domain: phys math

text:
T T Truong

Radon Transform on algebraic Cormack α-curves inversion formula

Keywords: Generalized Radon transforms, Integral transforms, Imaging science integral geometry

Radon transforms (RT) in R 2 , defined on simple integration manifolds such as straight lines or circles, have attracted extensive interest since they provide the reconstruction of physical density functions in various applicative inverse problems, via a closed form inversion formula. In an attempt to see whether RT on more involved integration manifolds would still admit closed form inversion formulas, A. M. Cormack has sought to invert RT defined on a class of nontrivial curves in the plane, called α-curves. They are generated by application of a specially conceived geometric transform on straight lines, so that the RT inversion process remains as close as possible to that of the RT on the lines. It was hoped that the inversion of RT on lines would show the way to invert the RT on α-curves. However Cormack could only establish a closed form inversion formula for RT on 1 m -curves, with m = 1, 2, .... In this paper, we generalize his formula to RT on algebraic α-curves, i.e. for α = m m with (m, m ), a pair of positive relatively prime integers and thereby give support to his initial conjecture. * The terminology was originally coined by M. Ein-Gal in [10] as consistency conditions on the set of linear equations he derived as equivalent formulation to the Line-RT.

Introduction

The Radon problem is accurately formulated in the title of J. Radon's seminal paper as "(On) the determination of functions from their integrals along certain manifolds" [START_REF] Radon | Über die Bestimmung von Funktionnen durch ihre Integralwerte längs gewisser Mannigfaltikeiten Ber[END_REF]. This problem may be viewed as an inverse problem in diverse fields of application such as medical imaging [START_REF] Kuchment | The Radon Transform and Medical Imaging[END_REF], industrial control [START_REF] Kershaw | The determination of the density distribution of a gas flowing in a pipe from mean density measurements[END_REF], geological prospection [START_REF] Beyklin | The inversion problem and applications of the generalized Radon transforms[END_REF] or astronomical research [START_REF] Bracewell | Strip integration in radio astronomy[END_REF], in which the data arise as integrals of some physical density function over curves (resp. surfaces) in two-(resp. three-) dimensional Euclidean space. The key issue is the recovery of this physical density function by an inverse formula from a complete integral data set. This is by no means an easy task in particular when the manifolds (curves or surfaces) are imposed by the measurement processes. The natural question to ask is to which extent the geometric properties of the integration manifolds affect the derivation of a Radon transform (RT) inverse formula. In view of this vast problem, it seems logical to tackle this question first in two dimensions.

The simplest Radon transform is defined on straight lines in R 2 . For our purposes in this paper, it will be called Line Radon Transform (or Line-RT). This Line-RT, which enjoys translational and rotational invariance, has been inverted initially by J. Radon himself in [START_REF] Radon | Über die Bestimmung von Funktionnen durch ihre Integralwerte längs gewisser Mannigfaltikeiten Ber[END_REF], and then later on by several other methods, see e.g. [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF]. The only inversion method exploiting rotational invariance is the one due to A. M. Cormack [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications I[END_REF]. In this approach, the Line-RT forward equation is replaced by an integral transform, called Circular Harmonic Transform (CH-T), which relates the circular harmonic components (CH-C) of the unknown function to its Line-RT CH-C. Cormack was able to obtain the exact inverse (CH-T) -1 in [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications I[END_REF]. However his solution, which satisfies the so-called the "hole theorem" [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF], turned out to be numerically unstable because "any noise in the Radon data will be amplified by the (CH-T) -1 kernel" at large integration variable values, as he put it in [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF][START_REF] Cormack | Radon's problem -Old and New[END_REF] ‡. To bypass this problem, Cormack has used orthogonal function expansions [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications II[END_REF] to express his solution. Much later in 1975, a stable (in the previous sense) inversion formula was obtained independently by R. M. Perry also in the framework of a CH-C treatment [START_REF] Perry | Reconstruction of a function by circular harmonic analysis of its Radon transform[END_REF]. Subsequently, another line of research was launched in 1973 by M. Ein-Gal [START_REF] Ein-Gal | The circular harmonic transform[END_REF], who made use of the Mellin transform to invert the CH-T and recovered the Cormack reconstruction formula. In this process, he found Consistency Conditions to be imposed on the data in the CH-T inversion. These conditions have already appeared in Cormack's earlier work [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications I[END_REF], but not used in his solution. A few years later E. W. Hansen [START_REF] Hansen | Theory of circular harmonic image reconstruction[END_REF], as well as J. Verly [START_REF] Verly | Circular and extended circular harmonic transforms and their relevance to image reconstruction from line integrals[END_REF] have reinvestigated the CH-T inversion procedure by Mellin transform and discovered that it is subjected to a choice of Bromwich integration contour in the complex plane, leading to causal and non-causal inverses of CH-T in the language of signal processing. Cormack's solution is identified to the causal and instable inverse whereas Perry's solution to the non-causal and stable inverse. It is only in 1984 that Cormack realized that Consistency Conditions on the data must be imposed in order to recover the Perry Line-RT closed form inversion formula [START_REF] Cormack | Radon's problem -Old and New[END_REF]. As pointed out later by Chapman and Cary [START_REF] Chapman | The circular harmonic Radon transform[END_REF], the above instability issue can be cured only by either going to orthogonal expansions (as in [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications II[END_REF]) or by using Consistency Conditions. But how the Perry's closed form inversion formula is obtained from Cormack's original solution remains to be established at this point.

As the Line-RT inversion problem seems to be somewhat settled, attention is then turned to more general RT inversion problems, in which integral manifolds are non-trivial curves in R 2 . Since there are no hints on how to proceed, A. M. Cormack has sought to explore a new idea. Since straight lines in R 2 have given rise to an invertible RT, why not seek to obtain an invertible RT defined on new curves, which are geometric transforms of straight lines? Of course this geometric transform should be so cleverly conceived that the new RT inversion process follows closely that of the Line-RT. Presumably this idea may have come up to his mind when he established the inversion of the RT on circles intersecting a fixed point, which are known to be just geometric inverses of straight lines in the plane, with inversion pole at a fixed point and arbitrary inversion modulus (see [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications I[END_REF]). The new integration curves are called by A. M. Cormack α-curves [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF]. They are so tailored as to produce an expression of ‡ This paper is not concerned by reconstruction stability estimates, but is aimed at the removal of this Radon data noise amplification by the (CH-T) -1 kernel. The question whether this removal is sufficient to prevent further noise propagation is also not addressed here and apparently it has not yet been considered in the current literature.

the α-curve Radon data almost identical to the Line-RT data. For α = 1 2 , 1, 2, 3, the α-curves are simple algebraic curves: parabola, straight line, one branch rectangular hyperbola, and one branch Humbert cubic [START_REF] Humbert | Sur le théorème d'Abel et quelques unes de ses applications géométriques[END_REF].

Cormack had hoped to establish a closed form inversion formula of these new α-curve RT, now called RT α for short. But his RT -1 α solution, which also satisfies the "hole-theorem", turned out to be plagued also by the above numerical instability. Only in 1984, could Cormack make a decisive progress step by establishing Consistency Conditions for the RT α with the aim to derive a closed form inversion formula for the RT α CH-T forward equation. Unfortunately he could do it only for α = 1 m , where m = 1, 2, .... He concluded his paper by briefly suggesting a mapping of an α-curve to a straight line as a alternative way to recover his RT 1 m inversion result. This mapping turns out to belong to the class of factorable mappings between two Riemannian spaces, which was introduced by V. P. Palamodov [START_REF] Palamodov | Reconstructive Integral Geometry[END_REF], on the basis of his rich experience on Radon transforms. In an unsophisticated formulation, a factorable mapping C from R 2 to R 2 is defined as a differentiable one-to-one mapping for points M → M = C(M ) and for smooth differentiable curves C → C = C(C), such that the line elements of the curve dl C (M ) and its C-transform dl C (M ) satisfy the following factorization property

dl C (M ) = h(M ) g(C) dl C (M ), (1)
where h(M ) is a function of M ∈ C and g(C) is a function of the curve C parameters. As a consequence, the Radon transform of a function f (M ) on curves C and the Radon transform of f (M ) on curves C are related by

C dl C (M ) f (M ) = g(C) C dl C (M ) h(M ) f (M ). (2)
This means that the invertibility of any one of the two RTs would imply the invertibility of the other. Several examples of this inversion property are given in [START_REF] Palamodov | Reconstructive Integral Geometry[END_REF]. In [START_REF] Palamodov | Reconstruction from Integral Data[END_REF], the RT α inversion is claimed to be deduced from the Line-RT inversion using the previous Cormack mapping for α = 1 m and for α = m, with m and m positive integers. However no new closed form inversion formula was reported in [START_REF] Palamodov | Reconstruction from Integral Data[END_REF].

The objective of this paper is to make further progress on this problem initiated by Cormack more than forty years ago, by establishing an exact closed form inversion formula for RT m m , where (m, m ) is a pair of relatively prime positive integers. This would prove Cormack's untold conjecture that the Radon transform on α-curves (as special geometric transforms of the straight line) are as invertible as the Line-RT. This paper is organized as follows. Section 2 gives a concise review of A. M. Cormack's work [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF][START_REF] Cormack | Radon's problem -Old and New[END_REF] on the α-curve Radon transform. This is necessary to understand how his approach has led him to the inverse formula of the Radon problem for α = 1 m , m = 1, 2, ..., but not for other values of α. Next, in section 3, we present a new form of Cormack's reconstruction kernel with which a formal inversion formula for general α can be derived. Section 4 gives the main result: a closed form inversion formula for the case α = m m , where (m, m ) is a pair of positive relatively prime integers. This formula shows that the full function reconstruction in polar coordinates is a direct sum of m partial angular function reconstructions because the original Fourier summation on angular components l ∈ Z is split into a direct sum of m partial Fourier summations on m equivalence classes l = (ml 0 , ml 1 + 1, ..., ml m-1 + m -1) where l j ∈ Z with j = 0, 1, ..., m -1. The paper ends with some possible follow-up research perspectives in the conclusion. In the Appendix are gathered some topics of interest but not directly relevant for the derivation of the RT m m inversion formula: i ) The algebraic nature of the α = m m Cormack curves when (m, m ) are relatively prime integers, ii ) The RT 1 m -Consistency Conditions as RT 1 m range characterization and iii ) The factorable mapping of the α-curve RT to the Line-RT, suggested by A. M. Cormack in [START_REF] Cormack | Radon's problem -Old and New[END_REF], as provider of a new derivation of the closed form inversion formula for RT 1 m . It seems that this factorable mapping has not been discussed in the literature.

Radon's problem on α-curves: a review of A. M. Cormack's work

This section reviews Cormack's derivation of the RT α Consistency Conditions, shows how they are taken into account in his former RT α inversion solution but fails to deliver a closed form inversion formula for general α, except for α = 1 m , m = 1, 2, ....

Circular Harmonic Component forward equation

For α ∈ R + A. M. Cormack, in [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF] has introduced and studied the Radon's problem on the family of α-curves with parameters (s,

φ) ∈ R + × [0, 2π] given in polar equation coordinates (r, θ) ∈ R + × [0, 2π] by r α cos α(θ -φ) = s α , with |θ -φ| < π 2α . (3)
The Radon transform of a smooth and rapidly decreasing function

f (r, θ) in R 2 is f α (s, φ), the α-curve integral of f (r, θ) f α (s, φ) = π/2α -π/2α
s dγ

(cos αγ) 1+ 1 α f s (cos αγ) 1 α , γ + φ . (4)
Because of rotational symmetry around the coordinate system origin O, it is most convenient to convert (4) into an integral equation for circular harmonic components (CH-C) f l (r) (resp. f α,l ) of f (r, θ) (resp. f α (s, φ)), defined by

f l (r) = 1 2π 2π 0 dθ e -ilθ f (r, θ), and f α,l (s) = 1 2π 2π 0 dφ e -ilφ f α (s, φ). (5)
(4) is now replaced by the CH-C "forward " equation

f α,l (s) = 2 ∞ s dr 1 -s 2α r 2α cos l α cos -1 s α r α f l (r), (6)
which is subsequently converted into a forward equation for new CH-C

F α,l (r ) = 1 α f l (r 1 α ) r 1 α -1 and F α,l (s ) = f α,l (r 1 α ), (7)
obtained by the substitutions of variables (r , θ ) = (r α , αθ) and parameters (s , φ ) = (s α , αφ), as

F α,l (s ) = 2 ∞ s dr cos l α cos -1 s r 1 -s 2 r 2 F α,l (r ). (8)
Originally Cormack has succeeded to invert this integral equation ( 8) in [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF] with the help of a remarkable integral identity listed in [START_REF] Hansen | A table of series and products[END_REF]

r2 r1 ds s cosh l α cosh -1 s r1 s 2 r 2 1 -1 cos l α cos -1 s r2 1 -s 2 r 2 2 = π 2 . ( 9
)
The solution is shown to be

F α,±l (r ) = - 1 πr ∞ r ds cosh l α cosh -1 s r s r 2 -1 d ds F α,±l (s ), (10)
for l = 0, 1, 2, .... From now on we shall concentrate on working with (10).

Consistency Conditions and consequences

Cormack has readily realized that the CH-C inversion kernel in [START_REF] Ein-Gal | The circular harmonic transform[END_REF] has the form

cosh l α cosh -1 x √ x 2 -1 = (x - √ x 2 -1) l α + (x + √ x 2 -1) l α 2 √ x 2 -1 , (11)
where x = s r . It behaves like (2x) l α -1 , for x → ∞ and large l. Hence "any noise in the data d ds F α,l (s ) for large s is badly propagated into the calculation of F l (r )" [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF][START_REF] Cormack | Radon's problem -Old and New[END_REF].

But simultaneously he also realized that, for α = 1 (or for the Line-RT), his reconstruction equation ( 10) is identical to the Perry's Line-RT inversion formula [START_REF] Perry | Reconstruction of a function by circular harmonic analysis of its Radon transform[END_REF] F 1,l (r ) = ( 12)

1 πr r 0 ds U l-1 s r d ds F 1,l (s ) - 1 πr ∞ r ds s r -s 2 r 2 -1 l s 2 r 2 -1 d ds F 1,l (s ),
where U l (x) is the second kind Chebyshev polynomial in x as defined in [START_REF] Oberhettinger | Formulas and Theorems for the Special Functions of Mathematical Physics 3rd Ed[END_REF] on page 257, provided that "Consistency Conditions" on the Line-RT data represented by

∞ 0 ds (s ) κ F l (s ) = 0, for κ = l -2, l -4, ... > -1, (13)
are taken into account in [START_REF] Ein-Gal | The circular harmonic transform[END_REF]. This is why his 1984 work [START_REF] Cormack | Radon's problem -Old and New[END_REF] is devoted to the derivation of the Consistency Conditions for the RT α , with which a new reconstruction kernel (without data noise propagation into F α,l (r ) at this level) is obtained.

Cormack's RT α -Consistency Conditions can be stated in an equivalent form as Proposition 2.1 Let κ > -1, the κ th radial moment of the circular harmonic component F α,l (s ) of the RT α data satisfies the condition

∞ 0 ds s κ d ds F α,l (s ) = 0, for κ = l α -1, l α -3, ... > -1. (14)
Proof: In the integral defining the κ-moment of d ds F α,l (s ), perform s -partial integration with zero boundary conditions at s = 0, ∞ to get the (κ -1) moment of F α,l (s ). Then replace F α,l (s ) by its integral expression [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications II[END_REF]. Next after exchanging the integration order between r and s , one ends up with an s -integral which a trigonometric integral listed in table [START_REF] Oberhettinger | Formulas and Theorems for the Special Functions of Mathematical Physics 3rd Ed[END_REF]. The zeros of Gamma functions in the integration result yield the quoted values of κ, see [START_REF] Cormack | Radon's problem -Old and New[END_REF] p. 35, which leads to [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF].

Proposition 2.2

The new form of the circular harmonic component inversion equation, after inclusion of Consistency Conditions, is the following

F α,l (r ) = 1 πr r 0 ds U α l s r d ds F α,l (s ) (15) - 1 πr ∞ r ds      V α l s r + s r -s 2 r 2 -1 l α 2 s 2 r 2 -1      d ds F α,l (s ),
where

(U α l (x), V α l (x)) =   k * l k=0 , ∞ k=k * l +1   (-1) k l α -1 -k k (2x) l α -1-2k . (16)
k * l is chosen so that the smallest exponent of (2x) in U α l (x) is larger than -1, and

l α -1 -k k = 1 k! l α -1 -k l α -1 -k -1 ... l α -1 -k -k + 1 ,( 17
)
is the binomial coefficient symbol as defined in [START_REF] Oberhettinger | Formulas and Theorems for the Special Functions of Mathematical Physics 3rd Ed[END_REF].

Proof: The proof can be found in [START_REF] Cormack | Radon's problem -Old and New[END_REF]. It starts with the decomposition of the CH-C inversion kernel in two parts, one of which is represented by a hypergeometric function, using the last formula on page 38 of [START_REF] Oberhettinger | Formulas and Theorems for the Special Functions of Mathematical Physics 3rd Ed[END_REF],

cosh l α cosh -1 x √ x 2 -1 = x - √ x 2 -1 l α 2 √ x 2 -1 + (2x) l α -1 F 2 -l α 2 , 1 -l α 2 ; 1 - l α ; 1 x 2 .( 18
)
Inserting the power series representation of the hypergeometric function (see Chap II of [START_REF] Oberhettinger | Formulas and Theorems for the Special Functions of Mathematical Physics 3rd Ed[END_REF]) but split this expansion in two parts according to the powers of (2x), we obtain the functions U α l (x) and V α l (x) as given by ( 16).

The RT α -Consistency Conditions (14) imply the vanishing of the integral

1 πr ∞ 0 ds U α l s r d ds F α,l (s ) = 0. ( 19
)
This is used to replace the unwanted behavior at large s -integration by the behavior near the origin 0 in F α,l (r ) as given by [START_REF] Hansen | A table of series and products[END_REF]. This is why this form (15) avoids data noise amplification as observed before in the original form and allows in principle a l re-summation before performing s -integration.

Inspection of [START_REF] Cormack | Radon's problem -Old and New[END_REF] shows that the (2x)-power series representing U α l (x) and V α l (x) appear as extensions of the power series representing (i) U l (x), the second kind Chebyshev polynomial in x of order l, see [START_REF] Oberhettinger | Formulas and Theorems for the Special Functions of Mathematical Physics 3rd Ed[END_REF] page 257,

(ii) (x- √ x 2 -1) l 2 √ x 2 -1
a known elementary algebraic function, in which the integer index l is replaced by real values l α . Unfortunately little is known about these functions, (see e.g. [START_REF] Ricci | A survey on pseudo-Chebyshev functions[END_REF]) and the Fourier series

{U α (x, γ), V α (x, γ)} = {U α 0 (x), V α 0 (x)} + l=1,2,... {U α l (x), V α l (x)} (e ilγ + e -ilγ ).
needed for a RT α closed form inversion formula, cannot be computed at this stage.

A closed form inversion formula for

α = 1 m with m = 1, 2, ... However Cormack did notice that, for α = 1 m , m = 1, 2, ... (or l α = lm integer), (i) U 1 m l (x) = U lm (x), the second kind Chebyshev polynomial in x of order lm , (ii) V 1 m l (x) = (x- √ x 2 -1) lm 2 √ x 2 -1
, a known elementary algebraic function.

They satisfy the so-called Pell's identity (see equation ( 16) in [START_REF] Cormack | Radon's problem -Old and New[END_REF])

T lm (x) √ x 2 -1 = U lm (x) + (x - √ x 2 -1) lm √ x 2 -1 , (20)
where T lm (x) = cosh(lm cosh -1 x) is the first kind Chebyshev polynomial of order lm in x. Then the full inversion of RT 1 m is given by

Proposition 2.3 For α = 1 m , m = 1, 2, ..., F 1 m (r , θ) has the closed form F 1 m (r , θ) = - 1 2π 2 r 2π 0 dφ ∞ 0 ds U m -1 s r T m s r -cos(θ -φ) ∂ ∂s F 1 m (s , φ), (21)
which has appeared as equation [START_REF] Ricci | A survey on pseudo-Chebyshev functions[END_REF] in [START_REF] Cormack | Radon's problem -Old and New[END_REF].

Proof: The RT 1 m CH-C inversion formula (15) becomes now after using ( 19)

F 1 m ,l (r ) = 1 πr r 0 ds U lm s r d ds F 1 m ,l (s ) - 1 πr ∞ r ds s r -s 2 r 2 -1 lm s 2 r 2 -1 d ds F 1 m ,l (s ). (22)
Inserting

d ds F 1 m ,l (s ) = 1 2π 2π 0 dφ e -ilφ ∂ ∂s F 1 m ,l (s , φ), (23)
in the θ-Fourier series

∞ l=-∞ F 1 m ,l (r ) e ilθ to reconstruct F 1 m (r , θ), we find F 1 m (r , θ) = 1 2π 2 r 2π 0 dφ r 0 ds U 1 m (s , θ -φ) ∂ ∂s F 1 m ,l (s , φ) (24)
-

1 2π 2 r 2π 0 dφ ∞ r ds V 1 m (s , θ -φ) ∂ ∂s F 1 m ,l (s , φ),
where

U 1 m (s , θ -φ) = ∞ l=0 U |l|m -1 s r (e il(θ-φ) + e -il(θ-φ) ), ( 25
) V 1 m (s , θ -φ) =          1 + ∞ l=i s r -s 2 r 2 -1 |l|m s 2 r 2 -1 (e il(θ-φ) + e -il(θ-φ) )          . ( 26
)
As Cormack has pointed out in [START_REF] Cormack | Radon's problem -Old and New[END_REF], after setting s r = cosh λ and U l-1 (x) = sin lµ sin µ , these series [START_REF] Truong | Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality[END_REF][START_REF] Truong | On geometric aspects of circular arcs Radon transforms for Compton scatter tomography[END_REF] are geometric series in e m σ±iγ or e -m σ±iγ with γ = (θ -φ) and σ = (λ, iµ). They can be easily summed with a convergence factor introduced by hand for series convergence and removed in the summation result afterwards, so that

U 1 m (s , θ -φ), V 1 m (s , θ -φ) = (-, +) U m -1 s r T m s r -cos(θ -φ) , (27)
This yields (21) §.

The closed form reconstruction formula for the original f (r, θ) is, by inverting ( 7)

f (r, θ) = - 1 2π 2 m r 2π 0 dφ ∞ 0 ds U m -1 m s r T m m s r -cos(θ -φ) ∂ ∂s f 1 m (s, φ). (28)
Note that the usual inversion formula for the Line-RT is recovered for m = 1 and the sintegral is understood as principal value. This review of Cormack's work on the Radon transform on α-curves shows why a new approach is needed when α = 1 m , m = 1, 2, ....

A new approach to the Radon Transform on α-curves (RT α )

In this section we introduce a new form of the CH-C reconstruction kernel appearing in [START_REF] Ein-Gal | The circular harmonic transform[END_REF], with which, after application of the RT α -Consistency Conditions, a formal closed form reconstruction of F (r , θ) emerges for arbitrary α > 0.

Let l α = (N l + δ l ) , where for every l integer, N l = l α is the largest integer in l α and δ l is its positive fractional part i.e. 0 < δ l < 1. Clearly both N l and δ l depend also on α, but we refrain from expressing this α-dependence to simplify the writing.

Lemma 3.1 Let λ = cosh -1 x with x = s r , the RT α CH-C inversion kernel can be put under the form

cosh l α λ sinh λ = (2x) δ l U N l -1 (x) + e λδ l -(2x) δ l U N l -1 (x) + cosh λδ l sinh λ e -N l λ , ( 29
)
where U N l -1 (x) is the second kind Chebyshev polynomial of order N l in x.

Proof: In (29) the left hand side is even in l, this means that we should have (N |l| , δ |l| ). But to save writing, we follow Cormack [START_REF] Cormack | Radon's problem -Old and New[END_REF] and keep on writing (N l , δ l ).

(i) Expand cosh (N l + δ l ) λ, use hyperbolic function representation for U N l -1 (x) and (ii) Observe that (2x) δ l U N l -1 (x) is a finite sum of powers of (2x) κ where κ = (N l -1 -2n + δ l ) are precisely the Consistency Conditions exponents of ( 14), since using the (2x) power expansion of U N l -1 (x) given in [START_REF] Oberhettinger | Formulas and Theorems for the Special Functions of Mathematical Physics 3rd Ed[END_REF], we have

(2x) δ l U N l -1 (x) = (2x) δ l N l -1 2 n=0 N l -1 -n n (2x) N l -1-2n . ( 30
)
By adding and subtracting (2x) δ l U N l -1 (x) in the obtained expression of the CH-C inversion kernel, one gets a sum of the three terms as given by (29).

The reconstructed CH-C F α,l (r ) of the RT α has now a new expression given by Lemma 3.2 Let F α,l (s ) be the s -derivative of F α,l (s ), then the reconstructed CH-C F α,l (r ) is given by

F α,l (r ) = 1 πr × (31) r 0 ds (2 s r ) δ l U N l -1 ( s r ) F α,l (s ) - ∞ r ds e λδ l U N l -1 ( s r ) + cosh δ l λ sinh λ e -N l λ F α,l (s ) ,
after application of the Consistency Conditions (14) (with λ = cosh -1 x and x = s r ). Proof: In [START_REF] Ein-Gal | The circular harmonic transform[END_REF], replace the original CH-C inversion kernel by its new form (29). Then use the RT α Consistency Conditions integral

∞ 0 ds (2x) δ l U N l -1 (x) F α,l (s ) = 0, (32)
to replace the first s -integral in the expression of F α,l (s ), i.e. replace

- 1 πr ∞ r ds (2 s r ) δ l U N l -1 ( s r ) F α,l (s ) by 1 πr r r0 ds (2 s r ) δ l U N l -1 ( s r ) F α,l (s ),
to get the result (31).

Theorem 3.3 Taking into account (31), a formal closed form reconstruction formula for F α (r , θ) reconstruction can be obtained as

F α (r , θ) = 1 2π 2 r 2π 0 dφ ∞ 0 ds U α (x, θ -φ) ∂ F α ∂s (s , φ), (33)
where the reconstruction kernel U α (x, θ -φ) is defined by its formal Fourier series

U α (x, γ) = ∞ l=0 (2 s r ) δ l U N l -1 ( s r ) e ilγ + e -ilγ . (34)
Proof: F α (r , θ) is given by its θ-Fourier series l∈Z F α,l (r ) e ilθ . Inserting (31), we find that F α (r , θ) is made up of two parts

F α (r , θ) = 1 2π 2 r 2π 0 dφ ∞ 0 ds U α (s /r , (θ -φ)) ∂ F α ∂s (s , φ) - 1 2π 2 r 2π 0 dφ ∞ r ds V α (s /r , (θ -φ)) + W α (s /r , (θ -φ)) ∂ F α ∂s (s , φ). ( 35
)
where V α (s /r , (θ -φ)) and W α (s /r , (θ -φ)) are generalizations of the sums in [START_REF] Truong | Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality[END_REF][START_REF] Truong | On geometric aspects of circular arcs Radon transforms for Compton scatter tomography[END_REF] appearing in the case α = 1 m . They are given by the formal Fourier series ¶

V α (x, γ) = ∞ l=0 e λδ l U N l -1 (x) e ilγ + e -ilγ , (36)
W α (x, γ) = 1 sinh λ 1 + ∞ l=1 e -N l λ cosh λδ l e ilγ + e -ilγ , (37)
where as before λ = cosh -1 x and x = s r . Right now the convergence of these series (35,36,37) cannot be asserted since for general α, the dependence of N l and δ l is unknown. However we have the following lemma Lemma 3.4 Setting cosh λ = x, the sum of V α (x, γ) and W α (x, γ) in (36,37) is just

V α (x, γ) + W α (x, (γ) = l∈Z cosh l α cosh -1 x sinh λ e ilγ = 0. ( 38
)
Proof: Formula (29) shows that

V α (x, (γ) + W α (x, (γ) = l∈Z cosh l α cosh -1 x sinh λ e ilγ .
This l-sum here is also a geometric series in e l λ α ±ilγ or in e -l λ α ±ilγ , which can be evaluated immediately as follows

l∈Z cosh l α λ sinh λ e ilγ = 1 2 sinh λ l∈Z (e l( λ α +iγ) + e l(-λ α +iγ) ) = 1 sinh λ -1 + 1 2 e -λ α -cos γ cosh λ α -cos γ + 1 2 e λ α -cos γ cosh λ α -cos γ = 0.( 39
)
At this step the kernel U α (x, γ) in (34) does not give rise to data noise amplification (in the sense of A. M. Cormack [START_REF] Cormack | Radon's problem -Old and New[END_REF]) since the RT α -Consistency Conditions have been correctly taken into account. However its defining Fourier series cannot be summed exactly for general α > 0. In section 4, it will be shown that when α = m m is an irreducible fraction a concrete new inversion formula can be obtained.

Inversion formula for α = m m with (m, m ) relatively prime integers

Originally in [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF], A. M. Cormack has introduced α-curves and β-curves. We have chosen to leave out the β-curves since they may be deduced from the α-curves by geometric inversion with inversion pole at the coordinate system origin O and unit inversion modulus. Moreover as geometric inversion is a Palamodov factorable mapping [START_REF] Palamodov | Reconstructive Integral Geometry[END_REF], Radon Transforms on β-curves inversion formulas can be deduced from their partner Radon Transforms on α-curves inversion formulas.

The RT α inversion formula (33) will be explicitly worked out for α = m m when (m, m ) are relatively prime integers. The result is expressed by the following theorem ¶ This form of these series is due to the application of the RTα-Consistency Conditions. As the CH-C F α,±l (r ) share the same kernel because of the cosh function, the factor (e ilγ + e -ilγ ) appears. There is no "hole theorem" as in [START_REF] Cormack | Radon's problem -Old and New[END_REF].

F m m (r , θ) = 1 2π 2 r 2π 0 dφ ∞ 0 ds U m m ( s r , θ -φ) ∂ F m m ∂s (s , φ). ( 40
)
where

U m m (x, γ) = -   m-1 j=0 (2x) j m m T j (cos γ)   U m -1 (x) T m (x) -T m (cos γ) , (41)
and T n (x) (resp. U n-1 (x)) are first kind (resp. second kind) Chebyshev polynomials of order n in x [START_REF] Oberhettinger | Formulas and Theorems for the Special Functions of Mathematical Physics 3rd Ed[END_REF].

Proof: The main job is the evaluation of U m m (x, γ), which is given by the series

U m m (x, γ) = ∞ l=0 (2x) δ l U N l -1 (x) e ilγ + e -ilγ . (42)
Now the Fourier summation on l ∈ Z in ( 42) is split into a direct sum over m groups of partial summations of the form l = (ml j + j) on l j ∈ Z for j = 0, 1, ..., (m -1), i.e.

.).(43)

Then F m m (r , θ) appears to be a direct sum of m partial reconstructions

F j m m (r , θ) + F m m ( r , θ) = (m-1) j=0 F j m m (r , θ). (44)
Yet each F j m m (r , θ) is reconstructed by a formula of type (33) with kernel U j m m (x, γ) of type (42), in which l is replaced by (ml j + j) and (i) N lj = m l j , where l j ∈ Z, (ii) δ lj = j m m , which is independent of l j . Since the factor (2x) δ l stands outside the l j -summation, the kernel U j m m (x, γ) can be evaluated exactly using the following lemma Lemma 4.2 The second kind Chebyshev polynomials U n-i (x) represented for 0 < x < 1 (resp. for 1 < x < ∞) by sin nµ sin µ (resp. by sinh nλ sinh λ ) satisfy the identity

∞ l=0 U lm -1 (x) e i(lm+j)γ = - e ijγ 2 U m -1 (x) T m (x) -T m (cos γ) , (45)
where (j, m, m ) are positive integers, γ ∈ [0, 2π] and T n (x) the first kind Chebyshev polynomial of order n in x.

Proof: The proof can be done either by the trigonometric or the hyperbolic representation. Here we use the trigonometric representation of

∞ l=0 U lm -1 (x) e i(lm+j)γ = (46) ∞ l=0 U lm -1 (x) e i(lm+j)γ = e ijγ 2i sin µ ∞ l=0 (e il(m µ+mγ) -e il(-m µ+mγ) ) = e ijγ 2i sin µ 1 1 -e im µ+imγ - 1 1 -e -im µ+imγ , = - e ijγ 2 U m -1 (cos µ) T m (cos µ) -T m (cos γ) .
Hence

U j m m (x, γ) = -(2x) j m m T j (cos γ) U m -1 (x) T m (x) -T m (cos γ) , (47)
Finally collecting all the j-partial results, equations (40,41) are established.

Clearly U m m (x, γ) is an extension of the Line-RT inversion kernel with m controlling the angle γ = (θ -φ) and m the scaled length s r . The recovery of the original function f (r, θ) can be done easily via [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications I[END_REF], but shall not be given here.

As illustrations, we give the results for m = 1, 2, 3 and m = 1.

(i) For m = 1, A. M. Cormack's formula in [START_REF] Cormack | Radon's problem -Old and New[END_REF] given by ( 28) is exactly recovered.

(ii) For m = 2, this result F 2 (r , θ) is new and corresponds to the Radon transform on the family of one branch rectangular hyperbolas in a π 2 -wide sector,

F 2 (r , θ) = - 1 2π 2 r 2π 0 ∞ 0 dφds 1 + 2 s r cos(θ -φ) s r -cos 2(θ -φ) ∂ F 2 ∂s (s , φ).
(iii) For m = 3, the curves are one-branch Humbert cubic [START_REF] Humbert | Sur le théorème d'Abel et quelques unes de ses applications géométriques[END_REF] in a π 3 -wide sector,

F 3 (r , θ) = - 1 2π 2 r 2π 0 ∞ 0 dφds 1 + 2 s r 1 3 cos(θ -φ) + 2 s r 2 3 cos 2(θ -φ) s r -cos 3(θ -φ) ∂ F 3 (s , φ)
∂s .

Conclusion and perspectives

In this paper we have established the closed form inversion formula for Radon transforms (RT) defined on non-trivial higher order algebraic α-curves in Euclidean plane (or RT α ) introduced by A. M. Cormack in [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF]. Originally Cormack has sought to see whether Radon transforms (RT) defined on more complicated curves than simple lines or circles can be exactly invertible. His basic idea is to use curves which are special geometric transforms of the line, with the hope that the Line-RT inversion would lead directly to the RT α inversion. He was able to prove his conjecture for α = 1 m with m = 1, 2, ..... This paper extends his results to α = m m -curves, where (m, m ) is a pair of relatively prime integers. The final inversion formula does bear a characteristic closeness to the Line-RT inversion formula as expected from the fact that the α-curves emanate from the straight line by geometric transforms. Hence Cormack's expectation is found to be supported in this case of algebraic α-curves and incite to work on more general RT α inversion formulas when α = m m . The availability of this exact closed form inversion formula for algebraic RT α may open the way to some new investigation directions such as Transform, introduced in [START_REF] Verly | Circular and extended circular harmonic transforms and their relevance to image reconstruction from line integrals[END_REF] (but not solved) by J. Verly, who did not realize that ECHT is related to RT α , (iii) Extension Cormack's α-mapping (subsection 7.3) to three classes of circles with constant power of the coordinate origin O in the plane [START_REF] Truong | Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality[END_REF][START_REF] Truong | On geometric aspects of circular arcs Radon transforms for Compton scatter tomography[END_REF] and search for closed form inverse formula for Radon transforms on these generalized α-curves. ). ( 49)

for α = m, s r m = T m (cos θ), x r = cos θ, ⇒ s r m = T m ( x r ). (50)
Using the T m (x) power expansion (see [START_REF] Oberhettinger | Formulas and Theorems for the Special Functions of Mathematical Physics 3rd Ed[END_REF] p. 257),

T m (x) = m 2 = [ m 2 ] k=0 (-1) k (m -k -1)! k!(m -2k)! 2 x r m-2k
, we see that (49) (resp. ( 50)) is an algebraic curve equation of order m (resp. m).

More generally the formal solutions of equations (48) are

θ = q 1 m cos -1 s r m m + q 2π m , q = 0, 1, ..., m -1, (51)
θ = p 1 m cos -1 x r + p 2π m , p = 0, 1, ..., m -1, (52)
where ( q , p ) = (±, ±). θ-elimination can be done by equating the cosines of θ from (51,52). Then the Cartesian equation of the (q, p)-branch of the sought curve arises as

cos 1 m cos -1 s r m m + q 2π m = cos 1 m cos -1 x r + p 2π m . (53)
The full curve is given by the set of the mm equations for all possible (q, p) pairs. The statement that this curve is algebraic of order mm requires too a lengthly proof to be given but its validity may be illustrated by the observation that the line x = constant intersects the curve at mm points whose y coordinates are given by equations (53).

m κ = (-1) κ ∞ 0 ds s κ f (s, φ + π) + ∞ 0 ds s κ f (s, φ). (54)
Switching to circular harmonic components f l (s) of f (s, φ), it becomes

m κ = l∈Z e ilφ (1 + (-1) κ+l ) ∞ 0 ds s κ f l (s). (55)
In this l-sum, only terms with l of same parity as κ will contribute and the corresponding s-integral is calculable in terms of the Gamma function (see [START_REF] Cormack | Radon's problem -Old and New[END_REF]), i.e.

∞ 0 ds s κ f l (s) = ∞ 0 dr r κ+1 f l (r) π Γ(κ + 1) 2 κ+1 Γ κ+l 2 + 1 Γ κ-l 2 + 1 .( 56
)
For κ = 2k even, there are an infinite number of even l which corresponds to poles of the Gamma function and this lead to no contribution to the l-sum in (55)

l = ±(κ + 2), ±(κ + 4), ±(κ + 6), ... (57)
The only values of l remaining are in the range (-κ) ≤ l ≤ (+κ). As this conclusion holds also for κ odd, we recover the Ludwig Line-RT characterization [START_REF] Ludwig | The Radon transform on Euclidean space[END_REF].

Proposition 7.2 A Ludwig's type of range characterization of the α-RT exists for α = 1 m . The κ-moment of the 1 m -RT data f α (s, φ) is a homogeneous polynomial in (cos φ, sin φ) of order (m (κ + 1) -1).

Proof: For α > 0 and κ ∈ N, the κ-moment of the α-RT data m κ α is the integral

m κ α = R ds s κ f α (s, φ), (58)
Using a standard δ-function of the α-curve and assuming Fubini's theorem to be valid, perform the s-integration to get

m κ α = R 2 r dθ dr r κ (cos(θ -φ)) κ+1 α -1 . (59)
If the exponent κ+1 α -1 is required to be a positive integer valued function of κ for fixed value of α, the unique solution is α

= 1 m , Then m κ 1 m
is a homogeneous polynomial in (cos φ, sin φ) of order (m (κ + 1) -1). For m = 1, we recover the case of the Line-RT. Cormack has suggested at the end of his paper [START_REF] Cormack | Radon's problem -Old and New[END_REF] to use the substitutions (r α , αθ) = (r , θ ) and (s α , αφ) = (s , φ ) to convert the α-curve equation in variables (r, θ) and parameters (s, φ) to the straight line equation in variables (r , θ ) and parameters (s , φ ). Then it is easy to see that f α (s, φ), the RT α of a function f (r, θ) (see (4)) will be converted to the Line-RT of the function

g α (r , θ ) = 1 α r 1-1 α f r 1 α , θ α . ( 60
) Its Line-RT transform g α (s, φ) = f α (s 1 α , φ α ) is then given by g α (s , φ ) = π 2 -π 2 s dγ cos 2 γ g α (r , γ + φ ). (61)
Consequently with complete Line-RT data g α (s , φ ), reconstruction of g α (r , θ ), via the well-known closed form Line-RT inversion formula, is given by (see e.g [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF][START_REF] Nievergelt | Elementary inversion of Radon's transform[END_REF]), Hence reversing to the original variables (r, θ), parameters (s, φ) and functions (f (r, θ), f α (s, φ)) this Line-RT closed form inversion formula would in principle yield a closed form inversion formula for the RT -1 α . However this cannot be done for arbitrary α. For α = m = 2, 3, ..., the complete Line-RT data parameterized by (s ∈ R + , φ ∈ [0, 2π]) is generated by an incomplete set of actual RT α data in (s ∈ R + , φ ∈ [0, 2π m ]) since φ = mφ. Since a set of incomplete actual data cannot faithfully reconstruct the original f (r, θ), this reconstruction formula becomes meaningless. However for α = 1 m with m = 1, 2, ..., this is not so, as shown below. Proposition 7.3 Under these assumptions, the required Line-RT inversion data parameterized by (s > 0, 0 < φ < 2π) appears as the direct sum of m copies of the physical RT α data parameterized by (s > 0, 0 < φ < 2πm ). Then using the 2π periodicity of the actual data (see (4)), the Line-RT inversion formula (54) reproduces the RT 1 m inversion formula of equation (28).

g α (r , θ ) = - 1 2π 2 r
Proof: Proceed with equation (62) and go back to the original variables (r, θ) and parameters (s, φ). As (66)

This j-sum in (66) is given by the following lemma.

Proof: The left-hand-side of (67) can be expressed as a logarithmic derivative of a product P m (ξ, γ)

where P m (ξ, γ) is a polynomial in ξ with highest order term ξ m and having m zeros at cos (γ-2πj) m , with j = 0, 1, ..., (m -1). Such a polynomial can be found as follows. Let Q m (ξ, cos γ) = (T m (ξ) -cos γ), where T m (ξ) is a first kind Chebyshev polynomial of order m . Next for any j = 0, 1, ..., (m -1), we can express cos γ as

cos γ = cos m γ -2πj m = T m cos γ -2πj m . ( 69
)
Thus we have P m (ξ, γ) = ( 12 ) m -1 Q m (ξ, cos γ). Then (67) is established by using the identity d dξ T m (ξ) = m U m -1 (ξ) (see [START_REF] Oberhettinger | Formulas and Theorems for the Special Functions of Mathematical Physics 3rd Ed[END_REF] p. 258).

Theorem 4 . 1

 41 The full closed form RT m m inversion formula for F m m (r , θ) is

  l∈Z (...l...) = l0∈Z (...(m l 0 )...)+ l1∈Z (...(m l 1 +1)...)+...+ lm-1∈Z (...(m l m-1 +(m-1))..

  (i) Search for new orthogonal function expansions arising from the RT -1 m m , as generalizations of those found by Cormack in the case of RT -1 1 m in [24], (ii) Inversion of the Extended Circular Harmonic Transform (ECHT) by Mellin

7. 3 .

 3 On Cormack's suggested mapping and a new derivation of RT 1 m inversion formula

∂ 1 m

 1 g α (s , φ ) ∂s ds = ∂ g α (s, φ) ∂s ds, s theorem to be valid, an exchange of s and φ integrations in (64) allows to rewrite the g 2π-periodicity of g α (s, φ) = g α (s, φ + 2π) (see subsection 3.1), we may re-express the kernel K 1 m (s, r, θ), after setting φ = (ψ + j2π) with j = 0, 1, ...(mcos θ-ψ-j2π m .

Lemma 7 . 4

 74 For given γ ∈ [0, 2π], the Chebyshev polynomials U m -1 (ξ) and T m (ξ) satisfy the identity(m -1) j=0 1 ξ -cos γ-j2π m = m U m -1 (ξ) T m (ξ) -cos γ .

  ln P m (ξ, γ), with P m (ξ, γ) =

  7.2. On RT α "Consistency Conditions" and their meaning As the RT α -Consistency Conditions * are crucial to the derivation of the RT α inverse formula, it is useful to see what they mean in the context of Cormack's 1984 results. In [27], it is shown that the Ludwig's Line Radon transform range characterization in polar coordinates are equivalent to the vanishing of a finite number of moments of a given radial component of the Line Radon data, although the name of Consistency Conditions has not been spelled out. Here we give a short proof of this equivalence in a form which can be extended to the α = 1 m curve Radon transform. Proposition 7.1 For the Line-RT (or RT 1 ), Cormack's Consistency Conditions imply Ludwig's characterization. Proof: The κ-moment of the Radon data f (s, φ) with κ ∈ N is by definition m κ = ∞ -∞ ds s κ f (s, φ). It may be put under the form

† To whom correspondence should be addressed (truong@cyu.fr)

§ In[START_REF] Cormack | Radon's problem -Old and New[END_REF], this formula was given with a missing global minus sign. Cormack, who had initiated this representation in[START_REF] Cormack | Radon's problem -Old and New[END_REF], did not use it in his CH-C inversion kernel.

+ This may be viewed as a generalization of the (even or odd) half-wave symmetry in signal theory which is represented by Fourier series only odd components.

Acknowledgements The author is grateful to the anonymous referees for their remarks and suggestions, as well as providing additional relevant references. They have greatly contributed to improve the quality of this paper.


Pierre Gaillard
email: pierre.gaillard@u-bourgogne.fr

Another rational solutions to the Burgers' equation

Keywords: Burgers equation PACS numbers : 33Q55, 37K10, 47.10A-, 47.35.Fg, 47.54.Bd

come

Another rational solutions to the Burgers' equation

Introduction

We consider the Burgers' equation which can be written as

u t + u xx + uu x = 0 (1)
where the subscripts x and t denote partial derivatives.

In 1915, Bateman [START_REF] Bateman | Some recent researches on the motion of fluids[END_REF] introduced this equation ( 1). This equation appears in different contexts in physics as in gas dynamics [START_REF] Xj | A mathematical model illustrating the theory of turbulence[END_REF], acoustics [START_REF] Sugimoto | Burgers equation with a fractional derivative: hereditary effects on nolinear acoustic waves[END_REF], heat conduction [START_REF] Bluman | The general similarity solution of the heat equation[END_REF], in soil water [START_REF] Su | Some recent researches on the motion of fluids[END_REF], in hydrodynamics turbulence [START_REF] Burgers | Mathematical examples illustating the relations occuring in the theory of turbulent fluid motion[END_REF][START_REF] Burgers | The non-linear diffusion equation: asymptotic solutions and statistical problems[END_REF][START_REF] Cole | On a quasi-linear parabolic equation occuring in aerodynamics[END_REF], in shock waves [START_REF] Whitham | Linear and nonlinear waves[END_REF],... The first solutions has been constructed by Bateman [START_REF] Bateman | Some recent researches on the motion of fluids[END_REF] in 1915. Other types of methods have been used to solve this equation. We can quote the exp-function method [START_REF] Ebaid | Exact solitary wave solution for some nonlinear evolution equations via exp-function method[END_REF], the tanh-coth method [START_REF] Wazwaz | Multiple front solutions for the Burgers equation and the coupled Burgers equations[END_REF], the groups actions on coset bundles [START_REF] Wolf | Nonlinear differential equations as invariants under group action on coset bundles: Burgers and Korteweg de Vries equation families[END_REF], the Cole-Hopf method [START_REF] Hopf | The partial differential equation u t + uu x = u xx[END_REF][START_REF] Ohwada | Cole-Hopf transformation as numerical tool for the Burgers equation[END_REF][START_REF] Vaganan | Cole-Hopf transformation for higher dimensional Burgers equation with variable coefficients[END_REF], the homotopy perturbation method [START_REF] Srivastava | 1+ n)-Dimensional Burgers' equation and its analytical solution: A comparative study of HPM[END_REF],...

We can quote some recent results in connection with this study as [START_REF] Gao | An analytical solution for two and three dimensional[END_REF], [START_REF] Yang | Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations[END_REF], [START_REF] Xia | Explicit solutions of higher dimensional Burgers equations[END_REF]. Rational solutions to the Burgers' equation are constructed in this paper. We give solutions as a quotient of a polynomial of degree n -1 in x and n -1 -n 2 in t by a polynomial of degree n in x and n 2 in t, [p] being the greater integer less or equal to p.

We explicitly build these solutions for orders between 1 and 20.

Rational solutions to the Burger's equation

We consider the following polynomials defined by

p n (x, t) = n k=0 x k k! (-t) n -k 2 n -k 2 ! 1 -n -k -2 n -k 2 , for n ≥ 0, p n (x, t) = 0 for n < 0.
(

) 2
With the choice of these polynomials, we have the following statement Theorem 2.1 The function v n defined by

v n (x, t) = -2 p n-1 (x, t) p n (x, t) , (3)
where p n are defined by previous relations [START_REF] Xj | A mathematical model illustrating the theory of turbulence[END_REF], is a solution to the Burgers' equation (1)

u t + u xx + uu x = 0.
Remark 2.1 In the following, we will call the solution v n , the solution of order n of the Burgers' equation (1).

Remark 2.2 More explicitly, the previous polynomials can be written as

p 2k (x, t) = n l=0 x 2l (2l)! (-t) k-l k -l! , for k ≥ 0, p 2k+1 (x, t) = n l=0 x 2l+1 (2l + 1)! (-t) k-l k -l! , for k ≥ 0, p n (x, t) = 0 for n < 0.
3 Explicit first order solutions All these rational solutions are singular. At each order, we see the appearence of curves of singularities. The patterns of singularities are lines or horseshoe type depending on the order of the solution.

First order solutions

Proposition 3.1 The function v defined by

v 1 (x, t) = 2 x ( 4
)
is a solution to the Burgers' equation (1).

Solutions of order two

Proposition 3.2 The function v 2 defined by

v 2 (x, t) = -4x -x 2 + 2 t , ( 5
)
is a solution to the Burgers' equation (1).

Solutions of order three

Proposition 3.3 The function v 3 defined by

v 3 (x, t) = 6 -x 2 + 2 t x (-x 2 + 6 t) , (6)
is a solution to the Burgers' equation (1).

Solutions of order four

Proposition 3.4 The function v 4 defined by

v 4 (x, t) = -8 x -x 2 + 6 t x 4 -12 x 2 t + 12 t 2 , ( 7
)
is a solution to the Burgers' equation (1).

Solutions of order five

Proposition 3.5 The function v 5 defined by

v 5 (x, t) = 10 x 4 -12 x 2 t + 12 t 2 x (x 4 -20 x 2 t + 60 t 2 ) , ( 8
)
is a solution to the Burgers' equation (1).

Solutions of order six

Proposition 3.6 The function v 6 defined by

v 6 (x, t) = -12 x x 4 -20 x 2 t + 60 t 2 -x 6 + 30 x 4 t -180 x 2 t 2 + 120 t 3 , ( 9
)
is a solution to the Burgers' equation (1).

Solutions of order seven

Proposition 3.7 The function v 7 defined by v 7 (x, t) = 14 -x 6 + 30 x 4 t -180 x 2 t 2 + 120 t 3 x (-x 6 + 42 x 4 t -420

x 2 t 2 + 840 t 3 ) , ( 10
)
is a solution to the Burgers' equation (1). 3.9 Solutions of order nine Proposition 3.9 The function v 9 defined by v 9 (x, t) = 18 x 8 -56 x 6 t + 840 x 4 t 2 -3360 x 2 t 3 + 1680 t 4 x (x 8 -72 x 6 t + 1512 x 4 t 2 -10080

x 2 t 3 + 15120 t 4 ) , ( 12
)
is a solution to the Burgers' equation (1).

Figure 9. Solution of order 9 to (1). -x 10 + 90 x 8 t -2520 x 6 t 2 + 25200 x 4 t 3 -75600 x 2 t 4 + 30240 t 5 x (-x 10 + 110 x 8 t -3960 x 6 t 2 + 55440 x 4 t 3 -277200

Solutions of order ten

Solutions of order eleven

x 2 t 4 + 332640 t 5 ) , (14)
is a solution to the Burgers' equation (1).

Solutions of order twelve

Proposition 3.12 The function v 12 defined by v 12 (x, t) = n(x, t) d(x, t) , n(x, t) = -24x(-x 10 +110 x 8 t-3960 x 6 t 2 +55440 x 4 t 3 -277200 x 2 t 4 +332640 t 5 ), d(x, t) = x 12 -132 tx 10 + 5940 t 2 x 8 -110880 t 3 x 6 + 831600 t 4 x 4 -1995840 t 5 x 2 + 665280 t 6 is a solution to the Burgers' equation (1). is a solution to the Burgers' equation (1).

Figure 13. Solution of order 13 to (1).

Solutions of order fourteen

Solutions of order fifthteen

Solutions of order sixteen

Proposition 3.16 The function v 16 defined by v 16 (x, t) = n(x, t) d(x, t) , n(x, t) = -32x(-x 14 + 210 tx 12 -16380 t 2 x 10 + 600600 t 3 x 8 -10810800 t 4 x 6 + 90810720 t 5 x 4 -302702400 t 6 x 2 + 259459200 t 7 ), d(x, t) = x 16 -240 tx 14 +21840 t 2 x 12 -960960 t 3 x 10 +21621600 t 4 x 8 -242161920 t 5 x 6 + 1210809600 t 6 x 4 -2075673600 t 7 x 2 + 518918400 t 8 is a solution to the Burgers' equation (1).

Figure 16. Solution of order 15 to (1).

Solutions of order seventeen

Proposition 3.17 The function v 17 defined by v 17 (x, t) = n(x, t) d(x, t) , n(x, t) = 34(x 16 -240 tx 14 +21840 t 2 x 12 -960960 t 3 x 10 +21621600 t 4 x 8 -242161920 t 5 x 6 + 1210809600 t 6 x 4 -2075673600 t 7 x 2 + 518918400 t 8 ), d(x, t) = x(x 16 -272 tx 14 +28560 t 2 x 12 -1485120 t 3 x 10 +40840800 t 4 x 8 -588107520 t 5 x 6 + 4116752640 t 6 x 4 -11762150400 t 7 x 2 + 8821612800 t 8 ) is a solution to the Burgers' equation [START_REF] Bateman | Some recent researches on the motion of fluids[END_REF]. is a solution to the Burgers' equation (1).

Figure 18. Solution of order 18 to (1).

Solutions of order nineteen

Solutions of order twenty

Conclusion

We have given an expression of rational solutions to the Burgers' equation involving particular polynomials.

In particular, we have constructed explicit solutions to the Burgers' equation for the orders n = 1 until n = 20. All these solutions are singular. We can classify them by the pattern of their singulatities.

The singularities of these solutions depend on the orders of the solutions. When we consider odd order solutions we have always the line x = 0 of singularities.

In the case of even order solutions n = 2p, the singularities form horseshoe patterns with p branches. It will be interesting to construct solutions of this equation depending on some real parameters.

Figure 1 .

 1 Figure 1. Solution of order 1 to (1).

Figure 2 .

 2 Figure 2. Solution of order 2 to (1).

Figure 3 .

 3 Figure 3. Solution of order 3 to (1).

Figure 4 .

 4 Figure 4. Solution of order 4 to (1).

Figure 4 .

 4 Figure 4. Solution of order 5 to (1).

Figure 6 .

 6 Figure 6. Solution of order 6 to (1).

Figure 7 .

 7 Figure 7. Solution of order 7 to (1).

3. 8

 8 Solutions of order eight Proposition 3.8 The function v 8 defined by v 8 (x, t) = -16x -x 6 + 42 x 4 t -420 x 2 t 2 + 840 t 3 x 8 -56 x 6 t + 840 x 4 t 2 -3360x 2 t 3 + 1680 t 4 , (11)is a solution to the Burgers' equation (1).

Figure 8 .

 8 Figure 8. Solution of order 8 to (1).

Proposition 3 . 10

 310 The function v 10 defined by v 10 (x, t) = 20x x 8 -72 x 6 t + 1512 x 4 t 2 -10080 x 2 t 3 + 15120 t 4 -x 10 + 90 x 8 t -2520 x 6 t 2 + 25200 x 4 t 3 -75600 x 2 t 4 + 30240 t 5 , (13) is a solution to the Burgers' equation[START_REF] Bateman | Some recent researches on the motion of fluids[END_REF].

Figure 10 .

 10 Figure 10. Solution of order 10 to (1).

Proposition 3 . 11

 311 The function v 11 defined byv 11 (x, t) = 22

Figure 11 .

 11 Figure 11. Solution of order 11 to (1).

Figure 12 .

 12 Figure 12. Solution of order 12 to (1).

Figure 14 .

 14 Figure 14. Solution of order 14 to (1).

Figure 15 .

 15 Figure 15. Solution of order 15 to (1).

Figure 17 .

 17 Figure 17. Solution of order 17 to (1).

Figure 19 .

 19 Figure 19. Solution of order 19 to (1).

  is a solution to the Burgers' equation[START_REF] Bateman | Some recent researches on the motion of fluids[END_REF].

Figure 20 .

 20 Figure 20. Solution of order 20 to (1).

  = -36x(x16 -272 tx 14 + 28560 t 2 x 12 -1485120 t 3 x 10 + 40840800 t 4 x 8 -588107520 t 5 x 6 + 4116752640 t 6 x 4 -11762150400 t 7 x 2 + 8821612800 t 8 ), d(x, t) = -x 18 + 306 tx 16 -36720 t 2 x 14 + 2227680 t 3 x 12 -73513440 t 4 x 10 + 1323241920 t 5 x 8 -12350257920 t 6 x 6 + 52929676800 t 7 x 4 -79394515200 t 8 x 2 + 17643225600 t 9

	3.18 Solutions of order eighteen
	Proposition 3.18 The function v 18 defined by v 18 (x, t) =	n(x, t) d(x, t)	,
	n(x, t)

  18 + 342 tx16 -46512 t 2 x 14 + 3255840 t 3 x 12 -126977760 t 4 x 10 + 2793510720 t 5 x 8 -33522128640 t 6 x 6 + 201132771840 t 7 x 4 -502831929600 t 8 x 2 + 335221286400 t 9 )

	is a solution to the Burgers' equation (1).

Proposition 3.19 The function v 19 defined by v 19 (x, t) = n(x, t) d(x, t) , n(x, t) = 38(-x 18 + 306 tx 16 -36720 t 2 x 14 + 2227680 t 3 x 12 -73513440 t 4 x 10 + 1323241920 t 5 x 8 -12350257920 t 6 x 6 + 52929676800 t 7 x 4 -79394515200 t 8 x 2 + 17643225600 t 9 ), d(x, t) = x(-x


Thomas Botzung

Pierre Nataf

Exact Diagonalization of SU(N ) Fermi-Hubbard Models

We show how to perform exact diagonalizations of SU(N ) Fermi-Hubbard models on L-sites clusters separately in each irreducible representation (irrep) of SU(N ). Using the representation theory of the unitary group U(L), we demonstrate that a convenient orthonormal basis, on which matrix elements of the Hamiltonian are very simple, is given by the set of semi-standard Young tableaux (or equivalently the Gel'fand-Tsetlin patterns) corresponding to the targeted irrep. As an application, we study the robustness of some SU(N ) phases predicted in the Heisenberg limit upon decreasing the on-site interaction on various lattices of size L ≤ 12 and for 2 ≤ N ≤ 6. In particular, we show that a long range color ordered phase emerges for moderate U for N=4 at filling 1/4 on the triangular lattice.

The Fermi-Hubbard model (FHM) is among the most important models in condensed matter [1][2][3]. In particular, the SU(2) FHM on the square lattice might describe the physics of electrons carrying a spin one half in cuprates superconductors [3][4][5] and has motivated numerous theoretical investigations [6,7]. Considering N , the number of degenerate orbitals, as an integer parameter of the models, a natural extension of the SU(N = 2) FHM is the SU(N ) FHM [8][9][10].

This higher symmetry group Hamiltonian, firstly introduced as a theoretical tool to provide an asymptotic description of spins 1/2 in the large N limit [11][12][13][14], can also describe some condensed matter systems like transition metal compounds [15,16] or graphene with SU(4) spin valley symmetry or in twisted bilayer [17]. Alternatively, Alkaline-earth cold atoms like 173 Yb or 87 Sr can simulate SU(N ) invariant FHMs for N up to 10 on various engineered optical lattices [18,19]. Besides, the continuous experimental achievements in this field [20][21][22][23][24][25] brought theoreticians to investigate these systems in order to look for exotic phases that would generalize their N = 2 counterpart.

Most of the theoretical investigations focused on the large on-site repulsion limit, where the atoms, in the Mott insulating phase, are described by SU(N ) Heisenberg models (HM) [26][27][28][29][30][31][32][33][34]. Depending on the lattices and on the number of colors N , different two-dimensional phases are predicted at T = 0 among which the SU(N ) plaquette phases [35][36][37][38][39][40] are cousins of the valence band states for spins 1/2, the Neel long-range color ordered (LRO) states [30,31] are analogous to the famous (π, π) (resp. 120 • ) Neel states existing on the square [41,42] (resp. the triangular [43,44]) lattice for N = 2, and diverse kinds of SU(N ) spin liquids [45][46][47][48][49][50][51][52][53] generalizing the Anderson Resonating Valence Band (RVB) states [4,54].

A part from the one-dimensional system where there is a Bethe ansatz solution [55], the theoretical investigation of these models, based on advanced numerical tools is challenging mainly because the dimension of the full Hilbert space on finite-size lattices increases exponentially, being equal to N L where L is the number of sites of the cluster, for filling 1/N (exactly one particle per site) in the SU(N ) HM.

However, it was realized that working in the SU(N ) singlets subspace, which contains the ground state (GS) in the antiferromagnetic case, is very advantageous as its dimension is much smaller than N L . For instance for N = 6 and L = 12, such a dimension is equal to 132 while N L ≡ 6 12 ≈ 2 × 10 9 . In addition, the exact diagonalization (ED) of the HM directly in the SU(N ) singlets subspace can be made easy on the basis of Standard Young Tableaux (SYT) using the orthogonal representation of the group of permutations S L [56,57]. It is crucial to extend this theory to the SU(N ) FHM as the dimension of the full Hilbert space is even larger, i.e equal to 2 N L . In fact, for N = 6, L = 12 at filling 1/6 the dimension of the singlets subspace for the SU(6) FHM is ≈ 14 × 10 6 while the full Hilbert space has dimension 2 72 ≈ 5 × 10 21 .

In this letter, we use the representation theory of the Lie group U (L) to show how to perform ED of the SU(N ) FHM directly in each irrep. After the description of the method, we apply the procedure to show some ED results on square/triangular clusters of size up to L = 12 and for N up to N = 6 to see how robust are some SU(N ) Mott insulating phases while decreasing the on-site repulsion.

The Hamiltonian for the SU(N ) FHM reads:

H = ⟨i,j⟩ -t ij E ij + h.c + U 2 L i=1 E 2 ii , (1)
where the t ij are the (possibly complex) hopping amplitude between sites i and j of a L-sites finite cluster, the on-site interaction amplitude is U . The SU(N ) invariant hopping terms

E ij = E † ji = N σ=1 c † iσ c jσ sat- isfy the commutation relation of the U (L) generators (∀1 ≤ i, j, k, l ≤ L) : [E ij , E kl ] = δ jk E il -δ li E kj , (2)
so that the Hamiltonian in Eq. ( 1), where the integer parameter N is hidden, can be seen as an element of the Lie algebra of the unitary group U (L) [START_REF]Or more precisely of its universal enveloping algebra since it contains some square of generators[END_REF]. It should be considered as the counterpart of the quantum permutation Hamiltonian for the SU(N )-invariant HM, i.e H = ⟨i,j⟩ J ij P ij + h.c, with P ij (resp. J ij ) the permutation (resp. coupling constant) between interacting sites i and j, for which the representation theory of the algebra of the group of permutations was used to perform ED directly and separately in each irrep of SU(N ) [56]. We remind that an irrep of SU(N ) is labelled by a Young tableau (YT), or shape α (See Fig. 1) the N rows of which representing N integers α = [α 1 , α 2 , ..., α N ] such that α 1 ≥ α 2 ≥ ... ≥ α N ≥ 0 and N i=1 α i = M , where M is the number of particles. For L = 1 site, the Mfermions wave-functions are fully antisymmetric in the exchange of colors, and belong to the irrep α which is a M-boxes single column α = [1, 1, ..α M = 1], and its dimension D M,N L=1 = N M simply corresponds to all the ways to choose M colors among N.

More generally, for L sites and M SU(N ) fermions, the Hilbert space dimension D M,N L is equal to:

α h ᾱ L L i=1 N ᾱ(i) = α d α N d ᾱ L , (3)
where the sums on both side run over all the YT α of maximum L columns and N rows. ᾱ = [ᾱ(1), ..., ᾱ(L)] is defined as the conjugate YT of α, transforming rows into columns (i.e flipping the shape by 45 • ), cf Fig. 1 for some examples. In the LHS of Eq. ( 3), ᾱ is a distribution of fermions: ᾱ(j) being the number of fermions (necessarily ≤ N ) on site j for 1 ≤ j ≤ L,

L i=1 N ᾱ(i)
is the number of states for such a distribution. The factor h ᾱ L , defined as h ᾱ L = L!/ N k=0 (n ᾱ k )!, where n ᾱ k = Cardinal{j ∈ 1; L /ᾱ(j) = k}, is the number of distributions corresponding to a given partition ᾱ while permuting the ᾱ(j) (or the site indices j) for 1 ≤ j ≤ L.

In the RHS of Eq. (3), d α N (resp. d ᾱ L ) stands for the dimension of the SU(N ) irrep α (resp. the U(L) irrep ᾱ) [START_REF]distinction is relevant when we include/exclude representation of generators with non vanishing trace like some combinations of Eii[END_REF], that we can calculate using existing formulas, like the hook length formulas [START_REF] Weyl | [END_REF][START_REF] De | Representation theory of the symmetric group[END_REF]. These dimensions are equal to the number of semi-standard Young tableaux (ssYT) of shape α (resp. ᾱ) filled with numbers from 1 to N (resp. L), since these latter form a basis of the SU(N ) or U(L) irrep. Given a U(L) irrep represented by some YT, a ssYT is filled up with integers numbers from 1 to L in non-descending order from left to right in any row (repetitions allowed), and in strictly ascending order (repetitions not allowed) from top to bottom in any column, cf Fig. 1 and Eq. ( 6) for some examples.

As we detail below, for the color-invariant SU(N ) FHM model in Eq. ( 1), the Hilbert space for the set of M SU(N ) fermions on a L-sites cluster can be decomposed, or color-factorized, following the RHS of the equation for the dimension D M,N L , i.e Eq. (3). In particular, targeting a given collective SU(N ) irrep α, we will need to diagonalize a matrix of dimension d ᾱ L , and there will be d α N independent copies (some multiplicity) of the corresponding spectrum in the full energy spectrum of the model. For instance, when M is a multiple of N , one important sector is the SU(N ) singlets sector as it usually L=12,N =6 ≈ 8.27 × 10 10 ). As N increases, it is more and more advantageous to implement the full SU(N ) symmetry, working in the SU(N ) singlets sector, and more generally in a sector of a given irrep α.

SU(2) Irrep ↵ = 6 fermions Total spin=0 ↵ = 4 fermions Total spin=1 Highest Weight State of U(4) irrep 1 1 2 2 3 3 1 1 2 3 Physical State or or a) b) = 2 2 2 0 2 2 2 2 2 2 = 2 1 1 0 2 1 1 2 1 2 1 p 2 + Application of Eij 1 2 3 4 d ↵ N = 3 independent sectors dimension d ↵ T L = 10 d ↵ N = 1 SU (2) singlets sector of of dimension d ↵ L = 10 of dimension d ↵ L = 15 sites Q z = 1 Q z = 1 Q z = 0 Q z = 0
For a given U(L) irrep ᾱ, the Highest Weight State (|hws⟩), uniquely (up to some similarity) and fully determines the irrep as one can generate the entire basis by applications of the generators E ij (for 1 ≤ i, j ≤ L). It is represented by the shape ᾱ filled with 1 for the first row (of length ᾱ(1)), 2 for the second row of length ᾱ(2)), etc...(cf Fig. 1). It is defined by the following properties:

E ii |hws⟩ = ᾱ(i) ∀i ∈ 1; L and E ij |hws⟩ = 0 for i < j. Crucially, in the D M,N L -dimensional space of M-fermions SU(N ) wave-functions, there are d α N orthonormal states |ϕ hws α,k ⟩ (k = 1 • • • d α N )
which have these properties and can then be represented by the same ssYT associated with the |hws⟩. For example, for the SU(N ) singlets irrep α S,M , there is only one state |ϕ hws α,1 ⟩ and it is the product of SU(N ) singlets for sites 1, 2, • • • M/N , with no particles on sites M/N + 1, • • • L (cf Fig. 1).

Introducing the SU(N ) total generators

Q a = L i=1 σ,σ ′ c † i,σ ′ λ σ ′ σ a c i,σ for a = 1 • • • N 2 -1, with (λ σ ′ σ a )
σ ′ σ the generalized Gell-Mann matrices for SU(N ), the states |ϕ hws α,k ⟩ are eigenvectors of the N -1 diagonal Q a (spanning the Cartan subalgebra) with different set of N -1 eigenvalues. Then, the d α N sectors generated by the (repeated) applications of the E ij (for 1 ≤ i, j ≤ L) on each |ϕ hws α,k ⟩ (for k ∈ 1; d α N ) will be invariant under the action of H in Eq. ( 1) since [Q a , E ij ] = 0, ∀1 ≤ i, j ≤ L and a ∈ 1; N 2 -1 . Moreover, each sector will independently represent the U(L) irrep ᾱ, i.e being in one to one correspondance with the set of ssYT of shape ᾱ. On such a basis, the matrix elements of the infinitesimal generators E p,p , which are the occupation numbers on site p for p = 1 • • • L and of E p-1,p (p = 2 • • • L), which generalize the lowering operator J -for SU (2), are very simple. They were found by Gelfand and Tsetlin [START_REF] Gelfand | [END_REF][START_REF] Vilenkin | Representation of Lie groups and special functions[END_REF] and expressed in the basis of what is now called the "Gelfand-Tsetlin" patterns (GTp) which are triangular arrangements of integers fully equivalent to the ssYT, the first row of which being the lengths ᾱ(1) ᾱ(2) • • • ᾱ(L) (See. Fig. 1 for some examples and [START_REF] Alex | [END_REF] for the systematic conversion from GTp to ssYT for a given shape ᾱ). Firstly, for p = 1 • • • L, the operators E p,p are diagonal in the basis of the ssYT, with coefficients equal to the number of occurrences of p inside each ssYT. It also means that we know readily from its content, the occupation numbers of a ssYT on each site (cf Fig. 1).

Secondly, calling |ν⟩ a ssYT, one has for p = 2 • • • L:

E p-1p |ν⟩ = p-1 j=1 a j p-1 F j p-1 |ν⟩, (4)
where the tableau operators F j p-1 transform the number p in the j th row in |ν⟩ into p -1. As for the coefficients a j p-1 , which vanishe in case such a transformation is not possible either because there is no p in the j th row of |ν⟩, either because the resulting tableau is not a proper ssYT, they read:

a j p-1 = p i=1 (l i,p -l j,p-1 ) p-2 i=1 (l i,p-2 -l j,p-1 -1) i̸ =j (l i,p-1 -l j,p-1 ) i̸ =j (l i,p-1 -l j,p-1 -1) 1/2 , ( 5
)
where l k,q = m k,qk with m k,q the length of the k th row of the sub-tableau that remains when we delete all the boxes containing numbers > q in |ν⟩. Thus, for the SU(4) adjoint irrep at filling 1/4 for L=12 (the basis has then 57972915 elements), we have for instance: .

Moreover, from successive applications of the commutation relations Eq. ( 2), and from E ij = E † ji , one gets the matrix representing H defined in Eq. 1 in the irrep ᾱ, which corresponds to the SU(N ) irrep α.

Finally, let's mention the large U limit at filling 1/N . Then, the ssYT with strictly more than one occurence of a site index are too high in energy, and we are left with the subset of SYTs, in agreement with the ED protocole developed for the HM [56]. Then, for each SYT |µ⟩ and for fixed i = 2 • • • L, a j i-1 is either 0, either a j i-1 = |d -1|/d for j the unique row where p stands in |µ⟩, with d the Manhattan distance between p and p -1. Then, the transposition P ii-1 being equal to -E ii-1 E i-1i + 1, we refind the Young rules for the orthogonal representations of the algebra of the group of permutations on the basis of SYT in a given irrep α [56,65]. We have applied this theory to study the FHM of Eq. ( 1) for uniform nearest neighbors hopping t ij ≡ t = 1 as a function of U at filling 1/N , starting from the Heisenberg limit (U → ∞) and diminishing U . The ground states of the SU(3) HM on the triangular lattice (TL) and of the SU(5) HM on the square lattice (SL) are both Neel LRO states, with a three sublattice ordering pattern for SU(3) [31,66,67] and a (chess) knight move pattern for SU(5) [56]. We give evidence of such orders by calculating the simple correlations patterns of the exact ground states of the FHM in Fig. 2 for U = 10. Moreover, the energy spectra plotted as a function of the quadratic Casimir C 2 [56,68] of the different irreps α c,d) and for the L = 4×3 periodic square lattice in (e) where we focused on N = 6. Note that the constant LU/2 has been withdrawn. In c), the Atos is reminiscent of the one revealing the four sublattice order in the HM with nnn couplings [28]. Inset of e): spin (resp. singlet) gap in blue (resp. red).

exhibit an Anderson tower of states (Atos) which reveals the continuous symmetry breaking of SU(3) (resp SU( 5)) as shown in Fig. 3 (resp. in [69]). We have checked the convergence in the limit U → ∞ within each irrep α of the eigenergies towards those of the HMs with the factor 2/U .

While diminishing U , the structure of the energy spectra stays the same up to U ∼ 2.5 (resp. U ∼ 1) for SU(3) on the L = 12 TL (resp. SU (5) on the L = 10 SL). Then, some energy plateau as a function of C 2 appears for smaller U, which is also true for SU(2) on the TL, as shown in Fig. 3. Such a system should be in the metallic phase for U ≲ 8.5, as expected from DMRG simulations on large cylinders [70], so that the plateau could be a signature of the metallic phase in the weak coupling limit. To further characterize the metallic phase and to locate its boundary, we show in Fig. 4 and in [69] the charge gap defined by: ∆

c = E 0 (M = L + 1) + E 0 (M = L -1) -2E 0 (M = L)
, where E 0 (M ) is the minimal energy for the lattice with M fermions, which implies the diagonalization over all the relevant M = L, L ± 1 boxes irreps ᾱ. It suggests that the metallic phase develops for U ≤ U c = 9.8(±0.4) for SU(3) on a TL and for U ≤ U c = 8.75(±0.15) for SU(5) on a SL, with apparently no intermediate phase between the latter and the LRO in the large U limit. The presence of an in-between phase might occur when the HM limit is not a LRO, like for SU(3) on the Honeycomb lattice [72]. The SU(4) FHM on the TL might enter into this category as the HM limit is a gapless QSL [28,40,52]. In fact, while there are four low lying energy SU(4) singlet states for U = 50 on the L = 12 sites TL, when decreasing the interaction, the singlets gap starts increasing and an emerging Atos appears for U ≲ 15 (cf Fig. 3 with both U = 12 and U = 50), which is similar to the one occurring in a pseudo HM with nextnearest neighbour (nnn) couplings on the same L = 12 TL cluster [28]. The nnn couplings, which are present at order 4 in t/U in the large U limit [51], were shown to stabilize a four sublattice order [28], also apparent in the correlation patterns of our Fig. 2. From the charge gap shown in Fig. 4, the boundary between this phase and the metallic phase occurs around U = U c = 11.2 ± 0.2. Like for the other systems, such a value changes for larger L due to finite size effects, as illustrated for SU(2), L = 12 and L = 16 in [69].

Finally, we have investigated the SU(6) FHM on the SL, as the HM limit is also not a LRO but a plaquette state [39]. Through the correlations pattern of the GS on a 4 × 3 periodic SL for U = 10 in Fig. 2, we found some evidence of the 6 sites plaquette state in the Mott phase, a feature confirmed by the presence of two low lying energy SU(6) singlet states, compatible with the periodic boundary conditions, for large U ( i.e U = 20 cf Fig. 3). When U decreases, the spin gap becomes smaller than the singlet gap, possibly revealing the Mott transition. However, with the current version of our code and with the limitation of our computational ressources, the necessary calculation of the charge gap was too demanding, leaving open both the question of the size of the metallic phase and the presence of some intermediate phase.

To conclude, we found an efficient protocole to perform ED of the FHM on L-sites clusters directly in each SU(N ) irrep, which uses the set of ssYT (or GT patterns) as a convenient basis with matrix elements of the U (L) group generators. This approach, which generalizes the use of SYT for SU(N ) in HMs [56], dramatically reduces the dimension of the matrices to diagonalize. We applied our method to study the survival of the SU(N ) Mott phases for N = 3 to N = 6 on TL and SL when the on-site interaction U decreases. In particular, we found an emerging intermediate LRO phase for SU(4) on the TL, reminiscent of the four sublattice order in the HM with nnn couplings [28]. In the future, one possible perspective would be the implementation of the ssYT basis in tensor networks/DMRG algorithm, in a fashion similar to what was done with the SYT for the HM [73,74].

Exact Diagonalization of SU(N ) Fermi-Hubbard Models -Supplemental Material -Thomas Botzung 1 , and Pierre Nataf In this supplemental material, we show complementary numerical results obtained via ED. In Sec. I, we focus on SU(5) on the square lattice with L=10 sites. For this specific case, we show some energy spectra and we detail the fitting procedure of the charge gap that we have used for all the systems under consideration in our manuscript. Then, in Sec. II, we consider the case SU(2) on a triangular lattice of L = 12 and L = 16 sites which illustrate the finite size effects on the spin/charge gaps. Finally, in Sec. III, we show the evolution of real space correlations while increasing the interaction U for SU(3) on the L=12 triangular lattice and for SU (5) on the square lattice.

I. SU(5) ON A SQUARE LATTICE

In this section, we show the data for SU(5) on a square lattice with L=10 sites at filling 1/5, that we have discussed in the main text. The long-range color ordered state with broken SU (5) symmetry is revealed by an Anderson tower of states (Atos), clearly visible on Fig. S1 (b) for U = 10. The situation is quite different at small U , as seen from Fig. S1 (a), where the Atos has disappeared and an energy plateau appears. In Fig. S1 (c), we plot the charge gap ∆ c to locate the boundary U = U c between the metallic and the LRO phases. We see that ∆ c opens rapidly around U ≈ 8.

To estimate the critical point roughly, we use a fit of the form ∆ c = a * Ub. The latter corresponds to the expected behavior of the charge gap at large U, since the energy cost to add one particle should be equal to U : using the definition of ∆ c one can readily show that for U → ∞, ∆ c = U . Then, such a linear function shown in dashed lines in the figures cross the x axis at U = U c . However, due to the small curvature of ∆ c , it depends on the values of U used for the fit; we took different sets belonging to U ∈ [40; 100]. With this method, we obtain 8.6 < U c < 8.9 with 0.992 < a < 0.997. In addition, in Fig. S1 (d), we show the spin gap, defined as the difference between the SU(5) adjoint and the SU(5) singlet minimal energies. The spin gap increases rapidly in the metallic phase, reaching its maximal value around U ≈ 8, close to U = U c , a feature also observed in the other cases studied here.

III. EVOLUTION OF CORRELATIONS WITH U

In this section, we study how the real space correlations, defined as ⟨P 1j ⟩ -1/N where P 1j ≡ -1 + E 1j E j1 , evolve while increasing the on-site interaction U. In Fig. S3, we show the correlations for two lattice geometries and different U. From left to right, we have U=1, U=5, U=10, and U=20. In (a), we consider the case SU(3) on a triangular lattice. We see that for U = 1, the correlations do not display the three-sublattice Neel order. The order becomes clearly visible at U = 5. In (b), the case SU(5) on a square lattice is shown. Here, the pattern compatible with LRO is visible at any U , although with smaller correlation strengths for small U . Note that while increasing the on-site interaction U , we do not observe the appearance of a new ( or differently configured) long-range orders. For completeness, we have checked that patterns do not evolve for U up to 50 (not shown). (5). They are defined as ⟨P1j⟩ -1/N where P1j ≡ -1 + E1jEj1, with the reference site 1 in black, and j the site indices being blue (resp. red) for positive (resp. negative) correlation, with area proportional to its absolute value.

Figure 1 .

 1 Figure 1. Examples of SU(N = 2) irrep α for a L = 4-sites cluster and M = 6 fermions in a) and M = 4 fermions in b). The Young tableaux (YT) are α = [33] ( resp. α = [31] ), representing singlets (resp. spin 1) states. We associate the conjugate YT, flipping the shape by 45 • , α = [33] → ᾱ = [222] for a) and α = [31] → ᾱ = [211] for b). We fill up ᾱ to get the Highest Weight State (|hws⟩) to which we associate d α N =2 physical states spanning the SU(N = 2) irrep α (d α N =2 different values of Qz) and which generate d ᾱ L -dimensional and independent sectors under the application of the operators Eij. See text for details.

Figure 2 .

 2 Figure 2. Correlations patterns of the ground states of the FHM at filling 1/N for U = 10 in a) c) and d) and for U = 12 in b).They are defined as ⟨P1j⟩ -1/N where P1j ≡ -1+E1jEj1, with the reference site 1 in black, and j the site indices being blue (resp. red) for positive (resp. negative) correlation, with area proportional to its absolute value. At the top, triangular lattice with L = 12 sites, for SU(3) (in a)) compatible with the three-sublattice Neel order[31,66,67], and for SU(4) (in b)) compatible with the four-sublattice Neel order[28]. c) SU(5) on the cluster √ 10× √ 10, pattern compatible with the (chess) knight move LRO[56]; d) SU(6) on the 3 × 4 cluster, compatible with the SU(6) plaquette state[39].

Figure 3 .

 3 Figure 3. Energy spectra of the FHM in Eq. 1 (with tij ≡ 1 at filling 1/N ) as a function of the quadratic casimir C2 for various values of U and N for the L = 12 periodic triangular lattice (in a,b,c,d) and for the L = 4×3 periodic square lattice in (e) where we focused on N = 6. Note that the constant LU/2 has been withdrawn. In c), the Atos is reminiscent of the one revealing the four sublattice order in the HM with nnn couplings[28]. Inset of e): spin (resp. singlet) gap in blue (resp. red).

  The scenario of successive LRO phases, with different antiferromagnetic orders as what has been pointed out in the SU(3) FHM on the SL[71], does not seem to occur here, since the correlation patterns are monotonic[69].

Figure 4 .

 4 Figure 4. (a) Charge gaps (∆c) for the FHM on the L = 12 sites triangular lattice for N=2,3 and 4. The dashed lines (cf [69] for the fitting procedure) cross the x axis at U ≃ 8.6 for N = 2, U ≃ 9.8 for N = 3 and U ≃ 11.2 for N=4 separating the small U metallic phase from the Mott insulators. b) Spin gaps (∆s) defined as the difference between the SU(N ) adjoint and the SU(N ) singlets minimal energies.

Figure S1 .

 S1 Figure S1. Energy spectra of the FHM in Eq. (1) (main text) (with tij = 1 at filling 1/N ) as a function of the quadratic casimir C2 for N = 5, for the L=10 periodic squared lattice for various values of U , (a) U = 0.5, (b) U = 10. In (b), the black line is a guide for the eye to show the presence of the Anderson tower of states. (c) Charge gap for the FHM on the L = 10 squared periodic lattice as a function of the one-site repulsion U . The dashed line (fit of the form ∆c = a * U -b) crosses the x axis at Uc ≈ 8.75, separating the small U metallic phase from the Mott insulators. (d) Spin gaps as a function of U . It exhibits a peak at U ≈ 8.

Figure

  Figure S2. (a) Charge gaps for the FHM on the L = 12 and L = 16 sites triangular lattice for N=2. The dashed lines correspond to a fit of the form ∆c = a * U -b. They cross the x axis at U = Uc = 8.6 ± 0.3 for L = 12 and at U = Uc = 7.9 ± 0.3 L = 16, separating the small U metallic phase from the Mott insulators. (b) Spin gaps for FHM on the L = 12 and L = 16 sites triangular lattice for N=2. A peak is present at U ≃ 8 in both cases.

Figure S3 .

 S3 Figure S3. Evolution of the correlations patterns of the ground states of the FHM at filling 1/N for various from left to right U = 1, 5, 10, 20. (a) On a triangular lattice SU(3). (b) On a square lattice SU(5). They are defined as ⟨P1j⟩ -1/N where P1j ≡ -1 + E1jEj1, with the reference site 1 in black, and j the site indices being blue (resp. red) for positive (resp. negative) correlation, with area proportional to its absolute value.

1 1

 1 Laboratoire de Physique et Modélisation des Milieux Condensés, Université Grenoble Alpes and CNRS, 25 avenue des Martyrs, 38042 Grenoble, France (Dated: September 19, 2023)

We acknowledge F. Mila for useful discussions. This work has been supported by an Emergence grant from the Institute of Physics of CNRS.