André Galligo
email: galligo@unice.fr

Budan Tables of Real Univariate Polynomials

Keywords: Budan-Fourier theorem, virtual roots, roots of real univariate polynomial, Budan table, genericity, representation of univariate polynomials, Hermite-Birkhoff interpolation, B-tree, stratification, incidence relations

The Budan table of f collects the signs of the iterated derivatives of f . We revisit the classical Budan-Fourier theorem for a univariate real polynomial f and establish a new connectivity property of its Budan table. We use this property to characterize the virtual roots of f (introduced by Gonzalez-Vega, Lombardi, Mahé in 1998); they are continuous functions of the coefficients of f . We also consider a property (P) of a polynomial f , which is generically satisfied, it eases the topological-combinatorial description and study of Budan tables. A natural extension of the information collected by the virtual roots provides alternative representations of (P)-polynomials; while an attached tree structure allows a finite stratification of the space of polynomials with fixed degree.

Introduction

Counting and localizing roots of univariate polynomials is a basic task in Algebra and Numerical analysis. Many known techniques and interesting historical notes can be found e.g. in the book [START_REF] Rahman | Analytic theory of polynomials[END_REF] of Rahman and Schmeisser. Established two centuries ago the Budan-Fourier theorem connects the number of real roots of a polynomial in an interval with the sign variations of a sequence and inspired the invention of Sturm sequences. It also gave rise to efficient and certified subdivision algorithms see [START_REF] Bochnack | Real Algebraic Geometry[END_REF] and [START_REF] Alkiviadis | Reflections on an pair of theorems by Budan and Fourier[END_REF].

Subdivision methods exploit the ordered structure of the real numbers, and are widely applied for calculating good approximations of solutions of polynomial equations or intersections of surfaces in many applied sciences. The analysis of their complexity and efficiency relies on the the geometric and algebraic nature of the inputs. The dictionary between invariants readable on equations and features of varieties, in complex algebraic geometry, is ultimately based on the fact that a polynomial of degree n admits n roots. This is not the case for real roots. A natural strategy for studying properties of real algebraic varieties is to consider simultaneously roots of iterated derivatives of the input. A progress was achieved by Gonzalez-Vega, Lombardi, Mahé in [START_REF] Gonzalez-Vega | Virtual roots of real polynomials[END_REF] when they introduced the concept of the n virtual roots of a degree n polynomial f (which depend continuously on the coefficients of f ), to provide a good substitute to its n complex roots. Their motivation was an attempt to prove Pierce-Birkhoff conjecture on the representation of C 0 -polynomial spline functions. Virtual roots were then studied by Coste, Lajous, Lombardi, Roy in [START_REF] Coste | Generalized Budan-Fourier theorem and virtual roots[END_REF], and recently by Bembé in his PhD thesis [START_REF] Bembé | Algebraic certificates for Budan's theorem[END_REF].

The table containing the signs of all the derivatives of a polynomial f is called, in this paper, Budan table. The table is an infinite rectangle contained in the real plane, formed by positive and negative rectangular blocks, separated by segments corresponding to roots of derivatives. This structure was used by various mathematicians for separating and labeling the different real roots of a polynomial, see [START_REF] Bochnack | Real Algebraic Geometry[END_REF]. However up to now, authors considered properties of either rows or columns of a table, or their sign variations but not the table "itself". Here, we adopt a 2D point of view, re-interpret known results and present new ones. We analyze the different admissible configurations of successive rows in such a table and deduce the 2D same-sign-connectivity of the positive (resp. negative) blocks components, then characterize the virtual roots using this data. We define a simple property, we call (P), on the roots of the iterative derivatives of a monic polynomial f , which is generically satisfied. Then, for a (P)-polynomial, i.e. assuming (P), we study and characterize the topological and combinatorial properties of the corresponding Budan table and virtual roots. We specialize results of Hermite-Birkhoff interpolation (see e.g. ( 21)) to our setting, to better analyze the relation between a (P)polynomial f and its virtual roots. We also introduce a finite discrete invariant, the B-tree of a (P)-polynomial f ; and initiate the study of the attached stratification by semi-algebraic sets of the space of (P)-polynomials. Then we extend most of the results to the general case when (P) is not necessarily satisfied.

The same-sign-connected components of the Budan table can be interpreted as plane surfaces delimited by discretized curves which can be approximated by continuous ones. This point of view was initiated in [START_REF] Bembé | A: Virtual Roots of Real Polynomials and Fractional Derivatives[END_REF] and in (12) using derivatives with non-integer orders, called fractional derivatives. We plan to develop it further in a future work [START_REF] Galligo | Deformation of Roots of Polynomials via Fractional Derivatives[END_REF].

The paper is organized as follows. Section 2 provides basic definitions and properties presented with our point of view. Section 3 introduces our generic assumption (P), then presents our results in this generic case and define our new invariant the B-tree; they are illustrated with simple examples; the proofs are collected in Section 4. Section 5 examines admissible configurations for Budan tables and other invariants in the non generic case and establishes continuity results. Section 6 concludes with some extensions and potential applications.

A partial and preliminary version of our approach was presented in the first part of our talk at the ISSAC'2011 conference, (and in the corresponding proceeding paper (6)).

Notations:

We denote by R the field of real numbers, by R[x] the ring of real univariate polynomial, by R[x] n the corresponding vector space when the degree is at most n; it is equipped with the L 2 norm on the coefficients in the monomial basis. We denote by f a monic polynomial of degree n, f := x n + i=n-1 i=0 a i x i , the set of coefficients (a i ) belongs to R n . The affine space E n of monic polynomials of degree n is identified with R n . For an integer m, we denote by f (m) the m-th derivative of f , with f (0) := f .

Basics on Budan tables

In this section, after the definition of a Budan table, we list classical results around Budan-Fourier theorem, the proofs are elementary and can be found e.g. in (24) chapter 10, section 1. We also recall results of (11; 9).

The simplest observation that we will exploit in our approach is the following elementary lemma.

Lemma 1. Assume that a is a root of f , then in a small enough interval ]a, a + h[, h > 0, f and its derivative f ′ have the same sign.

Proof. Express f (x) as a definite integral of f ′ (x) on a small interval ]a, a + h[ where f ′ keeps the same sign. 2 Definition 2. Let f be a monic univariate polynomial of degree n. The Budan table of f is a subset of the real plane equal to the union of n + 1 infinite rectangles of height one

L i := R × [i -1/2, i + 1/2[
for i from 0 to n, called rows. For i from 0 to n, each row L i is the union of a set of open rectangles (infinite on the left and right sides of the row) having the property that the sign of the (n-i)-th derivative of f is constant on each of them, they are separated by vertical segments, corresponding to roots of this derivative. We color in black the rectangles corresponding to negative values of the (n -i)-th derivative f (n-i) of f , and we color in gray the rectangles corresponding to positive values of f (n-i) .

Remark 3. (1) Since f is assumed monic, every infinite right rectangle of each row is gray. (2) Since f (n) is a positive constant, the row L 0 is a gray infinite rectangle.

(3) The first (infinite) rectangle of each row L i is alternatively gray or black, depending on the parity of i: it is gray if n -i is even.

(

(f ) (-1) k s 0 s sgn(f (m-1) ) s 1 s 1 s 1 sgn(f ′ ) (-1) k+1 s 0 s sgn(f (m) ) (-1) k s 2 0 s 2 ... ... 0 s ... ... 0 s 2 sgn(f (k-1) ) -s 0 s sgn(f (m+k-1) ) -s 2 0 s 2 sgn(f (k) ) s s s sgn(f (m+k) ) s 2 s 2 s 2
See (24) for a proof.

A classical "descriptor" attached to a Budan table is the function V f (x) of the real indeterminate x with values in the set of integers N, it counts the number of sign changes in the sequence formed by f and its derivatives evaluated at x. Definition 5. For a sequence (a 0 , . . . , a n ) ∈ (R \ {0}) n+1 the number of sign changes V(a 0 , . . . , a n ) is defined inductively in the following way:

V(a 0 ) := 0; V(a 0 , . . . , a i ) := V(a 0 , . . . , a i-1 ) if a i-1 a i > 0, V(a 0 , . . . , a i-1 ) + 1 if a i-1 a i < 0.
To determine the number of sign changes of a sequence (a 0 , . . . , a n ) ∈ R n+1 , delete the zeros in (a 0 , . . . , a n ) and apply the previous rule. (V of the empty sequence equals 0). We notice that the function V f is computed from the Budan table of f , but two different tables (of two polynomials f and g) may have the same function V f = V g . Therefore the Budan table is a finer invariant than V f attached to the polynomial f . We will study other invariants attached to f . Proposition 6 (Budan-Fourier theorem). Let f ∈ R[X] be monic of degree n. Then,

• V f (-∞) = n, V f (∞) = 0.
• Near a real root c of multiplicity k of f , which is not a root of another derivative f (l) of f with l > k, V f decreases by k when x moves from c -h to c + h, for sufficiently small positive h . • Near a real root c of multiplicity k of f (m) , which is not a root of another derivative f (l) of f with l > m + k or l < m, the following happens: If k is even, V f decreases by k.

If k is odd, V f decreases by the even integer k + s 1 s 2 , with the notations of Proposition 4.

• Near c, a real root of several non successive derivatives of f , V f decreases by the sum of the quantities corresponding to each of them.

• Near the other points of R, V f is constant. The function V f is decreasing on R. • For a, b ∈ R with a < b, the number of real roots of f in the interval ]a, b] counted with multiplicities is at most V f (a) -V f (b).
Moreover the difference between the number of roots and this bound is an even integer.

See ( 24) for a proof.

Definition 7. Let f be a monic real polynomial of degree n. We call virtual root of f any real value a where the function V f (x) jumps, and we call virtual multiplicity at a the integer value of the jump.

The previous proposition implies that there are n virtual roots of f counted with multiplicity. Each real roots of f is also a virtual root, the others virtual roots are called virtual non real root of f .

Moreover the following result obviously holds.

Proposition 8. Let f ∈ R[X] be monic of degree n. Let y 1 ≤ • • • ≤ y n the ordered virtual roots of f , repeated according to their multiplicities, and y 0 = -∞, y n+1 = ∞. Then we have for

1 ≤ r ≤ n + 1, with y r-1 = y r , x ∈ [y r-1 , y r [⇐⇒ V(f (x), f ′ (x), . . . , f (n) (x)) = n + 1 -r (resp. for r = 1 the interval x ∈] -∞, y 1 [).
The interest of the concept of virtual root introduced (with an equivalent definition) in [START_REF] Gonzalez-Vega | Virtual roots of real polynomials[END_REF] comes from the following important result.

Theorem 9 (( 11)). The virtual roots of a monic polynomial f depend continuously on the coefficients of f . See also another equivalent definition in [START_REF] Coste | Generalized Budan-Fourier theorem and virtual roots[END_REF]. In the sequel, we will see a geometric interpretation of virtual roots of f in terms of the same-color-connected components of the Budan table of f .

A generic case

In this section and the next one, we assume a condition (P), which is generically satisfied. We introduce and study several data, continuous or discrete, attached to f and its Budan table .   The proofs of the results are collected in Section 4.

(P)-polynomials

Definition 10. A polynomial g in R[x] satisfies condition (P) if and only if: each derivative of g has simple roots, and all these roots are pairwise distinct. A monic polynomial satisfying this condition will be called a (P)-polynomial.

Obviously, the parities of the number of real roots and degree of a (P)-polynomial are equal.

We now see that the property (P) is generically satisfied.

Proposition 11. Let f be a monic real polynomial of degree n. Let 1 and (a i ), 0 ≤ i ≤ n -1 be the ordered set of its coefficients in the monomial basis. For any positive ǫ, there exists g, another monic real polynomial of degree n, such that:

• g satisfies condition (P).

• |a i -b i | < ǫ, 0 ≤ i ≤ n -1,
where 1 and (b i ), 0 ≤ i ≤ n form the ordered set of the coefficients of g. • the number of real roots (counted with multiplicities) of f and g are equal. Moreover, the set H n of monic polynomials in E n , identified with R n , satisfying (P) form a semi-algebraic set of E n .

Generic Budan tables

Now, we determine the features of the Budan table B of a generic monic polynomial f with degree n. Definition 12. We say that a table B with (n + 1) rows L i , 0 ≤ i ≤ n numbered bottom-up, formed by rectangles of alternating colors, gray and black, separated by vertical segments, is a B table of degree n if it satisfies the following properties.

• The row L 0 is a gray infinite rectangle. The infinite rightmost rectangle of each row is gray. The first (infinite) rectangle of each row L i is alternatively gray or black, depending on the parity of i: it is gray if ni is even. • If i is even (resp. odd) the number of rectangles on the row L i is odd (resp. even).

• Let (l+1) be the number of rectangles of the top row L n , then l ≤ n and n-l is an even number 2p. There are l + p + 1 same-color-connected components of B. Each non first rectangle of L i , i > 0 is connected on the left to a rectangle of the same color of the row L i-1 . The l first rectangles of L n are in separated same-color-connected components.

There are p other connected components, bounded on the right, and surrounded by connected components of the opposite color. There is also a gray component unbounded on the right. • The previous item is true, replacing n by any m, 0 < m < n, and B by the table formed by the lower m + 1 rows. Moreover we say that a B table is a GB table if all the x-values of the end points of all the rectangles are pairwise distinct.

Theorem 13. Let f be a (P)-polynomial of degree n, and let m ≤ n be the number or real (simple) roots of f . Then m and n have the same parity, n = m + 2p and the Budan table of f is a GB table of degree n.

Example 1. Figures 1 and2 show two GB tables: one of degree 6 with m = 0, p = 3 and the other of degree 10 with m = 4, p = 3. The first one corresponds to the polynomial f := int(F, x)+200, where "int" denotes the integral and

F := x 5 -12x 4 +42x 3 -53x 2 -28x-2.
Proposition 14. Let f be a (P)-polynomial of degree n. Consider a gray (resp. black) connected component bounded on the right of the Budan table of f . The x value of the rightmost (i.e top-right) corner of this component is a virtual root of f . Let m ≤ n be the number or real (simple) roots of f , and let n -m = 2p. There are p virtual non real roots of f , their virtual multiplicity is two; each of them is a root of some derivative of f of positive order.

We recover that f admits n virtual roots, counted with multiplicities.

Definition 15. We call augmented virtual root of f the pair (y, k) formed by a virtual root of f and n minus the order of the derivative of f which vanishes at y, i.e. f (n-k) (y) = 0. Notice that k is the degree of that derivative.

Remark 16. The augmented virtual roots of f depend only on the Budan table B of f . Virtual roots of a GB table are well defined.

Proposition 17. Let f be a (P)-polynomial of degree n. By Rolle's theorem between two successive roots a < b of some derivative f (m) with 0 ≤ m ≤ n -2, (or in R if f (m) has no root), there is an odd number 2r + 1 of roots (X 1 < ... < X 2r+1 ) of the next derivative f (m+1) . Then the r roots with an even index (X 2 , ...X 2r ) are virtual non real roots of f . Similarly if a is the smallest (resp. the largest) root of f (m) , in the infinite interval ] -∞, a[ (resp. ]a, ∞[) there is an even number 2r of roots (X 1 < ... < X 2r ) of the next derivative f (m+1) . Then the r roots with an odd index (X 1 , ...X 2r-1 ), (resp. the k roots with an even index (X 2 , ...X 2r )) are virtual non real roots of f . For each augmented virtual non real root (y, k) of f , we have

f (k-1) (y)f (k+1) (y) > 0.
The following corollary explicits the relation between the virtual roots of f and f ′ .

Corollary 18. With the same hypothesis and notations.

The Budan table of the derivative f ′ of f is obtained by deleting the first row of the Budan table of f . Denote by (x i ); 1 ≤ i ≤ u the ordered set of real roots of f and by (y j ); 1 ≤ j ≤ v the ordered set of virtual roots of f with order 1; call U the ordered union of these 2 sets.

Then f ′ has u + 2v -1 real roots: the (y j ); 1 ≤ j ≤ v and a set (z r );

1 ≤ r ≤ u + v -1
of values in each of the u + v -1 bounded intervals limited by the successive elements of U . The virtual roots of order k of f ′ for k > 0 are the virtual roots of order k + 1 of f .

Hence, the virtual roots of f and f ′ satisfy the "classical" interlacing property. This was already noticed in (9).

Interpolation

Since it is monic, a (P)-polynomial f of degree n depends on n real coefficients, denoted by a 0 , ..., a n-1 . However it has "only" m real roots and p virtual non real roots, with n = m + 2p. In order to get the same number of parameters and constraints, we must attach to each of the p virtual non real roots another real value. We choose to attach to each augmented virtual root (y, k) with k > 0 the value f (n-k-1) (y), and so define a balanced system S of constraints.

Definition 19. We denote by S(f ) the system of n = m + 2p data formed by the m real roots (x i ), 1 ≤ i ≤ m, of f ; the p augmented virtual root (y j , k j ), 1 ≤ j ≤ p with k j > 0 and the p corresponding values w j := f (n-kj -1) (y j ), 1 ≤ j ≤ p.

Notice that S(f ) defines a system of affine equations in the coefficients of f . Two natural questions arise:

Is it true that two different (P)-polynomials f and g in H n , define different systems S(f ) = S(g) ?

Given a system S as above, does there always exists a (P)-polynomial f which satisfies these data ?

We will see that the answer to the first question is "Yes" while the answer to the second one is in general "No".

The first question is a so-called homogeneous Hermite-Birkhoff interpolation problem, Hermite-Birkhoff interpolation, (HB) in short, is an old subject comprehensively reviewed till 1983 in the book [START_REF] Lorentz | Birkhoff Interpolation[END_REF]. In contrast with Lagrange or Hermite interpolation, it does not always have a unique solution. In this theory, an important tool is the incidence matrix.

Incidence matrix

In HB theory, the list of constraints are written line by line, hence the rows of the incidence matrix correspond to real values where the functions are evaluated, and the columns correspond to degrees (or orders of differentiation). Since in our definition of Budan tables the roles of rows and columns are reversed, we will consider here the transposed of the usual HB incidence matrix. Yet the rows of the matrix are indexed (as usual) top-down.

Let h be the difference between two (P)-polynomials f and g having same augmented virtual roots and set of values w j : 1 ≤ j ≤ p, as above. Hence h has degree at most n -1, and its coefficients form n unknowns. It should satisfy n = m + 2p linear vanishing conditions: m corresponding to the m real roots of f and 2p corresponding to the p double virtual non real roots of f . We order the m + p distinct virtual roots y 1 < ... < y m+p of f . Definition 20. The incidence matrix E of a (P)-polynomial f only depends on the augmented virtual roots (y i , k i ) of f . It is the (n, m + p)-matrix E = (e j,i ) such that n of its entries are 1, and the others are 0 according to the following rule:

(1) If n -k i =0 then e 1,i = 1.

(2) If n -k i > 0 then e n-ki+1,i = 1 and e n-ki,i = 1.

(3) Otherwise e j,i = 0.

Example 2. For a (P)-polynomial of degree 5, having one real root y 2 and two augmented virtual roots (y 1 , 2) and (y 3 , 4), with y 1 < y 2 < y 3 ; the matrix E is:

           0 1 1 0 0 1 1 0 0 1 0 0 0 0 0           
In HB theory the following properties of the incidence matrix are extensively used.

Definition 21. E satisfies Polya condition if and only if M j > j for all j = 1...n; where M j denotes the sum of the entries in the first j + 1 rows of the incidence matrix E.

A maximal sequence of 1 in a column of E is called even (resp. odd) if the number of its elements is even (resp. odd). E is said conservative, if and only if every odd sequence of E is non supported. A maximal sequence of 1 in a column i of E starting at (j, i) is non supported if at least one of the two following sub-matrices of E (which can be void) does not contain any 1:

-The upper left sub-matrix of E formed by the rows 1...j and the columns 1...i -1 -The upper right sub-matrix of E formed by the rows 1...j and the columns i + 1...m + p.

In (3) it is proved that a homogeneous HB problem whose incidence matrix is both conservative and satisfies Polya condition, has only the trivial solution. See also [START_REF] Lorentz | Birkhoff interpolation[END_REF] for simpler proofs.

Proposition 22. The previously defined matrix E (attached to m + p augmented virtual roots) is conservative and satisfies Polya condition. The corresponding homogeneous HB problem has only the trivial solution.

Hence the answer to the first previous question is "Yes".

Proposition 23. Given a system S formed by pairs of real numbers and integers:

(x i , 0) with 1 ≤ i ≤ m, (y i , k i ) with 1 ≤ i ≤ p, k i ≤ n, the set of values w i with 1 ≤ i ≤ p , n = m + p;
there exist a unique monic polynomial F satisfying the conditions :

F (x i ) = 0, 1 ≤ j ≤ m+, F (n-ki) (y i ) = 0, 1 ≤ j ≤ p , w l := F (n-ki-1) (y i ), 1 ≤ i ≤ p. Remark 24. • The (y i , k i ) with 1 ≤ i ≤ m + p,
F := x 3 + a 2 x 2 + a 1 x + a 0 .
Require that y 1 := 0 is a root of F , y 2 := 1 is a root of F ′ , and the extra condition F (1) = w. The 3 constraints are:

a 0 = 0, 3 + 2a 2 + a 1 = 0, 1 + a 2 + a 1 = w.
The solution is:

a 0 = 0, a 1 = 2w + 1, a 2 = -w -2.
Then y 2 := 1 will be a virtual non real root of F , if and only if f (1)f ′′ (1) > 0. This means w(w -1) < 0, hence 0 < w < 1.

Notice that for w = 0, f admits a double real root at 1 and that for w = 1, f ′ admits a double real root at 1.

Hence the answer to the second question is "No".

Example 4. The generalized Vandermonde matrix corresponding to Example 2 is a (5, 5)-matrix with entries monomials in (y 1 , y 2 , y 3 ). Its determinant D(y 1 , y 2 , y 3 ) is a polynomial of degree 5 which does not vanish if the indeterminate are real and pairwise distinct.

D(y 1 , y 2 , y 3 ) = 12(y 1 -y 2 ) 2 ((y 1 + y 2 -2y 3 ) 2 + 2(y 1 -y 3 ) 2 ).
Fix a GB table of degree n, through the set of its m + p augmented virtual roots, it defines an incidence matrix E. Hence with a set of values (w l ) ∈ R p , it defines a system of constraints S. Proposition 25. The set W of values (w i ) ∈ R p such that the system S represents a (P)-polynomial, form a semi algebraic set (possibly empty) in R p . More precisely, if we fix the signs σ of the (w i ) ∈ R p such that the system S represents a (P)-polynomial, then the corresponding set W σ is a (possibly empty) open convex polytope in R p .

Realizable configurations

A question similar to interpolation is to ask if any GB table B of degree n is realizable, i.e. does there exists a (P)-polynomial f of degree n, which admits B as its Budan table ?   The answer is "No" because there are quantitative inequality between the roots of a polynomial and its derivative; see [START_REF] Peyser | On the Roots of the Derivative of a Polynomial with Real Roots[END_REF]. So we need to weaken the requirements.

V. Kostov,[START_REF] Kostov | A realization theorem about D-sequences[END_REF], proved that any sequence of numbers of roots of the derivatives, compatible with Rolle's theorem, is realizable. Then he investigated in a series of articles, (( 15), ( 16), ( 19)) arrangements of roots, compatible with Rolle's theorem. He proved that if we fix an order of derivation k, then all arrangements of roots of f and f k , compatible with Rolle's theorem are realizable. But he showed that in degree 4 any arrangement of the roots of all the derivatives, compatible with Rolle's theorem are not realizable; more precisely only 10 out of 12 are realized. There is also a related work by B. and M. Shapiro [START_REF] Shapiro | This strange and mysterious Rolle's theorem[END_REF].

An arrangement of the roots of all the derivatives, compatible with Rolle's theorem is equivalent to what we will call a qualitative B table.

Definition 26. Given a B table B of degree n as above, we call qualitative version of B the table obtained by ordering the roots of f and all its derivatives, say z i with 1 ≤ i ≤ N ; then replacing z i by i. There are a finite number of such qualitative B tables with fixed degree.

Since in Figures 1 and2, we did not mention the values z i , they are indeed examples of qualitative GB tables.

Hence we need to extract from a qualitative GB table discrete invariants in order to expect, with this weaker requirement, that all configurations are realizable.

A candidate can be the previous incidence matrix but we prefer the B-tree, defined in the next subsection, which keeps some of the information on the same-color-connected components.

B-tree

in this subsection we define our new invariant the B-tree. With the previous notations it is obtained, from a qualitative Budan table, by contacting the m + p same-colorconnected components to obtain a bi-color tree with m + p nodes and n + 1 leaves. Figures 3 and4 illustrate our construction with the trees corresponding to example 1.

Definition 27. The B-tree extracted from a qualitative B table B of degree n is a bicolor tree constructed on a grid [0..m + p + 3, 0..n] as follows. We position the n + 1 leaves at the points (0, i), 0 ≤ i ≤ n and color them alternatively in gray and black; (0, 0) is gray. The root of the tree is positioned at (m + p + 3, 0). The root of the tree, the nodes and the leaves are connected by colored edges which are the contraction of the the same-color-connected components of B.

Corollary 18

Apply Rolle's theorem.

Proposition 22

E is conservative because the only columns with one 1, have the 1 situated on the first row; the other columns have two 1. Polya condition is clearly satisfied for the first two rows, because if a polynomial f has no real root, then its derivative has a at least one real root, hence f has at least a virtual root of order 1. Let ρ j denote the number of real roots of f (j) , j ≥ 0. Then by Proposition 17, the number λ j , j > 0 , of virtual roots of order j of f is:

λ j = 1 2 (ρ j -ρ j-1 +1
). Each of them gives rise to two 1 in E. The count of 1 in the first j + 1 rows of E gives: M j = j + 1 2 (ρ j + ρ j-1 + 1). Hence M j > j.

Proposition 23

Indeed the corresponding generalized Vandermonde matrix is invertible, by Proposition 22.

Proposition 25

Consider p other extra constraints f (ki+1) (y i ) = z i , at the augmented non real virtual roots. The new system S ′ is solvable if and only if the 2p parameters (z i , w i ) satisfy a set of affine relations. And we also require for each i, 1 ≤ i ≤ p the non linear inequality z i .w i > 0. The solution is a union of convex polytopes whose vertices depend rationally on the y i , then we consider its projection on the (w i ) space R p . If we fix the signs σ of the (w i ), hence those of (z i ), the solutions correspond to the intersection of an affine space with an hyper-octant in R 2p , that we project on R p to get an open polytope.

Proposition 29

The proof of Proposition 25 gives semi-algebraicity by elimination of the (w i ). Θ T is a finite union of strata Σ T .

Proposition 30

The complex roots of f as well as the virtual roots of f depend continuously on the coefficients of f . But this dependency is analytic as far as the roots do not collide and create multiple roots. But this event cannot occur when f moves in a stratum Σ T , since it is formed by (P)-polynomials.

Without condition (P)

In this section we consider the case where the condition (P) is not necessarily satisfied. In order to generalize the concepts and results presented in Section 3, we need to slightly adapt the definition of a Budan table.

Multiplicities and simplified Budan table

Definition 31. We call simplified Budan table of f , the table obtained from the Budan table of f by skipping the segments between any two adjacent rectangles of the same color on a same row. In other words, for any integer k the roots of f (k) with even multiplicity are no longer indicated.

Theorem 32. Let f be a monic polynomial of degree n, and let m ≤ n be the number or real roots of odd multiplicity of f . Then m and n have the same parity, n = m + 2p and the simplified Budan table of f is a B table of degree n with m + p same-color-connected components.

Proof. Lemma 1 also applies when f or one of its derivatives admits multiple roots, and since we suppressed the segments corresponding to roots with even multiplicity, the endpoints of the rectangles only correspond to roots of f (i) , 0 ≤ i ≤ n -1 with odd multiplicities. So we can repeat the argument of the proof of Theorem 13 replacing the word "root" by the expression "root with odd multiplicity". 2 Proposition 33. Let f be a monic real polynomial of degree n. The x value of the rightmost upper segment of a connected component (either gray or black) of the simplified Budan table of f is a virtual root of f . The virtual multiplicities are counted as follows:

• the multiplicities of events appearing along a same x-value are added,

• the multiplicity of a simple root of f counts 1,

• the multiplicity of a simple virtual non real root (i.e. it is not a multiple root of a derivative of f ) counts 2, • the multiplicity of a multiple root of f of order k counts k,

• the multiplicity of a multiple virtual non real root which is a multiple root of order k of a derivative of f counts k if k is even, and otherwise k + s 1 s 2 with the notations of the second table of Proposition 4.

Definition 34. We call augmented virtual root of f a pair (y, k) formed by a virtual root of f and an integer 0 ≤ k ≤ n -1 which form the coordinates of the rightmost upper segment of a connected component (either gray or black) of the simplified Budan table of f . So there are m augmented virtual roots of type (y, n), and p augmented virtual roots of type (y, k), with k < n, such that n = m + 2p. Different augmented virtual roots of f can correspond to the same virtual root of f .

Incidence matrix and B-tree

The definitions of B-tree, decorated B-tree, qualitative version of a Budan table, and of the stratifications given in Section 3, apply verbatim for the simplified Budan table of a monic polynomial f , even if it (or one of its derivatives) admits a multiple root. Similarly we have: Definition 35. To a simplified Budan table B of degree n of a monic polynomial f corresponds an incidence matrix E = (e i,j ). Let m be the number of real roots of odd multiplicity of f and m + p be the number of augmented virtual roots, such that n = m + 2p. Then E is a (n, m + p) matrix with n entries equal to 1 and the others to 0, Example 5. We consider the monic polynomial f := (x -1) 4 g/20 of degree 7 such that g := 20x 3 + 10x 2 + 4x + 1 chosen such that we have at x = 0: f (0) > 0, f ′ (0) = 0, f (2) (0) = 0, f (3) (0) = 0, f (4) (0) > 0. The polynomial f also has a real root of order 4 at x = 1 and another simple real negative root.

The simplified Budan table of f is shown in Figures 5.

Deformations

Conjecture 1. Let f ∈ R[X] be monic of degree n and T be its decorated B-tree. Then, there exists a P-polynomial g of degree n with the same B-tree.

To prove that we need to control the deformations of f at the multiple roots of f and of its derivatives. The requirement is to deform f into F such that if x 0 is a root of f (k) of multiplicity r it gives rise near-by x 0 to no root of F (k) if r is even and to only one root of F (k) if r is odd. Example 6. For f as in example 5, the deformation F := x 7 +10.5x 6 +46.2x 5 +110.x 4 + 154.x 3 +126.x 2 +55.6x+10.4 of f gives the desired result. We constructed it by a sequence of deformations and integrations starting from the third derivative of f . Figures 6 and7 show the two families of graphs up to the third derivative. The stratifications Σ and Θ of H n extends to E n . The study of incidence relations among the strata of Θ (or Σ) is an interesting question, but out of the scope of the present paper. A first task could be to answer the following question. Question 3. Are the strata Θ T connected in E n ?

Conclusion

Extensions

In this paper, we have studied the same-sign-connected components of the Budan table of a univariate polynomial f , and extracted two types of information: the virtual roots which depend continuously on the coefficients of f and its Budan tree which is a finite discrete invariant attached to f . We related this construction to realizable configurations studied by V. Kostov and to Hermite-Birkhoff interpolation, which has received recently a renewed interest in Computer algebra, see [START_REF] Butcher | A: Polynomial algebra for Birkhoff interpolants[END_REF]. Then we defined a stratification of the space of polynomials and started its study. We left open several questions on the properties of the strata and their incidence relations. They provide directions for further investigations and experimentation.

There are other possibilities of extensions of the previous approach.

Algorithms for virtual roots isolation

In [START_REF] Galligo | A: Improved Budan-Fourier Count for Root Finding[END_REF], we exploited the intersection of Budan tables with grids to develop algorithms for virtual roots isolation, and consequently a new method for real roots isolation of a univariate polynomial.

Fewnomials

In ( 9), the Budan-Fourier theorem and the continuity property of the virtual roots, were generalized to the case of Fewnomials, with a modified set of differentiations depending on a sequence of functions f . More precisely, with our notations, our Lemma 2.1 and Proposition 2.4 are valid in that setting. Therefore our constructions can also be extended to that setting. In a joint paper with Mariemi Alonso Garcia (2),for Issac 2012, we applied the method presented in this article to develop real roots isolation algorithms for sparse polynomials.

"Circular" differentiation

Another extension could be to replace the input polynomial f (x) of degree n by its homogenization F (X, Y ) in degree n, and then set X = cos(t), Y = sin(t) to get a trigonometric polynomial G(t) depending on n coefficients, hence G is periodic and takes bounded values in R. In that case, the derivative G ′ (t) of G(t) with respect to t can be obtained similarly from another homogeneous polynomial. Then we can define a periodic generalization of the Budan table, bounded on the columns and unbounded on the rows, but we can truncate it. There is again a generic case and one can study the corresponding "virtual roots" and B-trees.

Two more directions of research

• Study the relationship between virtual roots in an interval and pairs of conjugate complex roots, which lie in a sector close to this interval as initiated in Obreschkoff's theorem, see [START_REF] Rahman | Analytic theory of polynomials[END_REF], chapter 10.

• Investigate what happens if the coefficients of the input polynomials follow Gaussian random distributions. See [START_REF] Galligo | Deformation of Roots of Polynomials via Fractional Derivatives[END_REF].

Fig. 1 .Fig. 2 .

 12 Fig. 1. A GB table of degree 6

Fig. 5 .

 5 Fig. 5. Budan table of example 5 according to the following rule. Let (i, k i ) be a node of the decorated B-tree then: If n -k i = 0 then e 1,i = 1. If n -k i > 0 then e n-ki,i = 1 and e n-ki+1,i = 1. Otherwise e i,j = 0. Again we have, with the same arguments: Proposition 36. The matrix E is conservative and satisfies Polya condition. The corresponding homogeneous HB problem has only the trivial solution.

Fig. 6 . 5 Fig. 7 .

 657 Fig.6. Graphs of the derivatives of example 5

  [START_REF] Butcher | A: Polynomial algebra for Birkhoff interpolants[END_REF] We are interested by the connected components of the union of the closures of the gray rectangles; and respectively for the black rectangles. It is clear that there is a gray connected component containing the infinite right rectangles of all rows. The other connected components (gray or black) are said bounded on the right.

	Proposition 4 (multiple roots). Let f be a monic real polynomial of degree n. If b is a
	root of multiplicity k of f with k ≤ n then for sufficiently small positive h, denoting by s
	the sign of f (k) (b), the columns of the Budan table of f near b are shown in the left part
	of the following figure.
	If c is a root of multiplicity k of f (m) with 0 < m < m + k ≤ n then, for sufficiently
	small positive h, denoting by s 1 the sign of f (m-1) (c), and by s 2 the sign of f (m+k) (c),
	the columns of the Budan table of f near c are shown in the right part of the following
	figure.
	b-h	b b+h	c-h	c c+h
	sgn

  are not necessarily the augmented virtual roots of F , as illustrated by the following Example 3.• The invertible matrix of the linear transformation is a generalization of a Vandermonde matrix, see the following Example 4.

Example 3. Consider the general degree 3 monic polynomial

Acknowledgements

We thank D. Bembé for his collaboration in the preparation of our joint proceeding paper (6). He has now other interests and did not participate to the elaboration of the present work. We also thank the anonymous referees for their careful reading of a previous version of this paper and their suggestions. The author was partially supported by the European ITN Marie Curie network SAGA.

We say that the B-tree is decorated if the coordinates (i, k i ), 1 ≤ i ≤ m + p of its nodes are given.

The obtained tree is rooted on the right of the picture and can be drawn (as usual) with short branches. From a node (i, n) corresponding to a simple root of f start two branches of different colors. From a node (i, k i ) with n -k i > 0 corresponding to a virtual non real root of f , start three branches of different colors: the two extremal ones have the same color as the one of its stem and the middle one is of opposite color.

All the degree n strict hyperbolic polynomials (i.e. having n distinct real roots) have the same decorated B-tree. Since they have several qualitative Budan table, we see that the decorated B-tree is a strictly weaker invariant.

So we may ask the two following questions about realizable configurations. Question 1. Let B be a GB table of degree n, does there exist a (P)-polynomial f of degree n, which admits the same decorated B-tree than B ? respectively the same B-tree as B ?

We conjecture that they are both true. Notice that the first question is stronger than prescribing the numbers of real roots in each row.

Definition 28. Let T (f ) and T (f ) denote the B-tree and the decorated B-tree associated to a (P)-polynomial f . For a fixed decorated B-tree T , respectively B-tree T , corresponding to a GB table of degree n, we consider the strata Σ T , respectively Θ T , of H n formed by all the (P)-polynomials f such that T (f ) = T , T (f ) = T . We denote by Σ and Θ these two stratification of H n .

Proposition 30. The virtual roots of a the polynomials f in a stratum Σ T of H n depend analytically on the coefficients of f . Question 2. Are the strata Σ T connected in H n ?

Proofs

This section provides the proofs of the statements of the previous section. In the proofs of each statement we keep the notations of that statement.

Proposition 11

It is precisely Lemma 10.2.2 of ( 24), its proof relies on a continuous representation of f by conjugated complex roots and Ostrowski continuity theorem, plus the use of discriminants.

Theorem 13

We consider the different conditions of Definition 12. Items 1 and 2 and the last condition are obvious. Let's prove item 3. By Lemma 1, each (non first) rectangle of row L i is connected to a rectangle of the same color on row L i-1 located on its left. So, by induction, any rectangle is same-colorconnected to one of the (n + 1) first infinite rectangles on the left side of the table. Each black connected component is bounded on its right; so it has a rightmost upper segment (call x its abscissa): if it is on L n , then x is a real root of f , else if it is on L i , i < n, it is surrounded below above and on the right by gray rectangles. In this last case, the black component stops at (x, i) and we also consider that the gray component above the segment coming from the left stops at (x, i -1) since it is in an affluent of the gray component coming from the right which continues upward. Therefore two components (a black one and a gray one) stop at the abscissa x. The same reasoning works for the gray components bounded on the right, exchanging the roles of gray and black. Let l denote the number of real roots of f and p denote the number of rightmost upper segment of a same-color-connected component bounded on the right. Since the gray component containing L 0 is unbounded, there are n other tails moving upward from the left side. Hence n = l + 2p.

Proposition 14

When a same-color-connected component stops, the jump in the Budan-Fourier count is 2 if the component is surrounded below above and on the right by rectangles of opposite color, while it is 1 if it stops on the top row.

Proposition 17

Here is the proof of the first assertion, the second is very similar. Assume without loss of generality that between a and b, f (m) is positive. Then by Lemma 1, on the left of x 1 , f (m+1) is also positive and changes sign at each of its successive root, so f (m+1) is positive at the right of X 2 . Between X 1 and X 2 , there are an odd number 2u + 1 of roots of f (m+2) , denoted by Y 1 , ..., Y 2u+1 . Then f (m+2) is positive at the right of Y 2u+1 . Hence the rectangle ]X 1 ..X 2 [ on L n-m is black and we deduce that it is surrounded (above, on its right and below) by gray rectangles.


S Barkhofen

M Bellec

U Kuhl

F Mortessagne

Disordered graphene and boron nitride in a microwave tight-binding analogue

Keywords: numbers: 42.70.Qs, 73.22.-f, 71.20.-b, 03.65.Nk

Experiments on hexagonal graphene-like structures using microwave measuring techniques are presented. The lowest transverse-electric resonance of coupled dielectric discs sandwiched between two metallic plates establishes a tight-binding configuration. The nearest-neighbor coupling approximation is investigated in systems with few discs. Taking advantage of the high flexibility of the discs positions, consequences of the disorder introduced in the graphene lattice on the Dirac points are investigated. Using two different types of discs, a boron nitride-like structure (a hexagonal lattice with a two-atom basis) is implemented, showing the appearance of a band gap.

I. INTRODUCTION

Graphene, due to its unusual and astonishing electronic and physical properties, is a subject of intense focus, especially since it was experimentally realized in 2005 by K. S. Novoselov, A. K. Geim and coworkers 1 , who were honored by the Nobel prize in physics in 2010. One of graphene's amazing features is the linear dispersion relation close to the K-points and the Dirac like Hamiltonian whose consequences can be seen in the density of states (for details on graphene see the reviews 2 and 3). Graphene is one of the candidates to take over from silicium based electronics. Graphene has a macroscopic ballistic carrier transport due to its high charge carrier mobility and a high magnetic susceptibility µ 4 . A problem is still that graphene in its pure form is metallic and has no band gap ∆E. Nevertheless a band gap can be induced by using bilayer graphene 5 , disorder 6,7 , lateral-superlattice, epitaxially grown graphene on top of a crystal like boron nitride [8][9][10] or by confinements 11,12 . Applying stress to graphene can induce effects on the electron transport like a magnetic field 13 . Thus graphene offers a big variety of interesting possibilities from both applied and fundamental points of view.

In real single layer graphene, it is quite difficult to prepare special shapes, induce precise structures of vacancies and so on. This can be easily performed in experiments with classical waves. For example in microwave analogue experiments, the Dirac point [14][15][16] , a topological phase transition [START_REF] Bellec | Topological transition of Dirac points in a microwave experiment[END_REF] , the global band structures [START_REF] Bittner | [END_REF] , and edge states 16,[START_REF] Bittner | [END_REF] have been observed. These states which are located only at the zig-zag edges can lead to selfguiding unidirectional electromagnetic transport if the hexagonal lattice is created by ferritic scatterers 19 . Edge states have also been found in acoustics 20 .

In this paper we mimic the established tight-binding form of graphene's Hamiltonian [21][22][23] with a microwave setup using dielectric discs between two metallic plates. In a previous publication we investigated the Dirac point in the density of states (DoS) 16 . Here we concentrate more on the experimental details of the realization of the tight-binding Hamiltonian (see Sec. III). We show the flexibility of the experimental ansatz by investigating two different cases. The first case is to introduce positional disorder, where for strong disorder the Dirac point (see Sec. V) vanishes. The second case is to realize a boronnitride like structure by using two different kinds of discs on the lattice (see Sec. VI). In this case a band gap opens.

II. THEORETICAL BACKGROUND

Graphene is a two-dimensional crystal of carbon atoms ordered in a regular hexagonal lattice, a so-called "honeycomb" structure (see Fig. 1). It was first completely described theoretically by Wallace in 1947 21 . The farreaching equivalence of graphene's Hamiltonian and the Dirac Hamiltonian for a free particle is depicted in Ref. 24. Using the order and the notation of Ref. 24 we first give a short repetition of a standard derivation of graphene's Hamiltonian to underline its equivalence to our experiment.

Graphene's characteristic honeycomb lattice structure is caused by the sp 2 -bond of the carbon atoms resulting in three σ-bonds in the plane including an angle of 120 • and a free p z -orbital perpendicular to the plane. The bonding length between the carbon atom is a b = 1.42 Å. The basis vectors of the hexagonal lattice -illustrated in Fig. 1 -are a 1 = √ 3/2, -1/2 a and a 2 = (0, 1) a, where a = √ 3 a b is the lattice constant. The vectors connecting the two sublattices are b

1 = 1/2 √ 3, 1/2 a, b 2 = -1/ √ 3, 0 a, and b 3 = 1/2 √ 3, -1/2 a.
In the tight-binding approximation we assume a basis of localized eigenfunction at the lattice sites imitating the p zorbitals. Thus we can write down the Hamiltonian H in second quantization which includes the position energy and the hopping term.

For the position term H pos we use a diatomic system (indicated in Fig. 1 by A andB) placed on the two triangular lattices A and B, which is realized e. g. in boron nitride. In the tight-binding model an atom of sort A only interacts with its three nearest neighbors that all belong to the lattice B (see Fig. 1). Second nearest-neighbor hopping is here neglected. The energy difference between electrons localized on lattice A and electrons on lattice B is called β = (ǫ A -ǫ B )/2. The associated part of the Hamiltonian can be written as

H pos = ǫ A n( A) + ǫ B n( B) = β A [n( A) -n( A + b 1 )] + ǫ A + ǫ B 2 ≡E0 N tot (1)
where n( A) and n( A + b 1 ) are the counting operators of electrons at positions A belonging to lattice A and at B = A+ b 1 to lattice B, respectively. They can be written using the annihilation (U , V ) and creation operators (U † ,

V † ) as n( A) = U † ( A)U ( A) and n( A + b 1 ) = V † ( A + b 1 )V ( A + b 1 ).
A is the sum over all atoms of lattice A. The additional term proportional to E 0 times the total number of electrons is only a constant and can be neglected in the further treatment. It corresponds to an energy shift to the mean energy of the two atoms (for graphene to the eigenenergy of one atom).

The second part of the Hamiltonian includes the hopping of electrons from one lattice site to one of the neighboring sites. The hopping parameter κ describes the probability for this process and corresponds to the nearest-neighbor coupling, which is accessible in the measurements (see Sec. III E).

H hop = κ A,i [U † ( A)V ( A + b i ) + V † ( A + b i )U ( A)](2)
with i = 1, 2, 3 for the three nearest neighbors in lattice B of an atom A. The spin of the electron is here neglected, as a spin is not existing in the microwave experiment. The entire Hamiltonian equals the sum of Eq. (1) and Eq. ( 2) and has the eigenenergies:

E( k) = ± β 2 + κ 2 e i k• b1 + e i k• b2 + e i k• b3 2 (3)
For the case of graphene (β = 0) the two bands with plus and minus signs are touching each other at the roots of the sum of the three exponential functions. These are given at K = 2π √ 3 a (1, √ 3) and K ′ = -K, the corners of the Brillouin zone. The dispersion relation is approximately linear close to these K-points with the so-called Fermi velocity v f as the proportionality constant:

E( K + q) ≈ ± √ 3 2 κ a ≡v f | q| for | q| ≪ | K| (4)
This equals the relativistic dispersion relation E( p) = ±c| p| for a massless particle, when v f plays the role of the speed of light, and leads to the common name "Dirac points" for K and K ′ . The strong connection to the Dirac theory is moreover given by the equivalence of the Dirac Hamiltonian and a certain form of graphene's Hamiltonian which can be derived via a basis transformation described in Ref. 24.

From Eq. ( 4) the density of states (DoS) in the vicinity of the Dirac points can be calculated

ρ(E) ≈ 4 √ 3π |E| κ 2 (5)
which is again linear.

In our experiments the DoS (strictly speaking the local density of states, LDoS) is an accessible quantity. In our previous work we have observed the linear dependence of the DoS close to the Dirac point 16 . In this paper we want to investigate the stability of the Dirac points against disorder (see Sec. V).

If the on-site energies are different, i.e. β = 0 like in boron nitride then a gap is generated at the Dirac point, where the gap width is given by 2β = ǫ A -ǫ B . An analogue to boron nitride is presented in Sec. VI.

III. EXPERIMENTAL REALISATION OF A TIGHT-BINDING SYSTEM A. Introducing microwave experiments

Microwave experiments have been proven to be an attractive implementation to perform quantum analogue measurements in the realm of "quantum chaos" since 20 years 25,[START_REF] Stöckmann | Quantum Chaos -An Introduction[END_REF] . Exploiting the formal analogy of the Helmholtz and the Schrödinger equation they have been used to investigate many different effects like eigenvalue distributions, nodal domains [START_REF] Kuhl | [END_REF]28 , affirmation of periodic orbit theory 29,30 , random matrix theory 31,32 and the random plane wave model 33 . Quantum mechanical currents and the visualisation of chaotic wave functions in billiard systems 34,35 have already been in the center of attention. Nowadays the area of application became much wider and reaches beyond the first investigations on chaotic billiards. For instance one-dimensional wave guides with introduced disorder have been suitable to check scattering theory phenomena. Effects of uncorrelated and correlated disorder have been observed [36][37][38] and localisation phenomena have been investigated in two dimensional systems using dielectric discs 39 . Photonic band gap materials (PBGM) (for a review see e. g. Ref. 40) assuming infinitely long cylinders have been suggested for two dimensional tight binding realizations [START_REF] Lidorikis | [END_REF] . Effects of disorder have been investigated theoretically 42 and implemented successfully experimentally 43 .

Transport of electrons in lattices is related to wave phenomena. As large interest in the special properties of the band structure of graphene occurred, microwave analogue experiments have been performed on graphene like structures using different kinds of realizations [14][15][16][START_REF] Bittner | [END_REF] .

Here we want to emphasize the possibilities of using dielectric discs with an high index of refraction 16 . This new field offers a high flexibility to realize random or symmetric lattice structures and investigate its characteristics by observing the local density of states at different positions.

B. The setup

For the investigation of tight-binding systems we need wave functions which live in a finite volume and decrease faster as 1 | r| 2 outside. This scenario can be considered as a potential well configuration. For the realization we chose dielectric discs (Temex-ceramics, E2000 series) with a dielectric permittivity ǫ r ≈ 36 and a relative permeability µ r = 1. They have a diameter of 8 mm and a height of 5 mm (see inset of Fig. 2).

These discs were previously used to realize disordered systems 39 . They are placed on a flat copper plate. Atop another metallic plate is positioned parallel to the bottom plate. The two metallic plates are separated by a distance h. Thus the cut-off frequency for the first TEmode in air due to the plates is given by ν cut = c 0 /(2h), where c 0 is the speed of light in air [START_REF] Jackson | Classical Electrodynamics[END_REF] . The investigated resonance frequency of the discs is smaller than ν cut guaranteing that the TE-waves are evanescent outside the disc. This assures additionally the two-dimensionality of the system [START_REF] Stöckmann | Quantum Chaos -An Introduction[END_REF] . Further descriptions about the disc eigenmodes are given in Sec. III C.

In Fig. 2 the reflection spectra are shown using two different kinds of antennas. The first is a pure dipole antenna, which consists of a vertical part only, exciting only TM modes (shown in orange). The second has an additional horizontal part which was soldered to the end of a standard dipole antenna (see inset) and can also excite the TE modes (spectrum shown in black). The sharp resonance marked by the arrow is the TE 1 resonance of the disc as it is not observable with a dipole antenna. Besides it is nicely separated from the neighboring resonances and has a quite flat background. Thus it is a suitable resonance for further investigations. We always make sure that the other resonances are still separated from the frequency range of this TE-resonance so that mixing or overlapping of TE and TM modes are avoided. Consequently we work in a one-level regime, where each disc brings in only a single resonance.

To mimic molecules or solids several discs have to be placed. As we want to mimic molecules consisting of one type of atoms, we have to ensure that each disc has the same eigenfrequency. One problem for this realization is the sensitivity of the resonance frequency once the top plate is very close to the top of the discs. This can be seen in Fig. 3, where the resonance frequency and width are shown in dependence of the height h t . h t is the distance between the top of the disc to the metallic top plate. At h t =0 mm the bottom plate is pressed against the upper plate using a step motor. Thereafter it is moved in steps of 0.1 mm. At the beginning this just corresponds to a release of the pressure and thus the resonance frequency does not change. Otherwise for small heights the change of the resonance frequency is large for even tiny variations of the height and than saturates for larger heights. Assuming height variation along the setup of a few tenth of a millimeter and a typical resonance width of Γ= 2 MHz a height h t above 5 mm ensures that the variation of resonance frequencies due to height variations stays within the resonance widths. Note that in Fig. 3 the resonance frequency is given in GHz whereas the width is given in MHz. The width decays for small distances and stabilises also but increases strongly once the eigenfrequency gets closer to the cut-off frequency. For all further measurements a height of h=16 mm (h t =11 mm) is chosen to guarantee sharp and stable resonances.

The discs are positioned by a motorized xy-stage that is controlled by a PC programme. Hence it is possible to create structures of a few hundreds of discs with an accuracy of the order of 0.1 mm. To ensure good and equivalent contact of the discs to the bottom plate the discs are pushed down while placing them.

C. Disc eigenmodes

The aim of our experiment is to realize a tight-binding system of localized functions with weak coupling to the outside. For the theoretical description, let's assume that h t =0 mm, i. e. the top plate touches the disc, the refractive index is n = 6, and the disc height is h = 5 mm. The TE 1 mode consists only of the z-component of the magnetic field B = (0, 0, B z ) and the perpendicular components of the electric field. For the given distance between top and bottom plates the cut-off frequency ν cut = 9.37 GHz for this first TE-mode. The z-dependence can be separated and if the resonance frequency is below the cut-off frequency the lowest TE resonance then looks like: where

B z (x, y, z, k) = B 0 × sin π nh z J 0 (k ⊥ r); r < r D α sin π h z K 0 (γr); r > r D (6)
k ⊥ = k 2 -π nh 2 , γ = π h
2 -k 2 , and r =

x 2 + y 2 is the distance from the center of the disc. J 0 and K 0 are Bessel functions and α is a constant to be determined from the continuity equations at the surface. At the resonance frequency of the disc, the wave number for r > r D is thus purely imaginary leading to an evanescent wave in the area outside the disc. Note that in the experiment there is a gap between the top plate and the discs, and this description is only valid approximately. We still use Eq. ( 6) but with an effective γ to be determined from the experiment.

To experimentally test the wave function inside the disc, we measured the reflection for a single disc as a function of the position of a perturber, i.e. a small metallic rod placed above the disc (see inset of Fig. 4). For this measurement the top plate is removed. The perturber induces a frequency shift which is proportional to the intensity of the wave at the position of the perturber. In Fig. 4 a monotonous resonance shift is observed confirming that the wave function corresponds to the lowest Bessel function J 0 . A direct coupling of the TE 1 mode to the TM 0 and TM 1 mode is strongly suppressed by the continuity conditions of the fields at the disc boundaries. Consequently the wave function in the discs can be identified with an electron wave localized at a lattice site.

D. Impact of the measuring antenna

For all the measurements we need to know the impact of the antenna on the disc eigenmode. Hence the coupling between two different objects, i. e. disc and antenna, is investigated. The pure presence of the antenna disturbs the symmetry of the system, but the coupling behavior can be quantified in a simple measurement series: we use a single disc with radius r D and increase its distance d a to the antenna. d a is the distance to the border of the disc and not to the center. Let us focus now on the shift of the resonance. The corresponding resonance frequencies are shown in Fig. 5. For smaller couplings, i. e. for larger distances, the resonance frequency becomes higher and its amplitude decreases. The resonance frequency approaches the pure eigenfrequency for vanishing coupling, which thus equals 6.67 GHz. The shift of the resonance due to the antenna is proportional to the wave function's intensity at the position of the antenna, i. e. at distance r D + d a . Thus we fit the following function

ν a (d a ) = κ a |K 0 [γ (r D + d a )] | 2 + ν 0 (7)
to the experimentally measured resonance frequency ν a .

In Fig. 5 the experimental values are shown. A good agreement between fit (see orange line) and data points (crosses) is found. We obtained the fit parameters as κ a =-13.89 GHz, γ=0.52 mm -1 , and ν 0 =6.67 GHz. It's worth noting that κ a also includes the global coupling characteristics of the antenna. For an infinite distance the pure eigenfrequency ν 0 = 6.67 GHz remains and defines the eigenenergy of an isolated disc. Let's now examine the disc-disc coupling.

E. Extracting the coupling parameter in two disc measurements

A system consisting of two discs with slightly different eigenfrequencies ν 0 and ν 0 + ∆ν 0 and a distance depending coupling κ 1 (d) can be described by the Hamiltonian

H(d) = ν 0 κ 1 (d) κ 1 (d) ν 0 + ∆ν 0 (8)
where the chosen basis (1, 0) and (0, 1) means "electron" in eigenstate of disc 1 with eigenfrequency ν 0 and eigenstate of disc 2 with eigenfrequency ν 0 +∆ν 0 , respectively. The difference ∆ν 0 of the eigenfrequencies includes the influence of the antenna, quantified in Sec. III D, and the discrepancy between the eigenfrequencies of the discs. The Hamiltonian has the eigenvalues

ν 1,2 (d) = ν 0 (d) + 1 2 ∆ν 0 ± 1 2 4 κ 1 (d) 2 + ∆ν 2 0 whose difference is given by ∆ν(d) = 4 κ 1 (d) 2 + ∆ν 2 0
The coupling parameter κ 1 can be estimated from the overlap of the wave function of the single discs. Using the evanescent waves (Eq. ( 6) for r > r D ) outside the discs the coupling is approximately given by

κ 1 (d) = κ 0 K 0 γ(r D + 1 2 d) 2 . ( 9
)
For the shift of the resonance we thus obtain To check it experimentally we varied the coupling between two discs by increasing their distance in steps of 0.2 mm and measured the splitting of the two resonances.

∆ν(d) = 4 κ 2 0 K 0 γ(r D + 1 2 d) 4 + ∆ν 2 0 (10)
Figure 6 shows the corresponding spectra for distances of 0.6 mm to 7.6 mm between the borders of the discs. With increasing distance the decrease of the resonance splitting due to weaker coupling is clearly observable.

After fitting the resonances with Lorentzians we can extract the resonance splitting depending on the distances of the discs (see Fig. 7 crosses). The experimental findings are described by Eq. ( 10) and by fitting we obtained κ 0 =1.35 GHz, γ=0.29 mm -1 , and ∆ν 0 =0.021 GHz.

In the next section we quantify the second nearestneighbor coupling in a three disc measurement.

F. Extracting the next nearest-neighbor coupling parameter

We performed an analogous measurement with three discs in a row whose distances will be equally increased in 0.2 mm steps. The antenna remains close to the central disc. For simplicity, we neglect in the modeling of such a situation the influence of the antenna and assume perfectly equal discs. This leads to a Hamiltonian like

H(d) =   ν 0 κ 1 (d) κ 2 (d) κ 1 (d) ν 0 κ 1 (d) κ 2 (d) κ 1 (d) ν 0  
with κ 1 the nearest-neighbor coupling and κ 2 the second nearest-neighbor coupling. From its three eigenvalues ν 1,2,3 the expressions for the coupling can be calculated:

κ 1 (d) = 2ν 2 1 + ν 1 ν 2 -ν 2 2 -5ν 1 ν 3 + ν 2 ν 3 + 2ν 2 3 3 √ 2 FIG. 8.
Coupling parameters for three disc measurement.

Upper figure shows the nearest-neighbor coupling κ1 with a fit of Eq. ( 9). The dashed line correspond to the coupling parameter κ1 from the two disc measurement. The lower figure presents the extracted values for the next nearest-neighbor coupling κ2.

κ 2 (d) = 1 3 (ν 1 -2 ν 2 + ν 3 )
We measured the spectra of the equidistant three disc system as a function of the border to border distance d and extracted the corresponding eigenfrequencies. From those eigenfrequencies we calculated the coupling parameters, which are presented in Fig. 8. Additionally we plot the extracted coupling coefficient κ 1 from the two disc measurement as dashed line. Overall the deviations are small. Note that we have neglected in the extraction of κ 1 for the three disc system any deviations of the resonance frequencies ∆ν 0 of the discs. coming from the different treatment of the differences of the eigenfrequencies of the discs ∆ν i . In the two disk problem they are taken into account properly, whereas for the three disc system they are incorporated in the three fit constants. Note that in the limit for infinite distances the coupling does not go to zero but to a constant value. The agreement of the coupling constants in case of the two and three disc system underlines the applicability of the tight binding approach. To justify the nearest neighbor coupling approximation we investigate the ratio |κ 2 /κ 1 |, which is presented in Fig. 9. The ratio is always smaller than 15 % thus the effects of next nearestneighbor couplings are small but not negligible. While generally ignored in graphene next nearest-neighbor and even next-next nearest-neighbor couplings are of the order of 5% and can be even larger in bilayer or doped graphene 22,[START_REF] Mcchesney | [END_REF]46 . Including higher neighbor couplings can for example shift the Dirac points or generate asymmetric band structures 22,[START_REF] Mcchesney | [END_REF]46 .

In contrast to the PBGM, which are typically realized by infinitely long dielectric cylinders, the transport between the discs is evanescent, whereas in PBGM the waves are freely propagating between the dielectric cylinders [START_REF] Lidorikis | [END_REF][42][43] . Thus higher order couplings are larger in the PBGM as the coupling is typically decreasing with d -2 , whereas in our case it decreases with the modified Bessel function |K 0 (d)| 2 .

G. Benzene-the main component of graphene

The next measurements were performed on a hexagonal cell. In the following it will be called "benzene" though only the 6 p z orbitals of the carbon atoms are imitated and their hydrogen atoms are ignored. For symmetry considerations it does not make any difference. First we check the predictions of group theory: as benzene has the dihedral symmetry D 6h we expect four resonances, consisting of two singlets and two doublets. The highest and lowest resonance are not degenerate. We measured spectra of benzene for different border to border distance d of the neighboring discs increasing in steps of 0.2 mm. The different spectra are shown in Fig. 10 and for each we find the expected 4 resonances. The degeneracy of the two central resonances was shown in Ref. 16. In case of benzene each disc has two nearest neighbors (κ 1 ), two next-nearest neighbors (κ 2 ) and one next-next-nearest neighbors (κ 3 ). From the spectra we extract the four resonances and calculated the coupling constants from the obtained resonances by

κ 1 = 1 6 (-ν 0 -ν 1 + ν 2 + ν 3 ) κ 2 = 1 6 (+ν 0 -ν 1 -ν 2 + ν ) κ 3 = 1 6 (+ν 0 + 2ν 1 -2ν 2 + ν 3 ) .
We obtain a fit function of the following form for κ 1

κ 1 (d) GHz = 2.28 K 0 0.33 4 + 1 2 d mm 2 + 0.0068.
The κ 1 for the two, three disc and the benzene measurement show only small deviations (typically below ±0.01 GHz). Again we have a look at the ratios of the three coupling parameters which are illustrated in Fig. 11. Note that the ratios κ 2 /κ 1 cannot be directly compared to the one obtained for the three disc measurement are not same. We observe that than 2 mm both ratios are than 15% (horizontal dashed line). As we want the influence of the higher order neighbor couplings to be small, we consequently work with disc distances above 2 mm and for the following measurements on graphene flakes we use d=4 mm.

Summarizing the basic measurements on few discs we conclude the following. The group theoretical predictions are fulfilled verifying the symmetry considerations. A consistent coupling behavior is found with small second and third nearest neighbor coupling. Thus we have realized a possibility to investigate tight-binding systems with a table top experiment, where several parameters can be varied easily. Now we proceed with hexagonal lattices.

Experimental results on graphene using the same setup have already been published in Ref. 16, where the Dirac point including the linear dispersion relation close to it have been observed. Additionally edge states at zigzag edges and the lack of edge states at armchair edges have been shown. Here we will proceed by showing the flexibility of the experiment and investigate the effect of positional (coupling) disorder on the Dirac point. Additionally it is possible to use different kinds of discs, where we will realize a correspondence to boron nitride by using two types of discs on the two triangular sublattices.

IV. LOCAL DENSITY OF STATES

As we want to relate our findings for larger systems to the density of states we derive here the relation between the measured reflection signal S 11 at an antenna to the LDoS.

In quantum mechanics the LDoS L( r, E) is defined by

L( r, E) ≡ n |φ n ( r)| 2 δ(E -E n ) (11)
where φ n ( r) is the n th eigenfunction at position r and can be expressed via the Green function G( r, r, E) of the system

L( r, E) = - 1 π Im lim ǫ→0 n |φ n ( r)| 2 E -E n + iǫ = - 1 π Im [G( r, r, E)] . (12)
To relate the Green function to the reflection R 2 , let's start with the probability T 2 for the incident wave to be transmitted via the antenna into the system and escape the system by other means, i.e. not via this antenna. In our case this corresponds to the absorption mainly in the top and bottom plate but also within the discs. By energy conservation it is connected to the reflection R 2 and given by

1 = R 2 + T 2 = |S 11 | 2 + T 2 T 2 = 1 -|S 11 | 2 = j d s (13)
where d s describes a contour around the antenna (assuming a two-dimensional setup). j is the two-dimensional quantum mechanical probability current

j = m Im (ψ * ∇ xy ψ)
which in electrodynamics corresponds to the twodimensional Poynting vector For TE-modes it is given by 44

S xy = - µ 2 ǫ ω Im (B * z ∇ xy B z ). (14)
We neglect the prefactors because only the proportionality of the current to Im(ψ * ∇ xy ψ) is important. Assuming a point-like source at R the wave function ψ = B z is proportional to the Green function G( r, R, E), as both are the solution of the Helmholtz differential equation with delta inhomogeneity. The proportionality coefficient Λ(E, R) depends on the coupling of the antenna to the system. Combining Eq. ( 12), Eq. ( 13) and Eq. ( 14) we obtain:

1 -|S 11 | 2 ∝ -|Λ(E, R)| 2 ×Im G * ( r, R, E)∇ xy G( r, R, E) d s ∝ |Λ(E, R)| 2 Im ξ β ( R, E) ∝ |Λ(E, R)| 2 L( R, E)
where the renormalised Green function

ξ β ( R, E) = lim r→ R G( r, R, E) - 1 2π ln | r -R| β (15)
removes the singularities at the antenna positions R. β is the scattering length of the antenna. A detailed derivation of the renormalized Green function can be found in Ref. 47. It is not clear how the coupling parameter Λ(E, R) depends on E or even on the LDoS. Therefore it is necessary to find a proper coupling theory. With the methods developed in Ref. 47 it is possible to derive an approximative description of the coupling for small cylindrical antennas showing an almost frequency independent prefactor Λ(E, R). Hence we finally obtain the relation between the LDoS and the reflection which is given by

L(R, E) ∝ 1 -|S 11 (R, E)| 2 (16)
Based on this knowledge we will plot in the following figures 1 -|S 11 | 2 on the y-axis.

V. DISORDERED GRAPHENE

Graphene's astonishing DoS is a consequence of its highly symmetric hexagonal lattice structure. But in the real world the perfect hexagonal symmetry of graphene will be distorted by different means. For examples external strain 13 , vacancies 6 , matching to substrates 48 , edges, and corners will destroy the symmetry. Additionally they can introduce disorder, e.g. via the position of vacancies in the bulk 6 or via the edges in finite systems 7 . Effects of Anderson localization have been numerically observed in transport through graphene nanoribbons with rough surfaces 49 . In a work on PBGM it was shown that a band gap can be quite stable against positional disorder if the original band structure comes from the Mie resonances 42 . This corresponds to our experimental realization apart from the fact that we have two touching bands at the Dirac point and not a proper gap.

One kind of perturbation which can be easily modeled by our setup is bulk disorder within the disc position. Starting from the perfect graphene lattice position of the discs r i,0 we create a new position r i,σ = r i,0 + ∆r i (cos θ i e x + sin θ i e y ), where Θ i is randomly chosen from {0,2π} giving the direction of the shift and ∆r i is randomly chosen from {0,σ} giving the length of the shift. e x and e y are the unit vectors in x and y direction. The positional disorder corresponds in the tight binding description to a disorder in the coupling.

We want to test how stable the Dirac point is against an introduced disorder. To reduce effects of edge and corner states we chose a hexagonal graphene flake as shown in the inset of Fig. 12. The antenna position where the reflection S 11 is measured is marked by a red cross. Starting with a regular lattice with a disc to disc distance of d=4 mm we add noise to every disc position as described above. We realized 4 disordered samples with a maximal distance to the original position of σ = 1, 2, 3, and 4 mm. The inset of Fig. 12 corresponds to the maximal value of σ=4 mm. The behavior of the LDoS is shown in Fig. 12. For σ = 1 mm we find a quite similar spectrum and can still see a strong reminiscence of the Dirac point in the LDoS (red arrow). For bigger shifts the structure disappears and the Dirac point becomes unobservable.

To investigate this in a more quantitative manner we perform averages over small frequency windows. The averaging windows are indicated by bars in Fig. 12 and are 13 MHz close to the Dirac point and 26 MHz inside the bands. To observe effects of the Dirac points in the averaged quantity, the averaging range around the Dirac point is chosen so small that a reduction of the DoS is still clearly observed. It is necessary to introduce the average as we are dealing with finite systems. Thus the experimental value at the Dirac point is dominated by the frequency distance of the closest resonance and its width. If small disorder is introduced we are not testing the stability of the global structure of the Dirac point but only the stability of the closest eigenvalue. By averaging locally we wash out this effect but still keeping the information of the diminution of the DoS close to the Dirac point.

In Fig. 13 on the upper part the averaged LDoS ( 1 -|S 11 | 2 ) is presented around the Dirac point (squares) and within the two bands (triangles and crosses). One observes with increasing disorder an increase at the Dirac point, whereas the averaged LDoS within the bands oscillates. At σ=2 mm the averaged values are of the same order, thus indicating the final vanishing of the Dirac point. That the Dirac point is quite stable against disorder perturbations is in agreement with the findings for PBGM 42 .

Additionally we performed numerical tight binding simulations on the same flake as in the experiment including more than only nearest neighbor couplings. We diagonalised the Hamiltonian where ν 0 = 6.65 GHz is the eigenfrequency of a single disc. The coupling matrix K contains the coupling elements κ. A disc is coupled to all neighbors which have a border to border distance d less than 18 mm. The coupling parameter κ is calculated by Eq. ( 9) using the specific disc distances d. The eigenfrequencies are obtained by diagonalizing the Hamiltonian. Then the DoS is calculated. On the lower part in Fig. 13 the averaged DoS for different frequency ranges are shown. The average is performed over the same frequency windows as in the experimental case. In contrast to the experiment the DoS is shown and not the LDoS. One observes also here that the density close to the Dirac point is increasing. Each step actually corresponds to one additional state inside the averaging region. Within the bands a decrease of the averaged density of states is found. Overall the Dirac point seems to be quite stable against the introduction of small white noise disorder.

H tb = ν 0 1 + K (17)

VI. IMITATING BORON NITRIDE

Boron nitride has a hexagonal lattice structure, but with two different sorts of atoms placed on the two triangular sublattices. Double layer of boron-nitride graphene lattices are a candidate to have on the one hand a band gap of similar energy than silicon and on the other hand keep the high charge carrier mobility of graphene 50 . Thus it is one candidate to substitute silicon and overcome especially problems creating high frequency transistors.

The energy difference β of the two types of atoms causes a widening of the band gap between the conducting and the valence band, corresponding to Eq. (3). We imitate this situation using two charges of discs with mean eigenfrequency of ν A =6.637 GHz respectively ν B =6.672 GHz and create the same lattice once with equal discs and once with the different discs. A comparison of the resulting LDoS is presented in Fig. 14. The orange curve, already presented in Ref. 16 (see Fig 4a )), corresponds to a graphene lattice with the Dirac point at ν A =6.637 GHz. The curves were shifted by their corresponding Dirac frequencies (ν 0 ) so that the central frequency for graphene and boron nitride coincide. The dashed horizontal line corresponds to the experimental background signal, mainly coming from the coupling of the antenna to the TM 0 mode and thus corresponds to the 0 for the DoS. The red dashed lines indicate the lin-ear behavior of the DoS close the Dirac curve shows the in of A and a corresponding of gap observed. shift due the fact that use two ent Thus the is be at GHz = 6.655 GHz, which is in good agreement with the experiment. The gap width is expected to be ∆ν g = |(6.637 -6.672)| GHz = 0.035 GHz, which corresponds to the difference between the dotted lines. It is in agreement with the experimentally observed gap.

VII. CONCLUSIONS

In this paper we present experimental results of hexagonal tight-binding configurations in a microwave setup. First, we show in detail how a tight-binding setup is realized. For setups with few discs we compare next-nearest neighbor coupling to the nearest neighbor coupling. The flexibility of the experiment allows to vary easily discs positions, i.e. the coupling, but also the 'on-site energies', i.e. eigenfrequency of the discs. We choose exemplarily to investigate disordered graphene. The disorder measurements show that the Dirac points seems to be quite stable against white noise disorder. As a second example showing the variation of the on-site energy we observe a band gap by using two different type of discs corresponding to a boron nitride sheet.

Exploiting the flexibility of the setup the realization of many different tight-binding structures are imaginable. Realizing different kinds of vacancies including disorder, transport properties along lattice axes. By lattice deformation one can either play with K and K ′ points 51,52 , which have been experimentally realized using a similar setup [START_REF] Bellec | Topological transition of Dirac points in a microwave experiment[END_REF] , or simulate pseudo-magnetic fields to generate associated Landau levels 53 .

FIG. 1 .

 1 FIG. 1. (Color online) Graphene as superposition of two triangular sublattices A and B with lattice vectors a1, a2 and the vectors b1, b2, b3 connecting nearest neighbors.

FIG. 2 .

 2 FIG. 2. (Color online) Spectrum of a single disc: Black line corresponds to a measurement with the TE-antenna shown in the inset; orange line shows the same measurement performed with a TM-antenna. The arrow marks the first TE-resonance which will be used for the further experiments (from Ref. 16).

FIG. 3 .

 3 FIG. 3. (Color online) Resonance frequency ν (upper) and width Γ (lower) of a single disc are shown in dependence of the distance ht between the metallic top plates and the top of the disc. The dashed blue line corresponds to the cut-off frequency at the corresponding height and the vertical dotted red line indicates the height at which the resonance frequency is above the cut-off frequency. The arrow marks the chosen distance ht = 11 mm between the top of the disc to the top plate for the experiments.

FIG. 4 .

 4 FIG. 4. (Color online)The spectra of a single disc are shown for different perturber positions. All spectra are obtained without top plate. Lowest (black) spectrum: no perturber, next (blue) spectrum: Perturber close to the border, the further up shifted the spectrum, the closer is the perturber placed to the disc's center. The spectra are shifted in y in steps of 0.05 for better visualisation. The inset on the upper right shows a photograph of the disc including the metallic perturber and the measuring antenna.

FIG. 5 .

 5 FIG. 5. (Color online) Resonance positions νexpcrosses) fit Eq. (7) (orange line) with γ=0.52 -1 , and ν0=6.67 GHz.

FIG. 6 .

 6 FIG.6. (Color online) Two disc spectra for increasing distance. They are up-shifted with increasing distance. The lowest spectrum corresponds to a distance of 0.6 mm, the topmost to 7.6 mm.

FIG. 7 .

 7 FIG. 7. (Color online) Resonance splitting of the two disc system depending on the disc distance (black crosses). The experimental data agrees well with the fit function Eq. (10) for values κ0 = 1.35 GHz, γ = 0.29 mm -1 and ∆ν0 = 0.021 GHz. The inset shows exemplary the spectrum for d = 3.3 mm (from Ref. 16).

FIG. 10 .

 10 FIG. 10. (Color online) Spectra of hexagonal cells (benzene)for different distances. The spectra are up-shifted with increasing distance. The lowest spectrum corresponds to a distance of 0.5 mm, increasing in steps of 0.2 mm.

FIG. 12 .

 12 FIG. 12. (Color online) The inset on the upper right shows the graphene flake of hexagonal shape with maximal deposition of σ=4 mm. The antenna position is marked by the red cross. The spectra of the hexagonal graphene flakes with different maximal random depositioning σ are shown. The lowest (black) spectrum corresponds to the unperturbed one (σ=0 mm). The spectra with σ=1, 2, 3 and 4 mm are shifted upwards correspondingly. The bars indicate the region of averaging used in Fig. 13.

FIG. 13 .

 13 FIG. 13. (Color online) On the upper figure the frequency averaged local density of states 1 -|S11| 2 around the Dirac point (squares) and within the bands at lower (crosses) and higher (triangles) frequencies than the Dirac point are shown for different maximal disorder σ. The windows for averaging are indicated in Fig. 12. On the lower part the numerically obtained averaged DoS is presented. Symbols are used correspondingly to the upper figure.

  FIG. 14. (Color online)Two spectra of the same lattice configuration, but consisting of one type of discs (graphene, orange) respectively two types of discs (boron-nitride, black). The disc close to the measuring antenna was not moved, to guarantee a consistent coupling. The dotted lines correspond to the eigenfrequencies of the two types of discs and indicate the band gap size. The spectra were shifted by its corresponding Dirac frequency ν0 such that the center of the bands are the same. Additionally the red dashed lines indicate the linear behavior of the DoS (see also Ref.16). The dashed horizontal line corresponds to the background signal of the experiment.

ACKNOWLEDGMENTS

This work was supported by the Deutsche Forschungsgemeinschaft via an individual grant and the Forschergruppe 760: Scattering systems with complex dynamics. S. B. thanks the LPMC at Nice for the hospitality during several long term visits and the University of Nice and the Fédération Döblin for financial supports.