in CaBaCo4O7 and in
+CaBaFe4O7 has asymmetric peaks around TC and TC2,
+respectively, which means increased dissipation on the
+magnetic domain walls at low-frequencies (below 1 kHz).
+In CaBaCo4O7, the broad symmetric peak at TC in the
+real part of χ⊥z is accompanied by an asymmetric peak
+in the imaginary part, and a step in Re{χ∥z}. Frequency
+dependence of the magnetic Im{χ⊥z} resembles to
+that of the dielectric Im{ϵ∥z}, however at much lower
+frequencies. In contrast to the pristine compounds, the
+antiferromagnetic CaBaFe2Co2O7 has no features in the
+dielectric constants in Fig. S9(b), and has only a small
+peak in the ac magnetic susceptibility in Fig. S9(e).
+Although the ac-χ has similar features to ϵ, which would
+suggest a strong connection between magnetic and lattice
+fluctuations, however, the magnetic fluctuations are at
+very low frequencies and the strength of the magnetic
+fluctuations decay quickly towards higher frequencies.
+Therefore, magnetic fluctuations alone cannot account
+for the increased phonon scattering observed in the
+optical measurements at significantly higher frequencies.
+As a conclusion, in CaBaCo4O7 and CaBaFe4O7, the
+electric and magnetic fluctuations are relevant only
+in the vicinity of the ferrimagnetic phase transitions,
+while the magnetic fluctuations are not relevant at
+optical frequencies. Therefore, the strong anharmonicity
+of phonon modes in the pristine compounds are not
+explained by these fluctuations.
+0
+100
+200
+300
+400
+10-1
+101
+103
+105
+107
+109
+1011
+0
+100
+200
+300
+400
+10-1
+101
+103
+105
+107
+109
+1011
+0
+100
+200
+300
+400
+10-1
+101
+103
+105
+107
+109
+1011
+360 380 400
+2
+4
+6
+�  (Ωcm)
+CaBaCo4O7
+TC
+(a)
+� ⊥z
+� ||z
+�  (Ωcm)
+TN
+CaBaFe2Co2O7
+(b)
+� ⊥z
+� ||z
+Temperature (K)
+� ⊥z
+�  (Ωcm)
+TC1
+TC2
+CaBaFe4O7
+(c)
+� ||z
+TS
+TS
+(d)
+FIG. S2.
+(Color online) Temperature dependence of the
+resistivity of (a) CaBaCo4O7,
+(b) CaBaFe2Co2O7,
+and
+(c) CaBaFe4O7, measured with currents parallel (ρ∥z) and
+perpendicular to the z-axis (ρ∥z).
+Panel (d) shows the
+resistivity of CaBaFe4O7 at TS. Note, that CaBaFe4O7 shows
+very little change in the resistivity, i.e.
+there is definitely
+no charge order-disorder type of transition accompanying the
+structural phase transition.
+LATTICE EXCITATIONS IN SWEDENBORGITES
+Swedenborgites, CaBaM4O7 (M=Co, Fe) are built
+up by alternating layers of triangular and kagom´e
+lattices of tetrahedrally coordinated transition metal
+ions.
+In general, these compounds realize a trigonal
+structure (P31c), with P63mc as the possible highest
+symmetry mother-structure. Note that the space group
+of the hexagonal manganites is P63cm. The irreducible
+representations of the infrared-active normal modes for
+the P63mc mother structure are:
+ΓIR = 9A1(z) + 12E1(xy).
+(S2)
+
+10
+Namely, for Eω ∥ z the reflectivity spectra may show
+9 non-degenerate A1 modes, and for Eω ⊥ z 12 doubly
+degenerate E1 modes. Note, that YBaCo3AlO7 in Fig. S6
+shows 9 modes for Eω ∥ z.
+At
+high-temperatures
+(above
+TS),
+the
+pristine
+CaBaFe4O7 and CaBaCo4O7, as well as the solid solution
+CaBaFe2Co2O7 at all temperatures have the trigonal
+structure, described by the P31c space group.
+The
+irreducible representations of the infrared-active phonons
+are:
+ΓIR = 12A1(z) + 25E(xy).
+(S3)
+Therefore, for Eω ∥ z and Eω ⊥ z, the reflectivity spectra
+may show 12 A1 and 25 E modes, respectively. For Eω ∥
+z in Fig. S5(b,d), CaBaFe2Co2O7 shows 12 modes.
+Below TS, CaBaCo4O7 and CaBaFe4O7 have the
+orthorombic Pbn21 structure and the infrared-active
+phonons are:
+ΓIR = 39A1(z) + 39B1(y) + 39B2(x).
+(S4)
+Namely, the reflectivity spectra should show 39 non-
+degenerate modes for all three polarizations of the
+electromagnetic radiation. For Eω ∥ z in Figs S3(b,d)
+and S4(b,d) we identify 22 and 31 phonon modes for
+CaBaCo4O7 and CaBaFe4O7, respectively.
+The lower
+number of phonons compared to the expected may come
+from accidental degenerations or from modes outside
+the spectral window with k=25 cm−1 cutoff energy. We
+note that low-energy orbital fluctuations of tetrahedral
+Fe2+ ions can also be active and mix among the phonon
+excitations.
+
+11
+100
+200
+300
+400
+500
+600
+700
+800
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+100
+200
+300
+400
+500
+600
+700
+800
+0
+100
+200
+300
+400
+100
+200
+300
+400
+500
+600
+700
+800
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+100
+200
+300
+400
+500
+600
+700
+800
+0
+100
+200
+300
+400
+(d)
+(c)
+(a)
+(b)
+ Reflectivity
+CaBaCo4O7
+        Eω ⊥ z
+       80K
+       90K
+     120K
+     150K
+     200K
+     250K
+     300K
+Conductivity (Ω-1cm-1)
+CaBaCo4O7
+        Eω ⊥ z
+Wavenumber (cm-1)
+ T = 10K
+       20K
+       30K
+       40K
+       50K
+       60K
+       70K
+#2
+#1
+#21
+ Reflectivity
+CaBaCo4O7
+         Eω || z
+#16
+Conductivity (Ω-1cm-1)
+CaBaCo4O7
+         Eω || z
+Wavenumber (cm-1)
+ T = 10K
+       20K
+       30K
+       40K
+       50K
+       60K
+       70K
+       80K
+       90K
+     120K
+     150K
+     200K
+     250K
+     300K
+#2
+#1
+#16
+#21
+FIG. S3. (Color online) Temperature dependence of the (a,b) optical reflectivity and (c,d) calculated optical conductivity
+spectra of CaBaCo4O7. Panels (a,c) and panels (b,d) show measurements for Eω ⊥ z and Eω ∥ z, respectively.
+100
+200
+300
+400
+500
+600
+700
+800
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+100
+200
+300
+400
+500
+600
+700
+800
+0
+50
+100
+150
+200
+250
+100
+200
+300
+400
+500
+600
+700
+800
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+100
+200
+300
+400
+500
+600
+700
+800
+0
+50
+100
+150
+200
+250
+(d)
+(c)
+(a)
+(b)
+ Reflectivity
+CaBaFe4O7
+        Eω ⊥ z
+Conductivity (Ω-1cm-1)
+CaBaFe4O7
+        Eω ⊥ z
+Wavenumber (cm-1)
+     212K
+     225K
+     250K
+     260K
+     270K
+     285K
+     300K
+ T = 10K
+       25K
+       50K
+       75K
+     100K
+     125K
+     150K
+     175K
+     200K
+#29
+ Reflectivity
+CaBaFe4O7
+         Eω || z
+#1
+#2
+#23
+     212K
+     225K
+     250K
+     260K
+     270K
+     285K
+     300K
+ T = 10K
+       25K
+       50K
+       75K
+     100K
+     125K
+     150K
+     175K
+     200K
+Conductivity (Ω-1cm-1)
+CaBaFe4O7
+         Eω || z
+Wavenumber (cm-1)
+#23
+#29
+#2
+#1
+FIG. S4. (Color online) Temperature dependence of the (a,b) optical reflectivity and (c,d) calculated optical conductivity
+spectra of CaBaFe4O7. Panels (a,c) and panels (b,d) show measurements for Eω ⊥ z and Eω ∥ z, respectively.
+
+12
+100
+200
+300
+400
+500
+600
+700
+800
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+100
+200
+300
+400
+500
+600
+700
+800
+0
+100
+200
+300
+400
+500
+100
+200
+300
+400
+500
+600
+700
+800
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+100
+200
+300
+400
+500
+600
+700
+800
+0
+50
+100
+150
+200
+(d)
+(c)
+(a)
+(b)
+ Reflectivity
+CaBaFe2Co2O7
+              Eω ⊥ z
+Conductivity (Ω-1cm-1)
+CaBaFe2Co2O7
+              Eω ⊥ z
+Wavenumber (cm-1)
+ T = 10K
+       25K
+       50K
+       75K
+     100K
+     125K
+     150K
+     175K
+     200K
+     225K
+     250K
+     300K
+ Reflectivity
+CaBaFe2Co2O7
+              Eω || z
+#1#2
+#5
+#10
+Conductivity (Ω-1cm-1)
+CaBaFe2Co2O7
+              Eω || z
+Wavenumber (cm-1)
+ T = 10K
+       25K
+       50K
+       75K
+     100K
+     125K
+     150K
+     175K
+     200K
+     225K
+     250K
+     300K
+#10
+#1#2
+#5
+FIG. S5. (Color online) Temperature dependence of the (a,b) optical reflectivity and (c,d) calculated optical conductivity
+spectra of CaBaFe2Co2O7. Panels (a,c) and panels (b,d) show measurements for Eω ⊥ z and Eω ∥ z, respectively.
+100
+200
+300
+400
+500
+600
+700
+800
+900
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+100
+200
+300
+400
+500
+600
+700
+800
+900
+0
+50
+100
+150
+200
+250
+100
+200
+300
+400
+500
+600
+700
+800
+900
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+100
+200
+300
+400
+500
+600
+700
+800
+900
+0
+50
+100
+150
+200
+250
+(d)
+(c)
+(a)
+(b)
+ Reflectivity
+YBaCo3AlO7
+         Eω ⊥ z
+Conductivity (Ω
+-1cm
+-1)
+YBaCo3AlO7
+         Eω ⊥ z
+Wavenumber (cm
+-1)
+ T = 10K
+       25K
+       50K
+       75K
+     100K
+     150K
+     200K
+     250K
+     300K
+ Reflectivity
+YBaCo3AlO7
+          Eω || z
+ T = 10K
+       25K
+       50K
+       75K
+     100K
+     150K
+     200K
+     250K
+     300K
+Conductivity (Ω
+-1cm
+-1)
+YBaCo3AlO7
+          Eω || z
+Wavenumber (cm
+-1)
+FIG. S6. (Color online) Temperature dependence of the (a,b) optical reflectivity and (c,d) calculated optical conductivity
+spectra of YBaCo3AlO7. Panels (a,c) and panels (b,d) show measurements for Eω ⊥ z and Eω ∥ z, respectively.
+
+13
+0
+5
+10
+15
+20
+25
+0.0
+0.1
+0.2
+0.3
+0
+5
+10
+15
+20
+25
+0
+1
+2
+3
+4
+5
+Reflectivity
+CaBaCo4O7
+CaBaFe2Co2O7
+CaBaFe4O7
+(a)
+CaBaCo4O7
+CaBaFe2Co2O7
+CaBaFe4O7
+Conductivity (103 Ω-1cm-1)
+Energy (eV)
+(b)
+FIG. S7.
+(Color online) (a) Hard UV reflectivity and (b)
+optical conductivity spectra of CaBaCo4O7, CaBaFe2Co2O7,
+and CaBaFe4O7 measured at T=300 K.
+
+14
+66
+67
+68
+69
+415
+420
+425
+CaBaCo4O7
+525
+530
+52
+53
+54
+0
+100
+200
+300
+1
+10
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+100
+200
+300
+1
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+CaBaFe2Co2O7
+555
+560
+565
+262
+264
+266
+80
+85
+90
+95
+0
+100
+200
+300
+1
+10
+0.0
+0.2
+0.4
+480
+485
+490
+CaBaFe4O7
+625
+626
+627
+64
+66
+68
+45
+50
+55
+#2
+ ω0 (cm-1)
+#16
+#21
+TC
+(a)
+#1
+�  (cm-1)
+Temperature (K)
+(g)
+×50
+S (106cm-2)
+×50
+(d)
+Temperature (K)
+�  (cm-1)
+(h)
+S (106cm-2)
+(e)
+#10
+TN
+(b)
+TC1
+#5
+#2
+ ω0 (cm-1)
+#1
+Temperature (K)
+�  (cm-1)
+(i)
+×10
+×10
+S (106cm-2)
+(f)
+(c)
+#23
+TC2
+#29
+ ω0 (cm-1)
+#2
+#1
+FIG. S8. (Color online) Temperature dependence of the fitted (a,b,c) phonon frequencies (ω0), (d,e,f) oscillator strengths (S),
+and (g,h,i) damping rates (γ) of CaBaCo4O7, CaBaFe2Co2O7, and CaBaFe4O7, respectively.
+
+15
+0
+100
+200
+300
+0
+5
+10
+15
+0
+100
+200
+300
+1.8
+2.0
+2.2
+2.4
+2.6
+5
+10
+15
+1.8
+2.0
+2.2
+2.4
+0
+100
+200
+300
+0
+10
+20
+0
+100
+200
+300
+0
+10
+20
+CaBaCo4O7
+0
+10
+20
+30
+CaBaFe2Co2O7
+0
+10
+20
+30
+0
+10
+20
+30
+0
+100
+200
+300
+0
+10
+20
+0
+25
+50
+75
+100
+0
+100
+200
+300
+0
+5
+10
+15
+20
+••100Hz
+••200Hz
+••500Hz
+••1kHz
+60
+63
+0
+1
+2
+Hω || z
+Temperature (K)
+� ||z (a.u.)
+Re{χ}
+H = 0kOe
+� ||z (a.u.)
+Temperature (K)
+Re{χ}
+Hω || z
+H = 0kOe
+•100Hz
+Hω⊥ z
+� ⊥z (a.u.)
+Im{χ}
+  ×5
+Re{χ}
+TC
+H = 0kOe
+•100Hz
+� ⊥z (a.u.)
+Re{χ}
+Hω ⊥ z
+(e)
+•100Hz
+TN
+H = 0kOe
+Eω || z
+ε||z
+Im{ε}
+  ×5
+Re{ε}
+••100Hz
+••1kHz
+••10kHz
+••100kHz
+ε||z
+Re{ε}
+Eω || z
+Im{ε}
+  ×5
+••10kHz
+••100kHz
+••100Hz
+Eω⊥ z
+Im{ε}
+  ×5
+ε⊥z
+••1kHz
+Re{ε}
+(a)
+ε⊥z
+Eω ⊥ z
+Im{ε} ×5
+Re{ε}
+(b)
+(d)
+CaBaFe4O7
+••100Hz
+••1kHz
+••10kHz
+••100kHz
+Im{ε} ×5
+Re{ε}
+ε⊥z
+(c)
+Eω ⊥ z
+Eω || z
+ε||z
+Re{ε}
+Im{ε} ×5
+TC2
+� ⊥z (a.u.)
+Hω ⊥ z
+Re{χ}
+(f)
+TC1
+•100Hz
+H = 0kOe
+••100Hz
+Temperature (K)
+� ||z (a.u.)
+Hω || z
+Im{χ}
+ ×10
+Re{χ}
+H = 3kOe
+(g)
+ 
+FIG. S9. (Color online) Temperature dependence of (a,b,c) the dielectric constant and (d,e,f) the ac magnetic susceptibility
+of CaBaCo4O7, CaBaFe2Co2O7, and CaBaFe4O7, respectively.
+Note that the imaginary part of the dielectric constant is
+multiplied by a factor of 5 for better visibility. (g) The inset shows a magnified region of the imaginary part of the ac-χ from
+panel (d).
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+Head-Free Lightweight Semantic Segmentation with Linear Transformer
+Bo Dong 1*, Pichao Wang 1 †, Fan Wang 2
+1 Alibaba Group
+{bo.dong.cst, pichaowang}@gmail.com; fan.w@alibaba-inc.com
+Abstract
+Existing semantic segmentation works have been mainly fo-
+cused on designing effective decoders; however, the com-
+putational load introduced by the overall structure has long
+been ignored, which hinders their applications on resource-
+constrained hardwares. In this paper, we propose a head-free
+lightweight architecture specifically for semantic segmenta-
+tion, named Adaptive Frequency Transformer (AFFormer).
+AFFormer adopts a parallel architecture to leverage proto-
+type representations as specific learnable local descriptions
+which replaces the decoder and preserves the rich image
+semantics on high-resolution features. Although removing
+the decoder compresses most of the computation, the accu-
+racy of the parallel structure is still limited by low com-
+putational resources. Therefore, we employ heterogeneous
+operators (CNN and Vision Transformer) for pixel embed-
+ding and prototype representations to further save compu-
+tational costs. Moreover, it is very difficult to linearize the
+complexity of the vision Transformer from the perspective
+of spatial domain. Due to the fact that semantic segmenta-
+tion is very sensitive to frequency information, we construct a
+lightweight prototype learning block with adaptive frequency
+filter of complexity O(n) to replace standard self atten-
+tion with O(n2). Extensive experiments on widely adopted
+datasets demonstrate that AFFormer achieves superior accu-
+racy while retaining only 3M parameters. On the ADE20K
+dataset, AFFormer achieves 41.8 mIoU and 4.6 GFLOPs,
+which is 4.4 mIoU higher than Segformer, with 45% less
+GFLOPs. On the Cityscapes dataset, AFFormer achieves 78.7
+mIoU and 34.4 GFLOPs, which is 2.5 mIoU higher than
+Segformer with 72.5% less GFLOPs. Code is available at
+https://github.com/dongbo811/AFFormer.
+Introduction
+Semantic segmentation aims to partition an image into sub-
+regions (collections of pixels) and is defined as a pixel-level
+classification task (Long, Shelhamer, and Darrell 2015; Xie
+et al. 2021; Zhao et al. 2017; Chen et al. 2018; Strudel et al.
+2021; Cheng, Schwing, and Kirillov 2021) since Fully Con-
+volutional Networks (FCN) (Long, Shelhamer, and Darrell
+*Work done during an internship at Alibaba Group.
+†Corresponding author; work done at Alibaba Group, and now
+affiliated with Amazon Prime Video.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+0
+256
+512
+768
+1024
+1280
+1536
+1792
+2048
+0
+50
+100
+150
+200
+250
+300
+350
+400
+FLOPs
+Input Scale
+ PSPNet
+ DeepLabV3+
+ SegFormer
+ AFFormer
+2.1x 1.8x
+2.1x
+2.5x
+3.0x
+3.6x
+4.4x
+5.3x
+11.3x
+5.6x
+81.0
+78.0
+75.0
+44.0
+40.0
+36.0
++4.4 mIoU
+…….
++2.5 mIoU
+…….
+SegFormer
+AFFormer
+Input Scale
+FLOPs
+ADE20K
+Cityscapes
+Figure 1: Left: Computational complexity under differ-
+ent input scales. Segformer (Xie et al. 2021) significantly
+reduces the computational complexity compared to tra-
+ditional methods, such as PSPNet (Zhao et al. 2017)
+and DeepLabV3+ (Chen et al. 2018) which have mo-
+bilenetV2 (Sandler et al. 2018) as backbone. However, Seg-
+former still has a huge computational burden for higher
+resolutions. Right: AFFormer achieves better accuracy on
+ADE20K and Cityscapes datasets with significantly lower
+FLOPs.
+2015). It has two unique characteristics compared to image
+classification: pixel-wise dense prediction and multi-class
+representation, which is usually built upon high-resolution
+features and requires a global inductive capability of im-
+age semantics, respectively. Previous semantic segmenta-
+tion methods (Zhao et al. 2017; Chen et al. 2018; Strudel
+et al. 2021; Xie et al. 2021; Cheng, Schwing, and Kirillov
+2021; Yuan et al. 2021b) focus on using the classification
+network as backbone to extract multi-scale features, and de-
+signing a complicated decoder head to establish the rela-
+tionship between multi-scale features. However, these im-
+provements come at the expense of large model size and
+high computational cost. For instance, the well-known PSP-
+Net (Zhao et al. 2017) using light-weight MobilenetV2 (San-
+dler et al. 2018) as backbone contains 13.7M parameters and
+52.2 GFLOPs with the input scale of 512×512. The widely-
+used DeepLabV3+ (Chen et al. 2018) with the same back-
+bone requires 15.4M parameters and 25.8 GFLOPs. The in-
+herent design manner limits the development of this field
+arXiv:2301.04648v1  [cs.CV]  11 Jan 2023
+
+and hinders many real-world applications. Thus, we raise the
+following question: can semantic segmentation be as simple
+as image classification?
+Recently vision Transformers (ViTs) (Liu et al. 2021; Lee
+et al. 2022; Xie et al. 2021; Strudel et al. 2021; Cheng,
+Schwing, and Kirillov 2021; Xu et al. 2021; Lee et al.
+2022) have shown great potential in semantic segmenta-
+tion, however, they face the challenges of balancing perfor-
+mance and memory usage when deployed on ultra-low com-
+puting power devices. Standard Transformers has computa-
+tional complexity of O(n2) in the spatial domain, where n
+is the input resolution. Existing methods alleviate this sit-
+uation by reducing the number of tokens (Xie et al. 2021;
+Wang et al. 2021; Liang et al. 2022; Ren et al. 2022) or
+sliding windows (Liu et al. 2021; Yuan et al. 2021a), but
+they introduce limited reduction on computational complex-
+ity and even compromise global or local semantics for the
+segmentation task. Meanwhile, semantic segmentation as a
+fundamental research field, has extensive application scenar-
+ios and needs to process images with various resolutions.
+As shown in Figure 1, although the well-known efficient
+Segformer (Xie et al. 2021) achieves a great breakthrough
+compared to PSPNet and DeepLabV3+, it still faces a huge
+computational burden for higher resolutions. At the scale
+of 512 × 512, although Segformer is very light compared
+to PSPNet and DeepLabV3+, it is almost twice as expen-
+sive as ours (8.4 GFLOPs vs 4.6 GFLOPs); at the scale of
+2048 × 2048, even 5x GFLOPs is required (384.3 GFLOPs
+vs 73.2 GFLOPs). Thus, we raise another question: can we
+design an efficient and lightweight Transformer network for
+semantic segmentation in ultra-low computational scenar-
+ios?
+The answers to above two questions are affirmative. To
+this end, we propose a head-free lightweight semantic seg-
+mentation specific architecture, named Adaptive Frequency
+Transformer (AFFormer). Inspired by the properties that
+ViT maintains a single high-resolution feature map to keep
+details (Dosovitskiy et al. 2021) and the pyramid structure
+reduces the resolution to explore semantics and reduce com-
+putational cost (He et al. 2016; Wang et al. 2021; Liu et al.
+2021), AFFormer adopts a parallel architecture to lever-
+age the prototype representations as specific learnable lo-
+cal descriptions which replace the decoder and preserves
+the rich image semantics on high-resolution features. The
+parallel structure compresses the majority of the compu-
+tation by removing the decoder, but it is still not enough
+for ultra-low computational resources. Moreover, we em-
+ploy heterogeneous operators for pixel embedding features
+and local description features to save more computational
+costs. A Transformer-based module named prototype learn-
+ing (PL) is used to learn the prototype representations, while
+a convolution-based module called pixel descriptor (PD)
+takes pixel embedding features and the learned prototype
+representations as inputs, transforming them back into the
+full pixel embedding space to preserve high-resolution se-
+mantics.
+However, it is still very difficult to linearize the complex-
+ity of the vision Transformer from the perspective of spatial
+domain. Inspired by the effects of frequency on classifica-
+tion tasks (Rao et al. 2021; Wang et al. 2020), we find that
+semantic segmentation is also very sensitive to frequency
+information. Thus, we construct a lightweight adaptive fre-
+quency filter of complexity O(n) as prototype learning to re-
+place the standard self attention with O(n2). The core of this
+module is composed of frequency similarity kernel, dynamic
+low-pass and high-pass filters, which capture frequency in-
+formation that is beneficial to semantic segmentation from
+the perspectives of emphasizing important frequency com-
+ponents and dynamically filtering frequency, respectively.
+Finally, the computational cost is further reduced by sharing
+weights in high and low frequency extraction and enhance-
+ment modules. We also embed a simplified depthwise con-
+volutional layer in the feed-forward network (FFN) layer to
+enhance the fusion effect, reducing the size of the two matrix
+transformations.
+With the help of parallel heterogeneous architecture and
+adaptive frequency filter, we use only one convolutional
+layer as classification layer (CLS) for single-scale feature,
+achieving the best performance and making semantic seg-
+mentation as simple as image classification. We demonstrate
+the advantages of the proposed AFFormer on three widely-
+used datasets: ADE20K, Cityscapes and COCO-stuff. With
+only 3M parameters, AFFormer significantly outperforms
+the state-of-the-art lightweight methods. On ADE20K, AF-
+Former achieves 41.8 mIoU with 4.6 GFLOPs, outperform-
+ing Segformer by 4.4 mIoU, while reducing GFLOPs by
+45%. On Cityscapes, AFFormer achieves 78.7 mIoU and
+34.4 GFLOPs, which is 2.5 mIoU higher than Segformer,
+with 72.5% less GFLOPs. Extensive experimental results
+demonstrate that it is possible to apply our model in compu-
+tationally constrained scenarios, which still maintaining the
+high performance and robustness across different datasets.
+Related Work
+Semantic Segmentation
+Semantic segmentation is regarded as a pixel classification
+task (Strudel et al. 2021; Xu et al. 2017; Xie et al. 2021).
+In the last two years, new paradigms based on visual Trans-
+formers have emerged, which enable mask classification via
+queries or dynamic kernels (Zhang et al. 2021; Li et al. 2022;
+Cheng, Schwing, and Kirillov 2021; Cheng et al. 2022). For
+instance, Maskformer (Cheng, Schwing, and Kirillov 2021)
+learns an object query and converts it into an embedding
+of masks. Mask2former (Cheng et al. 2022) enhances the
+query learning with a powerful multi-scale masked Trans-
+former (Zhu et al. 2021). K-Net (Zhang et al. 2021) adopts
+dynamic kernels for masks generation. MaskDINO (Li et al.
+2022) brings object detection to semantic segmentation, fur-
+ther improving query capabilities. However, all above meth-
+ods are not suitable for low computing power scene due
+to the high computational cost of learning efficient queries
+and dynamic kernels. We argue that the essence of these
+paradigms is to update pixel semantics by replacing the
+whole with individual representations. Therefore, we lever-
+age pixel embeddings as a specific learnable local descrip-
+tion that extracts image and pixel semantics and allows se-
+mantic interaction.
+
+DC-FFN
+AFF
+Add & Norm
+Restoring
+(i)
+Clustering
+(iii) Pixel Descriptor (PD)
+(ii) Prototype Learning (
+) 
+Positional 
+Encodings
+Add & Norm
+PL
+Clustering
+PD
+Image
+CLS
+Stem
+Pixel Classification
+PL
+Clustering
+PD
+PL
+Clustering
+PD
+PL
+Clustering
+PD
+…
+…
+…
+…
+Sharing
+AFF
+DC-FFN
+Stem
+Adaptive Frequency Filter
+Depthwise
+Feed-Forward Network
+Two Convolutional Layers
+CLS
+A Convolutional Layer
+Figure 2: An Overview of Adaptive Frequency Transformer (AFFormer). We first displays the overall structure of parallel
+heterogeneous network. Specifically, the feature F after patch embedding is first clustered to obtain the prototype feature G,
+so as to construct a parallel network structure, which includes two heterogeneous operators. A Transformer-based module
+as prototype learning to capture favorable frequency components in G, resulting prototype representation G′. Finally G′ is
+restored by a CNN-based pixel descriptor, resulting F ′ for the next stage.
+Efficient Vision Transformers
+The lightweight solution of vision Transformer mainly fo-
+cuses on the optimization of self attention, including follow-
+ing ways: reducing the token length (Wang et al. 2021; Xie
+et al. 2021; Wang et al. 2022) and using local windows (Liu
+et al. 2021; Yuan et al. 2021a). PVT (Wang et al. 2021)
+performs spatial compression on keys and values through
+spatial reduction, and PVTv2 (Wang et al. 2022) further re-
+places the spatial reduction by pooling operation, but many
+details are lost in this way. Swin (Liu et al. 2021; Yuan
+et al. 2021a) significantly reduce the length of the token
+by restricting self attention to local windows, while these
+against the global nature of Transformer and restrict the
+global receptive field. At the same time, many lightweight
+designs (Chen et al. 2022; Mehta and Rastegari 2022) in-
+troduce Transformers in MobileNet to obtain more global
+semantics, but these methods still suffer from the square-
+level computational complexity of conventional Transform-
+ers. Mobile-Former (Chen et al. 2022) combines the par-
+allel design of MobileNet (Sandler et al. 2018) and Trans-
+former (Dosovitskiy et al. 2021), which can achieve bidi-
+rectional fusion performance of local and global features far
+beyond lightweight networks such as MobileNetV3. How-
+ever, it only uses a very small number of tokens, which is
+not conducive to semantic segmentation tasks.
+Method
+In this section, we introduce the lightweight parallel hetero-
+geneous network for semantic segmentation. The basic in-
+formation is first provivided on the replacement of semantic
+decoder by parallel heterogeneous network. Then, we intro-
+duce the modeling of pixel descriptions and semantic fre-
+quencies. Finally, the specific details and the computational
+overhead of parallel architectures are discussed.
+Parallel Heterogeneous Architecture
+The semantic decoder propagates the image semantics ob-
+tained by the encoder to each pixel and restores the lost de-
+tails in downsampling. A straightforward alternative is to
+extract image semantics in high resolution features, but it
+introduces a huge amount of computation, especially for vi-
+sion Transformers. In contrast, we propose a novel strategy
+to describe pixel semantic information with prototype se-
+mantics. For each stage, given a feature F ∈ RH×W ×C,
+we first initial a grid G ∈ Rh×w×C as a prototype of the
+image, where each point in G acts as a local cluster center,
+and the initial state simply contains information about the
+surrounding area. Here we use a 1 × C vector to represent
+the local semantic information of each point. For each spe-
+cific pixel, because the semantics of the surrounding pixels
+are not consistent, there are overlap semantics between each
+cluster centers. The cluster centers are weighted initialized
+in its corresponding area α2, and the initialization of each
+cluster center is expressed as:
+G(s) =
+n
+�
+i=0
+wixi
+(1)
+where n = α × α, wi denotes the weight of xi, and α is
+set to 3. Our purpose is to update each cluster center s in
+the grid G instead of updating the feature F directly. As
+h × w ≪ H × W, it greatly simplifies the computation.
+Here, we use a Transformer-based module as prototype
+learning to update each cluster center, which contains L lay-
+ers in total, and the updated center is denoted as G′(s). For
+each updated cluster center, we recover it by a pixel descrip-
+tor. Let F ′
+i denote the recovered feature, which contains not
+only the rich pixel semantics from F, but also the prototype
+semantics collected by the cluster centers G′(s). Since the
+cluster centers aggregate the semantics of surrounding pix-
+
+200 175 150 125 100
+75
+50
+25
+5
+10
+15
+20
+25
+30
+35
+40
+mIoU
+Filter Radius
+Filtered Image
+Figure 3: The effect of different frequency components on
+semantic segmentation. We use the cut-edge method Seg-
+former (Xie et al. 2021) to evaluate the impact of frequency
+components on semantic segmentation on the widely used
+ADE20K dataset (Zhou et al. 2017). The image is trans-
+formed into the frequency domain by a fast Fourier trans-
+form
+(Heideman, Johnson, and Burrus 1984), and high-
+frequency information is filtered out using a low-pass op-
+erator with a radius. Removing high-frequency components
+at different levels results the prediction performance drops
+significantly.
+els, resulting in the loss of local details, PD first models local
+details in F with pixel semantics. Specifically, F is projected
+to a low-dimensional space, establishing local relationships
+between pixels such that each local patch keeps a distinct
+boundary. Then G′(s) is embedded into F to restore to the
+original space feature F ′ through bilinear interpolation. Fi-
+nally, they are integrated through a linear projection layer.
+Prototype Learning by Adaptive Frequency Filter
+Motivation
+Semantic segmentation is an extremely com-
+plex pixel-level classification task that is prone to category
+confusion. The frequency representation can be used as a
+new paradigm of learning difference between categories,
+which can excavate the information ignored by human vi-
+sion (Zhong et al. 2022; Qian et al. 2020). As shown in
+Figure 3, humans are robust to frequency information re-
+moval unless the vast majority of frequency components are
+filtered out. However, the model is extremely sensitive to
+frequency information removal, and even removing a small
+amount would result in significant performance degrada-
+tion. It shows that for the model, mining more frequency
+information can enhance the difference between categories
+and make the boundary between each category more clear,
+thereby improving the effect of semantic segmentation.
+Since feature F contains rich frequency features, each
+cluster center in the grid G also collects these frequency in-
+formation. Motivated by the above analysis, extracting more
+beneficial frequencies in grid G helps to discriminate the
+attributes of each cluster. To extract different frequency fea-
+tures, the straightforward way is to transform the spatial do-
+main features into spectral features through Fourier trans-
+form, and use a simple mask filter in the frequency domain
+H Groups
+Dynamic Low-pass Filters
+N Groups
+…
+     
+     
+     
+     
+…
+Dynamic High-pass Filters
+Weight 
+Sharing
+Frequency 
+Aggregation
+ 
+Frequency Similarity Kernel
+       
+M Groups
+Aggregation
+Convolution
+Upsampling
+Figure 4: Structure of the adaptive frequency filter in pro-
+totype learning. The prototype as learnable local descrip-
+tion utilizes frequency component similarity kernel to en-
+hance different components while combining efficient and
+dynamic low-pass and high-pass filters to capture more fre-
+quency information.
+to enhance or attenuate the intensity of each frequency com-
+ponent of the spectrum. Then the extracted frequency fea-
+tures are converted to the spatial domain by inverse Fourier
+transform. However, Fourier transform and inverse trans-
+form bring in additional computational expenses, and such
+operators are not supported on many hardwares. Thus, we
+design an adaptive frequency filter block based on the vanilla
+vision Transformer from the perspective of spectral correla-
+tion to capture important high frequency and low frequency
+features directly in the spatial domain. The core components
+are shown in Figure 4 and the formula is defined as:
+AF F (X) = ||Dfc
+h (X)||H
+�
+��
+�
+corr.
++ ||Dlf
+m(X)||M + ||Dhf
+n (X)||N
+�
+��
+�
+dynamic filters
+,
+(2)
+where Dfc
+h , Dlf
+m(X) and Dhf
+n (X) denote the frequency
+similarity kernel with H groups to achieve frequency com-
+ponent correlation enhancement, dynamical low-pass filters
+with M groups and dynamical high-pass filters with N
+groups, respectively. || · || denotes concatenation. It is worth
+noting that these operators adopt a parallel structure to fur-
+ther reduce the computational cost by sharing weights.
+Frequency Similarity Kernel (FSK)
+Different frequency
+components distribute over in G, and our purpose is to se-
+lect and enhance the important components that helps se-
+mantic parsing. To this end, we design a frequency similar-
+ity kernel module. Generally, this module is implemented
+by the vision Transformer. Given a feature X ∈ R(hw)×C,
+with relative position encoding on G through a convolution
+layer (Wu et al. 2021). We first use a fixed-size similarity
+kernel A ∈ RC/H×C/H to represent the correspondence be-
+tween different frequency components, and select the impor-
+tant frequency components by querying the similarity ker-
+nel. We treat it as a function transfer that computes the keys
+K and values V of frequency components through a linear
+
+layer, and normalizes the keys across frequency components
+by a Softmax operation. Each component integrates a simi-
+larity kernel Ai,j, which is computed as:
+Ai,j = ekiv⊤
+j /
+n
+�
+j=1
+eki,
+(3)
+where ki represents the i-th frequency component in K,
+vj represents the j-th frequency component in V . We also
+transform the input X into the query Q through a linear
+layer, and obtain the component-enhanced output through
+interactions on the fixed-size similarity kernel.
+Dynamic Low-Pass Filters (DLF)
+Low-frequency com-
+ponents occupy most of the energy in the absolute image and
+represent most of the semantic information. A low-pass fil-
+ter allows signals below the cutoff frequency to pass, while
+signals above the cutoff frequency are obstructed. Thus, we
+employ typical average pooling as a low-pass filter. How-
+ever, the cutoff frequencies of different images are different.
+To this end, we control different kernels and strides in multi-
+groups to generate dynamic low-pass filters. For m-th group,
+we have:
+Dlf
+m(vm)) = B(Γs×s(vm)),
+(4)
+where B(·) represents bilinear interpolation and Γs×s de-
+notes the adaptive average pooling with the output size of
+s × s.
+Dynamic High-Pass Filters (DHF)
+High-frequency in-
+formation is crucial to preserve details in segmentation. As
+a typical high-pass operator, convolution can filter out irrel-
+evant low-frequency redundant components to retain favor-
+able high-frequency components. The high-frequency com-
+ponents determine the image quality and the cutoff fre-
+quency of the high-pass for each image is different. Thus,
+we divide the value V into N groups, resulting vn. For each
+group, we use a convolution layer with different kernels to
+simulate the cutoff frequencies in different high-pass filters.
+For the n-th group, we have:
+Dhf
+n (vn)) = Λk×k(vn),
+(5)
+where Λk×k denotes the depthwise convolution layer with
+kernel size of k ×k. In addition, we use the Hadamard prod-
+uct of query and high-frequency features to suppress high
+frequencies inside objects, which are noise for segmentation.
+FFN helps to fuse the captured frequency information, but
+owns a large amount of calculation, which is often ignored
+in lightweight designs. Here we reduce the dimension of the
+hidden layer by introducing a convolution layer to make up
+for the missing capability due to dimension compression.
+Discuss
+For the frequency similarity kernel, the compu-
+tational complexity is O(hwC2). The computational com-
+plexity of each dynamic high-pass filter is O(hwCk2),
+which is much smaller than that of frequency similarity
+kernel. Since the dynamic low-pass filter is implemented
+by adaptive mean pooling of each group, its computational
+complexity is about O(hwC). Therefore, the computational
+complexity of a module is linear with the resolution, which
+Table 1: Comparison to state of the art methods on
+ADE20K with resolution at 512 × 512. Here we use
+the
+Segformer
+as
+the
+baseline
+and
+report
+the
+per-
+centage growth. MV2=MobileNetV2, EN=EfficientNet,
+SV2=ShuffleNetV2.
+Model
+#Param.
+FLOPs
+mIoU
+FCN-8s
+9.8M
+39.6G
+19.7
+PSPNet (MV2)
+13.7M
+52.2G
+29.6
+DeepLabV3+ (MV2)
+15.4M
+25.8G
+38.1
+DeepLabV3+ (EN)
+17.1M
+26.9G
+36.2
+DeepLabV3+ (SV2)
+16.9M
+15.3G
+37.6
+Lite-ASPP
+2.9M
+4.4G
+36.6
+R-ASPP
+2.2M
+2.8G
+32.0
+LR-ASPP
+3.2M
+2.0G
+33.1
+HRNet-W18-Small
+4.0M
+10.2G
+33.4
+HR-NAS-A
+2.5M
+1.4G
+33.2
+HR-NAS-B
+3.9M
+2.2G
+34.9
+PVT-v2-B0
+7.6M
+25.0G
+37.2
+TopFormer
+5.1M
+1.8G
+37.8
+EdgeViT-XXS
+7.9M
+24.4G
+39.7
+Segformer (LVT)
+3.9M
+10.6G
+39.3
+Swin-tiny
+31.9M
+46G
+41.5
+Xcit-T12/16
+8.4M
+21.5G
+38.1
+ViT
+10.2M
+24.6G
+37.4
+PVT-tiny
+17.0M
+33G
+36.6
+Segformer
+3.8M
+8.4G
+37.4
+AFFormer-tiny
+1.6M(-58%)
+2.8G(-67%)
+38.7(+1.3)
+AFFormer-small
+2.3M(-41%)
+3.6G(-61%)
+40.2(+2.8)
+AFFormer-base
+3.0M(-21%)
+4.6G(-45%)
+41.8(+4.4)
+is advantageous for high resolution in semantic segmenta-
+tion.
+Experiments
+Implementation Details
+We validate the proposed AFFormer on three publicly
+datasets: ADE20K (Zhou et al. 2017), Cityscapes (Cordts
+et al. 2016) and COCO-stuff (Caesar, Uijlings, and Fer-
+rari 2018). We implement our AFFormer with the PyTorch
+framework base on MMSegmentation toolbox (Contributors
+2020). Follow previous works (Cheng, Schwing, and Kir-
+illov 2021; Zhao et al. 2017), we use ImageNet-1k to pre-
+train our model. During semantic segmentation training, we
+employ the widely used AdamW optimizer for all datasets
+to update the model parameters. For fair comparisons, our
+training parameters mainly follow the previous work (Xie
+et al. 2021). For the ADE20K and Cityscapes datasets, we
+adopt the default training iterations 160K in Segformer,
+where mini-batchsize is set to 16 and 8, respectively. For the
+COCO-stuff dataset, we set the training iterations to 80K and
+the minibatch to 16. In addition, we implement data augmen-
+tation during training for ADE20K, Cityscapes, COCO-stuff
+by random horizontal flipping, random resizing with a ratio
+of 0.5-2.0, and random cropping to 512×512, 1024×1024,
+512 × 512, respectively. We evaluate the results with mean
+Intersection over Union (mIoU) metric.
+
+Table 2: Comparison to state of the art methods on
+Cityscapes val set. The FLOPs are test on the resolution of
+1024 × 2048. Meanwhile, we also report the percentage in-
+crease compared to Segformer.
+Model
+#Param.
+FLOPs
+mIoU
+FCN
+9.8M
+317G
+61.5
+PSPNet (MV2)
+13.7M
+423G
+70.2
+DeepLabV3+ (MV2)
+15.4M
+555G
+75.2
+SwiftNetRN
+11.8M
+104G
+75.5
+EncNet
+55.1M
+1748G
+76.9
+Segformer
+3.8M
+125G
+76.2
+AFFormer-tiny
+1.6M(-58%) 23.0G(-82%)
+76.5(+0.3)
+AFFormer-small
+2.3M(-41%) 26.2G(-79%)
+77.6(+1.4)
+AFFormer-base
+3.0M(-21%) 34.4G(-73%)
+78.7(+2.5)
+Comparisons with Existing Works
+Results on ADE20K Dataset.
+We compare our AF-
+Former with top-ranking semantic segmentation methods,
+including CNN-based and vision Transformer-based mod-
+els. Following the inference settings in (Xie et al. 2021), we
+test FLOPs at 512×512 resolution and show the single scale
+results in Table 1. Our model AFFormer-base improves by
+5.2 mIoU under the same computing power consumption as
+Lite-ASPP, reaching 41.8 mIoU. At the same time, by reduc-
+ing the number of layers and channels, we obtain AFFormer-
+tiny and AFFormer-small versions to adapt to different com-
+puting power scenarios. For the lightweight and efficient
+Segformer (8.4 GFLOPs),our base version (4.6 GFLOPs)
+also gain 4.4 mIoU using half the computing power and
+the tiny version (2.4 GFLOPs) with only 1/4 the computing
+power improving 1.3 mIoU. Only 1.8 GFLOPs are needed
+for the lighter topformer, but our base version has 2.1M less
+parameters (5.1M vs 3M) with 4.0 higher mIoU.
+Results on Cityscapes Dataset.
+Table 2 shows the results
+of our model and the cutting-edge methods on Cityscapes.
+Although the Segformer is efficient enough, due to its
+square-level complexity, we only use 30% of the compu-
+tational cost to reach 78.7 mIoU, which is 2.5 mIoU im-
+provement with a 70% reduction in FLOPs. Meanwhile, we
+report the results at different high resolutions in Table 3. At
+the short side of {512, 640, 768, 1024}, the computational
+cost of our model is 51.4%, 57.5%, 62.5% and 72.5% of
+that of Segformer, respectively. Meanwhile, the mIoU are
+improved by 1.6, 1.9, 1.2 and 2.5, respectively. The higher
+the input resolution, the more advantageous of our model in
+both computational cost and accuracy.
+Results on COCO-stuff Dataset.
+COCO-stuff dataset
+contains a large number of difficult samples that collected
+in COCO. As show in Table 4, although complex decoders
+(e.g., PSPNet, DeepLabV3+) can achieve better results than
+LR-ASPP (MV3), they bring a lot of computational cost.
+Our model achieves an accuracy of 35.1 mIoU while only
+taking 4.5 GFLOPs, achieving the best trade-off.
+Ablation Studies
+All the ablation studies are conducted on ADE20K dataset
+with AFFormer-base unless otherwise specified.
+Rationalization of Parallel Structures.
+Parallel architec-
+ture is the key to removing the decoder head and ensuring
+accuracy and efficiency. We first adjust the proposed struc-
+ture to a naive pyramid architecture (denoted as “w/o PD”)
+and a ViT architecture (denoted as “w/o PL”) to illustrate the
+advantages of the parallel architecture. Specifically, the “w/o
+PD” means removing PD module and keeping only PL mod-
+ule, while the “w/o PL” does the opposite. As shown in Ta-
+ble 5, the setting “w/o PD” reduces 2.6 mIoU due to the lack
+of high-resolution pixel semantic information. The “w/o PL”
+structure without the pyramid structure has a significant re-
+duction in accuracy due to few parameters and lack of rich
+image semantic information. It also demonstrates that our
+parallel architecture can effectively combine the advantages
+of both architectures.
+Advantages of Heterogeneous Structure.
+The purpose
+of the heterogeneous approach is to further reduce the com-
+putational overhead. The PL module is adopted to learn
+the prototype representation in the clustered features, and
+then use PD to combine the original features for restoration,
+which avoids direct calculation on the high-resolution origi-
+nal features and reduce the computational cost. It can be seen
+from Table 6 that when the parallel branch is adjusted to the
+pixel description module (denote as “All PD”), which means
+that the prototype representation is learned by PD module.
+The model size is only 0.6M, and the FLOPs are reduced by
+2.5G, but the accuracy is reduced by 14.3 mIoU. This is due
+to the PD lacks the ability to learn great prototype represen-
+tations. In contrast, after we replace the PD module with the
+PL module (denote as “All PL”), the FLOPs are increased
+by 2.4G, but there is almost no difference in accuracy. We
+believe that the PD module is actually only a simple way to
+restore the learned prototype, and the relatively complex PL
+module saturates the model capacity.
+Advantages of Adaptive Frequency Filter.
+We use two
+datasets with large differences, including ADE20K and
+Cityscapes, to explore the core components in adaptive fre-
+quency filter module. The main reason is that the upper limit
+of the ADE20K dataset is only 40 mIoU, while the upper
+limit of the Cityscapes is 80 mIoU. The two datasets have
+different degrees of sensitivity to different frequencies. We
+report the benefits of each internal component in the Table 7.
+We find that DHF alone outperforms DLF, especially on the
+Cityscapes dataset by 2.6 mIoU, while FSK is significantly
+higher than DLF and DHF on ADE20K. This shows that
+ADE20K may be more inclined to an intermediate state be-
+tween high frequency and low frequency, while Cityscapes
+needs more high frequency information. The combined ex-
+periments show that the combination of the advantages of
+each component can stably improve the results of ADE20K
+and Cityscapes.
+Frequency Statistics Visualization.
+We first count the
+characteristic frequency distribution of different stages, as
+shown in Figure 5. It can be found that the curves of G2
+and F2 almost overlap, indicating that the frequencies after
+clustering are very similar to those in the original features.
+The same goes for G3 and F3. Whereas, the learned proto-
+
+Table 3: Speed-accuracy tradeoffs at different scales on Cityscapes.
+Model
+size
+FLOPs
+mIoU
+Segformer (3.8M)
+512 × 1024
+17.7G
+71.9
+AFFormer-base (3.0M)
+512 × 1024
+8.6G(-51.4%)
+73.5(+1.6)
+Segformer (3.8M)
+640 × 1280
+31.5G
+73.7
+AFFormer-base (3.0M)
+640 × 1280
+13.4G(-57.5%)
+75.6(+1.9)
+Segformer (3.8M)
+768 × 1536
+51.7G
+75.3
+AFFormer-base (3.0M)
+768 × 1536
+19.4G(-62.5%)
+76.5(+1.2)
+Segformer (3.8M)
+1024 × 2048
+125G
+76.2
+AFFormer-base (3.0M)
+1024 × 2048
+34.4G(-72.5%)
+78.7(+2.5)
+Table 4: Comparison to state of the art meth-
+ods on COCO-stuff. We use a single-scale
+results at the input resolution of 512 × 512.
+MV3=MobileNetV3
+Model
+#Param.
+FLOPs
+mIoU
+PSPNet (MV2)
+13.7M
+52.9G
+30.1
+DeepLabV3+ (MV2)
+15.4M
+25.9G
+29.9
+DeepLabV3+ (EN)
+17.1M
+27.1G
+31.5
+LR-ASPP (MV3)
+–
+2.37G
+25.2
+AFFormer-base
+3.0M
+4.6G
+35.1
+Table 5: Ablation studies on the parallel structure.
+Setting
+#Param.
+FLOPs
+mIoU
+w/o PD
+2.78G
+2.98G
+39.2
+w/o PL
+0.42G
+1.65G
+19.5
+Parallel
+3.0G
+4.6G
+41.8
+Table 6: Ablation studies on heterogeneous architecture.
+Setting
+#Param.
+FLOPs
+mIoU
+All PD
+0.6M
+1.85G
+27.4
+All PL
+3.6M
+7.0G
+41.6
+Heterogeneous
+3.0M
+4.6G
+41.8
+Table 7: Ablation studies on frequency aware statistics.
+Setting
+#Param.
+FLOPs
+ADE20K
+Cityscapes
+DLF
+2.4M
+3.6G
+38.7
+75.7
+DHF
+2.6M
+3.9G
+39.3
+78.3
+FSK
+2.9M
+4.2G
+40.5
+75.3
+DLF + DHF
+2.7M
+3.9G
+41.1
+77.8
+DLF + FSK
+2.8M
+4.2G
+40.0
+76.2
+DHF + FSK
+2.9M
+4.3G
+41.2
+77.3
+Whole
+3.0M
+4.6G
+41.8
+78.7
+type representation after frequency adaptive filtering signifi-
+cantly improves the contained frequency information. After
+PD restoration, different frequency components can be em-
+phasized in different stages. As shwon in Figure 6, we also
+analyze the frequency effects of the core components in the
+AFF module. As expected, DLF and DHF show strong low-
+pass and high-pass capabilities, respectively, as FSK does.
+At the same time, we also found that the important frequency
+components screened and enhanced by FSK are mainly con-
+centrated in the high frequency part, but the frequency signal
+is more saturated than that of DHF. This also shows that the
+high-frequency component part is particularly important in
+the semantic segmentation task, because it emphasizes more
+on the boundary details and texture differences between ob-
+jects. Meanwhile, according to the analysis in Table 7 (the
+effects of ADE20K and Cityscapes have been steadily im-
+proved), each core component has its own advantages, and
+the AFF module shows strong robustness in various types
+and complex scenes.
+Speed and Memory Costs.
+Meanwhile, we report the
+speed on the Cityscapes dataset in Table 8. We can find that
+the proposed model improves by 10 FPS and performs much
+better than Segformer on such high-resolution Cityscapes
+images.
+Table 8: The FPS is tested on a V100 NVIDIA GPU with a
+batch size of 1 on the resolution of 1024x2048.
+Model
+FPS
+mIoU
+Segformer
+12
+76.2
+AFFormer
+22
+78.7
+Figure 5: Frequency analysis of stage-2 (left) and stage-3
+(right).
+Input
+DHF
+DLF
+FSK
+DHF
+Input
+DLF
+FSK
+Figure 6: Frequency analysis of the core components in PL
+module.
+Conclusion
+In this paper, we propose AFFormer, a head-free lightweight
+semantic segmentation specific architecture. The core is to
+learn the local description representation of the clustered
+prototypes from the frequency perspective, instead of di-
+rectly learning all the pixel embedding features. It removes
+the complicated decoder while having linear complexity
+Transformer and realizes semantic segmentation as simple
+as regular classification. The various experiments demon-
+strate that the AFFormer owns powerful accuracy and great
+stability and robustness at low computational cost.
+
+8.0
+7.0
+Log amplitude
+6.0
+5.0
+4.0
+3.0
+2.0
+0.0l
+0.2πl
+0.4π
+0.6㎡l
+0.8Tl
+1.0l
+Frequency8.0
+7.0
+6.0
+ Log amplitude
+5.0
+4.0
+3.0
+2.0
+1.0
+0.0
+0.0l
+0.2 
+0.4
+0.6
+0.8
+1.0π
+Frequency0
+5
+10
+15
+20
+25
+30
+0
+5 
+10
+15
+20
+25
+304.0
+2.0
+0.0
+-2.0
+-4.0
+-6.0-
+0.0
+0.2π
+0.4π
+0.6π
+0.8π
+1.0π
+Frequency0
+5
+10
+15
+20
+25
+30
+5 
+10
+15
+20
+25
+300
+5
+10
+15
+20
+25
+30
+0
+5 
+10
+15
+20
+25 -
+300
+5
+10
+15
+20
+25
+30
+0
+5 -
+10
+15
+20
+25
+30Acknowledgements
+This work was supported by Alibaba Group through Alibaba
+Research Intern Program.
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+DiffSTG: Probabilistic Spatio-Temporal Graph Forecasting
+with Denoising Diffusion Models
+Haomin Wen 1 Youfang Lin 1 Yutong Xia 2 Huaiyu Wan 1 Roger Zimmermann 2 Yuxuan Liang � 2
+Abstract
+Spatio-temporal graph neural networks (STGNN)
+have emerged as the dominant model for spatio-
+temporal graph (STG) forecasting. Despite their
+success, they fail to model intrinsic uncertainties
+within STG data, which cripples their practicality
+in downstream tasks for decision-making. To this
+end, this paper focuses on probabilistic STG fore-
+casting, which is challenging due to the difficulty
+in modeling uncertainties and complex ST depen-
+dencies. In this study, we present the first attempt
+to generalize the popular denoising diffusion prob-
+abilistic models to STGs, leading to a novel non-
+autoregressive framework called DiffSTG, along
+with the first denoising network UGnet for STG
+in the framework. Our approach combines the
+spatio-temporal learning capabilities of STGNNs
+with the uncertainty measurements of diffusion
+models. Extensive experiments validate that Diff-
+STG reduces the Continuous Ranked Probabil-
+ity Score (CRPS) by 4%-14%, and Root Mean
+Squared Error (RMSE) by 2%-7% over existing
+methods on three real-world datasets.
+1. Introduction
+Humans enter a world that is inherently structured, in which
+a myriad of elements interact with each other both spatially
+and temporally, resulting in a spatio-temporal composition.
+Spatio-Temporal Graph (STG) is the de facto most popular
+tool for injecting such structural information into the formu-
+lation of practical problems, especially in smart cities. In
+this paper, we focus on the problem of STG forecasting, i.e.,
+predicting the future signals generated on a graph given its
+historical observations, such as traffic prediction (Li et al.,
+2018), weather forecasting (Simeunovi´c et al., 2021), and
+taxi demand estimation (Yao et al., 2018). To facilitate
+understanding, a sample illustration is given in Figure 1(a).
+1School of Computer and Information Technology, Beijing Jiao-
+tong University, Beijing, China 2School of Computing, National
+University of Singapore, Singapore. Correspondence to: Yuxuan
+Liang <yuxliang@outlook.com>.
+Preprint. Under review.
+STG Forecasting
+2
+Problem Definition
+h
+T
+V D
+h
+X
+ 
+
+p
+T
+V D
+p
+X
+ 
+
+STG Model
+➢ Given the historical spatial-temporal graph (STG) to predict the future STG.
+➢ Stochastic Prediction
+Problem
+Definition
+Related Work
+Motivation
+Solution
+Experiment
+Stochastic STG Forecasting    
+t
+Prediction
+(a) Spatio-Temporal Graph
+(b) Probabilistic Prediction
+Time
+1t
+2t
+3t
+…
+…
+Space
+Forecast
+History
+Figure 1. Illustration of probabilistic STG forecasting.
+Recent techniques for STG forecasting are mostly determin-
+istic, calculating future graph signals exactly without the
+involvement of randomness. Spatio-Temporal Graph Neural
+Networks (STGNN) have emerged as the dominant model
+in this research line. They resort to GNNs for modeling spa-
+tial correlations among nodes, and Temporal Convolutional
+Networks (TCN) or Recurrent Neural Networks (RNN) for
+capturing temporal dependencies. Though promising, these
+deterministic approaches still fall short of handling uncer-
+tainties within STGs, which considerably trims down their
+practicality in downstream tasks for decision-making. For
+example, Figure 1(b) depicts the prediction results of passen-
+ger flows in a metro station. In the black box, the determin-
+istic method cannot provide the reliability of its predictions.
+Conversely, the probabilistic method renders higher uncer-
+tainties (see the green shadow), which indicates a potential
+outbreaking of passenger flows in that region. By knowing a
+range of possible outcomes we may experience and the like-
+lihood of each, the traffic system is able to take operations
+in advance for public safety management.
+While prior endeavors on stochastic STG forecasting were
+conventionally scarce, we are witnessing a blossom of prob-
+abilistic models for time series forecasting (Rubanova et al.,
+2019; Salinas et al., 2020; Rasul et al., 2021). Denoising
+Diffusion Probabilistic Models (DDPM) (Ho et al., 2020)
+are one of the most prevalent methods in this stream, whose
+key insight is to produce the future samples by gradually
+transforming a noise into a plausible prediction through a
+denoising process. Unlike vanilla unconditional DDPMs
+that were originally designed for image generation, such
+transformation function between consecutive steps is condi-
+tioned on the historical time series readings. For example,
+TimeGrad (Rasul et al., 2021) sets the LSTM-encoded rep-
+resentation of the current time series as the condition, and
+arXiv:2301.13629v1  [cs.LG]  31 Jan 2023
+
+100
+groud-truth
+06
+deterministic
+80 -
+probabilistic
+70
+60 -
+50-
+40-
+30 -
+20
+10
+0
+5
+10
+15
+20Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+estimates the future regressively. CSDI (Tashiro et al., 2021)
+directly utilizes observed values as the condition to model
+the data distribution.
+However, the above probabilistic time series models are still
+insufficient for modeling STGs. Firstly, they only model
+the temporal dependencies within a single node, without
+capturing the spatial correlations between different nodes.
+In reality, objects are correlated with each other spatially, for
+example, nearby sensors in a road network tend to witness
+similar traffic trends. Failing to encode such spatial depen-
+dencies will drastically deteriorate the predictive accuracy
+(Yu et al., 2018; Wu et al., 2019). Secondly, the training and
+inference of existing probabilistic time series models, e.g.,
+Latent ODE (Rubanova et al., 2019) and TimeGrad, suffer
+notorious inefficiency due to their sequential nature, thereby
+posing a hurdle to long-term forecasting.
+To address these issues, we generalize the popular DDPMs
+to spatio-temporal graphs for the first time, leading to a
+novel framework called DiffSTG, which couples the spatio-
+temporal learning capabilities of STGNNs with the uncer-
+tainty measurements of DDPMs. Targeting the first chal-
+lenge, we devise a simple yet effective module (UGnet) as
+the denoising network of DiffSTG. As its name suggests,
+UGnet leverages a Unet-based architecture (Ronneberger
+et al., 2015) to capture multi-scale temporal dependencies
+and GNN to model spatial correlations. Compared to ex-
+isting denoising networks in standard DDPMs, our UGnet
+performs more accurate denoising in the reverse process
+by virtue of capturing ST dependencies. To overcome the
+second issue, our DiffSTG produces future samples in a
+non-autoregressive fashion. In other words, our framework
+efficiently generates multi-horizon predictions all at once,
+rather than producing them step by step as what TimeGrad
+did. In summary, our contributions lie in three aspects:
+• We hit the problem of probabilistic STG forecasting from
+a score-based diffusion perspective with the first shot. Our
+DiffSTG can effectively model the complex ST dependen-
+cies and intrinsic uncertainties within STG data.
+• We develop a novel denoising network called UGNet
+dedicated to STGs for the first time. It contributes as a
+new and powerful member of DDPMs’ denoising network
+family for modeling ST-dependencies in STG data.
+• We empirically show that DiffSTG reduces the Continu-
+ous Ranked Probability Score (CRPS) by 4%-14%, and
+Root Mean Squared Error (RMSE) by 2%-7% over exist-
+ing probabilistic methods on three real-world datasets.
+The rest of this paper is organized as follows. We delineate
+the concepts of DDPM in Section 2. The formulation and
+implementation of the proposed DiffSTG are detailed in Sec-
+tion 3 and 4, respectively. We then examine our framework
+and present the empirical findings in Section 5. Lastly, we
+introduce related arts in Section 6 and conclude in Section 7.
+2. Denoising Diffusion Probabilistic Models
+Given samples from a data distribution q(x0), Denoising
+Diffusion Probabilistic Models (DDPM) (Ho et al., 2020)
+are unconditional generative models aiming to learn a model
+distribution pθ(x0) that approximates q(x0) and is easy to
+sample from. Let xn for n = 1, · · · , N be a sequence
+of latent variables from the same sample space of x0 (de-
+noted as X). DDPM are latent variable models of the form
+pθ(x0) =
+�
+pθ(x0:N)dx1:N. It contains two processes,
+namely the forward process and the reverse process.
+Forward process. The forward process is defined by a
+Markov chain which progressively adds Gaussian noise to
+the observation x0:
+q(x1:N|x0) =
+N
+�
+n=1
+q(xn|xn−1),
+(1)
+where q(xn|xn−1) is a Gaussian distribution as
+q(xn|xn−1) = N(xn;
+�
+1 − βnxn−1, βnI),
+(2)
+and {β1, · · · , βN} is an increasing variance schedule with
+βn ∈ (0, 1) that represents the noise level at forward step n.
+Unlike typical latent variable models such as the variational
+autoencoder (Rezende et al., 2014), the approximate pos-
+terior q(x1:N|x0) in diffusion probabilistic models is not
+trainable but fixed to a Markow chain depicted by the above
+Gaussian transition process.
+Let ˆαn = 1 − βn and αn = �N
+n=1 ˆαn be the cumulative
+product of ˆαn, a special property of the forward process is
+that the distribution of xn given x0 has a close form:
+q(xn|x0) = N(xn; √αnx0, (1 − αn)I),
+(3)
+which can also be expressed as xn = √αnx0 + √1 − αnϵ
+by the reparameteriztioin trick (Kingma & Welling, 2013),
+with ϵ ∈ N(0; I) as a sampled noise. The above property
+allows us to directly sample xn at any arbitrary noise level
+n, instead of computing the forward process step by step.
+Reverse process. The reverse process denoises xN to re-
+cover x0 recurrently. It also follows a Markov chain but
+with learnable Gaussian transitions starting with p(xN) =
+N(xN; 0, I), which is defined as
+pθ(x0:N) = p(xN)
+1
+�
+n=N
+pθ(xn−1|xn).
+(4)
+Then, the transition between two nearby latent variables is
+denoted by
+pθ(xn−1|xn) = N(xn−1; µ��(xn, n), σθ(xn, n)),
+(5)
+with shared parameters θ. Here we choose the same param-
+eterization of pθ(xn−1|xn) as in (Ho et al., 2020) in light
+
+Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+of its promising performance on image generation:
+µθ(xn, n) = 1
+αn
+�
+xn −
+βn
+√1 − αn
+ϵθ (xn, n)
+�
+,
+(6)
+σθ(xn, n) = 1 − αn−1
+1 − αn
+βn,
+(7)
+where ϵθ(X ×R) → X is a trainable denoising function that
+decides how much noise should be removed at the current
+denoising step. The parameters θ are learned by solving the
+following optimization problem:
+min
+θ
+L(θ) = min
+θ
+Ex0∼q(x0),ϵ∼N (0,I),n ∥ϵ − ϵθ (xn, n)∥2
+2 .
+Since we already know x0 in the training stage, and recall
+that xn = √αnx0 + √1 − αnϵ by the property as men-
+tioned in the forward process, the above training objective
+of unconditional generation can be specified as
+min
+θ
+L(θ) = min
+θ
+E
+��ϵ − ϵθ
+�√αnx0 +
+√
+1 − αnϵ, n
+���2
+2 .
+(8)
+This training objective can be viewed as a simplified ver-
+sion of loss similar to the one in Noise Conditional Score
+Networks (Song & Ermon, 2019; 2020). Once trained,
+we can sample x0 from Eq. (4) and Eq. (5) starting from
+the Guassian noise xN. This reverse process resembles
+Langevin dynamics, where we first sample from the most
+noise-perturbed distribution and then reduce the noise scale
+step by step until we reach the smallest one. We provide
+details of DDPM in Appendix A.1.
+3. DiffSTG Formulation
+Let G = {V, E, A} represent a graph with V nodes, where
+V, E are the node set and edge set, respectively. A ∈
+RV ×V is a weighted adjacency matrix to describe the graph
+topology. For V = {v1, . . . , vV }, let xt ∈ RF ×V denote
+F-dimentional signals generated by the V nodes at time t.
+Given historical graph signals xh = [x1, · · · , xTh] of Th
+time steps and the graph G as inputs, STG forcasting aims
+at learning a function F to predict future graph signals xp,
+formulated as:
+F : (xh; G) → [xTh+1, · · · , xTh+Tp] := xp,
+(9)
+where Tp is the forecasting horizon. In this study, we focus
+on the task of probabilistic STG forecasting, which aims to
+estimate the distribution of future graph signals.
+As introduced in Section 1, on the one hand, current de-
+terministic STGNNs are capable of capturing the spatial-
+temporal correlation in STG data, while failing to model the
+uncertainty of the prediction. On the other hand, diffusion-
+based probabilistic time series forecasting models (Rasul
+et al., 2021; Tashiro et al., 2021) have powerful abilities
+in learning high-dimensional sequential data distributions,
+while incapable of capturing spatial dependencies and facing
+efficiency problems when applied to STG data.
+To this end, we generalize the popular DDPM to spatio-
+temporal graphs and present out a novel framework called
+DiffSTG for probabilistic STG forecasting in this section.
+DiffSTG couples the spatio-temporal learning capabilities
+of STGNNs with the uncertainty measurements of diffusion
+models.
+3.1. Conditional Diffusion Model
+The original DDPM is designed to generate an image from
+a white noise without condition, which is not aligned with
+our task where the future signals are generated conditioned
+on their histories. Therefore, for STG forecasting, we first
+extend the DDPM to a conditional one by making a few
+modifications to the reverse process. In the unconditional
+DDPM, the reverse process pθ(x0:N) in Eq. (4) is used to
+calculate the final data distribution q(x0). To get a con-
+ditional diffusion model for our task, a natural approach
+is adding the history xh and the graph structure G as the
+condition in the reverse process in Eq. (4). In this way, the
+conditioned reverse diffusion process can be expressed as
+pθ(xp
+0:N|xh, G) = p(xp
+N)
+1
+�
+n=N
+pθ(xp
+n−1|xp
+n, xh, G).
+(10)
+The transition probability of two latent variables in Eq. (5)
+can be extended as
+pθ(xp
+n−1|xp
+n, xh, G)
+= N(xp
+n−1; µθ(xp
+n, n|xh, G), σθ(xp
+n, n|xh, G)).
+(11)
+Furthermore, the training objective in Eq. (8) can be rewrit-
+ten as a conditional one:
+min
+θ
+L(θ) = min
+θ
+Exp
+0,ϵ
+��ϵ − ϵθ
+�
+xp
+n, n|xh, G
+���2
+2 .
+(12)
+3.2. Generalized Conditional Diffusion Model
+In Eq. (10)-(12), the condition xh and denoising target xp
+are separated into two sample space xh ∈ X h and xp ∈ X p.
+However, they are indeed extracted from two consecutive
+periods. Here we propose to consider the history xh and fu-
+ture xp as a whole, i.e., xall = [xh, xp] ∈ RF ×V ×T , where
+T = Th + Tp. The history can be represented by masking
+all future time steps in xall, denoted by xall
+msk. So that the
+condition xall
+msk and denoise target xall share the same sam-
+ple space X all. Thus, the masked version of Eq. (10) can be
+rewritten as
+pθ(xall
+0:N|xall
+msk, G) = p(xall
+N )
+1
+�
+n=N
+pθ(xall
+n−1|xall
+n , xall
+msk, G).
+(13)
+
+Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+The masked version of Eq. (12) can be rewritten as
+min
+θ
+L(θ) = min
+θ
+Exall
+0 ,ϵ
+���ϵ − ϵθ
+�
+xall
+n , n|xall
+msk, G
+����
+2
+2 .
+(14)
+Compared with the formulation in Eq. (10)-(12), this new
+formulation is a more generalized one which has the fol-
+lowing merits. Firstly, the loss in Eq. (14) unifies the recon-
+struction of the history and estimation of the future, thus the
+historical data can be fully utilized to model the data distri-
+bution. Secondly, the new formulation unifies various STG
+tasks in the same framework, including STG prediction,
+generation, and interpolation (Li & Revesz, 2004).
+Training. In the training process, given the conditional
+masked information xall
+msk, graph G and the target xall
+0 , we
+sample noise targets xall
+n
+= √αnxall
+0
++ √1 − αnϵ, and
+then train ϵθ by the loss function in Eq. (14). The training
+procedure of DiffSTG is presented in Algorithm 1.
+Algorithm 1 Training of DiffSTG
+1: Input: distribution of training data q(xall
+0 ), number of diffu-
+sion step N, variance schedule {β1, · · · , βN}, graph G.
+2: Output: Trained denoising function ϵθ
+3: repeat
+4:
+n ∼ Uniform({1, · · · , N}), xall
+0
+∼ q(xall
+0 )
+5:
+Constructing the masked signals xall
+msk according to ob-
+served values
+6:
+Sample ϵ ∼ N(0, I) where ϵ’s dimension corresponds to
+xall
+0
+7:
+Calculate noisy targets xall
+n = √αnxall
+0 + √1 − αnϵ
+8:
+Take gradient step ∇θ∥ϵ − ϵθ(xall
+n , n|xall
+msk, G))∥2
+2 accord-
+ing to Eq. (14)
+9: until converged
+Inference. As outlined in Algorithm 2, the inference pro-
+cess utilizes the trained denoising function ϵθ to sample
+xall
+n−1 step by step according to Eq. (13), under the guidance
+of xall
+msk and G.
+Algorithm 2 Sampling of DiffSTG
+1: Input: Historical graph signal xh, graph G, trained denoising
+function ϵθ
+2: Output: Future forecasting xp
+3: Construct xall
+msk according to xh
+4: Sample ϵ ∼ N(0, I) where ϵ’s dimension corresponds to
+xall
+msk
+5: for n = N to 1 do
+6:
+Sample xall
+n−1 using Eq. (13) by taking xall
+msk and G as
+condition
+7: end for
+8: Take out the forecast target in xall
+0 , i.e., xp
+9: Return xp
+4. DiffSTG Implementation
+After introducing the DiffSTG’s formulation, we implement
+it via elaborately-designed model architecture, which is
+illustrated in Figure 2. At the heart of the model is the pro-
+posed denoising network (UGnet) ϵθ in the reverse diffusion
+process, which performs accurate denoising with the ability
+to effectively model ST dependencies in the data.
+4.1. Denoising Network: UGnet
+Denoising network ϵθ in previous works can be mainly
+classified into two classes, Unet-based architecture (Ron-
+neberger et al., 2015) for image-related tasks (Rombach
+et al., 2022; Voleti et al., 2022; Ho et al., 2020), and
+WaveNet-based architecture (van den Oord et al., 2016) for
+sequence-related tasks (Kong et al., 2020; Liu et al., 2022b;
+Kim et al., 2020). These networks consider the input as ei-
+ther grids or segments, lacking the ability to capture spatio-
+temporal correlations in STG data. To bridge this gap, we
+propose a new denoising network ϵθ(X all × R|X all
+msk, G) →
+X all, named UGnet. It adopts an Unet-like architecture in
+the temporal dimension to capture temporal dependencies
+at different granularities (e.g., 15 minutes or 30 minutes),
+and utilizes GNN to model the spatial correlations.
+Specifically, as shown in Figure 2, UGnet takes xall
+msk, xall
+n ,
+n, G as inputs, and outputs the denoised noise ϵ. It first
+concatenates xall
+n ∈ RF ×V ×T and xall
+msk ∈ RF ×V ×T in the
+temporal dimension to form a new tensor �xall
+n ∈ RF ×V ×2T ,
+which is projected to a high-dimensional representation
+H ∈ RC×V ×2T by a linear layer, where C is the projected
+dimension. Then H is fed into several Spatio-temporal
+Residual Blocks (ST-Residual Blocks for short), with each
+capturing temporal dependencies and spatial dependencies,
+respectively. Let Hi ∈ RC×V ×Ti (where H0 = H) denote
+the input of the i-th ST-Residual Block, where Ti is the
+length of time dimension.
+Temporal Dependency Modeling. As shown in Figure 7,
+at each ST-Residual Block, Hi is fed into a Temporal Con-
+volution Network (TCN) (Bai et al., 2018) for modeling
+temporal dependence, which is a 1-D gated causal convo-
+lution of K kernel size with padding to get the same shape
+with input. The convolution kernel ΓT ∈ RK×Ct
+in×Ct
+out
+maps the input Hi to outputs Pi, Qi ∈ RCt
+out×V ×Ti with
+the same shape. Formally, the temporal gated convolution
+can be defined as
+ΓT (Hi) = Pi ⊙ σ(Qi) ∈ RCt
+out×V ×Ti,
+(15)
+where ⊙ is the element-wise Hadamard product, and σ is
+the sigmoid activation function. The item σ(Qi) can be
+considered a gate that filters the useful information of Pi
+into the next layer. We denote the output of TCN as Hi.
+Spatial Dependency Modeling. Graph Convolution Net-
+works (GCNs) are generally employed to extract highly
+meaningful features in the space domain (Zhou et al., 2020).
+The graph convolution can be formulated as
+ΓG(Hi) = σ
+�
+Φ
+�
+Agcn, Hi
+�
+Wi
+�
+,
+(16)
+
+Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+UGnet
+Solution：Model V3: Masked Conditional STG Diffusion for Prediction and Interpolation
+Problem
+Definition
+Related Work
+all
+all
+1
+(
+|
+)
+n
+n
+q x
+x −
+all
+all
+ll
+ms
+1
+a
+k
+(
+|
+,
+, )
+n
+n
+p
+x
+x
+x
+
+−
+Forward Diffusion Process
+Time
+…
+…
+Space
+…
+…
+…
+…
+all
+msk
+(
+, )
+x
+Time
+…
+…
+Space
+Reverse Denoising Diffusion Process
+…
+Noise Schedule
+…
+…
+…
+Conditional Noise 
+Predictor UGnet
+all 
+nx
+condition:
+all
+0x
+all
+1
+nx −
+all
+nx
+all
+N
+x
+DiffSTG
+Time
+…
+…
+Space
+Forecast
+History
+Time
+…
+…
+Space
+…
+…
+…
+…
+…
+…
+n
+
+G
++
+Temporal Conv
+Temporal Conv
+Graph Conv
+Layer Norm
+Up/Down-Sample
+emb
++
+ST-Residual Block
+n
+all 
+nx
+all
+msk
+x
+Concatenate
+ST-Residual Block
+ST-Residual Block
+ST-Residual
+Block
+ST-Residual Block
+ST-Residual Block
+FC
+H
+i
+C V T
+i
+ 
+
+e( )
+n
+Figure 2. Illustration of proposed DiffSTG and denoising network UGnet.
+where Wi ∈ RCg
+in×Cg
+in denotes a trainable parameter and σ
+is an activation function. Φ(·) is an aggregation function that
+decides the rule of how neighbors’ features are aggregated
+into the target node. In this work, we do not focus on
+developing the function Φ(·). Instead, we use the form
+in the most popular vanilla GCN (Kipf & Welling, 2017)
+that defines a symmetric normalized summation function
+as Φgcn
+�
+Agcn, Hi
+�
+= AgcnHi, where Agcn = D− 1
+2 (A +
+I)D− 1
+2 ∈ RV ×V is a normalized adjacent matrix of graph G.
+I is the identity matrix and D is the diagonal degree matrix
+with Dii = �
+j(A + I)ij. Note that we reshape the output
+of the TCN layer to Hi ∈ RV ×Cg
+in, where Cg
+in = Ti × Ct
+out,
+and fed this node feature Hi to GCN.
+Noise Level Embedding.
+As shown in the right part
+of Figure 2, like previous diffusion-based models (Rasul
+et al., 2021), we use positional encodings of the noise level
+n ∈ [1, N] and process it using a transformer positional
+embedding (Vaswani et al., 2017):
+e(n) =
+�
+. . . , cos
+�
+n/r
+−2d
+D
+�
+, sin
+�
+n/r
+−2d
+D
+�
+, . . .
+�T
+,
+(17)
+where d = 1, · · · , D/2 is the dimension number of the
+embedding (set to 32), and r is a large constant 10000. For
+more details about UGnet, please refer to Appendix A.2.
+4.2. Sampling Acceleration
+From a variational perspective, a large N (e.g., N = 1000
+in (Ho et al., 2020)) allows results of the forward process
+to be close to a Gaussian distribution so that the reverse de-
+noise process started with Gaussian distribution becomes a
+good approximation. However, large N makes the sampling
+low-efficiency since all N iterations have to be performed
+sequentially. To accelerate the sampling process, we adopt
+the sampling strategy in (Song et al., 2020), which only sam-
+ples a subset {τ1, · · · , τM} of M diffusion steps. Formally,
+the accelerated sampling process can be denoted as
+xτm−1 = √ατm−1
+�
+xτm−√
+1−ατmϵ(τm)
+θ
+√ατm
+�
++
+�
+1 − ατm−1 − σ2τm · ϵ(τm)
+θ
++ στmϵτm,
+(18)
+where ϵτm
+∼ N(0, I) is standard Gaussian noise in-
+dependent of xn.
+And στm controls how stochas-
+tic
+the
+denoising
+process
+is.
+We
+set
+σn
+=
+�
+(1 − αn−1) / (1 − αn)
+�
+1 − αn/αn−1 for all diffusion
+steps, to make the generative process become a DDPM.
+When the length of the sampling trajectory is much smaller
+than N, we can achieve significant increases in computa-
+tional efficiency. Moreover, note that the data in the last k
+few reverse steps xall
+i
+(i ∈ {1, . . . , k}) can be considered
+a good approximation of the target. Thus we can also add
+them as samples, reducing the number of the reverse diffu-
+sion process from S to S/k, where S is the required sample
+number to form the data distribution.
+4.3. Comparsion among Different Approaches
+We give the overview of related models in Figure 3: i) De-
+terministic STGNNs calculate future graph signals exactly
+without the involvement of randomness. While the vanilla
+DDPM is a latent variable generative model without con-
+dition; ii) To estimate the data distribution from a trained
+model, i.e., getting S samples, TimeGrad runs S × Tp × N
+diffusion steps for the prediction of all future time steps,
+where N, Tp is the diffusion step, prediction length, respec-
+tively; iii) Compared with current diffusion-based models
+for time series, DiffSTG 1) incorporates the graph as the
+condition so that the spatial correlations can be captured,
+and 2) is a non-autoregressive approach with �S × �
+N dif-
+fusion steps to get the estimated data distribution, where
+�S = S/k < S and �
+N = M < N.
+…
+𝑥h
+𝑥p
+STGNN
+STGNNs
+DDPM
+𝑥0
+𝑥𝑛
+𝑥𝑁
+…
+…
+…
+ℎ𝑡−2
+ℎ𝑡−1
+��0
+𝑡−1
+𝑇𝑝
+RNN
+TimeGrad
+DiffSTG
+…
+𝑥0
+𝑡
+𝑥𝑛−1
+𝑡
+𝑥𝑛𝑡
+𝑥𝑁
+𝑡
+…
+…
+𝑥0
+p
+𝑥𝑛−1
+p
+𝑥𝑛
+p
+𝑥𝑁
+p
+𝑥h ,
+(          )
+…
+…
+…
+…
+…
+condition:
+Figure 3. Overview of different models.
+
+Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+Table 1. Experiment Results. Smaller MAE, RMSE, and CRPS indicate better performance.
+Method
+AIR-BJ
+AIR-GZ
+PEMS08
+MAE
+RMSE
+CRPS
+MAE
+RMSE
+CRPS
+MAE
+RMSE
+CRPS
+Latent ODE (Rubanova et al., 2019)
+20.61
+32.27
+0.47
+12.92
+18.76
+0.30
+26.05
+39.50
+0.11
+DeepAR (Salinas et al., 2020)
+20.15
+32.09
+0.37
+11.77
+17.45
+0.23
+21.56
+33.37
+0.07
+CSDI (Tashiro et al., 2021)
+26.52
+40.33
+0.50
+13.75
+19.40
+0.28
+32.11
+47.40
+0.11
+TimeGrad (Rasul et al., 2021)
+18.64
+31.86
+0.36
+12.36
+18.15
+0.25
+24.46
+38.06
+0.09
+MC Dropout (Wu et al., 2021)
+20.80
+40.54
+0.45
+11.12
+17.07
+0.25
+19.01
+29.35
+0.07
+DiffSTG (ours)
+17.88
+29.60
+0.34
+10.95
+16.66
+0.22
+17.68
+27.13
+0.06
+Improvement
+4.1%
+7.1%
+5.6%
+1.5%
+2.4%
+4.3%
+7.0%
+7.6%
+14.3%
+5. Experiments
+We conduct extensive experiments to evaluate the effective-
+ness of our proposed DiffSTG on three real-world datasets
+and compare it with other probabilistic baselines.
+5.1. Dataset and Experiment Settings
+Datasets. In the experiments, we choose three real-world
+datasets from two domains, including a traffic flow dataset
+PEMS08 (Song et al., 2020), and two air quality datasets
+AIR-BJ and AIR-GZ (Yi et al., 2018). The PEMS08 dataset
+records the traffic flow collected by sensors deployed on the
+road network. The air quality datasets AIR-BJ and AIR-GZ
+consist of one-year PM2.5 readings collected by air quality
+monitoring stations in two metropolises (i.e., Beijing and
+Guangzhou) in China, respectively. Statistics of the datasets
+are shown in Table 2. More details on the datasets are
+provided in Appendix A.3.
+Table 2. Details of all datasets.
+Dataset
+Nodes
+F
+Data Type
+Time interval
+#Samples
+PEMS08
+170
+1
+Traffic flow
+5 minutes
+17,856
+AIR-BJ
+34
+1
+PM2.5
+1 hour
+8,760
+AIR-GZ
+41
+1
+PM2.5
+1 hour
+8,760
+Implementation Details. As for hyperparameters, we set
+the batch size as 8 and use the Adam optimizer with a learn-
+ing rate of 0.002, which is halved every 5 epochs. For CSDI
+and DiffSTG, we adopt the following quadratic schedule
+for variance schedule: βn =
+�
+N−n
+N−1
+√β1 + n−1
+N−1
+√βN
+�2
+.
+We set the minimum noise level β1 = 0.0001 and hidden
+size C = 32 and search the number of the diffusion step N
+and the maximum noise level βN from a given parameter
+space (N ∈ [50, 100, 200], and βN ∈ [0, 1, 0.2, 0.3, 0.4]),
+and each model’s best performance is reported in the ex-
+periment. For other baselines, we utilize their codes and
+parameters in the original paper. For all datasets, the history
+length Th, and prediction length Tp are both set to 12. All
+datasets are split into the training, validation, and test sets
+in chronological order with a ratio of 6:2:2. The models are
+trained on the training set and validated on the validation
+set by the early stopping strategy. The source code will be
+released after the review process.
+5.2. Performance Comparison
+The efforts of stochastic models for probabilistic STG fore-
+casting were traditionally scarce. Hence, we compare our
+model with baselines in the field of probabilistic time se-
+ries forecasting, including Latent ODE (Rubanova et al.,
+2019), DeepAR (Salinas et al., 2020), TimeGrad (Rasul
+et al., 2021), CSDI (Tashiro et al., 2021), and a recent STG
+probabilistic forecasting method MC Dropout (Wu et al.,
+2021). We choose the Continuous Ranked Probability Score
+(CRPS) (Matheson & Winkler, 1976) as an evaluation met-
+ric, which is used to measure the compatibility of an es-
+timated probability distribution with an observation. We
+also report MAE and RMSE of the deterministic forecast-
+ing results by averaging S (set to 8 in our paper) generated
+samples. More details are provided in Appendix A.3.
+In Table 1, DiffSTG outperforms all the probabilistic base-
+lines: it reduces the CRPS by 5.6%, 4.3%, and 14.3% on
+the three datasets compared to the most competitive base-
+line in each dataset, respectively. Distributions in DeepAR
+and Latent ODE can be viewed as some types of low-rank
+approximations of the target, which naturally restricts their
+capability to model the true data distribution. TimeGrad out-
+performs LatentODE due to its DDPM-based architecture
+with tractable likelihoods that models the distribution in a
+general fashion. CSDI is a diffusion-based model originally
+proposed for time series imputation, thus performing worse
+in our forecasting tasks. MC Dropout achieves the second
+best performance on MAE and RMSE in most datasets, due
+to its strong ability in modeling the ST correlations. Our
+DiffSTG yields the best performance in both deterministic
+and probabilistic prediction, revealing that it can preserve
+the spatio-temporal learning capabilities of STGNNs as well
+as the uncertainty measurements of the diffusion models.
+Inference Time. Table 3 reports the average time cost per
+prediction of two diffusion-based forecasting models. We
+observe that TimeGrad is extremely time-consuming due to
+its recurrent architecture. DiffSTG (with M=100 and k=1)
+achieves 40× speed-up compared to TimeGrad, which stems
+from its non-autoregressive architecture. The accelerated
+sampling strategy achieves 3∼4× speed-up beyond Diff-
+STG (M=100, k=1). We also find that when S is large, one
+can increase k for efficiency without loss of performance.
+See Appendix A.4 for more details.
+
+Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+Case
+AIR-GZ
+AIR-GZ
+PEMS08
+PEMS08
+PM 2.5
+PM 2.5
+Traffic Flow
+Traffic Flow
+Reconstruction
+of  history
+Prediction
+of future
+(a)
+(b)
+(c)
+(d)
+(e)
+AIR-GZ
+AIR-GZ
+AIR-GZ
+PM 2.5
+PM 2.5
+PM 2.5
+Geographic 
+Location
+Figure 4. Example of probabilistic spatio-temporal graph forecasting for air quality and traffic dataset.
+Table 3. Time cost (by seconds) of TimeGrad and DiffSTG in AIR-
+GZ (Th = 12, Tp = 12, N = 100). S is the number of samples.
+Method
+S = 8
+S = 16
+S = 32
+TimeGrad (Rasul et al., 2021)
+9.58
+128.40
+672.12
+DiffSTG (M=100, k=1)
+0.24
+0.48
+0.95
+DiffSTG (M=40, k=1)
+0.12
+0.20
+0.71
+DiffSTG (M=40, k=2)
+0.07
+0.12
+0.21
+Visualization. We plot the predicted distribution of dif-
+ferent methods to investigate their performance intuitively.
+We choose the AIR-GZ and PEMS08 for demonstration,
+and more examples on other datasets can be found in Ap-
+pendix A.5. We have the following observations: 1) Fig-
+ure 4(a) shows that DiffSTG can capture the data distribution
+more precisely than DeepAR; 2) in Figure 4(b), where the
+predictions of both DeepAR and DiffSTG cover the observa-
+tions, DiffSTG provides a more compact prediction interval,
+indicating the ability to provide more reliable estimations.
+3) Note that the model also needs to learn to reconstruct
+the history in the loss of Eq. (14), we also illustrate the
+model’s capability in history reconstruction in Figure 4(c);
+4) Figure 4(d) draws the prediction result by a deterministic
+method STGCN (Yu et al., 2018) and DiffSTG. In the red
+box of Figure 4(d), the deterministic method fails to give an
+accurate prediction. In contrast, our DiffSTG model renders
+a bigger area (which covers the ground truth) in that region,
+indicating that the data therein is coupled with higher un-
+certainties. Such ability to accurately provide uncertainty
+can be of great help for practical decision-making; 5) More-
+over, as shown in Figure 4(e), we illustrate the estimated
+distribution of DiffSTG on three stations, to illustrate its
+spatial dependency learning ability. Compared with station
+29, the estimated distribution of station 4 is more similar to
+station 1, which is reasonable because the air quality of a
+station has stronger connections with its nearby neighbors.
+Equipped with the proposed denoising network UGnet, the
+model is able to capture the ST correlations, leading to more
+reliable and accurate estimation.
+5.3. Ablation Study
+We conduct an ablation study on the AIR-GZ dataset to
+verify the effect of each component. Figure 5 illustrates
+the results. Firstly, removing the spatial dependency learn-
+ing in UGnet (w/o Spatial) brings considerable degenera-
+tion, which validates the importance of modeling spatial
+correlations between nodes. Secondly, when turning off
+the temporal dependency learning in UGnet (w/o Tempo-
+ral), the performance drops significantly on all evaluation
+metrics. Thirdly, we detach the Unet-based structure in
+UGnet and only use one TCN block (w/o U-structure) for
+feature extraction, the performance degrades dramatically,
+which demonstrates the merits of a Unet-based structure in
+capturing ST-dependencies at different granularities.
+DiffSTG
+w/o Spatial
+w/o Temporal
+DiffSTG
+w/o Spatial
+w/o Temporal
+w/o Spatial
+w/o Temporal
+w/o U-structure
+Figure 5. Ablation Study.
+5.4. Hyperparameter Study
+In this section, we examine the impact of several crucial
+hyperparameters on DiffSTG. Specifically, we report the
+performance on AIR-GZ under different variance sched-
+ules (i.e., the combination of βN and diffusion step N) and
+hidden size C.
+For different variance schedules {β1, . . . , βN}, we set
+β1 = 0.0001 and let βN and N be from two search spaces,
+where N ∈ [50, 100, 200] and βN ∈ [0.1, 0.2, 0.3, 0.4]. A
+variance schedule can be specified by a combination of βN
+
+node:4
+100
+90
+80
+70
+60
+50
+40
+30
+0
+5
+10
+15
+20node:29
+90
+80
+70
+60
+50
+40
+30
+0
+5
+10
+15
+202980
+60
+40
+20
+0
+0
+5
+10
+15
+20node:29
+550
+500
+450
+400
+350
+300
+5
+10
+15
+20node:1
+550
+500
+450
+400
+350
+300
+0
+5
+10
+15
+2080
+60
+40
+20
+0
+5
+10
+15
+20DiffSTG 90% interval
+DeepAR 90% interval
+observationsobservationsDiffSTGDiffSTG
+STGCN
+.
+observationsnode:1
+120
+100
+80
+60
+40
+0
+5
+10
+15
+2011.50
+0.240
+22.0
+11.40
+21.0
+0.235
+11.30
+20.0
+0.230
+19.0
+11.20
+18.0
+0.225
+17.0
+11.10
+0.220
+16.0
+11.00
+15.0
+0.215
+MAE
+RMSE
+CRPSProbabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+(a) The effect of variance schedule
+(b) The effect of hidden size
+Figure 6. Influence of hyperparameters.
+and N. The results are shown in Figure 6. We note that
+the performance deteriorates rapidly when N = 50 and
+βN = 0.1. In this case, the result of the forward process
+is far away from a Gaussian distribution. Consequently,
+the reverse process starting with Gaussian distribution be-
+comes an inaccurate approximation, which heavily injures
+the model performance. When N gets larger, there is a
+higher chance of getting a promising result.
+Figure 6 shows the results of DiffSTG with N = 100, and
+βN = 0.4 vs. different hidden size C, from which we
+observe that the performance first slightly increases and
+then drops with the increase in hidden size. Compared with
+the variance schedule, the model’s performance is much less
+sensitive to the hidden size. We also investigated the effect
+of other hyperparameters in Appendix A.4.
+5.5. Limitations
+Though promising in probabilistic prediction, DiffSTG still
+has a performance gap compared with current state-of-the-
+art STGNNs in terms of deterministic forecasting. Table 4
+shows the deterministic prediction performance of DiffSTG
+(by averaging 8 in generate samples) and four deterministic
+methods, including DCRNN (Li et al., 2018), STGCN (Yu
+et al., 2018), STGNCDE (Choi et al., 2022), and GMSDR
+(Liu et al., 2022a). While DiffSTG is competitive with most
+probabilistic methods, it is still inferior to the state-of-the-art
+deterministic methods. Different from deterministic meth-
+ods, the optimization goal of DiffSTG is derived from a vari-
+ational inference perspective (see details in Appendix. A.1),
+where the learned posterior distribution might be inaccurate
+when the data samples are insufficient. We have similar ob-
+servations in other DDPM-based models, such as TimeGrad
+and CSDI, as shown Table 1. We leave improving DiffSTG
+to surpass those deterministic methods in future work.
+Table 4. Comparison with deterministic methods. Lower MAE,
+and RMSE indicate better performance.
+Method
+AIR-BJ
+AIR-GZ
+PEMS08
+MAE
+RMSE
+MAE
+RMSE
+MAE
+RMSE
+DCRNN
+16.99
+28.00
+10.23
+15.21
+18.56
+28.73
+STGCN
+19.54
+30.51
+11.05
+16.54
+20.15
+30.14
+STGNCDE
+19.17
+29.56
+10.51
+16.11
+15.83
+25.05
+GMSDR
+16.60
+28.50
+9.72
+14.55
+16.01
+24.84
+DiffSTG
+17.88
+29.60
+11.04
+16.75
+17.68
+27.13
+6. Related Work
+Spatio-temporal Graph Forcasting.
+Recently, a large
+body of research has been studied on spatio-temporal fore-
+casting in different scenarios, such as traffic forecasting (Yu
+et al., 2018; Wu et al., 2019; Guo et al., 2021; Peng et al.,
+2020; Ji et al., 2022) and air quality forecasting (Liang et al.,
+2022). STGNNs have become dominant models in this field,
+which combine GNN and temporal components (e.g., TCN
+and RNN) to capture the spatial correlations and tempo-
+ral features, respectively. However, most existing works
+focus on point estimation while ignoring quantifying the
+uncertainty of predictions. To fill this gap, this paper devel-
+ops a conditional diffusion-based method that couples the
+spatio-temporal learning capabilities of STGNNs with the
+uncertainty measurements of diffusion models.
+Score-based Generative Models. The diffusion model that
+we adopt belongs to score-based generative models (please
+refer to Section 2 for more details), which learn the gra-
+dient of the log-density with respect to the inputs, called
+Stein Score function (Hyv¨arinen & Dayan, 2005; Vincent,
+2011). At inference time, they use the gradient estimate to
+sample the data via Langevin dynamics (Song & Ermon,
+2019). By perturbing the data through different noise levels,
+these models can capture both coarse and fine-grained fea-
+tures in the original data. Which, leads to their impressive
+performance in many domains, such as image (Ho et al.,
+2020), audio (Kong et al., 2020; Chen et al., 2020), graph
+(Niu et al., 2020) and time series (Rasul et al., 2021; Tashiro
+et al., 2021).
+Time Series Forecasting. Methods in time series forecast-
+ing can be classified into two streams: i) deterministic meth-
+ods, including transformer-based approaches (Zhou et al.,
+2021; 2022) and RNN-based models (Che et al., 2018); and
+ii) probabilistic methods such as popular diffusion-based
+models (Rasul et al., 2021; Hernandez & Dumas, 2022; Li
+et al., 2023; Chang et al., 2023).
+7. Conclusion and Future Work
+In this paper, we propose a novel probabilistic framework
+called DiffSTG for spatio-temporal graph forecasting. To
+the best of our knowledge, this is the first work that general-
+izes the DDPM to spatio-temporal graphs. DiffSTG com-
+bines the spatio-temporal learning capabilities of STGNNs
+with the uncertainty measurements of diffusion models.
+Moreover, unlike previous diffusion-based models designed
+for the image or sequential data, we devise the first denoising
+network UGnet for capturing the spatial and temporal cor-
+relations in STG data. Extensive experiments demonstrate
+the effectiveness and efficiency of our proposed method. A
+direction of future work is to apply DiffSTG to other STG
+learning tasks, such as spatio-temporal graph imputation.
+
+Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
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+
+Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+A. Appendix
+A.1. Details of DDPM
+We introduce the details of denoising diffusion probabilistic
+models in this section.
+Diffusion probabilistic models are latent variable models
+that consist of two processes, namely the forward process
+and the reverse process. The forward process is a fixed
+Gaussian transition process as defined in Eq. (1) and Eq. (2).
+The reverse process is a learnable Gaussian transition pro-
+cess defined in Eq. (4) and Eq. (5). Then, the parameters θ
+are learned by minimizing the negative log-likelihood via
+the variational lower bound (ELBO):
+minθ Eq(x0) [− log pθ(x0)]
+≤ − log pθ (x0) + DKL (q (x1:N | x0) ∥pθ (x1:N | x0))
+= − log pθ (x0) + Ex1:N∼q(x1:N|x0)
+�
+log
+q(x1:N|x0)
+pθ(x0:N)/pθ(x0)
+�
+= − log pθ (x0) + Eq
+�
+log q(x1:N|x0)
+pθ(x0:N) + log pθ (x0)
+�
+= Eq(x0:N)
+�
+log q(x1:N|x0)
+pθ(x0:N)
+�
+:= LELBO.
+(19)
+We can further decompose LELBO into different terms ac-
+cording to the property of Markov chains:
+LELBO
+= Eq(x0:N )
+�
+log q(x1:N |x0)
+pθ(x0:N )
+�
+= Eq
+�
+log
+�N
+n=1 q(xn|xn−1)
+pθ(xN ) �N
+n=1 pθ(xn−1|xn)
+�
+= Eq
+�
+− log pθ (xN) + �N
+t=2 log
+q(xn|xn−1)
+pθ(xn−1|xn) + log
+q(x1|x0)
+pθ(x0|x1)
+�
+= Eq
+�
+log q(xN |x0)
+pθ(xN ) + �N
+t=2 log
+q(xn−1|xn,x0)
+pθ(xn−1|xn) − log pθ (x0|x1)
+�
+= Eq[DKL (q (xN | x0) ∥pθ (xN))
+�
+��
+�
+LN
++ �N
+t=2 DKL (q (xn−1 | xn, x0) ∥pθ (xn−1 | xn))
+�
+��
+�
+Ln−1
+− log pθ (x0 | x1)
+�
+��
+�
+L0
+].
+(20)
+By the property in Eq. (3), (Ho et al., 2020) show that
+the forward process posterior when conditioned on x0, i.e.,
+q(xn−1|xn, x0) is tractable, formulated as
+q (xn−1 | xn, x0) = N
+�
+xn−1; ˜µ (xn, x0) , ˜βnI
+�
+(21)
+where
+˜µn (xn, x0) =
+√αn−1βn
+1 − αn
+x0 +
+√αn (1 − αn−1)
+1 − αn
+xn,
+(22)
+and
+˜βn = 1 − αn−1
+1 − αn
+βn.
+(23)
+So far, we can see that each term in LELBO (except for
+L0) calculates the KL Divergence between two Gaussian
+distributions, therefore they can be computed in closed form.
+LN is constant that can be ignored in training because q has
+no learnable parameters and xN is a Gaussian noise. (Ho
+et al., 2020) models L0 using a separate discrete decoder.
+Especially, the loss term of Lt (t ∈ {2, · · · , T}), have the
+following closed form:
+Ex0,ϵ
+�
+β2
+n
+2Σθαn (1 − αn)
+��ϵ − ϵθ
+�√αnx0 +
+√
+1 − αnϵ, n
+���2�
+,
+which can be further simplified by removing the coefficient
+in the loss term, formulated as
+Ex0,ϵ
+���ϵ − ϵθ
+�√αnx0 +
+√
+1 − αnϵ, n
+���2�
+.
+A.2. Details of UGnet
+We propose a novel denoising network to effectively capture
+spatio-temporal correlations in STG data, named UGnet.
+It adopts an Unet-like architecture in the temporal dimen-
+sion and can also process the graph as the condition. Unet
+structure can capture features at different levels because its
+Convolutional Neural Networks (CNN) kernels gradually
+merge low-level features into high-level features. Similarly,
+in the context of spatio-temporal forecasting, we naturally
+have different granularities in the temporal dimension (e.g.,
+15 minutes, 30 minutes, and 1 hour). Therefore, an intuitive
+way is to adopt the idea in Unet that gradually reduce the
+shape in the temporal dimension and reverse it back so that
+temporal features at different levels can be well captured.
+Doing so also brings the model the ability to scale up to
+large STG.
+Specifically, as shown in Figure
+7, UGnet ϵθ(X all ×
+R|X all
+msk, G) → X all takes xall
+msk, xall
+n , n, G as input, and
+outputs the denoised noise ϵ. We first concatenate xall
+n ∈
+RF ×V ×T and xall
+msk ∈ RF ×V ×T in the temporal dimen-
+sion to form a new matrix �xall
+n ∈ RF ×V ×2T . UGnet con-
+tains several Spatio-temporal Residual Blocks (ST-Residual
+Block for short) with two types: the down-residual block and
+the up-residual block. The down-residual blocks gradually
+reduce the shape in the temporal dimension (i.e., increase
+the temporal granularity). While the up-residual blocks
+gradually convert the temporal granularity back to the level
+that is the same as the input. Both blocks contain the same
+residual block that can capture both spatial and temporal
+correlations in the data with the help of the graph structure.
+Here, we introduce the details of the ST-Residual block. We
+first project input xall
+n ∈ RF ×V ×T into a high-dimensional
+representation H ∈ RC×V ×2T by a linear layer, where C
+is the projected dimension. Let Hi ∈ RC×V ×Ti denote the
+input of the i-th ST-Residual Block, where Ti is the length
+of the time dimension, and H0 = H.
+
+Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+1
+5
+Solution：How to capture the ST-correlation in p_theta? 
+Problem
+Definition
+Related Work
+Motivation
+Solution
+Experiment
+concatenate
+ST-Residual Block
+ST-Residual Block
+ST-Residual
+Block
+ST-Residual Block
+ST-Residual Block
+FC
+all 
+nx
+msk
+nx
+n
+
+Gated Causal Convolution
+Gated Causal Convolution
+Graph Convolution
+Layer Norm
+Up / Down-Sample
++
+emb
++
+ST-Residual 
+Block
+n
+Hi
+( , , )
+F V T
+( , , )
+F V T
+H
+( , ,2 )
+F V
+T
+( , , )
+C V T
+( , ,
+/ 2)
+C V T
+( , , )
+C V T
+( , , )
+C V T
+( , ,
+)
+i
+C V T
+( , ,2 )
+F V
+T
+( , , )
+F V T
+Figure 7. The architecture of denoising network UGnet. It adopts
+an Unet-like structure to model both spatial and temporal depen-
+dencies at different temporal granularities, conditioned on the noise
+level and given graph structure.
+Temporal Dependence Modeling. As shown in Figure
+7, Hi is first fed into a Temporal Convolution Network
+(TCN) (Bai et al., 2018) for modeling temporal dependence,
+which is a 1-D gated causal convolution with K kernel
+size with padding to get the same shape with input. The
+convolution kernel ΓT ∈ RK×Ct
+in×Ct
+out maps the input
+Hi to two outputs Pi, Qi with the same shape Pi/Qi ∈
+RCt
+out×V ×Ti. As a result, the temporal gated convolution
+can be defined as,
+ΓT (Hi) = Pi ⊙ σ(Qi) ∈ RCt
+out×V ×Ti := Hi,
+(24)
+where ⊙ is the element-wise Hadamard product, and σ is the
+sigmoid activation function of GLU. The item σ(Qi) can
+be considered a gate that filters the useful information of Pi
+into the next layer. Furthermore, residual connections are
+implemented among stacked temporal convolutional layers
+to further exploit the full input times horizon.
+Spatial Dependence Modeling. Graph convolution net-
+work (GCN) is employed to directly extract highly meaning-
+ful features and patterns in the space domain. The input of
+GCN is the node feature matrix, which is reshaped output of
+the TCN layer in our case, denoted as Hi ∈ RV ×Cg
+in, where
+Cg
+in = Ti×Ct
+out). A general formulation (Zhou et al., 2020)
+of a graph convolution can be denoted as
+ΓG(Hi) = σ
+�
+Φ
+�
+Agcn, Hi
+�
+Wi
+�
+,
+(25)
+where Wi ∈ RCg
+in×Cg
+in denotes a trainable parameter and σ
+is an activation function. Φ(·) is an aggregation function that
+decides the rule of how neighbors’ features are aggregated
+into the target node. In our work, we do not focus on how to
+develop an elaborately designed function Φ(·). Therefore,
+we use the form in the most popular vanilla GCN (Kipf &
+Welling, 2017) that defines a symmetric normalized sum-
+mation function as Φgcn
+�
+Agcn, Hi
+�
+= AgcnHi, where
+Agcn = D− 1
+2 (A + I)D− 1
+2 ∈ RV ×V is a normalized adja-
+cent matrix. I is the identity matrix and D is the diagonal
+degree matrix with Dii = �
+j(A + I)ij.
+A.3. Details of datasets and baselines
+Datasets. PEMS08 is a traffic flow dataset collected by
+the Caltrans Performance Measurement System (PeMS).
+It records the traffic flow recorded by sensors (nodes) de-
+ployed on the road network. In this work, we use the dataset
+extracted by STSGCN (Song et al., 2020). The traffic net-
+works (adjacency matrix) for these datasets are constructed
+according to the actual road network. If the two sensors are
+on the same road, the two points are considered connected
+in the spatial network.
+Air quality datasets were collected by our system, containing
+the PM2.5 readings from air quality monitoring stations. The
+system details can be found in (Yi et al., 2018). AIR-BJ
+records data from 34 stations in Beijing from 2019/01/01 to
+2019/12/31. And AIR-GZ records data from 41 stations in
+Guangzhou from 2017/01/01 to 2017/12/31. We build the
+spatial correlation matrix A using the distance between two
+stations.
+Probabilistic baselines. The following methods are imple-
+mented as baselines for probabilistic STG forecasting:
+• Latent ODE (Rubanova et al., 2019). It defines a proba-
+bilistic generative process over time series from a latent
+initial state, which can be trained with variational infer-
+ence.
+• DeepAR (Salinas et al., 2020), which utilizes a Gaussian
+distribution to model the data distribution;
+• TimeGrad (Rasul et al., 2021), which is an auto-regressive
+model that combines the diffusion model with an RNN-
+based encoder;
+• CSDI (Tashiro et al., 2021), which is a diffusion-based
+non-autoregressive model first proposed for multivariate
+time series imputation. We mask all the future signals to
+adapt CSDI to our task.
+• MC Dropout (Wu et al., 2021), which is developed based
+on MC Dropout (Gal et al., 2017) for probabilistic spatio-
+temporal forecasting.
+Deterministic baselines. We choose some popular and
+state-of-the-art methods for comparison:
+• DCRNN (Li et al., 2018): Diffusion Convolutional Re-
+current Neural Network integrates diffusion convolution
+with sequence-to-sequence architecture to learn the repre-
+sentations of spatial dependencies and temporal relations.
+• STGCN (Yu et al., 2018): Spatial-Temporal Graph Con-
+volution Network combines spectral graph convolution
+with 1D convolution to capture spatial and temporal cor-
+relations.
+
+Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+• STGNCDE (Choi et al., 2022). Spatio-Temporal Graph
+Neural Controlled Differential Equation introduces two
+neural control differential equations (NCDE) for process-
+ing spatial and sequential data, respectively, which can
+be considered as an NCDE-based interpretation of graph
+convolutional networks.
+• GMSDR (Liu et al., 2022a): Graph-based Multi-Step
+Dependency Relation improves RNN by explicitly taking
+the hidden states of multiple historical time steps as the
+input of each time unit.
+Metrics. We choose the Continuous Ranked Probability
+Score (CRPS) (Matheson & Winkler, 1976) as the metric to
+evaluate the performance of probabilistic prediction, which
+is commonly used to measure the compatibility of an esti-
+mated probability distribution F with an observation x:
+CRPS(F, x) =
+�
+R
+(F(z) − I{x ≤ z})2 dz,
+(26)
+where I{x ≤ z} is an indicator function which equals one
+if x ≤ z, and zero otherwise. Smaller CRPS means better
+performance.
+In addition, we leverage Mean Absolute Error (MAE)
+and Root Mean Squared Error (RMSE) to evaluate
+the performance of deterministic prediction.
+Let Y
+be the label,
+and
+ˆY
+denote the predictive result.
+MAE(Y, ˆY ) =
+1
+|Y |
+�|Y |
+i=1
+���Yi − ˆYi
+��� , and RMSE(Y, ˆY ) =
+�
+1
+|Y |
+�|Y |
+i=1
+�
+Yi − ˆYi
+�2
+, where a smaller metric means bet-
+ter performance.
+A.4. Additional results and experiments
+Effect of the number of generated samples S. We inves-
+tigate the relationship between the number of samples S
+and the performance in Figure 8. It shows the effect of prob-
+abilistic forecasting, as well as the effect on deterministic
+forecasting. From which we can see that five or ten samples
+are enough to estimate good distributions. While increasing
+the number of samples further improves the performance,
+the improvement becomes marginal over 32 samples.
+Effect of k. Recall that k is the number of utilized samples
+in the last few diffusion steps when sampling S samples.
+We provide the results of k = 1 and k = 2 in Figure 8, in
+which we have several interesting observations: i) When S
+is large enough (i.e., S > 32), the performance of k = 2 is
+almost the same as k = 1, and the sample speed of k = 2 is
+1.5 times faster than k = 1; ii) When the number of reverse
+diffusion processes (i.e., S/k) is settled, a large k can in-
+crease sample diversity thus leading to better performance,
+especially when S is small.
+0
+25
+50
+75
+100
+S
+0.200
+0.225
+0.250
+0.275
+0.300
+0.325
+CRPS
+k=1
+k=2
+0
+25
+50
+75
+100
+S
+11
+12
+13
+MAE
+k=1
+k=2
+0
+25
+50
+75
+100
+S
+0.0
+0.5
+1.0
+1.5
+Time
+k=1
+k=2
+Figure 8. The effect of the number of generated samples.
+In light of the above results, we give the following recom-
+mendations for the combination of S and k: 1) when S
+is small, a small k is recommended to increase the sam-
+ple diversity for better performance; 2) when S is large,
+one can increase k for efficiency without much lose of the
+performance.
+A.5. Additional examples of probabilistic forecasting
+This section illustrates various probabilistic forecasting ex-
+amples to show the characteristic of different methods. Note
+that the scales of the y-axis depend on the stations.
+We compare DiffSTG with DeepAR and TimeGrad for se-
+lected stations of AIR-BJ in Figure 10-11. The geographic
+distribution of stations is shown in Figure 9. We select two
+groups of nodes according to their spatial location. Nodes
+in the first group are far away from each other, including
+nodes 0, 2, 17, and 20. While nodes in the second group are
+close to each other, including nodes 7, 8, 9, and 18.
+For the comparison in Figure 10, while TimeGrad fails to
+capture the data distribution, DiffSTG computes reason-
+able probabilistic forecasting for a majority of the stations.
+For the comparison in Figure 11, DiffSTG provides tighter
+uncertainty than DeepAR. And in both Figure 10 and Fig-
+ure 11, we can see that DiffSTG tends to provide similar
+estimated distribution for stations nearby, which is reason-
+able since the air quality of a station is strongly correlated
+with its neighbors. The above examples further illustrate
+that DiffSTG can effectively learn the spatial and tempo-
+ral dependency in STG, thus providing more reliable and
+accurate estimations than others.
+Figure 9. Geographic distribution of stations on AIR-BJ.
+
+20
+0
+18
+17
+7
+2
+98Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+Figure 10. Comparison of probabilistic STG forecasting between TimeGrad and DiffSTG for air quality dataset (AIR-BJ).
+
+Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models
+Figure 11. Comparison of probabilistic STG forecasting between DeepAR and DiffSTG for air quality dataset (AIR-BJ).
+

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@@ -0,0 +1,1137 @@
+arXiv:2301.01027v1  [math.OA]  3 Jan 2023
+Detecting ideals in reduced crossed product C*-algebras of
+topological dynamical systems
+Are Austad and Sven Raum
+Abstract. We introduce the ℓ1-ideal intersection property for crossed product C∗-algebras. It is
+implied by C∗-simplicity as well as C∗-uniqueness. We show that topological dynamical systems of
+arbitrary lattices in connectedLie groups, arbitrary lineargroups overthe integers in a numberfield
+and arbitrary virtually polycyclic groups have the ℓ1-ideal intersection property. On the way, we
+extend previous results on C∗-uniqueness of L1-groupoid algebras to the general twisted stetting.
+1
+Introduction
+Crossed products associated with topological dynamical systems are among the prime sources of
+examples in the theory of C∗-algebras. In recent years, amenable dynamical systems received abun-
+dant attention in the context of Elliott’s classification programme (see e.g. [HWZ15; Sza15; DPS15;
+KS20]), while reduced group C∗-algebras were put in the spotlight by breakthrough results on C∗-
+simplicity [KK17; Bre+17; Ken20; Haa16].
+A foundational problem about crossed product C∗-algebras concerns their ideal structure. For
+tame dynamical systems it is possible to give a complete description of the primitive ideal space of
+the associated crossed product in terms of induced primitive ideals, thanks to the Mackey machine
+[Ros94; EW08]. For wilder dynamical systems and for group C∗-algebras, it is the question of sim-
+plicity that received most attention. Following seminal work on simplicity of group C∗-algebras
+[KK17; Bre+17], a satisfactory characterisation of topological dynamical systems whose crossed
+product C∗-algebras are simple could be obtained in [Kaw17]. This line of research even led to
+complete results about simplicity of C∗-algebras associated with étale groupoids [Bor19; Ken+21].
+One important insight from the study of the ideal structure of groupoid C∗-algebras was the insight
+originating from [Tom92] that specific subalgebras have the potential to detect ideals.
+Much fewer results are available for dynamical systems that neither are tame nor give rise to
+simple crossed products. However, the idea of employing subalgebras to detect ideals had surfaced
+earlier in a completely different context. In work on abstract harmonic analysis and representation
+theory of solvable Lie groups, the concept of C∗-uniqueness of L1-convolution algebras was intro-
+duced in the late 1970’s and early 1980’s [Boi+78; Boi84]. This notion can be reformulated as an
+ideal intersection property for the inclusion L1(G) ⊆ C∗(G). While for exponential solvable Lie
+groups Boidol could establish conclusive results [Boi80], there have been no noteworthy advances
+in the investigation of C∗-uniqueness of discrete groups beyond the positive results for virtually
+nilpotent groups and some metabelian groups in [Boi+78]. Nevertheless, it is considered an open
+question whether every amenable discrete group is C∗-unique [LN04, Remark 3.6]. In some recent
+work starting with [GMR18], variations of C∗-uniqueness replacing the ℓ1-convolution algebra of
+a discrete group by its complex group algebra have been considered. Results from [AK19; Ale19;
+Sca20] create an unclear image of which groups might have this property termed algebraic C∗-
+uniqueness.
+last modified on January 4, 2023
+MSC 2020 classification: 46L05, 37B02, 22E40, 20G30, 20F16
+Keywords: ideal intersection property, crossed product C∗-algebra, topological dynamical system, lattices in Lie groups,
+linear groups, polycyclic groups
+
+The aim of this article is to introduce and study a new ideal intersection property for crossed
+product C∗-algebras, which can be established for a large class of examples including all C∗-simple
+groups andall C∗-unique groups. Atpresent, we have no example of a topological dynamical system
+Γ ↷ X that fails the following ℓ1-ideal intersection property: every non-zero ideal of the C∗-algebraic
+crossed product C0(X)⋊redΓ has non-zero intersection with the ℓ1-crossed product C0(X)⋊ℓ1 Γ.
+The next two theorems describe non-amenable and amenable examples of groups for which we can
+establish the ℓ1-ideal intersection property.
+Theorem A (See Corollary 6.6, Corollary 6.8 and Corollary 6.9). Let Γ be either an acylindri-
+cally hyperbolic group, a lattice in a connected Lie group or a linear group over integers in a number field.
+Then every action of Γ on a locally compact Hausdorff space has the ℓ1-ideal intersection property.
+Weremarkthat(acylindrically)hyperbolicgroupsandtheiractionontheirGromovboundary, map-
+ping class groups, and algebraic actions of arithmetic groups fall in the scope of our theorem. For
+amenable groups the ℓ1-ideal intersection property coincides with the notion of C∗-uniqueness, so
+that we obtain the first major enlargement of the class of groups for which this property is known.
+Theorem B (See Corollary 6.10). Every action of a locally virtually polycyclic group on a locally com-
+pact Hausdorff space has the ℓ1-ideal intersection property. In particular, every locally virtually polycyclic
+group is C∗-unique.
+We remark that also metabelian groups can be covered by our methods, as noted in Remark 6.11.
+Thus our results cover and extend all previously known examples of C∗-unique groups. In order to
+obtain the results above, we establish a general criterion on groups to satisfy the ℓ1-ideal intersec-
+tion property for all their dynamical systems. It combines assumptions that relate to C∗-simplicity
+with conditions on the structure of amenable subgroups.
+Theorem C (See Theorem 6.5). Let Γ be a discrete group such that the following three conditions hold
+for every finitely generated subgroup of Λ ≤ Γ:
+• the Furstenberg subgroup of every subgroup of Λ equals its amenable radical,
+• the Tits alternative holds for Λ, and
+• there is l ∈ N such that every solvable subgroup of Λ is polycyclic of Hirsch length at most l.
+Then every action of Γ on a locally compact Hausdorff space has the ℓ1-ideal intersection property.
+The proof of Theorem C employs groupoid techniques in order to set up an induction scheme.
+The assumptions on amenable subgroups are exploited to construct a twisted groupoid, which is
+analysed from the point of view of the L1-ideal intersection property for groupoids, previously
+studied in [AO22]. Ultimately our induction argument reduces the maximal possible Hirsch length
+of polycyclic subgroups. The following generalisation of work from [AO22] allows us to analyse the
+twisted groupoids we obtain when applying our strategy. It might be of independent interest.
+Theorem D (See Corollary 4.5). Let E be a twist over a second-countable locally compact étale Haus-
+dorff groupoid G. Assume that there is a dense subset D ⊆ G(0) such the fibres (IG
+x ,IE
+x ) have the ℓ1-ideal
+intersection property for all x ∈ D. Then (G,E) has the L1-ideal intersection property.
+2
+
+Structure of the article
+In Section 2, we introduce necessary background material and fix notation. In Section 3 we for-
+mally define the ℓ1-ideal intersection property for twisted C∗-dynamical systems and show in par-
+ticular that it is closed under directed unions of groups. In Section 4 we study the L1-ideal inter-
+section property for twisted groupoids. In Section 5 we present certain twisted group C∗-algebras
+as twisted groupoid C∗-algebras, which is a key ingredient for the subsequent Section 6, where we
+obtain our main results.
+Acknowledgements
+ThefirstauthorgratefullyacknowledgesthefinancialsupportfromtheIndependentResearchFund
+Denmark through grant number 1026-00371B. The second author was supported by the Swedish
+Research Council through grant number 2018-04243. The authors would like to thank the organ-
+isers of the 28th Nordic Congress of Mathematicians in Aalto, where this project was initiated.
+They are grateful to Matthew Kennedy for interesting discussions about the ℓ1-ideal intersection
+property and to Magnus Goffeng for asking whether beyond group algebras also crossed product
+C∗-algebras could be investigated with the present techniques. We thank Becky Armstrong for
+clarifying conversations on the role of the second-countability assumption in her work.
+2
+Preliminaries
+Convention 2.1. All groups in this article are discrete unless otherwise specified.
+2.1
+Virtually polycyclic groups
+A group Γ is called polycyclic if there exists a subnormal series
+1 = Γ0 ⊴ Γ1⋯ ⊴ Γn−1 ⊴ Γn = Γ
+such that each of the factor groups Γi/Γi−1 is cyclic. The group is called poly-Z if each of the factor
+groups are isomorphic to Z.
+It is known that polycyclic groups are precisely the solvable groups for which every subgroup
+is finitely generated, see [Seg83, Chapter 1, Proposition 4]. This fact will be extensively used in the
+present article. We also recall from [Seg83, Chapter 1, Proposition 2] that a group Γ is virtually
+polycyclic if and only if it is (poly-Z)-by-finite.
+The Hirsch length of a virtually polycyclic group Γ is the number of infinite cyclic factors in a
+subnormal series with cyclic or finite factors. It is denoted by h(Γ). See [Seg83, Chapter 1, Part
+C] for a discussion of the Hirsch length and its properties. In particular, we will make use of the
+following properties:
+• If Λ ≤ Γ, then h(Λ) ≤ h(Γ). Equality holds if and only if [Γ ∶ Λ] < ∞.
+• If Λ ⊴ Γ is normal, then h(Γ) = h(Λ) + h(Γ/Λ).
+3
+
+2.2
+C*-uniqueness
+A group Γ is called ℓ1-to-C∗-unique or just C∗-unique for short if ℓ1(Γ) has a unique C∗-norm. It
+is clear that every C∗-unique group is amenable and that a group Γ is C∗-unique if and only if ℓ1(Γ)
+has non-zero intersection with every non-zero ideal of C∗(Γ). The following result is a special case
+of [Boi+78, Satz 2].
+Theorem 2.2. Every finitely generated group of polynomial growth is C∗-unique.
+In this work we will only need to apply Theorem 2.2 in the special case of finitely generated
+torsion-free abelian groups, that is groups isomorphic with Zn for some n ∈ N. For the sake of a
+self-contained presentation, we give a short and direct proof in this case.
+Proof of Theorem 2.2 for Zn. It suffices to show that ℓ1(Zn) ⊆ C∗(Zn) has the ideal intersection
+property. Consider the Schwartz algebra
+S(Zn) = {f ∈ ℓ1(Zn) ∣ ∀k ∈ N ∶ f(x)∣x∣k → 0 as x �→ ∞}
+and the Fourier isomorphism F∶C∗(Zn) �→ C(Tn). Then F(S(Zn)) = C∞(Tn) is the algebra of
+smooth functions and it suffices to show that C∞(Tn) ⊆ C(Tn) has the ideal intersection property.
+If I = {f ∈ C(Tn) ∣ f∣A ≡ 0} for some proper closed subset A ⊆ Tn is an ideal in C(Tn), then there
+is a non-zero smooth function f ∈ C∞
+c (Tn ∖ A) ⊆ I. So I ∩ C∞(Tn) ≠ 0.
+2.3
+C*-simplicity
+In this section we recall some terminology from the theory of C∗-simple groups, and prove one
+result which is needed for our work and can be directly deduced from the literature. A group Γ
+is called C∗-simple if C∗
+red(Γ) is simple. It is clear that every C∗-simple group has the ℓ1-ideal
+intersection property.
+As proven in [Ken20], a group Γ is C∗-simple if and only if the stabiliser URS (uniformly recur-
+rent subgroup) of its Furstenberg boundary ∂FΓ is trivial. We call a group in this URS a Furstenberg
+subgroup, and by abuse of notation will talk about the Furstenberg subgroup of Γ. Recall also that
+the amenable radical R(Γ) is the largest amenable normal subgroup of Γ. We need the following
+result that is not explicitly stated in the literature.
+Proposition 2.3. Let Γ be a group whose Furstenberg subgroup is the amenable radical. Then Γ/R(Γ)
+is C∗-simple.
+Proof. It follows from [Kaw17, Corollary 8.5] that the C∗-algebra generated by the image of the
+quasi-regular representation with respect to a Furstenberg subgroup is simple. So by assumption
+C∗
+red(Γ/R(Γ)) = λΓ/R(Γ)(C∗
+red(Γ)) is simple.
+2.4
+Twisted C*-dynamical systems
+A twisted C∗-dynamical system is a tuple (A,Γ,α,σ) where A is a C∗-algebra, Γ is a group and
+α∶Γ → Aut(A), σ∶Γ × Γ → U(M(A)) are maps satisfying
+αg1 ○ αg2 = Ad(σ(g1,g2)) ○ αg1g2
+σ(g1,g2)σ(g1g2,g3) = αg1(σ(g2,g3))σ(g1,g2g3)
+σ(g1,e) = σ(e,g1) = 1
+4
+
+for all g1,g2,g3 ∈ Γ. The special case of A = C corresponds exactly to a group Γ with the choice of
+a 2-cocycle in Z2(Γ,S1).
+Denotingby π∶A → B(H)the universal representation ofA, the reducedtwistedcrossedprod-
+uct associated with (A,Γ,α,σ) is the C∗-subalgebra A ⋊α,σ,red Γ ⊆ B(H ⊗ ℓ2(Γ)) generated by
+the elements πα(a)λσ(g) for a ∈ A, g ∈ Γ, where πα∶A → B(H ⊗ ℓ2(Γ)) is given by
+πα(a)(ξ ⊗ δg) = π(α−1
+g (a))ξ ⊗ δg
+and λσ is the twisted regular representation given by
+λσ(γ)(ξ ⊗ δg) = π(σ((γg)−1,γ))ξ ⊗ δγg .
+Here a ∈ A, γ,g ∈ Γ and ξ ∈ H. If A = C, then the reduced twisted crossed product C∗-algebra is
+equal to the reduced twisted group C∗-algebra where the twist is given by the 2-cocycle associated
+to the twisted C∗-dynamical system.
+The ∗-algebra generated by the elements πα(a)λσ(g) for a ∈ A, g ∈ Γ can be equipped with
+the ℓ1-norm
+∥∑
+g∈Γ
+πα(ag)λσ(g)∥
+ℓ1
+= ∑
+g∈Γ
+∥ag∥.
+Its completion with respect to this norm is the twisted ℓ1-crossed product A ⋊α,σ,ℓ1 Γ.
+Convention 2.4. Below we will need to consider restrictions of a twisted C∗-dynamical system
+(A,Γ,α,σ) to subgroups Λ ≤ Γ. For notational ease, we will denote the restrictions of α and σ by
+the same symbols.
+2.5
+Twisted groupoid algebras
+For material on étale Hausdorff groupoids and groupoid twists we refer the reader to [SSW20]. We
+recall the definition of a twist over a groupoid and its convolution algebra, which will be used in
+this article.
+Definition 2.5. Let G be an étale groupoid. A twist over G is a sequence
+G(0) × S1
+E
+G
+i
+q
+where G(0) × S1 is the trivial group bundle with fibres S1, where E is a locally compact Hausdorff
+groupoid with unit space i(G(0) ×{1}), and i and q are continuous groupoid homomorphisms that
+restrict to homeomorphisms of unit spaces, such that
+• i is injective,
+• E is a locally trivial G-bundle, that is for every point α ∈ G there is an open neighbourhood
+U that is a bisection and on which there exists a continuous section S∶U �→ E satisfying
+q ○S = idU, and such that the map (α,µ) ↦ i(r(α),µ)S(α) is a homeomorphism of U ×S1
+onto q−1(U)
+• i(G(0) × T) is central in E, that is i(r(g),µ)g = gi(s(g),µ) holds for all g ∈ E and µ ∈ S1,
+and
+5
+
+• q−1(G(0)) = i(G(0) × S1).
+Notation 2.6. A twist as in the definition above will be denoted by (E,i,q) or simply by E if no
+confusion is possible. Further, we will frequently identify the unit space of E and G.
+Given a twist q∶E ↠ G over a locally compact étale Hausdorff groupoid G we write
+C(G,E) ∶= {f ∈ Cc(E) ∣ f(µ ⋅ g) = µf(g) for all g ∈ E and µ ∈ S1},
+which becomes a ∗-algebra when equipped with the following convolution product and involution
+[Kum86, Proposition 5]. We consider the action S1 ↷ C × E given by µ(z,g) = (µz,µg) and let
+C×S1 E be the quotient, which is a complex line bundle over G. It carries a partially defined product
+(that is, it is a small category) given by [z1,g1][z2,g2] = [z1z2,g1g2] for any pair of composable
+elements g1,g2 ∈ E. The space C(G,E) is isomorphic with the space of sections Γ(C ×S1 E → G)
+by mapping f ∈ C(G,E) to the section q(g) ↦ [f(g),g]. This is well-defined since (f(µg),µg) =
+(µf(g),µg) = µ(f(g),g) holds. The space Γ(C ×S1 E) carries the natural involution f ∗(g) =
+f(g−1) and the convolution product
+f1 ∗ f2(g) = ∑
+g1g2=g
+f1(g1)f2(g2),
+for f,f1,f2 ∈ Γ(C ×S1 E) and g ∈ G.
+Remark 2.7. The attentive reader will have noticed that our conventions for C(E,G) slightly differ
+from the usual requirement that f(µg) = µf(g). This goes hand-in-hand with the divergence from
+Kumjian’s convention µ(z,g) = (µz,µg). Our choice of conventions is justified by the following
+example, which is the basis for understanding the construction presented in Section 5.
+Given a discrete group Γ and a cocycle σ ∈ Z2(Γ,S1) one associates the central extension S1 ↪
+Γ ×σ S1 ↠ Γ, where the product in Γ ×σ S1 is given by (γ1,µ1)(γ2,µ2) = (γ1γ2,σ(γ1,γ2)µ1µ2).
+We observe that (Γ,Γ×σS1) is a twisted groupoid and one expects an identification C(Γ,Γ×σ S1) ≅
+C[Γ,σ]. This is the case with our conventions, while the usual conventions yield the anticipated
+natural isomorphism C(Γ,Γ ×σ S1) ≅ C[Γ,σ].
+Let us elaborate. For γ ∈ Γ, we define the section fγ(γ′) = δγ,γ′[1,γ,1] ∈ C ×S1 (Γ ×σ S1).
+Observe that the function in C(Γ,Γ ×σ S1) associated with it is the unnatural map (γ,µ) ↦ µ. We
+show that the map γ ↦ fγ is σ-multiplicative. Indeed, for γ1,γ2,γ′ ∈ Γ we make the calculation
+fγ1 ∗ fγ2(γ′) =
+∑
+γ′=g1g2
+fγ1(g1)fγ2(g2)
+= δγ′,γ1γ2[1,γ1,1][1,γ2,1]
+= δγ′,γ1γ2[1,γ1γ2,σ(γ1,γ2)]
+= δγ′,γ1γ2[σ(γ1,γ2),γ1γ2,1]
+= σ(γ1,γ2)δγ′,γ1γ2[1,γ1γ2,1]
+= σ(γ1,γ2)fγ1γ2(γ′).
+This justifies our conventions sufficiently.
+Convention 2.8. If Γ is a group, then a twist over Γ is the same as an extension 1 → S1 → E →
+Γ → 1 and hence up to choice of a section Γ → E the same as an element in H2(Γ,S1). In view
+of Remark 2.7, in some situations we continue to use the notation C∗
+red(Γ,E) for the associated
+twisted group C∗-algebra.
+6
+
+We will also use the following completions of the twisted groupoid algebra C(G,E). We denote by
+• L1(G,E) its I-norm completion, which is a Banach ∗-algebra,
+• C∗(G,E) the enveloping C∗-algebra of L1(G,E), and
+• C∗
+red(G,E) the reduced C∗-algebra completion of L1(G,E).
+For a locally compact étale Hausdorff groupoid G we denote the interior of its isotropy groupoid by
+IG. For x ∈ G(0), let IG
+x be the group appearing in the fibre x of IG. It has been shown in [Arm22,
+Corollary 2.11] that for a twist E over a locally compact étale Hausdorff groupoid G
+• the interior of the isotropy subgroupoid IE is a twist over the interior of the isotropy sub-
+groupoid IG, and
+• for each x ∈ G(0), the isotropy group IE
+x is a twist over the isotropy group IG
+x .
+We will apply the following result on the ideal intersection property for twisted groupoid C∗-
+algebras associated with a twist over a locally compact étale Hausdorff groupoid and its restriction
+to the interior of the isotropy bundle. We summarise results from [Arm22, Proposition 6.1 and
+Theorem 6.3], which generalised previous work in the untwisted case published in [Bro+16].
+Theorem 2.9 ([Arm22]). Let E be a twist over a second-countable locally compact étale Hausdorff
+groupoid G. There is an injective ∗-homomorphism ι∶C∗
+red(IG,IE) → C∗
+red(G,E) such that
+ι(f)(g) =
+⎧⎪⎪⎨⎪⎪⎩
+f(g)
+if g ∈ IE
+0
+if g /∈ IE ,
+for all f ∈ C(IG,IE) and all g ∈ E. The image of ι has the ideal intersection property in C∗
+red(G,E).
+3
+The ℓ1-ideal intersection property for twisted C*-algebraic
+dynamical systems: basic results
+We will in later sections need to consider the ideal intersection property in the setting of twisted
+crossed products. In this section we therefore define the ideal intersection property in this gener-
+ality, before deriving some useful reformulations and results which come in handy later.
+Definition 3.1. A twisted C∗-dynamical system (A,Γ,α,σ) is said to have the ℓ1-ideal intersec-
+tion property if every non-zero ideal A ⋊α,σ,red Γ has non-zero intersection with A ⋊α,σ,ℓ1 Γ.
+In situations, where partof the twistedaction is trivial, e.g. fortwistedgroup C∗-algebras associated
+with a pair (Γ,σ) or untwisted crossed products associated with an action Γ ↷ X, we simplify
+notation and say that (Γ,σ), respectively Γ ↷ X has the ideal intersection property.
+Remark 3.2. Let us put the notion introduced in the previous definition into context.
+• Every twisted C∗-dynamical system with a simple crossed product trivially has the ℓ1-ideal
+intersection property. Such systems arise from C∗-simple groups [BK18].
+7
+
+• For amenable twisted C∗-dynamical system (A,Γ,α,σ) the ℓ1-ideal intersection property
+is equivalent to C∗-uniqueness of A⋊α,σ,ℓ1 Γ. This can be inferred from the fact that reduced
+and universal crossed products of such systems coincide combined with [Bar83, Proposition
+2.4]. Alternatively, Proposition 3.3 below can be employed.
+The following reformulation shows that the ℓ1-ideal intersection property for twisted
+C∗-dynamical systems is a question of minimality of the reduced C∗-algebra norm on A ⋊ℓ1,σ Γ. It
+will be frequently used without further reference.
+Proposition 3.3. Let (A,Γ,α,σ) denote a twisted C∗-dynamical system. The following conditions are
+equivalent.
+(i) (A,Γ,α,σ) has the ℓ1-ideal intersection property.
+(ii) If a ∗-homomorphism into a C∗-algebra π∶A ⋊α,σ,red Γ → B is injective on A ⋊α,σ,ℓ1 Γ then it is
+injective itself.
+(iii) The reduced C∗-norm on A ⋊α,σ,ℓ1 Γ is minimal.
+Proof. Suppose that (A,Γ,α,σ) has the ℓ1-ideal intersection property and let π∶A ⋊α,σ,red Γ → B
+be injective on A ⋊α,σ,ℓ1 Γ. Then ker π ∩ A ⋊α,σ,ℓ1 Γ = {0} implies that ker π = 0. So π itself is
+injective.
+Assume next that Item (ii) holds and let ν∶A ⋊α,σ,ℓ1 Γ → R≥0 be a C∗-norm dominated by the
+reduced C∗-norm. Denoting by B = A ⋊α,σ,ℓ1 Γ
+ν the completion, the natural ∗-homomorphism
+π∶A⋊α,σ,red Γ → B is faithful on the ℓ1-crossed product. The assumption implies that π is injective
+and henceforth isometric. Thus, ν is equal to the reduced C∗-norm.
+Now assume that Item (iii) holds and let I ⊴ A⋊α,σ,red Γ be a non-zero ideal with quotient map
+π∶A⋊α,σ,redΓ → B. The assumption allows us to infer that ker π∩A⋊α,σ,ℓ1Γ ≠ {0}. Since I = ker π
+and I was arbitrary, we conclude that (A,Γ,α,σ) has the ℓ1-ideal intersection property.
+Remark 3.4. The analogue of the minimality of the reduced C∗-norm featuring in Item (iii) of
+Proposition 3.3 has previously been introduced for algebraic group rings in [AK19] under the name
+C∗
+r -uniqueness.
+We next show that the ℓ1-ideal intersection property is closed under directed unions, in the follow-
+ing precise sense.
+Proposition 3.5. Let (A,Γ,α,σ) be a twisted C∗-dynamical system. Assume that Γ = ⋃i∈I Γi is a
+directed unionsuch that (A,Γi,α,σ)hastheℓ1-ideal intersectionpropertyfor all i ∈ I. Then(A,Γ,α,σ)
+has the ℓ1-ideal intersection property.
+Proof. We employ the characterisation in Item (ii) of Proposition 3.3 of the ℓ1-ideal intersection
+property. Let π∶A⋊α,σ,red Γ → B be a ∗-homomorphism whose restriction to the ℓ1-crossed prod-
+uct is injective. The assumptions imply that for all i ∈ I the restriction π∣A⋊α,σ,redΓi is injective and
+thus isometric. Since ⋃i∈I A ⋊α,σ,red Γi is dense in A ⋊α,σ,red Γ, the result follows.
+8
+
+4
+The L1-ideal intersection property for twisted groupoids
+and their isotropy bundles
+In this section we prove the L1-ideal intersection property for certain twisted groupoids, in the
+same spirit as [AO22] did for cocycle twisted groupoids. In view of applications in Section 6, our
+statements are established in slightly greater generality.
+Definition 4.1. A twisted locally compact étale Hausdorff groupoid (G,E) is said to have the L1-
+ideal intersection property if every non-zero ideal of C∗
+red(G,E) has non-zero intersection with
+L1(G,E).
+The next result shows that in order to establish the L1-ideal intersection property for a twisted
+groupoid, it suffices to study its isotropy bundle. It is a direct consequence of Armstrong’s results
+recalledin Section 2.5, andits analogue in the contextofC∗-uniqueness of cocycle twistedgroupoid
+C∗-algebras was obtained in [AO22, Proposition 3.2].
+Proposition 4.2. Let G be a second-countable locally compact étale Hausdorff groupoid and let E be a
+twist over G. If (IG,IE) has the L1-ideal intersection property, then so does (G,E).
+Proof. Let I ⊴ C∗
+red(G,E) be a non-zero ideal. Appealing to the work of [Arm22] described in
+Theorem 2.9 and identifying C∗
+red(IG,IE) with its image in C∗
+red(G,E), we find a that J = I ∩
+C∗
+red(IG,IE) is a non-zero ideal. By assumption of the proposition, we can thus conclude that
+0 ≠ J ∩ L1(IG,IE) ⊆ I ∩ L1(G,E)
+which completes the proof of the proposition.
+In the remainder of this section we aim to prove that if sufficiently many fibres of the isotropy
+bundle have the ℓ1-ideal intersection property, then the full isotropy bundle has the L1-ideal inter-
+section property. Following the same strategy as in [AO22], we achieve this by decomposing any
+C∗-completion of L1(IG,IE) as a C∗-bundle over G(0). We will need the following lemma in or-
+der to describe the fibres of this bundle. It generalises [AO22, Lemma 3.4], but we give a shorter
+proof which applies in greater generality, which is later needed in the proof of Theorem 6.5. Given
+a groupoid G, we call x ∈ G(0) strongly fixed if Gx = IG
+x .
+Lemma 4.3. Let E be a twist over a locally compact étale Hausdorff groupoid G. Assume that x ∈ G(0)
+is a strongly fixed point and denote by resx∶L1(G,E) �→ L1(Gx,Ex) the restriction map and by Ix its
+kernel. Then resx is a continuous ∗-homomorphism which induces an isometric ∗-isomorphism between
+L1(G,E)/Ix and L1(Gx,Ex). Further, Ix is the ideal generated by C0(G(0) ∖ {x}).
+Proof. It is clear that resx is continuous and in order to show that it induces an isometric
+∗-isomorphism, it suffices to show that resx∣C(G,E) factors through to an isometry with dense image.
+We first prove density. Let fx ∈ C(Gx,Ex) be arbitrary. Considering Ex ⊆ E as a closed subset
+and making use of local compactness of the latter, Tietze’s theorem provides some function ˜fx ∈
+Cc(E) such that ˜fx∣Ex = fx. Define
+f(g) = ∫
+S1
+µ ˜fx(µg)dµ.
+9
+
+Then f ∈ C(G,E) holds thanks to invariance of the Haar measure, and f∣Ex = fx by S1-equivariance
+of fx. This proves density of the image.
+Given f ∈ C(G,E)andε > 0 there is a neighbourhoodU ⊆ G(0) of xsuchthatsuppf∩s−1(U) ⊆
+IE and ∣∥resy(f)∥−∥resx(f)∥∣ < ε for all y ∈ U. Since G(0) is locally compact, by Tietze’s theorem
+there is g ∈ C(G(0)) with 0 ≤ g ≤ 1, g∣G(0)∖U ≡ 1 and g(x) = 0. Then f ∗ g ∈ Ix and we find that
+∥f + Ix∥ ≤ ∥f − f ∗ g∥ ≤ sup
+y∈U
+∥resy(f)∥ ≤ ∥resx(f)∥ + ε.
+Further,
+∥resx(f)∥ = inf
+h∈Ix ∥resx(f + h)∥ ≤ inf
+h∈Ix ∥f + h∥ = ∥f + Ix∥.
+It remains to show that Ix is equal to the ideal J generated by C0(G(0) ∖{x}) in L1(G,E). If f ∈ Ix
+and ε > 0, there is ˜f ∈ C(G,E) such that ∥f − ˜f∥I < ε. Thus ∥resx( ˜f)∥ < ε and hence we find as
+above g ∈ C0(G(0) ∖ {x}) such that ∥ ˜f − ˜f ∗ g∥I < ε. This implies that ∥f − ˜f ∗ g∥ < 2ε. Since
+˜f ∗ g ∈ J and ε > 0 was arbitrary, this finishes the proof.
+We are now ready to prove the main result of this section, which generalises [AO22, Theorem 3.1].
+It is stated and proven in the generality needed for Theorem 6.5. Extending usual conventions and
+accepting zero-fibres, for a ∗-homomorphism C0(X) → Z(M(A)), we denote by Ax the quotient
+of A by the ideal generated by the image of C0(X ∖ {x}).
+Theorem 4.4. Let E be a twist over a second-countable locally compact étale Hausdorff groupoid G. As-
+sume that there is a dense subset D ⊆ G(0) such that ℓ1(IG
+x ,IE
+x ) ⊆ C∗
+red(IG
+x ,IE
+x) has the ideal inter-
+section property for all x ∈ D. Let π∶C∗
+red(G,E) → A be a ∗-homomorphism into a C∗-algebra. If
+πx∶C∗
+red(IGx ,IEx ) → π(C∗
+red(IG,IE))x restricts to an injection of ℓ1(IGx ,IEx ) for all x ∈ D, then π is
+injective.
+Proof. Let π∶C∗
+red(G,E) → A and D ⊆ G(0) be as in the statement of the theorem. Without loss
+of generality, we may assume that π is non-degenerate. By Theorem 2.9, it suffices to show that
+π∣C∗
+red(IG,IE) is injective. Since πx∣ℓ1(IG
+x ,IEx ) is injective for all x ∈ D, it is in particular non-zero,
+so that density of D ⊆ G(0) implies that π∣C0(G(0)) is injective. Hence B = π(C∗
+red(IG,IE)) is a
+C0(G(0))-algebra. Denote by B = (Bx)x the upper semi-continuous C∗-bundle associated with it
+by [Nil96, Theorem 2.3], which recovers B as the algebra of sections B ≅ Γ0(B).
+By Lemma 4.3, we obtain the following commutative diagram upon taking quotients by the
+ideal generated by C0(G(0) ∖ {x}) in each algebra of its top row.
+L1(IG,IE)
+C∗
+red(IG,IE)
+B
+ℓ1(IGx ,IEx )
+C∗
+red(IGx ,IEx )
+Bx
+πx
+For x ∈ D, the ∗-homomorphism ℓ1(IG
+x ,IE
+x ) → Bx is injective and (IG
+x ,IE
+x ) has the ℓ1-ideal in-
+tersection property. So πx is an isomorphism of C∗-algebras and as such an isometry. Let now
+f ∈ Γ0(B) be an element in the image of L1(IG,IE). Then
+∥f∥B = sup
+x∈G(0) ∥f(x)∥Bx ≥ sup
+x∈D
+∥f(x)∥Bx = sup
+x∈D
+∥f(x)∥C∗
+red(IG
+x ,IEx ) = ∥f∥C∗
+red(IG,IE)
+since the regular representations of (IG,IE) are continuous by construction [Kum86, Section 2].
+10
+
+Corollary 4.5. Let E be a twist over a second-countable locally compact étale Hausdorff groupoid G.
+Assume that there is a dense subset D ⊆ G(0) such that (IG
+x ,IE
+x ) has the ℓ1-ideal intersection property for
+all x ∈ D. Then (G,E) has the L1-ideal intersection property.
+Proof. Let π∶C∗
+red(G,E) → A be a ∗-homomorphism that is injective on L1(G,E) and write
+B = π(C∗
+red(IG,IE)). In order to prove injectivity of π, by Theorem 4.4, it suffices to check that
+πx∶C∗
+red(IGx ,IEx ) → Bx is injective when restricted to ℓ1(IGx ,IEx ) for all x ∈ G(0). By Lemma 4.3,
+taking the quotient by the ideal generated by C0(G(0) ∖ {x}) in the inclusion L1(G,E) ↪ B, we
+indeed obtain the desired inclusion ℓ1(IGx ,IEx ) ↪ Bx, which finishes the proof.
+5
+Groupoid C*-algebras from abelian normal subgroups
+In this section we describe a twisted groupoid associated with an inclusion of a normal abelian
+subgroup into a discrete group endowed with an S1-valued 2-cocycle. This construction should be
+folklore, but has not been presented explicitly to our knowledge.
+Definition 5.1. Let A ⊴ Γ be a normal abelian subgroup of a discrete group. A cocycle σ ∈
+Z2(Γ,S1) is A-admissible if it satisfies
+• σ∣A×A ≡ 1, and
+• σ(γ,a)σ(γa,γ−1) = 1 = σ(a,γ−1)σ(γ,aγ−1) for all γ ∈ Γ and a ∈ A.
+Let A ⊴ G and σ be as above. Write Λ = Γ/A and consider the action Λ
+α↷ A given by
+αλ(a) = γaγ−1 for γA = λ. Since A is abelian, this is well-defined. Denote by G = Λ ⋉ ˆA the
+transformation groupoid associated with the dual action of α. Further, let Γ ⋉σ (S1 × ˆA) be the
+twisted transformation groupoid whose product is given by
+(γ1,µ1,γ2χ)(γ2,µ2,χ) = (γ1γ2,µ1µ2σ(γ1,γ2),χ)
+for γ1,γ2 ∈ Γ, µ1,µ2 ∈ S1 and χ ∈ ˆA. and consider
+N = {(a−1,χ(a),χ) ∣ a ∈ A,χ ∈ ˆA} ⊆ Γ ⋉σ (S1 × ˆA).
+The following lemma describes a twisted groupoid associated to the tuple (Γ,A,σ).
+Lemma 5.2. The set N ⊆ Γ ⋉σ (S1 × ˆA) is a closed normal subgroupoid. Further, Γ ⋉σ (S1 × ˆA)/N is
+a twist over G.
+Proof. It follows from the fact that evaluation of characters in ˆA is continuous, that N is closed.
+Further, it is multiplicatively closed since σ∣A×A ≡ 1 and the calculation
+(a−1,χ(a),χ)−1 = (a,χ(a)σ(a−1,a),χ) = (a,χ(a−1),χ)
+for a ∈ A and χ ∈ ˆA shows that N is also closed under inverses. So it is a closed subgroupoid of
+Γ ⋉σ (S1 × ˆA). We next check normality of N. Thanks to centrality of S1 it suffices to observe for
+a ∈ A, γ ∈ Γ and χ ∈ ˆA that
+(γ,1,χ)(a−1,χ(a),χ)(γ−1,1,γχ) = (γa−1γ−1,χ(a)σ(γ,a−1)σ(γa−1,γ−1),γχ)
+= (γa−1γ−1,γχ(γaγ−1)),γχ).
+11
+
+We now want to show that the quotient E = Γ ⋉σ (S1 × ˆA)/N is a twist over G. The inclusion
+{e} × S1 × ˆA ⊆ Γ ⋉σ (S1 × ˆA) descends to an inclusion i∶S1 × ˆA �→ E since N ∩ ({e} × S1 × ˆA) =
+{(e,1)} × ˆA. Further, the projection onto the first and last component Γ ⋉σ (S1 × ˆA) �→ Γ × ˆA
+induces a continuous quotient map q∶E �→ Γ/A ⋉ ˆA = G. It is clear that i(S1 × ˆA) is central in
+E and that q−1({eA} × ˆA) = i(S1 × ˆA). What remains to be shown is that E is locally trivial. Let
+(γA,χ0) ∈ G and consider the open bisection U = {γA} × ˆA. The map S∶U �→ E∶(γA,χ) ↦
+(γ,1,χ) is continuous and satisfies q ○ S = idU. Further,
+q−1(U) = {[γa,µ,χ] ∈ E ∣ µ ∈ S1,a ∈ A,χ ∈ ˆA}
+= {[γ,µ,χ] ∈ E ∣ µ ∈ S1,χ ∈ ˆA}
+is naturally isomorphic with S1 × U.
+Let us introduce some notation in order to refer to the twisted groupoid just constructed.
+Definition 5.3. Given a group Γ with a normal abelian subgroup A and an A-admissible 2-cocycle
+σ ∈ Z2(Γ,S1), we denote the associated twisted groupoid by
+G(Γ,A,σ) = Γ/A ⋉ ˆA
+E(Γ,A,σ) = Γ ⋉σ (S1 × ˆA)/{(a−1,χ(a),χ) ∣ a ∈ A,χ ∈ ˆA}.
+We next identify the twisted group algebras associated to (Γ,σ) with the twisted groupoid algebra
+associated with a normal abelian subgroup A ⊴ Γ for which σ is admissible. This proposition
+generalises the identification described in Remark 2.7.
+Proposition 5.4. Let A ⊴ Γ be an abelian normal subgroup of a discrete group and σ ∈ Z2(Γ,S1) an A-
+admissible cocycle. Let (G,E) = (G(Γ,A,σ),E(Γ,A,σ)) be the associated twisted groupoid and write
+elements of C ×S1 E as equivalence classes [z,γ,µ,χ] with z ∈ C, γ ∈ Γ, µ ∈ S1 and χ ∈ ˆA. Given γ ∈ Γ
+define the following section of C ×S1 E ↠ G:
+fγ(gA,χ) =
+⎧⎪⎪⎨⎪⎪⎩
+[1,γ,1,χ]
+if gA = γA
+0
+otherwise.
+Then the map γ ↦ fγ
+(i) extends to a contractive embedding ℓ1(Γ,σ) ↪ L1(G,E), which
+(ii) extends to an isomorphism C∗
+red(Γ,σ) ↪ C∗
+red(G,E).
+Proof. We first show that the map γ → fγ is σ-twisted multiplicative. For γ1,γ2,g ∈ Γ and χ ∈ ˆA,
+we find that
+fγ1 ∗ fγ2(gA,χ) =
+∑
+(g1A)(g2A)=gA
+fγ1(g1A,g2χ)fγ2(g2A,χ)
+=
+∑
+(g1A)(g2A)=gA
+g1A=γ1A, g2A=γ2A
+[1,g1,1,g2χ][1,g2,1,χ]
+=
+⎧⎪⎪⎨⎪⎪⎩
+[1,γ1,1,γ2χ][1,γ2,1,χ] = [1,γ1γ2,σ(γ1,γ2),χ]
+if gA = γ1γ2A
+0
+otherwise
+= σ(γ1,γ2)fγ1γ2(gA,χ).
+12
+
+Since fe is the neutral element for the convolution product, this shows that the map γ ↦ fγ extends
+to a unital ∗-homomorphism C[Γ,σ] → L1(G,E).
+We next show that this ∗-homomorphism extends to a contraction ℓ1(Γ,σ) → L1(G,E). To
+this end, we need to identify the functions ˜fγ ∈ C(G,E) associated with fγ. We claim that
+˜fγ([g,µ,χ]) =
+⎧⎪⎪⎨⎪⎪⎩
+µχ(g−1γ)
+if γA = gA
+0
+otherwise.
+Indeed, for γA = gA, µ ∈ S1 and χ ∈ ˆA we calculate
+[µχ(g−1γ),g,µ,χ] = [χ(g−1γ),γγ−1g,1,χ] = [χ(g−1γ),γ,χ(γ−1g),χ] = [1,γ,1,χ].
+Take now ∑γ∈Γ cγuγ ∈ C[Γ,σ]. Then
+sup
+χ∈ ˆ
+A
+∥ ∑
+γ∈Γ
+cγ ˜fγ∥ℓ1(Gχ) = sup
+χ∈ ˆ
+A
+∑
+gA∈Γ/A
+∣∑
+γ∈Γ
+cγ ˜fγ([g,1,χ])∣
+≤ sup
+χ∈ ˆ
+A
+∑
+gA∈Γ/A
+∣ ∑
+γ∈gA
+cγχ(γ−1g)∣
+≤ ∑
+γ
+∣cγ∣.
+Similarly, we obtain that
+sup
+χ∈ ˆ
+A
+∥ ∑
+γ∈Γ
+cγ ˜fγ∥ℓ1(Gχ) = sup
+χ∈ ˆ
+A
+∑
+gA∈Γ/A
+∣∑
+γ∈Γ
+cγ ˜fγ([g,1,g−1χ])∣
+≤ sup
+χ∈ ˆ
+A
+∑
+gA∈Γ/A
+∣ ∑
+γ∈gA
+cγχ(gγ−1)∣
+≤ ∑
+γ
+∣cγ∣.
+Together, these calculations show that ∥∑γ cγ ˜fγ∥I ≤ ∥∑γ cγuγ∥ℓ1(Γ). So indeed, we obtain a con-
+traction ℓ1(Γ,σ) → L1(G,E).
+We now show that the contraction above extends to a ∗-isomorphism C∗
+red(Γ,σ) ≅ C∗
+red(G,E).
+This will imply in particularthatthe map ℓ1(Γ,σ) → L1(G,E)is injective. Considerthe conditional
+expectation E∶C∗
+red(G,E) → C( ˆA) given by restriction of functions in C(G,E). Further, denote by
+∫ dχ the Haar integral on ˆA. We observe that for every γ ∈ Γ, we have
+∫ dχ ○ E(fγ) = ∫
+ˆ
+A
+˜fγ([e,1,χ])dχ
+=
+⎧⎪⎪⎨⎪⎪⎩
+∫ ˆ
+A χ(γ)dχ
+if γ ∈ A
+0
+otherwise
+=
+⎧⎪⎪⎨⎪⎪⎩
+1
+if γ = e
+0
+otherwise.
+13
+
+This shows that we obtain an isometric ∗-homomorphism C∗
+red(Γ,σ) → C∗
+red(G,E) and it remains
+to argue that it has dense image. To this end it suffices to show that for every gA ∈ Γ/A and every
+section f∶G → C×S1 E supported on {gA}× ˆA lies in the image of C∗
+red(Γ,σ). Let f ˆ
+A∶ ˆA → C be the
+unique continuous function such that f(gA,χ) = [f ˆ
+A(χ),g,1,χ] for all χ ∈ ˆA. We can identify
+f ˆ
+A with an element in C(G,E), and find that f = fg ∗ f ˆ
+A, which finishes the proof.
+Let us next describe the isotropy groups and the associated twists. Recall that for a group action
+Γ ↷ X and x ∈ X, the subgroup Γ○x = {γ ∈ Γ ∣ ∃U open ∶ x ∈ U,γ∣U = idU} is the neighbourhood
+stabiliser of x in Γ.
+Proposition 5.5. Let A ⊴ Γ be an abelian normal subgroup and σ ∈ Z2(Γ,S1) an A-admissible cocycle.
+Let (G,E) be the associated twisted groupoid. Then the fibre of (IG,IE) at χ ∈ ˆA is given by the quotient
+Γ○χ ×σ S1/N → Γ○χ/A obtained from Γ○χ ×σ S1 → Γ○χ/A by dividing out the normal subgroup N =
+⟪(a,χ(a) ∣ a ∈ A⟫ ⊴ Γ○χ ×σ S1.
+Furthermore, given a section s∶Γ○χ/A → Γ○χ and the associated 2-cocycle ρ ∈ Z2(Γ○χ/A,A), we define
+a section ˜s∶Γ○
+χ/A → Γ○
+χ ×σ S1/N by ˜s(h) = [s(h),1]. Then the associated S1-valued 2-cocycle is
+(χ ○ ρ) ⋅ (σ ○ (s × s)).
+Proof. It is clear that IGχ = Γ○χ/A. We can thus calculate the fibre
+IE
+χ = {[γ,µ,χ] ∣ γ ∈ Γ○
+χ,µ ∈ S1} ≅ Γ○
+χ ×σ S1/N .
+Now fix a section s∶Γ○
+χ/A → Γ○
+χ and define ˜s(h) = [s(h),1] as in the statement of the theorem.
+For h1,h2 ∈ Γ○χ/A, using the fact that χ is fixed by Γ○χ, we find that
+˜s(h1)˜s(h2) = [s(h1),1][s(h2),1]
+= [ρ(h1,h2)s(h1h2),σ(s(h1),s(h2))]
+= [s(h1h2)(s(h1h2)−1ρ(h1,h2)s(h1h2)),σ(s(h1),s(h2))]
+= [s(h1h2),(χ ○ ρ(h1,h2)) ⋅ (σ(s(h1),s(h2)))]
+= (χ ○ ρ(h1,h2)) ⋅ (σ(s(h1),s(h2)))˜s(h1h2).
+This shows that (χ ○ ρ) ⋅ (σ ○ (s × s)) is indeed a 2-cocycle and that it is the extension cocycle
+associated with ˜s.
+6
+Proof of the main results
+In this section we prove all main results described in the introduction. We start with three lemmas,
+which will be used in the proof of Theorem 6.5.
+Lemma 6.1. Let Γ be a group whose subgroups all have the ℓ1-ideal intersection property. Then any
+action of Γ on a locally compact Hausdorff space has the ℓ1-ideal intersection property.
+Proof. This directly follows from Corollary 4.5 applied to the transformation groupoid Γ⋉X with
+a trivial twist.
+Lemma 6.2. Let Γ be finite-by-(C∗-simple) and σ ∈ Z2(Γ,S1). Then (Γ,σ) satisfies the ℓ1-ideal inter-
+section property.
+14
+
+Proof. Let F ⊴ Γ be a finite normal subgroup such that Λ = Γ/F is C∗-simple and let σ ∈ Z2(Γ,S1).
+After a choice of section s∶Λ → Γ satisfying s(e) = e, we infer from [PR89, Theorem 4.1] that
+C∗
+red(Γ,σ) ≅ C[F,σ] ⋊α,ρ,red Λ, where the twisted crossed product is defined with respect to the
+maps
+α∶Λ → Aut(C[F,σ])∶αh(uf) = σ(s(h),f)σ(s(h)fs(h)−1,s(h))us(h)fs(h)−1
+ρ∶Λ × Λ → U(C[F,σ])∶
+σ(h1,h2) = σ(s(h1),s(h2))σ(s(h1)s(h2)s(h1h2)−1,s(h1h2))us(h1)s(h2)s(h1h2)−1 .
+Inspection of the proof of [PR89, Theorem 4.1] shows that moreover the inclusion ℓ1(Γ,σ) ⊆
+C∗
+red(Γ,σ) is isomorphic with the inclusion of twisted crossed products C[F,σ] ⋊α,ρ,ℓ1 Λ ⊆
+C[F,σ] ⋊α,ρ,red Λ. So it suffices to show that C[F,σ] ⊆ C[F,σ] ⋊α,ρ,red Λ satisfies the ideal in-
+tersection property.
+Since C[F,σ] is finite dimensional, it is a multi-matrix algebra and hence the twisted
+C∗-dynamical system (C[F,σ],Λ,α,ρ) decomposes as a direct sum of Λ-simple dynamical sys-
+tems, say C[F,σ] ≅ ⊕n
+i=1 Ai. We can apply [BK18, Corollary4.4] to infer that Ai⋊α,ρ,redΛ is simple.
+So ideals of C[F,σ] ⋊α,ρ,red Λ are precisely of the form
+I = ⊕
+i∈S
+(Ai ⋊α,ρ,red Λ)
+for some subset S ⊆ {1,... ,n}. If I ∩ C[F,σ] = {0}, then S = ∅ follows, which in turn implies
+I = 0. This finishes the proof of the lemma.
+For the next lemma recall the notion of admissible cocycles from Definition 5.1.
+Lemma 6.3. Let A ⊴ Γ be a normal finitely generated abelian subgroup and let σ ∈ Z2(Γ,Z/nZ). There
+is a finite index characteristic subgroup B ≤ A and a B-admissible cocycle ρ ∈ Z2(Γ,Z/nZ) equivalent
+to σ.
+Proof. Denote by o = ∣Tors(A)∣ the order of the torsion subgroup of A and let B ≤ A be the in-
+tersection of all its finite index subgroups of index o. Then B has finite index, since A is finitely
+generated, and B is characteristic in A. Also B is a finitely generated torsion-free abelian group
+so that the isomorphism H2(B) ≅ B ∧ B together with the universal coefficient theorem in coho-
+mology imply that σ∣B×B ∈ Z2(B,Z/nZ) is equivalent to a bicharacter. Specifically, there is a map
+ϕ∶B → Z/nZ such that (b1,b2) ↦ σ(b1,b2)−ϕ(b1b2)+ϕ(b1)+ϕ(b2) is a bicharacter. Extending
+ϕ to a map ˜ϕ∶Γ → Z/nZ, we may replace σ by an equivalent 2-cocycle ρ satisfying
+ρ(γ1,γ2) = σ(γ1,γ2) − ˜ϕ(γ1γ2) + ˜ϕ(γ1) + ˜ϕ(γ2).
+Let i be the index of the finite index subgroup {b ∈ B ∣ ∀b′ ∈ B ∶ σ(b,b′) = σ(b′,b) = 0} ≤ B.
+We denote by C the intersection of all subgroups of B with index i, which is of finite index and
+characteristic in B. Consider now the central extension
+Z/nZ ↪ ˜Γ ↠ Γ
+associated with ρ. Since C is torsion-free, its preimage in ˜Γ is isomorphic with C ⊕ Z/nZ in such
+a way that the action of Γ on it is given by αγ(c,k) = (γcγ−1,σ(γ,c) + σ(γc,γ−1)) for all γ ∈ Γ,
+c ∈ C. In particular, since Z/nZ has exponent n, we find that
+(γcnγ−1,σ(γ,cn) + σ(γcn,γ−1)) = αγ((cn,0)) = αγ((c,0))n = ((γcγ−1)n,0).
+15
+
+This implies that the subgroup D = ⟨cn ∣ c ∈ C⟩ ≤ C satisfies ρ(γ,d) + ρ(γd,γ−1) = 0 for all γ ∈ Γ
+and d ∈ D. By definition D ≤ C is characteristic. Further it has finite index, because C is finitely
+generated abelian.
+The next definition describes the groups for which we prove the ℓ1-ideal intersection property in
+the subsequent theorem.
+Definition 6.4. We denote by U the class of all discrete groups Γ such that the following three
+conditions hold for every finitely generated subgroup of Λ ≤ Γ:
+• the Furstenberg subgroup of every subgroup of Λ equals its amenable radical,
+• the Tits alternative holds for Λ, and
+• there is l ∈ N such that every solvable subgroup of Λ is polycyclic of Hirsch length at most l.
+We are now ready to prove the main theorem of this work.
+Theorem 6.5. Let Γ be a group from the class U, let X be a locally compact Hausdorff space and let
+Γ ↷ X be an action by homeomorphisms. Further, let σ ∈ Z2(Γ,S1) be a 2-cocycle taking values in a
+finite subgroup of S1. Then (X,Γ,σ) has the ℓ1-ideal intersection property.
+Proof. By Lemma 6.1, it suffices to consider the case where X is a point, that is twisted group
+C∗-algebras.
+The statement is clear for finite groups. For an induction, fix l ≥ 1 and assume that the ℓ1-ideal
+intersection property holds for all 2-cocycles with values in a finite subgroup of S1 on groups in U
+whose polycyclic subgroups all have Hirsch length at most l − 1. Let Γ be a group in U all whose
+polycyclic subgroups have Hirsch length at most l ≥ 1, and let σ ∈ Z2(Γ,S1) be a cocycle with
+values in a finite subgroup of S1, say Z/nZ ⊆ S1. Thanks to Proposition 3.5, we may assume that Γ
+is finitely generated. Let ν∶ℓ1(Γ,σ) → R≥0 be a C∗-norm dominated by ∥ ⋅ ∥red. We denote by A =
+ℓ1(Γ,σ)
+ν the completion with respect to ν. Since Γ ∈ U, its amenable radical is virtually polycyclic.
+If it is finite, we infer from Proposition 2.3 that Γ itself is finite-by-(C∗-simple). So Lemma 6.2 can
+be applied. Otherwise, its maximal polycyclic subgroup Λ ≤ R(Γ) is infinite. Let d be the derived
+length of Λ and observe that Λ(d−1) is a finitely generated abelian group. By Lemma 6.3, there is
+a finite index characteristic subgroup A ≤ Λ(d−1) and an A-admissible cocycle ρ ∈ Z2(Γ,Z/nZ)
+equivalent to σ. Since equivalence of cocycles preserves the isomorphism class of twisted group
+algebras, we may assume that ρ = σ.
+Observe that all the inclusions A ≤ Λ(d−1) ≤ Λ ≤ R(Γ) ≤ Γ are characteristic and hence A ≤ Γ
+is characteristic. In particular, A is normal in Γ.
+Denote by (G,E) the twisted groupoid constructed from (Γ,A,σ) as in Definition 5.3. By
+Proposition 5.4, there is a commutative diagram
+ℓ1(Γ,σ)
+C∗
+red(Γ,σ)
+A
+L1(G,E)
+C∗
+red(G,E)
+≅
+π
+16
+
+We need to prove that π is injective.
+Since finitely generated abelian groups are C∗-unique by Theorem 2.2, the restriction of π to
+C( ˆA) is injective. Let D = Tors( ˆA) be the torsion subgroup of ˆA, which is dense, because A
+is a free abelian group. By Theorem 4.4 it suffices to prove that for all χ ∈ D the induced map
+πχ∶C∗
+red(IG
+χ,IE
+χ) → π(C∗
+red(IG,IE))χ is injective on ℓ1(IG
+χ,IE
+χ).
+Fix χ ∈ D and consider the neighbourhood stabiliser Γ○χ = {g ∈ Γ ∣ ∃U ∋ χ∶g∣U = idU} for the
+action Γ ↷ ˆA. Observe that A ≤ Γ○
+χ. By Proposition 5.5, the inclusion ℓ1(IG
+χ,IE
+χ) ⊆ C∗
+red(IG
+χ,IE
+χ)
+is isomorphic with ℓ1(Γ○χ/A,(χ ○ ρ) ⋅ (σ ○ (s × s))) ⊆ C∗
+red(Γ○χ/A,(χ ○ ρ) ⋅ (σ ○ (s × s))), where
+s∶Γ○
+χ/A → Γ○
+χ is a section and ρ ∈ Z2(Γ○
+χ/A,A) the associated extension cocycle. We write ˜σ =
+(χ ○ ρ) ⋅ (σ ○ (s × s).
+Let (H,F) be the groupoid associated with (Γ○χ,A,σ), where by abuse of notation we still keep
+the notation σ instead of writing σ∣Γ○χ×Γ○χ. We have an inclusion of twisted groupoids (IG,IE) ↪
+(H,F) ↪ (G,E). Let I ⊴ π(C∗
+red(H,F)) be the ideal generated by C0( ˆA∖{χ}) and observe that
+we have a commutative diagram
+π(C∗
+red(IG,IE))
+π(C∗
+red(IG,IE))χ
+π(C∗
+red(H,F))
+π(C∗
+red(H,F))/I
+≅
+We write B = π(C∗
+red(H,F)) and B/I = Bχ. By Lemma 4.3, the kernel of the restriction map
+resχ∶L1(H,F) → ℓ1(Hχ,Fχ) ≅ ℓ1(Γ○
+χ/A, ˜σ) is the ideal J generated by C0(H(0)∖{χ}) = C0( ˆA∖
+{χ}). Using the fact that we have a commutative diagram
+ℓ1(Γ○χ,σ)
+L1(H,F)
+ℓ1(Γ○
+χ/A, ˜σ)
+resχ
+we infer that the injection ℓ1(Γ○χ,σ) ↪ B ⊆ A when dividing by J ∩ ℓ1(Γ○χ,σ) and I = π(J)
+descends to an injection ℓ1(Γ○χ/A, ˜σ) ↪ Bχ. So the induction hypothesis can be applied, since A
+being infinite, the Hirsch length of every subgroup of Γ○
+χ/A is at most l − 1. So we have shown that
+we have a commutative diagram
+ℓ1(Γ○
+χ/A, ˜σ)
+Bχ
+ℓ1(IGχ,IEχ)
+π(C∗
+red(IG,IE))χ
+≅
+≅
+πχ
+which implies what we had to show.
+We now describe several classes of groups to which Theorem 6.5 applies. Our first application
+concerns the large class of acylindrically hyperbolic groups. We remark that the ℓ1-ideal intersec-
+tion property for their group algebras can be deduced directly from Lemma 6.2, while the general
+statement for dynamical systems could be deduced using solely Lemma 6.1 and C∗-uniqueness of
+virtually cyclic groups.
+17
+
+Corollary 6.6. Let Γ be an acylindrically hyperbolic group and Γ ↷ X an action on a locally compact
+Hausdorff space. Then C0(X) ⋊ℓ1 Γ ⊆ C0(X) ⋊red Γ has the ideal intersection property.
+Proof. In orderto apply Theorem 6.5, we needto checkall conditions ofDefinition 6.4. By [DGO17,
+Theorem 2.35] combined with Proposition 2.3 the first condition is satisfied. The second and third
+conditions are satisfied thanks to [Osi16, Theorem 1.1], which shows that subgroups of acylindri-
+cally hyperbolic groups are virtually cyclic or contain a copy of the free group.
+In order to obtain our next class of examples to which our main result applies, we need the fol-
+lowing result, which is folklore. We refer the reader unfamiliar with Lie theory to [OV90, Table 9,
+p. 312-317] for the classification of simple real Lie algebras and their rank, which by definition is
+the dimension of a maximal R-diagonalisable Lie subalgebra.
+Proposition 6.7. Let Γ be a lattice in a connected Lie group. Then there is l ∈ N such that every solvable
+subgroup of Γ is virtually polycyclic and has Hirsch length at most l.
+Proof. Let G be a connected Lie group in which Γ is a lattice. By [Pra76, Lemma 6], there is a normal
+subgroup Λ ⊴ Γ suchthatΛ is virtually a lattice in a connectedsolvable Lie group andΓ/Λ is a lattice
+in a connected semisimple Lie group with trivial centre and without compact factors. By [Rag72,
+Proposition 3.7] every lattice in a connected simply connected solvable Lie group is polycyclic of
+Hirsch length bounded by the dimension of the Lie group. Since every connected solvable Lie group
+is a quotient by a central discrete subgroup of its universal cover, the conclusion applies to lattices
+in arbitrary connected solvable Lie groups. So we may assume for the rest of the proof that Γ is a
+lattice in a connected semisimple Lie group G with trivial centre and without compact factors.
+Passing to a finite index subgroup of Γ, there are direct product decompositions G = ∏n
+i=1 Gi
+and Γ = ∏n
+i=1 Γi such that Γi ≤ Gi is an irreducible lattice [Rag72, Theorem 5.22]. It hence suffices
+to consider the case where Γ ≤ G is already irreducible. Assuming that G is locally isomorphic
+with SO+(n,1) or SU(n,1), the group Γ acts on the hyperbolic boundary of G. Thus, every solv-
+able subgroup of G is virtually cyclic, finishing the proof in this case. Assume that G is not locally
+isomorphic with either SO+(n,1) or SU(n,1). Then the arithmeticity theorems of Margulis for
+lattices in semisimple Lie groups of higher rank presented in [Mar91, Chapter IX] and [Zim84,
+Theorem 6.1.2], and the arithmeticity theorem for simple Lie groups of rank one locally isomor-
+phic with Sp(n,1) or F4(−20) by Corlette [Cor92] and Gromov-Schoen [GS92] applies to show that
+Γ is virtually linear over Z. Say it virtually embeds into GLn(Z). Now [DFO13, Proposition 2.9]
+says that there is l = l(n) such that every solvable subgroup of GLn(Z) is polycyclic of Hirsch
+length at most l.
+Corollary 6.8. Let Γ be a lattice in a connected Lie group. Then any action of Γ on a locally compact
+Hausdorff space has the ℓ1-ideal intersection property.
+Proof. In order to apply Theorem 6.5, we have to check all conditions of Definition 6.4. Let Λ be the
+amenable radical of Γ. Then by [Pra76, Lemma 6], we infer that Λ is virtually a lattice in a connected
+solvable Lie group and that Γ/Λ is a lattice in a semisimple Lie group with trivial centre and without
+compact factors. Since Lie groups with trivial centre are linear, [Bre+17, Theorem 6.9] implies that
+Γ/Λ is C∗-simple. So the first condition of Definition 6.4 is verified thanks to Proposition 2.3.
+Also, the Tits alternative for linear groups in characteristic zero [Tit72] shows that every amenable
+subgroup of Γ/Λ is virtually solvable. Since Λ is virtually solvable, this shows that every amenable
+subgroup of Γ is virtually solvable. This checks the second condition of Definition 6.4. In order to
+verify the last one, we can apply Proposition 6.7.
+18
+
+A variation of the core arguments in the previous theorem, also covers many linear groups.
+Corollary 6.9. Let Γ be a linear group over the integers of a number field. Then any action of Γ on a
+locally compact Hausdorff space has the ℓ1-ideal intersection property.
+Proof. Let Γ be as in the statement of the theorem. We have to check all three conditions of
+Definition 6.4. The first condition is satisfied thanks to, Proposition 2.3 combined with [Bre+17,
+Theorem 6.9]. The second condition holds thanks to the Tits alternative for linear groups in char-
+acteristic zero [Tit72]. The last condition holds thanks to [DFO13, Proposition 2.9].
+Our final class of examples to which Theorem 6.5 applies are virtually polycyclic groups, and more
+generally locally virtually polycyclic groups, which are precisely those groups whose finitely gen-
+erated subgroups are virtually polycyclic. We state the result in terms of C∗-uniqueness.
+Corollary 6.10. Every locally virtually polycyclic group is C∗-unique.
+Proof. By Proposition 3.5, it suffices to show that every virtually polycyclic group satisfies the
+conditions of Definition 6.4. The first condition is satisfied since virtually polycyclic groups are
+amenable. The second condition holds, since every subgroup of a polycyclic group is polycyclic.
+Finally, the Hirsch length is monotone for inclusions of groups, so that the last condition is also
+satisfied. Now Theorem 6.5 applies.
+Remark 6.11. It would be interesting to understand whether all linear groups have the ℓ1-ideal in-
+tersection property. We expect that a positive answer can be obtained. However, the groupoid
+techniques employed in the present work will likely not be sufficient to prove such a result for two
+reasons. First, there need not be any torsion points in the dual of an abelian group, so that an induc-
+tion like in the proof of Theorem 6.5 cannot be performed. Second, following the strategy of the
+present work, there is no clear induction variable available for solvable groups which are not poly-
+cyclic. The derived length is not suitable. Indeed, the induction step in the proof of Theorem 6.5
+only divides out a (possibly proper) subgroup of the last term in the derived series.
+Concrete examples of solvable, non-polycyclic groups can nevertheless be covered by our
+present methods. The arguments presented show that metabelian groups have the ℓ1-ideal intersec-
+tion property, since each such group is an inductive limit of semi-direct products A⋊Zni for some
+monotone sequence of natural numbers (ni)i. We don’t give any details of the argument. Many
+metabelian groups are already known to have the ℓ1-ideal intersection property by [Boi+78, p. 11,
+Korollar].
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+DK-5230 Odense
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+Sven Raum
+Department of Mathematics
+Stockholm University
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+SE-114 19 Stockholm
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+raum@math.su.se
+22
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