Oscar Pozo

Cécile Repellin

Adolfo G Grushin

Quantization in Chiral Higher Order Topological Insulators: Circular Dichroism and Local Chern Marker

The robust quantization of observables in units of universal constants is a hallmark of topological phases. We show that chiral higher order topological insulators (HOTIs), bulk insulators with chiral hinge states, present two unusual features related to quantization. First, we show that circular dichroism is quantized to an integer or zero depending on the orientation of the sample. This probe locates the hinge states, and can be used to distinguish different types of chiral HOTIs. Second, we find that the average of the local Chern marker over a single surface, an observable related to the surface Hall conductivity known to be quantized in the infinite slab geometry, is nonuniversal for a finite surface. This is due to a nonuniversal contribution of the hinge states, previously unaccounted for, that distinguishes surfaces of chiral HOTIs from Chern insulators. Our findings are relevant to establish higher order topology in systems such as the axion insulator candidate EuIn2As2, and cold atomic realizations.

Introduction.-The quantization of global observables is a quantum mechanical signature that distinguishes trivial and topological matter.

For example, twodimensional (2D) insulators without time-reversal symmetry present a Hall conductance that is quantized to Ce 2 /h where C is an integer known as the Chern number. In contrast, experiments that show a quantized observable in three dimensions (3D) remain scarce; the sole examples are the observation of a quantized optical rotation in 3D time-reversal invariant topological insulators [1,2], and the quantization of a photocurrent in chiral topological semimetals [3].

Our first result is that 3D chiral higher order topological insulators (HOTIs) have a quantized circular dichroism. HOTIs are a subclass of topological insulators [28,29] for which a d-dimensional bulk topological invariant protects metallic boundary modes in dimension d -2 or lower. More specifically, 3D chiral HOTIs [6, 8-10, 13, 14, 16, 18-21, 24] are fully gapped with only conducting chiral hinge states. The protection is guaranteed by combinations of crystalline and discrete symmetries, but symmetry alone does not necessarily dictate which hinges have a boundary mode.

Circular dichroism is an optical probe that is sensitive to topological properties [30][31][32][33][34][35][36][37][38][39][40]. It is the differential absorption of left and right circularly polarized light and its frequency integral is determined by the dc Hall conductivity through a sum rule [32,41]. For 2D Hall insulators this integral is quantized to the Chern number for periodic boundary conditions [37,39], and vanishes upon including the edge state contribution [32,37,38]. To filter out the edge contribution, it is possible to use optical selection rules in Landau levels [38], or, in periodically driven cold-atomic gases, a trap-release protocol [37] or a harmonic trap as was demonstrated experimentally [42]. In 3D we show that the circular dichroic absorption of chiral HOTIs restricted to a frequency range within the FIG. 1. The frequency integrated in-gap absorption difference between left-and right-circularly polarized light (red and green lines, respectively) is quantized to C depending on the polarization plane. The cubes depict chiral HOTIs with hinge modes protected by (a) C z energy gap but across the whole sample is determined by an integer. The integer depends on the orientation of the sample and thus may be used to distinguish different types of chiral HOTIs (see Fig. 1).

In addition to bulk properties, the topological character of topological insulators can emerge as a quantized surface property. The surface Hall conductivity of a chi-ral HOTI is quantized in units of e 2 /2h, half of the conductance quantum [19,43]. This quantization is inherited from the Hall conductivity of a gapped topological insulator surface state [44,45] and can be derived in the infinite slab geometry [19,46]; each surface Dirac cone contributes with ±e 2 /2h to the total surface Hall conductivity.

Our second result revises such quantization of surface observables; we show that for finite samples, the hinge states modify the surface quantization of the Hall conductivity by a nonuniversal factor. We use a real space version of the Berry curvature, known as the local Chern marker [47]. It measures the local (space resolved) contributions to the Hall conductivity. In 2D insulators, the bulk average of the local Chern marker is quantized and equal to the Chern number C of the corresponding periodic system. For open boundary conditions, its average over the whole sample vanishes. It means that the total contribution from edge states is equal and opposite in sign to the bulk contribution [37,38,47]. We find that this does not happen for a finite surface of chiral HOTIs due to the spatial distribution of the local Chern marker in the hinge states, which distinguishes surfaces of HOTIs from purely 2D Chern insulators, despite their spectral similarities [24].

Models.-We consider a generic tight-binding model of chiral HOTIs on the cubic lattice [6,13,18]

Ĥ0 = M + J 3 i=1 cos(k i a) τ 3 σ 0 + 3 i=1 λ i sin(k i a) τ 1 σ i + + D cos(k 1 a) -cos(k 2 a) τ 2 σ 0 + τ 0 B • σ , (1)
where k i are the cartesian momentum components, a is the lattice spacing, σ i , τ i are two sets of Pauli matrices acting on the spin and orbital degrees of freedom and σ 0 , τ 0 are 2 × 2 identity matrices. At half-filling the first two terms describe a time-reversal invariant 3D topological insulator, with anisotropic Fermi velocities if λ i are chosen to be different. The third term, proportional to D, breaks C z 4 and time-reversal (T ) symmetries, but preserves their combination. The last term proportional to B is a Zeeman term that breaks T but preserves inversion symmetry (I). The Hamiltonian Eq. ( 1) interpolates between a topological insulator (D = |B| = 0), a chiral HOTI protected by the rotoinversion symmetry C z 4 I [Fig. 1 (a)] and a chiral HOTI protected by I [Fig. 1 (b)]. We choose the parameters that maximize the bulk gap and therefore the localization of hinge states. Specifically, M/J = 2, λ i /J = 1 (all three velocities are the same unless otherwise stated) and |B|/J = 0.5, with D/J = 0 (D/J = 1) and B (1, 1, 1) ( (0, 0, 1)), for I-(C z 4 I-) symmetric HOTIs (see also [48] sec. A). Quantized circular dichroism of chiral HOTIs.-To first order in perturbation theory, the absorption rate Γ(ω) of a circular periodic excitation of intensity E 2 polarized in the xy plane at frequency ω is (see [48] sec. B for details)

Γ ± (ω) 2πE 2 = n,m m| (x ± iŷ) |n 2 δ (t) (E m -E n -ω), (2)
where |n (|m ) are occupied (unoccupied) states with energy E n (E m ) and δ (t) ( ) ≡ (2 /πt) sin 2 ( t/2 )/ 2 (analogous definitions hold for other cartesian planes). We assume half-filling and discuss changing the chemical potential in [48] sec B. In the long time limit δ (t) ( ) is the Dirac delta function, so that Eq. (2) becomes the familiar Fermi's Golden Rule. The integrated differential rate of absorption is defined as

∆Γ(ω max ) ≡ ωmax 0 dω [Γ + (ω) -Γ -(ω)] /2. (3)
In the limit where ω max → ∞ it simplifies to [32,37] ∆Γ(∞) = -2πiE 2 Tr Qx, P ŷ ≡ E 2 all ri C xy (r i ).

(

) 4
This equality-valid in two and three dimensionsexpresses the dichroic signal as the sum of the local Chern marker C xy (r i ) defined as a trace of the complete set of Wannier states {|r i }. P ( Q) projects onto the occupied (unoccupied) eigenstates, and x, ŷ are position operators.

In a 2D insulator with periodic boundary conditions the average of the local Chern marker is equal to the Chern number [47,49]. Generically, for a finite system with open boundary conditions the average of C xy (r i ) over the whole sample vanishes due to the trivial character of the fiber bundle [47], leading to ∆Γ(∞) = 0. To show this mathematically recall that the trace of the commutator in Eq. ( 4) vanishes in a finite vector space.

Our main result is that the integrated in-gap circular dichroism is finite and quantized; by choosing the frequency cutoff in Eq. ( 3) to be ω max = E g , where E g is the gap of the infinite slab geometry, ∆Γ(E g ) isolates the quantized hinge state contribution.

To reach this result, consider first periodic boundary conditions in the polarization plane, in which case only bulk to bulk transitions contribute and ∆Γ(∞) = ∆Γ b→b . From Eq. ( 4) ∆Γ b→b is proportional to the integral of the Chern marker, which is not an integer in general [37]. For a 2D insulator with area A, ∆Γ b→b = CE 2 A where C is the Chern number [37]. For a 3D chiral HOTI slab with surface area A only the two surfaces parallel to the polarization plane contribute to ∆Γ b→b ; their surface Dirac cones contribute with ±1/2 each, resulting in a total of

∆Γ b→b = CE 2 A , ( 5
)
where C is the sum of these contributions, and thus an integer [19]. We have checked this expectation numerically (see [48] sec. B).

In the more realistic case of open boundary conditions there are four types of n → m transitions contributing to Eq. ( 4): from bulk to bulk (b → b), bulk to edge (b → e), edge to bulk (e → b) and edge to edge (e → e) states. The sum of all transitions must satisfy

∆Γ(∞) = n,m∈{e,b} ∆Γ n→m = 0 . ( 6
)
We have evaluated the relative sizes of the different contributions numerically using Eq. (2) in the long time limit for the Iand C 4 I-symmetric HOTIs [see Fig. 2 (c)]. In all cases the edge-bulk and bulk-edge transitions become negligible as we increase the system size; this is related to the exponentially small overlap between the surface and bulk states (see [48] sec. B). In virtue of Eq. ( 6) the bulk-bulk and edge-edge transitions are equal and opposite, and due to Eq. ( 5) their respective thermodynamic values are ±CE 2 A. Thus, restricting the integration of the circular dichroic signal to in-gap absorption singles out edge-edge transitions, which guarantees the quantization of ∆Γ(E g ).

In Figs. 2 (a) and (b) we show the circular dichroism calculated by finite time evolution in the frequency range ω ∈ (0, E g ) for Iand C z 4 I-symmetric HOTIs. By evaluating ∆Γ(E g ) for N tot = N 3 sites with N = 17 for xy polarization we obtain C = -0.73 and -0.86 for the Iand C z 4 I-symmetric HOTIs, respectively. As N grows, C approaches exact quantization as seen in Fig. 2 (c). For yz polarization, we obtain C = -0.73 and 0 for the Iand C z 4 I-symmetric HOTIs, respectively. An experimentally useful property of circular dichroism is that it depends on the type of chiral HOTI and the polarization plane. This follows from Fig. 2 and is depicted schematically in Fig. 1. For the I-symmetric HOTI, the circular dichroic signal is always nonzero, and quantized in the thermodynamic limit. However, for the C z 4 I-symmetric HOTI, C vanishes for xz and yz polarized light, but it remains finite for xy polarization. Indeed, in the C z 4 I HOTI the projection of the hinge states onto the xy plane results in an orientable closed path akin to the edge state of a Chern insulator. This path contributes an integer C = 0 to the circular dichroic signal, in contrast to the xz and yz projections, where no such path exists. On the contrary, in the I-symmetric HOTI, all cartesian polarizations lead to orientable closed paths, and the circular dichroic signal is quantized. In general, the quantization is determined by the projection of the 3D hinge state onto the polarization plane, which might be non-Cartesian. In this plane, each orientable closed path contributes an integer C = 0 to the circular dichroic signal normalized by its area, and with a sign dictated by the orientation of the path.

Our characterization scheme relies on the in-gap (ω < E g ) integration of the dichroic signal. Although ab initio methods may be able to approximate E g , it is desirable to determine it experimentally. This is possible by measuring ∆Γ(ω max ) as a function of ω max , as illustrated Fig. 2 (d). We expect a singularity at ω max = E g , caused by the activation of bulk-to-bulk transitions. The additional singularities at ω max < E g in Fig. 2 (d) are due to the discretization of hinge states in finite size, and will not affect the extraction of the gap in the thermodynamic limit.

Local Chern marker of chiral HOTIs.-So far we have taken advantage of the energy resolution of circular dichroism to predict a quantized dichroic signal. Can sacrificing energy resolution in favor of a local measurement lead to a universal quantized observable? In infinite slabs the answer is affirmative: the Hall conductivity of the surface of a chiral HOTI is quantized in units of ±e 2 /2h [43,44,51]. It stems from the quantization to ±1/2 [19,46] of the local Chern marker C xy (r i ) [47], the dimensionless operator defined in Eq. ( 4), see also [48] sec. B and C. We show next that the situation is different for finite surfaces: the hinge contribution renders the surface average of the local Chern marker nonuniversal. Our results carry over to other choices of position operators in Eq. ( 4).

We have calculated the local Chern marker using a recent approach [52] based on the kernel polynomial method [53] implemented using the Kwant package [50]. It allows us to reach sizes beyond exact diagonalization, and up to N tot = N 3 sites with N = 28. For concreteness, we focus on the distribution of C xy (r i ) for the C z 4 Isymmetric HOTI, shown in Fig. 3 (a). C xy (r i ) is close to zero deep in the bulk of the cube. The largest contribution to C xy (r i ) comes from the xy surfaces and decays fast with increasing distance to the surface.

Consider the top xy surface of the HOTI, as represented in Fig. 3 (b). Far from the hinges, we expect the local Chern marker to approach -1/2, since the Hall conductivity is quantized to -e 2 /2h in the infinite slab geometry [19,43,44,51] (see also [48] sec. B and C). Our numerical results indeed show this convergence as a function of system size at the central points of the surface (see Fig. 3 a) and b). Remarkably, the average of the local Chern marker over a single surface of the HOTI does not vanish (see Fig. 3 d)). This is in stark contrast with 2D Chern insulators (see Fig. 3 c)) and highlights the 3D character of the HOTI.

The average of the local Chern marker over the surface of a HOTI is finite, but is it still universal? Note that the sum of all hinge contributions must exactly compensate the bulk, a fact we have numerically checked. However, C z 4 I symmetry only imposes that hinges related by this symmetry have equal weight. In Fig. 3 d) we show that the Chern marker can be distributed inhomogeneously over the horizontal and vertical hinges by choosing λ x = λ y ≡ λ = λ z . These parameters introduce hopping anisotropy but do not break the C z 4 I symmetry that protects the topological state. As N increases, and up to numerical precision (see also [48] sec. C), different parameters lead to a different distribution of the local Chern marker along the hinges. This means that although the total hinge contribution is fixed and equal to one, it can be unevenly distributed over the horizontal and vertical hinges. This in turn, implies that the surface average of the Chern marker is a non-universal number.

Discussion. We have shown that the in-gap circular dichroism, a spatially averaged, frequency integrated quantity is quantized and can differentiate different types of chiral HOTIs. Although experimentalists currently measure polarization rotations [1,2], we propose integrated circular dichroism as an advantageous alternative to measure quantized optical responses; importantly, it does not depend on the substrate's details as long as it preserves time-reversal symmetry. Specifically, it could probe higher-order topology in EuIn 2 As 2 [54], CrI 3 /Bi 2 Se 3 /MnBi 2 Se 4 heterostructures [55], and MnBi 2 Te 4 [56][57][58], all of which are predicted to host chiral hinge states. In particular, a quantized signal can be expected for a sample of EuIn 2 As 2 , illuminated with light polarized in a plane perpendicular to any of the crystal axis, and integrated up to the gap energy (E g ∼ 10meV from ab-initio calculations [54]).

These energies are comparable to those already used to measure quantized optical rotation [1]. Moreover, the quantization of circular dichroism applies to interacting systems [39], which suggests the possibility to distinguish different interacting HOTIs [59]. Lastly, quantized circular dichroism [37] has been measured in ultra-cold atomic Chern insulators [42] opening up the possibility to measure quantization in synthetic chiral HOTIs.

We have also found that the surface average of the local Chern marker, a spatially resolved quantity integrated over all energies, is non-universal in chiral HOTIs due to the hinge states, in sharp contrast to the infinite slab geometry and to a 2D Chern insulator. As a consequence, an experiment aiming to characterize a chiral HOTI based on its surface properties must distinguish bulk and hinge contributions. In solid state systems the local Chern marker is related to the orbital magnetization [60]. In cold atom experiments, it could be measured using a quantum gas microscope [61,62]. Mechanical [63] and photonic topological metamaterials [64] may offer additional avenues to measure this quantity.

We expect our work to motivate new experiments that can push the ongoing effort to measure quantization in 3D [1][2][3]65], and topology in real space [61,63]. It calls to revisit 3D topological markers [66][67][68] to complete our understanding of topological matter in real space.

The chiral HOTIs we study in the main text, and also Chern insulators, present one dimensional chiral metallic states at their hinges and edges. These states are fingerprints of their non-trivial topology. Through exact diagonalization, one can solve the tight-binding model (Eq. ( 1) in the main text) and compute the eigenstates {|n } in the Wannier basis {|r i }. These metallic states are identified with the eigenstates energetically welllocated inside the insulating gap, and the local density

ρ n (r i ) = | r i |n | 2 , ( 7
)
can be used to check their dimensionality and position. The hinge states of the Iand C z 4 I symmetric HOTIs studied in the main text are shown in Fig. 4.

The slab geometry is achieved assuming periodic boundary conditions in two of the three spatial directions, leading to three different slab geometries. There are no hinges in the slab geometry, therefore the energy spectrum is fully gapped. The slab gap E g used in the main text is taken to be the minimum gap between the three possible slab geometries.

B. FURTHER DETAILS ON THE COMPUTATION OF THE CIRCULAR DICHROISM Computation of the absorption rate

In the main text we presented the expression of the absorption rate, Γ ± (ω) (Eq. ( 2)), to analyze how circular dichroism is quantized in chiral HOTIs. Here we explain the necessary steps to determine such expression.

Consider a chiral HOTI subjected to an optical signal at frequency ω polarized in the xy plane. Then the full Hamiltonian at time t is given by

Ĥ± = Ĥ0 + 2E [cos(ωt)x ± sin(ωt)ŷ] , (8)
where Ĥ0 is the HOTI Hamiltonian defined in Eq. ( 1) of the main text, E is the amplitude of the electric field, ± indicates left-or right handed polarization and x, ŷ are position operators. Part of this light is absorbed by the electrons of the system, bringing them from the initial ground state |Ψ 0 to an excited state |Ψ(t) . The probability of excitation is

P ± (t) = 1 -| Ψ(t)|Ψ 0 | 2 , (9)
If the electric field amplitude is small enough, the wave function |Ψ(t) can be computed using time-dependent perturbation theory up to first order. The resulting absorption rate Γ ± (ω) = P ± (t)/t is Eq. ( 2) in the main text. For long enough times this absorption rate is given by Fermi's Golden Rule

Γ ± (ω) 2πE 2 = n,m m| (x ± iŷ) |n 2 δ(E m -E n -ω), (10)
where |n (|m ) are occupied (unoccupied) states and E n (E m ) their energies. The total circular dichroism is obtained by taking the difference (Γ + -Γ -)/2 and integrating over all frequencies as defined in the main text. This difference can be expressed as ∆Γ(∞) = -2iπE 2 Tr Qx, P ŷ ≡ E 2 all ri C xy (r i ) , (11) where P ( Q) is the projector over occupied (unoccupied) states. Independently of dimensionality, the total absorption rate is proportional to the total Hall conductivity σ αβ of the sample [37] where αβ is the polarization plane of the optical signal.

Layer-resolved Chern number

In the case of periodic boundary conditions in the polarization plane αβ, let us resolve the right hand side of Eq. ( 11) along the direction γ perpendicular to the polarization plane. We show this layer-resolved Chern number C αβ (γ) in Fig. 5 (a) and (c) for the C z 4 I-symmetric HOTI. In the infinite slab geometry, the Hall conductivity can be calculated following [19], by summing the layer-resolved Chern number over the direction of propagation γ. Fig. 5 (a) and (c) show that the only sizable contributions are due to the surface states which contribute ±1/2, and extend as much as two layers from the surface. This justifies our choice in Fig. 3 (a) in the main text where we chose the two last layers as our surface. Summing over the whole slab we find that the contributions of opposite surfaces either cancel, leading to C = 0 [see Fig. 13) with Ntot = N × N sites computed using exact diagonalization. Edge-bulk and bulk-edge transitions give a negligible contribution, while we see the convergence of the bulk-bulk and edge-edge contributions towards their quantized thermodynamic limit value.

The results in Fig. 5, and using Eq. (11) show that for periodic boundary conditions, where only bulk-bulk transitions are present, the circular dichroism is quantized to

∆Γ b→b = CE 2 A . (12)
as we use in the main text.

Edge-bulk and bulk-edge contributions in Chern insulators

The quantization of bulk responses is only ensured in the thermodynamic limit. It is in such case that bulk states of a finite system are independent of boundary conditions and resemble those of periodic boundary conditions. As a consequence, our finite size results do not show an exact quantization of circular dichroism but rather a trend as we approach the thermodynamic limit. Due to the 2D character of Chern insulators, it is possible to calculate the absorption rate of larger systems with exact diagonalization [52,70]. This allows us to show the quantization of edge-bulk, bulk-bulk and edge-edge contributions in the thermodynamic limit to larger precision (see Fig. 6). The Chern insulator model we have considered is

H = M -t 2 i=1 cos (k i a) σ 3 + λ 2 i=1 sin(k i a)σ i , ( 13
)
where M/t = 3/2 and λ/t = -1/2.

Effect of a finite chemical potential

In the main text we have assumed half-filling, and thus the chemical potential µ sits exactly at the center of the gap. This assumption made the edge-bulk and bulk-edge contributions equivalent in Fig. 2 b) of the main text. In general these two types of transitions can be different. However, our results remain unchanged so long as the chemical potential remains in the gap. Out of all possible transitions, the bulk-bulk transitions are still quantized if the chemical potential is within the gap. The edge-bulk and bulk-edge contributions vanish, independent of the chemical potential. Therefore, the sum of all edge-edge transitions must exactly compensate the sum of bulk-bulk transitions with an equal and opposite Chern number. Summing over all edge-edge transitions implies integrating ∆Γ(E g ), as for the case where µ = 0.

C. FURTHER DETAILS ON THE COMPUTATION OF THE LOCAL CHERN MARKER

In Fig. 3 of the main text, we see that C xy (r i ) reaches its thermodynamic value much faster in the bulk than in the hinges. This is related to the contrasting scaling of correlation functions in these two regions: states are respectively exponentially and algebraically localized in the bulk and the hinges. As a result, at a given bulk point, the main contribution to the local Chern marker is determined by the nearest points [71][72][73], while it involves larger non-local contributions for the hinges. A more formal way to understand this is to consider the expression of the local Chern marker in terms of the density correlator

ρ n (r i , r j ) = r i |n n|r j , (14)
which reads [73] C

xy (r i ) = -4π rj ,r k E l <0 Em>0 En<0 Im [ρ l (r i , r j )ρ m (r j , r k )ρ n (r k , r i )] x j y k . (15)
Since the magnitude of ρ n (r i , r j ) is proportional to the modulus of the density of the state |n at r i and at r j , bulk-edge and edge-bulk contributions are suppressed here as in the case of circular dichroism. Because of the long-range density correlations within the hinge, the local Chern marker at a hinge point r i receives significant contributions from other hinge points (r j , r k ), even though they are not spatially close.

Convergence of the hinge contribution

As shown in Fig. 3 (a) and (b) in the main text at a point r i in the top and bottom surface of the C z 4 Isymmetric HOTI C xy (r i ) is quantized to 1/2, similar to how a bulk point in a Chern insulator defined by Eq. ( 13) is quantized to ±1 [74]. In contrast, the value of the local Chern marker at a hinge point need not be quantized. We analyze the non-negligible local Chern marker contributions, which come from the surface bulk (b), the horizontal (h) and the vertical (v) hinges (see Fig. 3 of the main text). Note that, due to the C z 4 I-symmetry, the contribution of all vertices is the same and we can democratically associate one vertex to each hinge. In an finite system, these contributions verify the following sum rule

0 = ri C xy (r i ) = 2C b + 4C h + 4C v . ( 16
)
C b is quantized to ±1/2N s where N s is the number of surface area points defined in blue in Fig. 3 (a) of the main text (in the thermodynamic limit N s N 2 ). The sum C h + C v is thus quantized (in units of N s ), but C h and C v are not forced by C z 4 I symmetry to be quantized independently. We note this is unlike the Chern insulator Eq. ( 13), which is invariant under C z 4 , implying that the Chern marker at each edge is forced to be C edge = -C/4.

Finally, for Fig. 3 (d) we computed the local Chern marker using the efficient numerical approximation based on the Kernel Polynomial Method [75] implemented via the Kwant Python package [50]. It allows us to reach larger system sizes than the ones achievable with exact diagonalization. The projector P is approximated as a polynomial in powers of Ĥ of degree inversely proportional to the broadening parameter indicated in the main text (see [52] for details). We have checked convergence by observing that a broadening of 10 -3 reproduces the local Chern marker results from exact diagonalization for small N , and that our results change by less than 0.1% when the broadening is reduced by a factor of 10.

FIG. 2 .

 2 FIG. 2. Distinguishing C z 4 I-and I-symmetric HOTIs with circular dichroism. (a), (b) In-gap absorption of the circular signal at frequency ω for left (red) or right (green) polarization or the difference between the two (blue), for polarization planes xy and yz. We choose Ntot = N 3 sites, with N = 17, tJ/ = 200. In the I HOTI, the integral of the blue curve in (a) is C = -0.73, -0.73 for xy and yz polarization, respectively. In contrast, in the C z 4 I HOTI, (b), we find C = -0.86, 0. (c) Contributions from edge (e) and bulk (b) states as a function of system size, illustrating the convergence to a quantized (or vanishing) value. (d) Integrated dichroic signal, Eq. (3), as a function of integration cutoff. The position of the gap Eg is indicated by a vertical line. The absorption Γ was obtained using Eq. (2) [(a),(b)] or its approximation in the long-time limit [(c),(d)]. See [48] sec. B for more details.

FIG. 3 .

 3 FIG. 3. (a) Local Chern marker Cxy(ri) of the C z 4 I-symmetric HOTI. In the top surface (hinge excluded, highlighted in blue) it is quantized to -1/2. The enlargement of this surface cut (b) shows the difference with the local Chern marker distribution in a Chern insulator (c). (d) Local Chern marker integrated over one vertical (Cv) or horizontal (C h ) hinge, or over the top surface hinge excluded (C b ) or included (Cs = 2C h + C b ). The Fermi velocities of the HOTI model are either isotropic (λz/λ = 1) or anisotropic (λz/λ = 0.6, 0.8). For these calculations, we have considered surface and hinges with a two-site depth and normalized over the area of the blue region. The data in d) have been calculated using the Kwant package[50], with a broadening of 10 -3 (see also[48] sec. C).

FIG. 4 .

 4 FIG. 4. Local electronic density of (a) the I symmetric HOTI and (b) the C z 4 I symmetric HOTI defined in Eq. (1) of the main text.

FIG. 5 .

 5 FIG. 5. Layer-resolved [(a),(c)] and layer-summed [(b),(d)] Chern number of the C z 4 I-symmetric HOTI in the zy [(a),(b)] and xy [(c),(d)] polarization planes, for a finite slab with periodic boundary conditions in the polarization plane. The Chern number summed over all layers is proportional to the circular dichroic signal ∆Γ b→b . The curves are calculated following [19] using the open source code PythTB [69] and a slab thickness of N = 12.

  5 (b)], or add up to a quantized value C = -1 [see Fig. 5 (d)].

FIG. 6 .

 6 FIG.6. Circular dichroic signal of the 2D Chern insulator Eq. (13) with Ntot = N × N sites computed using exact diagonalization. Edge-bulk and bulk-edge transitions give a negligible contribution, while we see the convergence of the bulk-bulk and edge-edge contributions towards their quantized thermodynamic limit value.

I-or (b) I symmetries. The light green and dark blue surfaces signal positive and negative values of the local Chern marker. Orange hinges represent gapless chiral states.

Acknowledgements We are thankful to B. A. Bernevig, D. Benoît, F. Jean-Jacques, F. de Juan, T. Neupert, P. San Jose, I. Souza, D. Varjas, and M. A. H. Vozmediano for support and useful conversations. We are thankful to N. Goldman for his critical reading of the manuscript. A. G. G. is supported by the ANR under the grant ANR-18-CE30-0001-01 and the European Unions Horizon 2020 research and innovation programme under grant agreement No 829044. C.R. is supported by the Marie Sklodowska-Curie program under EC Grant agreement No. 751859. O.P. is supported by an FPU predoctoral contract from MINECO No. FPU16/05460 and the Spanish MECD Grant No. FIS2014-57432-P.


Euripidis Loukis
email: eloukis@aegean.gr

Niki Kyriakou

Manolis Maragoudakis

Using Government Data and Machine Learning for Predicting Firms' Vulnerability to Economic Crisis

Keywords: economic crisis, economic recession, data analytics, predictive analytics, artificial intelligence, machine learning

The COVID-19 pandemic is expected to lead to a severe recessionary economic crisis with quite negative consequences for large numbers of firms and citizens; however, this is an 'old story': recessionary economic crises appear repeatedly in the last 100 years in the market-based economies, and they are recognized as one of the most severe and threatening weaknesses of them. They can result in closure of numerous firms, and decrease of activities of many more, as well as poverty and social exclusion for large parts of the population, and finally lead to political upheaval and instability; so they constitute one of the most threatening and difficult problems that governments often face. For the above reasons it is imperative that governments develop effective public policies and make drastic interventions for addressing these economic crises. Quite useful for these interventions can be the prediction of the vulnerability of individual firms to recessionary economic crisis, so that government can focus its attention as well as its scarce economic resources on the most vulnerable ones. In this direction our paper presents a methodology for using existing government data in order to predict the vulnerability of individual firms to economic crisis, based on Artificial Intelligence (AI) Machine Learning (ML) algorithms. Furthermore, a first application of the proposed methodology is presented, based on existing data from the Greek Ministry of Finance and Statistical Authority concerning 363 firms for the economic crisis period 2009-2014, which gives encouraging results.

Introduction

The COVID-19 pandemic is expected to lead to a severe recessionary economic crisis with quite negative consequences for large numbers of firms and citizens, as containment and social distancing policies adopted in most countries will on one hand 'flatten' the medical curve of virus spread (which is quite positive), but on the other hand will 'steepen' the resulting recession curve (which will have quite negative economic and social consequences) [START_REF] Baldwin | Mitigating the COVID economic crisis: Act fast and do whatever it takes[END_REF]. However, this is an 'old story': recessionary economic crises of various intensities and durations appear repeatedly in the last 100 years in market-based economies, and are recognized as one of the most severe and threatening weaknesses of them, having quite negative consequences for the whole economy and society [START_REF] Keeley | From Crisis to Recovery -The Causes, Course and Consequences of the Great Recession[END_REF][START_REF] Knoop | Recessions and Depressions: Understanding Business Cycles -2nd edition[END_REF][START_REF] Allen | Financial Crises and Recession in the Global Economy -Fourth Edition[END_REF][START_REF] Santana | Turnaround strategies for companies in crisis: Watchout the causes of decline before firing people[END_REF][START_REF] Izsak | The impact of the crisis on research and innovation policies[END_REF][START_REF] Arvanitis | Firm Characteristics and the Cyclicality of R&D Investments[END_REF]. The fluctuations that economic activity usually exhibits, and also some critical events, such as banking crises (like the one that gave rise to the 2008 Global Financial Crisis), epidemics (like the COVID-19 one), large increases in the prices of important goods (e.g. oil or gas), etc., can lead to significant economic recessions, meant as serious contractions of economic activity, resulting in economic crises. These contractions of economic activity can lead to big reductions of firms' production, procurement, investment, innovation and employment [START_REF] Santana | Turnaround strategies for companies in crisis: Watchout the causes of decline before firing people[END_REF][START_REF] Izsak | The impact of the crisis on research and innovation policies[END_REF][START_REF] Arvanitis | Firm Characteristics and the Cyclicality of R&D Investments[END_REF], and also closure of numerous firms, with serious social consequences, such as increase of unemployment, poverty and social exclusion, resulting in political unrest and instability. For these reasons economic crises constitute one of the most severe and threatening problems that governments often face and have strong pressure to deal with them effectively in order at least to avoid catastrophic consequences. So it is imperative that governments develop effective public policies and make drastic interventions for addressing these economic crises. The most usual of these interventions is the provision of various kinds of support to the most vulnerable firms in the beginning of such crises, or even earlier, when an economic crisis is in sight (i.e. when there are serious forecasts that an economic crisis is going to occur, like the ones currently made by serious economists and economic research centers that the COVID-19 pandemic will give rise to a severe recessionary economic crisis [START_REF] Baldwin | Mitigating the COVID economic crisis: Act fast and do whatever it takes[END_REF]). This support usually includes the provision to firms of low-interest loans, subsidies, education or consulting services, co-finance of investments or employees' payroll, tax reductions, etc. However, since the available government resources (e.g. financial, human, etc.) for providing such support to firms are usually limited, much below than the ones required for meeting the high demands (from the numerous firms that submit applications for receiving this support), a selection among the applicant firms has to be made; it is based on some predefined criteria, which usually concern firm's sector and location (according to existing sectoral and regional priorities), as well as sales and profits in the last 3-5 years, market share, export activities, debts, etc. It would be very good if to these criteria could be added firm's expected vulnerability to the crisis, as an additional criterion, aiming to provide more assistance and support to the most vulnerable firms; this can increase substantially the effectiveness of such interventions, making them more focused, and improve public value generation from them towards the reduction of the negative consequences of recessionary economic crises. For the above reasons it would be quite useful if we can make a prediction of the vulnerability of individual firms to economic crisis.

Given the high importance of firm-level economic crisis vulnerability prediction it is necessary to exploit for this purpose to the highest possible extent: a) existing firms' data from economic crisis periods in various government agencies (such as Ministries of Finance and Statistical Authorities); and b) the most sophisticated algorithms for 'learning' crisis vulnerability prediction models from such data, and especially the most widely used Artificial Intelligence (AI) algorithms for such 'predictive analytics': the Machine Learning (ML) ones [START_REF] Russell | Artificial Intelligence: A Modern Approach -3rd Edition[END_REF][START_REF] Murphy | Machine learning : a probabilistic perspective[END_REF][START_REF] Witten | Data mining -practical machine learning tools and techniques[END_REF][START_REF] Blum | Foundations of Data Science[END_REF][START_REF] Siegel | Predictive Analytics[END_REF]. In this direction our paper presents a methodology for using existing government data in order to predict the vulnerability of individual firms to economic crisis, based on their particular characteristics (concerning strategic directions, processes, human and technological resources, structures) using ML. Furthermore, a first application of the proposed methodology is presented, based on existing data from the Greek Ministry of Finance and Statistical Authority concerning 363 firms for the economic crisis period 2009-2014, which gives encouraging results.

In the following section 2 the background of our methodology is outlined, while in section 3 the methodology is described, followed by its abovementioned application in secion 4. The final section 5 summarizes conclusions and proposes directions for future research.

Background

Economic Crises

As mentioned in the Introduction one of the most serious problems of market-based economies are the economic recessions that repeatedly appear, which cause significant disruptions and problems to the economy and the society in general, and have to be addressed by government through effective public policies including appropriate interventions [START_REF] Keeley | From Crisis to Recovery -The Causes, Course and Consequences of the Great Recession[END_REF][START_REF] Knoop | Recessions and Depressions: Understanding Business Cycles -2nd edition[END_REF][START_REF] Allen | Financial Crises and Recession in the Global Economy -Fourth Edition[END_REF][START_REF] Santana | Turnaround strategies for companies in crisis: Watchout the causes of decline before firing people[END_REF][START_REF] Izsak | The impact of the crisis on research and innovation policies[END_REF][START_REF] Arvanitis | Firm Characteristics and the Cyclicality of R&D Investments[END_REF]. The economic crises have negative both short-term as well as mediumand long-term consequences for the economy and the society. The short-term consequences include reductions of the demand for many goods and services, resulting in serious decrease of firms' sales, production and profits, leading to reductions in personnel employment (thus increasing unemployment) and materials' procurement (thus propagating the crisis towards the suppliers' sectors, etc., spreading the crisis to large parts of the economy). Furthermore, during economic crises firms usually reduce capital investment in production equipment, ICT, buildings, etc., and also in product, service and process innovations; this reduces the degree of renewal and improvement of their equipment, products, services and operation, as well as the exploitation of emerging new technologies, which causes serious medium-and long-term consequences for their efficiency and competitiveness. Considerable research has been conducted concerning the negative impact of such recessionary economic crises on different aspects of firms' activities and performance, as well as the factors that affect the magnitude of this negative impact [START_REF] Latham | Contrasting Strategic Response to Economic Recession in Start-Up versus Established Software Firms[END_REF][START_REF] Latham | Economic recessions, strategy, and performance: a synthesis[END_REF][START_REF] Wickramasinghe | HRM practices during the global recession (2008-2010) -Evidence from globally distributed software development firms in Sri Lanka. Strategic Outsourcing[END_REF][START_REF] Ahmed | Impact of operational and marketing capabilities on firm performance: Evidence from economic growth and downturns[END_REF][START_REF] Lai | Are the HR responses of small firms different from large firms in times of recession[END_REF][START_REF] Burger | Determinants of firm performance and growth during economic recession: The case of Central and Eastern European count[END_REF][START_REF] Arvanitis | Factors explaining ICT expenditure behaviour of Greek firms during the economic crisis 2009-2014. 7th International Conference on eDemocracy[END_REF]. It has concluded that the above negative consequences of the economic crises differ significantly among firms, and depend critically on their individual characteristics, such as their human and technological resources, their strategic orientations, etc. Some firms have superior human and technological resources, as well as management sturtures, that allow them to be more efficient and effective during difficult crisis periods; these firms are able to offer higher value-for-money products and services during the crisis, and in general have higher capacity to make the required adaptations to the crisis conditions, which make them less vulnerable and more resilient to the crisis. Such firms have less negative consequences on their sales revenue, and therefore on their employment, procurement, as well as capital investment and innovation. For some other firms the opposite holds. Therefore it would be interesting to investigate the capabilities of AI/ML algorithms to predict firm's vulnerability to recessionary economic crisis based on its particular characteristics, e.g. concerning its human and technological resources, management structures, etc.

Artificial Intelligence

Artificial Intelligence (AI) includes a group of techniques that enable computers to perform tasks of higher intelligence, approaching the human one, by learning from their environment, and then using the knowledge they have obtained from it for taking or proposing action [START_REF] Craglia | Artificial Intelligence -A European Perspective, EUR 29425 EN[END_REF][START_REF] Duan | Artificial intelligence for decision making in the era of Big Data -evolution, challenges and research agenda[END_REF][START_REF]Artificial Intelligence in Society[END_REF]. The most representative and widely used AI techniques are definitely the Machine Learning (ML) ones [START_REF] Russell | Artificial Intelligence: A Modern Approach -3rd Edition[END_REF][START_REF] Murphy | Machine learning : a probabilistic perspective[END_REF][START_REF] Witten | Data mining -practical machine learning tools and techniques[END_REF]. They enable exploiting historic data we possess for a number of units (e.g. individuals, firms, etc.) concerning the value of an important dependent variable (usually an outcome one), and also a set of independent variables (that might be possible causes of this outcome or factors affecting it), by processing them through various algorithms, and finally extracting knowledge from them, usually having the form of a model or a set of rules, concerning the relationships between the independent variables and the dependent one (this is usually referred to as 'model training', while the historic data used for this purpose as 'training data'). This knowledge can be used then on one hand for gaining deeper understanding and insight about these relationships, and on the other hand for predicting the value of the dependent variable for new units, based on the known values of the independent variables, which can be quite useful for supporting relevant decisions or optimizing actions.

Initially AI was used successfully in the private sector, and these 'success stories' generated high levels of interest to exploit AI techniques in the public sector as well, in order to automate or support more sophisticated mental tasks than the simpler routine ones automated or supported by the traditional operational IS of government agencies [START_REF] Eggers | AI-augmented government. Using cognitive technologies to redesign public sector work[END_REF][START_REF] Desouza | Learning from public sector experimentation with artificial intelligence[END_REF][START_REF] Desouza | Delivering Artificial Intelligence in Government: Challenges and Opportunities[END_REF][START_REF] Sun | Mapping the challenges of Artificial Intelligence in the public sector: Evidence from public healthcare[END_REF][START_REF] Desousa | How and where is artificial intelligence in the public sector going? A literature review and research agenda[END_REF]. Some first research has been conducted concerning the exploitation of AI in a variety of public sector thematic domains, such as education, policing, justice, public health, transport, etc. [START_REF] Rockoff | Can you recognize an effective teacher when you recruit one?[END_REF][START_REF] Chandler | Predicting and preventing shootings among at-risk youth[END_REF][START_REF] Kang | Where not to eat? Improving public policy by predicting hygiene inspections using online reviews[END_REF][START_REF] Camacho-Collados | A Decision Support System for predictive police patrolling[END_REF][START_REF] Ku | A decision support system: Automated crime report analysis and classification for e-government[END_REF][START_REF] Kouziokas | The application of artificial intelligence in public administration for forecasting high crime risk transportation areas in urban environment[END_REF]. However, this research has revealed and investigated only a small part of the large potential of AI use in government. Therefore, much more research is required in order to exploit more this potential, in order to reveal and investigate more opportunities and methodologies for AI exploitation in government, in a wide variety of public sector thematic domains. Our study makes a contribution in this direction, by developing a methodology for using AI (ML algorithms) in a highly important domain of government activity: the intervention of government for addressing the quite threatening recessionary economic crises that repeatedly appear in marketbased economies and cause major disruptions and problems.

The Proposed Methodology

As mentioned in section 2.1 previous research concerning the negative impact of recessionary economic crises on firms has found that its magnitude differs significantly among firms, depending on their particular characteristics, such as their human and technological resources, management structures, etc. Furthermore, previous manage-ment science research has identified the main elements of a firm that determine its performance; so we can expect that these main elements might determine to a considerable extent firm's performance during the crisis in coping with the difficult recession conditions as well, and therefore the degree of its resilience/vulnerability to the crisis. Several conceptualizations of the main elements of a firm have been developed, the most widely validated and used among them being definitely the 'Leavitt's Diamond' framework [START_REF] Leavitt | Applied organization change in industry: structural, technical, and human approaches[END_REF]. According to it the most important elements of a firm that determine its performance are: a) task (= the strategies as well as the administrative and production processes of the firm); b) people (= the skills of firm's human resources of the firm); c) technology (= the technologies used for implementing the above processes); and d) structure (= the organization of the firm in departments, and the communication and coordination patterns them). An extension of it has been developed subsequently, which analyses the above 'task' element into 'strategy' and 'processes' [START_REF] Scott-Morton | The Corporation of the 1990s[END_REF]. Therefore we expect that firm's characteristics concerning the above five main elements (strategy, processes, people, technology and structure) might be good predictors of its crisis resilience/vulnerability. Government traditionally collects large amounts of data concerning firms' economic performance (the Taxation Authorities) and also firms' chacteristics concerning human resources, use of various technologies (e.g. ICT or various production technologies), organization, innovation and other strategic orientations, etc (the Statistical Authorities). It is quite useful to exploit these existing government data from economic crisis periods in order to construct firm-level crisis resilience/vulnerability prediction models based on firm's characteristics. In particular, using government firm-level data from an economic crisis period concerning: -on one hand firms' resilience/vulnerability to the crisis (e.g. data from Taxation Authorities concerning decrease of sales revenue, profitability, employment, etc. during the crisis) -and on the other hand characteristics of the same firms concerning the above main elements (e.g. relevant data collected from Statistical Authorities), We can train/construct prediction models of the resilience/vulnerability of a firm to recessionary economic crisis (to be used as dependent variable Y), based on the above characteristics of it (to be used as independent variables Xi); for this purpose we can use several different alternative ML algorithms [START_REF] Russell | Artificial Intelligence: A Modern Approach -3rd Edition[END_REF][START_REF] Murphy | Machine learning : a probabilistic perspective[END_REF][START_REF] Witten | Data mining -practical machine learning tools and techniques[END_REF], such as Decision Trees, Random Forests, Gradient Boosted Trees, Support Vector Machines, Generalized Linear Modelling, etc., and finally select among them the one providing the highest prediction accuracy. The constructed model can be used for predicting the value of the resilience/vulnerability of an individual firm to future economic crisis Y (taking into account that all economic crises, despite their differences, have in common their main characteristics: economic recession, contraction of economic activity and decrease of demand for products and services of most sectors) for which we have the values of the abovementioned characteristics Xi. The structure of the proposed methodology is shown in Fig. 1. Our methodology will use existing government data from two main sources: I) Data from Taxation Authorities for a large number of firms concerning the decrease of sales revenue, profits or other measures of economic performance during the economic crisis II) Data from Statistical Authorities about characteristics of the same firms concerning the above five main elements of them: -Strategic orientations: characteristics concerning the degree of adoption of the main strategies described in relevant strategic management literature [START_REF] Johnson | Exploring Corporate Strategy[END_REF], such as cost leadership, differentiation, focus, innovation, export, etc.

training
-Processes: characteristics of firm's processes, such as complexity, efficiency, formality, flexibility, etc. -Human Resources: characteristics concerning the general education/skills level of firm's human resources (e.g. shares of firm's personnel having tertiary education, vocational/technical education, etc.), as well as the possession of specific skills concerning important technologies (e.g. concerning ICTs or various production technologies), the provision of various kinds of training, etc.

-Technology: characteristics concerning the use of the main enterpsise ICTs, such as Enterprise Resource Planning (ERP), Customer Relationships Management (CRM), Supply Chain Management (SCM), Business Intelligence/Business Analytics (BI/BA) systems, Collaboration Support (CS), e-sales, etc. systems, and also social media, cloud computing, or other technologies, or the use of various production technologies.

Train Model

ML Algorithm

Prediction

Model Use Model prediction of Y -Structure: characteristics concerning various aspects of the management structure of the firm, such as its main structural design (functional, product/service based, geographic, matrix), degree of differentiation, specialization, centralization-decentralization, or the use of organic structural forms (such as teamwork or job rotation) [START_REF] Jones | Organizational Theory, Design, and Change[END_REF][START_REF] Donaldson | The Contingency Theory of Organization[END_REF]. Furthermore, we can also include general data about each firm, such as sector, degree of competiveness in this sector (in comparison with the other competitor firms), etc.

Application

A first application of the proposed methodology has been made, using firm-level data for the period 2009-2014 from the Ministry of Finance -Taxation Authorities as well as the Statistical Authority of Greece, in order to construct a prediction model of the most important measure of firm's vulnerability to the crisis, the reduction of sales revenue due to economic crisis, based on firm's characteristics. In particular, we used data provided by the Ministry of Finance -Taxation Authorities concerning the percentage of sales revenue reduction due to the economic crisis in the period 2009 -2014 (dependent variable) of 400 Greek firms; these data were discretized by the Ministry of Finance (in order to avoid providing detailed firm's sales revenue data) into a variable with 13 possible discrete values (SALREV_RED): increase by more than 100%; increase by 80-100%; increase by 60-80%; increase by 40-60%; increase by 20-40%; increase by 1-20%; unchanged sales; decrease by 1-20%; decrease by 20-40%; decrease by 40-60%; decrease by 60-80%; decrease by 80-100%; decrease by more than 100%. For 363 of these firms from the Statistical Authority we found data concerning 43 characteristics of them (independent variables), which concern their strategic orientations (12 variables), human resources (9 variables), technology (16 variables), structure (1 variable) and also general characteristics (5 variables). The definitions -corresponding questions for our dependent and independent variables are shown in the Appendix. These 363 firms cover a wide range of sectors and sizes: 40.2% of them were from manufacturing sectors, 9.4% from constructions, and 50.4% from services sectors; also, 52.6% of them were small, 36.1% medium and 11.3% large ones.

Using the above data we constructed prediction models of SALREV_RED based on the abovementioned 43 firm characteristics. Since the dependent variable is ordinal, with a large number of values [START_REF] Latham | Contrasting Strategic Response to Economic Recession in Start-Up versus Established Software Firms[END_REF] and equal distances between them, it approaches a continuous variable, we have constructed and compared prediction models of SALREV_RED using the main alternative ML algorithms proposed by relevant literature [START_REF] Russell | Artificial Intelligence: A Modern Approach -3rd Edition[END_REF][START_REF] Murphy | Machine learning : a probabilistic perspective[END_REF][START_REF] Witten | Data mining -practical machine learning tools and techniques[END_REF] for the case that the dependent variable is continuous: the classical regression (termed as 'Generalized Linear Model') and also four alternative more advanced statistical methods: -Decision Trees, which splits the initial training set into more homogeneous sub-sets using the most appropriate independent variables, and finally construct a decision tree with internal 'decision nodes' concerning some of the independent variables (the most useful as predictors) -actually tests concerning these independent variables -as well as leaf 'terminal nodes' that include predicted values of the dependent variable [START_REF] Murphy | Machine learning : a probabilistic perspective[END_REF][START_REF] Tan | Introduction to Data Mining -2nd Edition[END_REF]; -Random Forests, which creates a pool of decision trees based on su-sets of the training data set, and then gets the prediction from each of them and finally selects the best prediction by means of 'voting' of these decision trees [START_REF] Breiman | Random Forests[END_REF][START_REF] Tan | Introduction to Data Mining -2nd Edition[END_REF]; -Gradient Boosted Trees, which is based on boosting prediction accuracy through the combination of a learning algorithm in a series, in order to achieve a 'strong learner' from many sequentially connected 'weak learners'; usually the the weak learners are decision trees, and consequtive decision trees are constructed, with each new tree attempting to minimize the errors of the previous tree [START_REF] Friedman | Stochastic Gradient Boosting[END_REF]; -Support Vector Machines, which mqp each training data item as a point in n-dimensional space, where n is number of independent variables; then, the algorithm tries to find hyper-plane that best fits the available training data [START_REF] Tan | Introduction to Data Mining -2nd Edition[END_REF][START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction[END_REF].

For each of these five ML algorithms we calculated its mean absolute error, which is according to relevant ML literature the recommended measure of prediction accuracy for the case that the dependent variable is continuous, using the k-fold cross-validation procedure (we selected k=10), which includes the following steps [START_REF] Russell | Artificial Intelligence: A Modern Approach -3rd Edition[END_REF][START_REF] Murphy | Machine learning : a probabilistic perspective[END_REF][START_REF] Witten | Data mining -practical machine learning tools and techniques[END_REF][START_REF] Tan | Introduction to Data Mining -2nd Edition[END_REF]: a) the data set is divided randomly in k equally sized partitions, b) then we estimate the prediction model k times, in each of them using one of these segments as test data, and the remaining ones as training data, c) and finally the absolute prediction error is estimated as the mean of the prediction errors over the test sets of the above k runs.

In Fig. 2 we can see the estimated mean absolute prediction errors of these five ML algorithms. We can see that the examined algorithms have similar levels of prediction performance/error (ranging from 1.422 to 1.572), with the Decision Trees algorithm exhibiting the lowest mean absolute error (1.422). This is a satisfactory prediction performance, given the small size of the dataset we have used (data from 363 firms), so using a larger dataset (which is feasible, as Ministries of Finance -Taxation Authorities and Statistical Authorities have such data for quite large numbers of firms) can result in a smaller mean absolute error, and therefore an even more accurate prediction of the sales revenue reduction of a firm during an economic crisis.

Conclusions

In the previous sections has been described the development of a methodology for exploiting existing government data for economic crisis periods, in order to construct prediction models of individual firms' vulnerability to economic crises, based on Artificial Intelligence (AI) Machine Learning (ML) algorithms. In particular, it uses existing data for economic crisis periods, on one hand from Taxation Authorities (concerning firms' sales revenue, profits, employment, etc. decrease), and on the other hand from Statistical Authorities (concerning human and technological resources, structures, processes, strategic orientations, etc. of these firms); from these data prediction models can be trained/constructed, which enable the prediction of the vulnerability of a firm to future economic crises (e.g. in terms of sales revenue, profitability, employment, etc. decrease), based on firm's chatacteristics (concerning human and technological resources, structures, processes, strategic orientations, etc.). The prediction of the degree of vulnerability of a firm to economic crisis can be useful as a criterion in firm support programmes of government agencies, for the selection of the applicant firms that will receive support, as part of multi-criteria evaluation of applications, in combination with other selection criteria usually used in such programmes, such as sector, geographic location, sales, profits, market share, export activities, debts, etc. The research described in this paper has interesting implications for both research and practice. With respect to research it creates new knowledge in the emerging areas of government data analytics and government AI exploitation, concerning a highly important domain of economic government activity/intervention, aiming to address the quite threatening recessionary economic crises that repeatedly appear in market-based economies and cause major disruptions and problems. With respect to practice it proposes a methodology of using existing historic government data for constructing prediction models of the vulnerability that individual firms will exhibit to future recessionary economic crises. These prediction models can be very useful to government agencies that implement firm support programs in the beginning of economic crises, or even earlier when crises are in sight, in order to use more effectively their available financial resources.

Future research is required towards: i) further application of the proposed methodology using larger datasets, in various different national contexts experiencing economic crises of different intensities, and also at sectoral level (using also sector specific independent variables); ii) investigation of the prediction performance of other ML algorithms, and especially Deep Learning ones; iii) analysis of the legal aspects of the practical application of the proposed methodology by government agencies, and especially with respect to the new EU GDPR (as the data used by our methodology have been collected by government for quite different purposes).

Appendix. Definitions/questions of the dependent and independent variables

Figure 1 :

 1 Figure 1: Structure of the proposed methodology

Figure 2 :

 2 Figure 2: Mean absolute prediction errors of the five ML algorithms

CL_PAAS

  To what extent you use PaaS (Platform as a Service = remote use of the above plus database management systems and application develop-ment tools/ environments/languages through the Internet)? (five levels ordinal variable) CL_SAAS To what extent you use SaaS (Software as a Service = use through the Internet of remote application software that run on provider's systems)? (five levels ordinal variable) Independent Variables -Structure ORG Over the last three years did your firm use organic structural forms of work organization (such as teamwork and job rotation) ? (binary) Independent Variables -General SECT Firm's sector COMP_PROF How good was the financial performance of your firm over the last three years in comparison with your competitors in terms of profitability? (five levels ordinal variable) COMP_SALR How good was the financial performance of your firm over the last three years in comparison with your competitors in terms of sales revenue? (five levels ordinal variable) COMP_MS How good was the financial performance of your firm over the last three years in comparison with your competitors in terms of market share? (five levels ordinal variable) COMP_ROI How good was the financial performance of your firm over the last three years in comparison with your competitors in terms of ROI (return on investment)? (five levels ordinal variable)

firms' data

	Y	X1	X2	...	XN
	... ...	...	...	...
	...	...	...	...	...
	...	...	...	...	...
		New firm's Data
	Y	X1	X2	...	XN
	?	...	...	...	...

  SM_OPCOTo what extent do you use social media in order to collect customers' opinions, comments and complaints about your products or services? (five levels ordinal variable) SM_IMPS To what extent do you use social media in order to collect ideas for improvements or innovations in your product or services? (five levels ordinal variable) SM_PERS To what extent do you use social media in order to search for and find personnel? (five levels ordinal variable) SM_INTC To what extent do you use social media in order to support the internal exchange of information and co-operation among the employees of your firm? (five levels ordinal variable) SM_IPAR To what extent do you use social media in order to support the external exchange of information and co-operation with other firms (e.g. part-ners, suppliers, customers, etc.)? (five levels ordinal variable) CLOUD Do you use cloud computing? (binary) CL_IAAS To what extent you use IaaS (Infrastructure as a Service = use of remote computing power and storage through the Internet)?

		your firm (continuous)
	EMPL_VOCT	Percentage of vocational/technical education graduates in the per-sonnel of your firm (continuous)
	EMPL_HIGH	Percentage of high school graduates in the personnel of your firm
		(continuous)	Variable
	SALREV_RED EMPL_ELEM EMPL_COM	Percentage of elementary school graduates in the personnel of Total percentage of change of your sales (increase or decrease) your firm (continuous) during the economic crisis of 2009-2014 Independent Variables -Strategic Orientations What percentage of the employees of your firm use computer in their work (e.g. PC, terminal, or laptop)? (continuous)
	STRAT_CL EMPL_INTRA STRAT_DIF EMPL_INTER EMPL_ICT STRAT_INNOV	To what extent does your business strategy include low prices in What percentage of the employees of your firm uses the intranet (internal network) of the firm in their work the? (continuous) comparison with the competition? (five levels ordinal variable) What percentage of the employees of your firm uses Internet in To what extent does your business strategy include high quality of their work? (continuous) products/services in comparison with the competition? (five lev-Percentage of qualified ICT personnel in the workforce of your els ordinal variable) firm (continuous) To what extent does your business strategy include introduction of new products /services (with significant innovations) ? (five Independent Variables -Technology
	INNOV_PRS D_ERP INNOV_PROC D_CRM NEW_PS_P D_SCM	levels ordinal variable) To what extent are Enterprise Resource Planning (ΕRP) systems Over the last three years did your firm introduce product innova-used in your firm? (five levels ordinal variable) tions (=new or significantly improved products)? (binary) To what extent are Customer Relationship Management (CRM) Over the last three years did your firm introduce process innova-sys-tems used in your firm? (five levels ordinal variable) tions (=new or significantly improved processes)? (binary) To what extent are Supply Chain Management (SCM) systems (= What percentage of your total sales revenue (turnover) in 2014 systems that support the electronic exchange of information with came from new products/services that were introduced in the mar-customers, suppliers and business partners, such as inventory lev-ket during the three previous years ? (continuous) els, orders, production, shipments, invoices, etc.) used in your
		What percentage of the total sales revenue (turnover) in 2014 firm? (five levels ordinal variable)
	IMPR_PS_P D_BIBA	came from products /services that you had introduced before To what extent are Business Intelligence/Business Analytics sys-2012, but were improved significantly over the last three years? (continuous) tems (= systems that support advanced forms of processing busi-ness data, which lead to the creation of useful reports, as well as
	INN_PRSD	Did you introduce methods/process innovation in the goods pro-various types of models that aim at the support of decision-mak-duction or services' delivery processes in the last three years? (bi-ing -this can be either a separate software, or a module of an
		nary) ERP or CRM system) used in your firm? (five levels ordinal vari-
	INN_SSWM	Did you introduce methods/process innovations in the sales, ship-able) ment or warehouse management processes? (binary) To what extent are Collaboration support systems (= systems that
	INN_SUPP D_CS	Did you introduce methods/process innovations in the support processes (e.g. in the equipment maintenance ones) (binary) support the internal collaboration between employees of the firm, and/or external collaboration with customers, suppliers and part-
	R&D	Did your firm conduct R&D (Research and Development) in the ners, offering capabilities of sharing various forms of content (e.g.
		last three years? (binary) text files, images), forum, instant messaging (and other forms of
	EXP_P	Percentage of exports in firm's sales revenue in 2014 (contin.) communication), project management, etc.) used in your firm?
		(five levels ordinal variable)
	E-SAL	Independent Variables -Human Resources Do you conduct online sales of products/services through the In-
	EMPL SM_SPRO	ternet? (binary) Number of employees at the end of 2014 (including any tempo-rary employees, part-time, etc., who should be counted as full To what extent do you use social media for sales promotion? (five
		time equivalents) (continuous) levels ordinal variable)
	EMPL_TERT	Percentage of tertiary education graduates in the personnel of