P Van Isacker

A geometry for the shell model

Keywords: numbers: 21, 60, Cs, 21, 30, Fe

A geometric interpretation is given of matrix elements of a short-range interaction between states that are written in terms of aligned neutron-proton pairs.

I. INTRODUCTION

In this contribution matrix elements of the nucleon-nucleon interaction between shellmodel states are studied from a geometric point of view. An approach of this kind goes back to the seminal study of Schiffer [1]. The interaction between two nucleons can be expressed in terms of the angle between the angular momenta of the nucleons. Furthermore, the strength of the nucleon-nucleon interaction has a universal dependence on this angleuniversality that can be shown to result from the short-range character of the effective interaction [2]. This geometric interpretation of the nucleon-nucleon interaction is by now an accepted feature of nuclear structure; a discussion of it can be found in textbooks on nuclear physics, see e.g. Sect. 4.2 of Ref. [START_REF] Casten | Nuclear Structure from a Simple Perspective[END_REF].

The geometry of the shell model can be extended from two-to four-nucleon configurations [START_REF] Van Isacker | [END_REF]. In this contribution the geometric interpretation of the neutron-proton (np) interaction is examined for a four-nucleon configuration when written in terms of aligned np pairs.

II. MATRIX ELEMENTS OF THE NEUTRON-PROTON INTERACTION IN A 2N-2P BASIS

States for two neutrons with angular momentum J ν and two protons with angular momentum J π , coupled to total angular momentum J, can be written as |J ν J π ; J ≡ |j ν j ν (J ν ), j π j π (J π ); J , which, for all even values of J ν and J π allowed by angular-momentum coupling, form a complete and orthogonal basis. Matrix elements of the np interaction in the 2n-2p basis are

J ν J π ; J| Vνπ |J ν J π ; J = 4[J ν ][J ν ][J π ][J π ] R (2R + 1)      j ν j π J π J ν R j π J j ν j ν j π J π J ν      V R νπ , (1)
with [x] ≡ √ 2x + 1, and where V R νπ ≡ j ν j π ; R| Vνπ |j ν j π ; R are the np two-body matrix elements and the symbol in square brackets is a 12j coefficient of the second kind, which is known as a sum of products of four 6j coefficients [START_REF] Yutsis | The Theory of Angular Momentum (Israel Program for Scientific Translations[END_REF].

The expression (1) is used in Ref. [START_REF] Kim | [END_REF] to demonstrate the crucial role played by the np matrix element V R νπ with aligned neutron and proton angular momenta, that is, for R = j ν + j π . This generic feature of the np interaction results from angular momentum coupling [i.e., from the 12j coefficient in Eq. ( 1)] as well as from the matrix element in the aligned configuration, which is moderately to strongly attractive for any reasonable np interaction. The importance of the aligned np matrix element raises the question whether a shell-model approximation can be formulated in terms of aligned np pairs, as recently proposed in Ref. [START_REF] Cederwall | [END_REF]. (See Sect. 4.3 of Ref. [8] for a review of this debate.) This question is also addressed in Ref. [START_REF] Kim | [END_REF] for a 2n-2p system, and more generally for kn-kp systems in N = Z nuclei in Ref. [9], with the conclusion that the wave functions of many, but certainly not all, yrast states have a dominant component in terms of aligned np pairs. Here I examine whether a description in terms of aligned np pairs gives rise to a geometry of the shell model.

III. MATRIX ELEMENTS OF THE NEUTRON-PROTON INTERACTION IN A NP-NP BASIS

An alternative basis exists in terms of np pairs of the form |J 1 J 2 ; J ≡ |j ν j π (J 1 ), j ν j π (J 2 ); J . If J 1 and J 2 acquire all values allowed by angular-momentum coupling, this defines an overcomplete basis of states, which are not necessarily normalized, as indicated by the double bracket. Matric elements in the np-np basis can be obtained with the help of the transformation

|J 1 J 2 ; J = - Jν even Jπ even      j ν j π J 1 j ν j π J 2 J ν J π J      |J ν J π ; J , (2)
where the symbol in square brackets is a unitary 9j coefficient [START_REF] Talmi | Simple Models of Complex Nuclei. The Shell Model and the Interacting Boson Model[END_REF]. Note that the basis states on the right-hand side are orthogonal and normalized while those on the left-hand side are not. In the application of interest one constructs states in terms of a single np pair such that J 1 = J 2 ≡ J, in which case the overlap matrix element is

J2 ; J| J2 ; J = 1 + (-) J 4      1 -      j ν j π J j π j ν J J J J           , (3)
which is zero for odd values of J. This shows that the np pair behaves as a boson since two identical np pairs can only coupled to even J. Henceforth it is assumed that J is even. The matrix elements of the np interaction in the np-np basis are

J2 ; J| Vνπ | J2 ; J = V J νπ -2      j ν j π J j π j ν J J J J      V J νπ +(2 J + 1) 2 R (2R + 1)      j ν j π J J R j ν J j π j ν j π J J      V R νπ . (4)
For aligned np pairs, J max = j ν + j π , the unitary 9j coefficient in the overlap (3) and in the matrix element ( 4) is much smaller than 1 and can be neglected. [For J = 2J max the unitary 9j coefficient equals 1 but this angular momentum is excluded by the Pauli principle-see Eq. ( 3).] The sum over the 12j coefficients, however, cannot be neglected. To a good approximation the matrix elements of the np interaction between normalized states are therefore

J 2 max ; J| Vνπ |J 2 max ; J ≈ 2V Jmax νπ + 2(2J max + 1) 2 R (2R + 1)      j ν j π J max J max R j ν J j π j ν j π J max J max      V R νπ . (5)
Not surprisingly, one finds that, in a basis constructed out of aligned np pairs, the main contribution to the 2n-2p configuration stems from the matrix element V Jmax νπ . There are however corrections to this dominant contribution, which depend on the total angular momentum J and on the strengths of the np interaction with R = J max .

IV. EXPRESSIONS IN THE HIGH-J AND THE LOW-J LIMITS

For a schematic short-range interaction the expressions in the np-np basis simplify.

Specifically, for a surface delta interaction (SDI) classical approximations to the sum in Eq. ( 5) can be worked out for aligned np pairs, J max = j ν + j π . With the method of Ref. [START_REF] Van Isacker | [END_REF], based on the classical limit of 3nj coefficients [START_REF] Wigner | Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra[END_REF], the matrix element (5) can be reduced to

J 2 max ; J| V SDI νπ |J 2 max ; J ≈ 2V Jmax νπ - 3a 0 + a 1 π sin θ + (-) ν +jν + π +jπ a 0 -a 1 π tan θ , (6)
� ν = ��/� � π = ��/� � = � � � �� �� �� � � � � ν = ��/� � π = ��/� � = � � � �� �� �� � � � � ν = ��/� � π = ��/� � = � � � �� �� �� � � �
FIG. 1: Illustration of the 12j-coefficient approximation for j ν = 21/2, j π = 19/2 and J = 0, 2, 4, as a function of R. The dots are the exact expression on the left-hand side and the curve is its approximation on the right-hand side of Eq. ( 8).

where a 0 (a 1 ) is the isoscalar (isovector) strength of the np interaction and θ is the angle between the angular momenta J max of the two np pairs that couple to total angular momentum

J, θ = arccos J(J + 1) -2J max (J max + 1) 2J max (J max + 1) . (7)
The classical approximation ( 6) is reasonable except for low angular momentum J, in particular for J = 0 or θ = π when it diverges. Unlike in Ref. [START_REF] Van Isacker | [END_REF] one is interested in the case J = 0 since it corresponds to the energy of the ground state. The following approximate expression can be derived for the 12j coefficient under the assumption that J is low and that the np pair is aligned, J max = j ν + j π :

(J max + 1) 2 (2J max + 1)      j ν j π J max J max R j ν J j π j ν j π J max J max      ≈ exp -4R(R + 1) -J(J + 1) + R(R + 1)J(J + 1) + 4(j ν -j π ) 2 2(J max + 1) . ( 8
)
This approximation is accurate for J = 0 but deteriorates rapidly for increasing values of J, unless the single-particle angular momenta are very large (see Fig. 1). The low-J and high-J approximations to the matrix element (5) are illustrated in Fig. 2.

V. THE EXAMPLE OF 128 CD

Let me now illustrate the different approximations with the example of the nucleus 128 Cd, with two neutron (proton) holes in ν0h 11/2 (π0g 9/2 ) with respect to the core 132 Sn. The (�)

�� + × �� + � � �  νπ ��� �� + × �� + � � � �� �� �� �� -� � ������� �������� � ������ (�) �� -× �� -� � �  νπ ��� �� -× �� -� � � �� �� -� � ������� �������� � ������ FIG. 2:
The exact expression for the 2n-2p matrix element (5) of the np interaction (dots) compared the high-J (red) and low-J (blue) approximations of Eqs. ( 6) and ( 8), respectively. The comparison is carried out for (a) a SDI with a 0 = a 1 = 0.25 and for j ν = 21/2 and j π = 19/2 and ν and π even, and (b) a SDI with a 0 = 0.75 and a 1 = 0.25, and for j ν = h 11/2 and j π = g 9/2 . The relevant two-body np matrix elements are derived from the realistic interaction jj45pna [13],

constant contribution 2V Jmax νπ is not included. �� ��� �� �� ���� ������ (�) (�) (�) (�) (�) � + ( � + ) ( � + ) ( � + ) ( � + ) ( �� + ) � + � + � + � + � + �� + -� -� -� -� -� -� -� ������ (���)
and the nn and pp interactions are taken from 130 Sn and 130 Cd, respectively. The results of a shell-model calculation with this interaction in the (ν0h 11/2 ) -2 (π0g 9/2 ) -2 model space are shown in Fig. 3(a), and are seen to agree with the observed energies. The jj45pna interaction can therefore be considered as a realistic basis for further approximations. The approximation (b) consists of keeping only the np matrix element V 10 - νπ and putting all others (nn, pp and np) to zero. As already pointed out by Zamick and Escuderos [14], this clearly is inadequate. It is essential to keep the nn and pp interactions and therefore, in order to the test the influence of the np interaction, in the approximations (c)-(e) the realistic nn and pp interactions are taken without modification. In Fig. 3(c) are shown the results obtained in the (ν0h 11/2 ) -2 (π0g 9/2 ) -2 model space with the exact nn and pp interactions but with a single non-zero matrix element V 10 - νπ . This is the approximation advocated in Ref. [START_REF] Kim | [END_REF]; it is indeed remarkable that with a single component of V R νπ that much of the np correlations can be accounted for. The results (d) are obtained with Eq. ( 5) and equivalent expressions for the nn and pp interactions (not given here). This is the aligned-np-pair approximation [9]: take the full, realistic interaction within the restricted space constructed out of aligned np pairs, which in this example consists of a single state. Comparison of (a) and (d) shows that the aligned-np-pair approximation is adequate for low J but fails for J = 8 and 10.

Finally, the results (e) are obtained with the np interaction in the low-J approximation (8) for J ≤ 4 and the classical approximation (6) for J ≥ 4 with a 0 = 0.5 and a 1 = 0.13, strengths obtained from a fit of the SDI to the jj45pna interaction. The results for J = 4 are essentially indistinguishable in the two approximations.

VI. CONCLUDING REMARK

It is important to distinguish between the following two approximations: (i) The full shell-model interaction is considered in the restricted space constructed out of aligned np pairs; (ii) Only the component of the np interaction in the aligned configuration is taken and diagonalized in the full shell-model space. The two approximations are not equivalent and have their respective merits and problems. If approximation (i) is made for a shortrange interaction, a simple geometric interpretation can be obtained, as argued in this contribution. Approximation (ii), on the other hand, can be shown to give rise to a partially solvable problem for the four-nucleon system [15].

FIG. 3 :

 3 FIG. 3: The experimental spectrum of 128 Cd compared to the shell-model approximations (a)-(e) discussed in the text.

Acknowledgement

I acknowledge fruitful discussions with Yung Hee Kim, Maurycy Rejmund and Antoine

Lemasson.


Jean-Noël Vittaut

Jean Méhat

Fast Instantiation of GGP Game Descriptions Using Prolog with Tabling

We present a method to instantiate game descriptions used in General Game Playing with a Prolog interpreter using tabling. Instantiation is a crucial step for speeding up the interpretation of game descriptions and increasing the playing strength of general players. Our method allows us to ground almost all of the game descriptions present on the GGP servers in a time that is compatible with the common time settings of the GGP competition. It instantiates more rapidly than previous published methods.

INTRODUCTION

General Game Playing (GGP) aims at conceiving programs capable of playing a large variety of games without knowing the rules in advance. The Game Description Language (GDL) [START_REF] Love | General game playing: Game description language specification[END_REF] has been used to communicate the rules of the game to be played at the beginning of a match in the General Game Playing competition since 2005. GDL allows to describe combinatorial perfect information games. It has also been extended to handle incomplete and imperfect information games (GDL-II). It uses first order logic and is similar to Datalog with negation as failure. Its syntax consists of Lisp S-expressions. A game is described with a set of facts and rules; a few keywords are reserved for logic and game-specific features [START_REF] Love | General game playing: Game description language specification[END_REF].

Fast interpretation of GDL is important because it can significantly improve the strength of a player. An approach to speed up a reasoner is to ground the rules and build a Propositional Net [START_REF] Genesereth | General Game Playing[END_REF].

We use the tabling engine built in a Prolog interpreter. Tabling consists in storing answers for subgoals and reusing them whenever the same subgoal is called again. At the cost of a modification of the unification process, it avoids redundant sub-computations and deals with infinite loops. We use here the tabling as implemented in the YAP Prolog interpreter because of its performance, its availability and our familiarity with this interpreter [START_REF] Rocha | A tabling engine for the YAP prolog system[END_REF], [START_REF] Rocha | Dynamic mixed-strategy evaluation of tabled logic programs[END_REF].

INSTANTIATION OF GDL RULES

Cleaning step The instantiation starts with a cleaning step where the explicit or operator, which has been deprecated in GDL since 2007, is removed by rewriting all the rules into a disjuctive form.

Adding input and base predicates The input and base predicates are used to enumerate all the moves and facts that may be used in any reachable game state. We add them to the game descriptions when they are missing.

These predicates are a recent addition to GDL and we suppose they were introduced to facilitate the instantiation of game descriptions. Actually, different descriptions of these predicates can lead to dramatic differences between grounded game descriptions, for instance the input predicate of the Breakthrough game description is defined more lazily on the Tiltyard server than on the Stanford server. It leads to a grounded description that contains 20 times more rules.

We compute them using either an iterative method equivalent to the one used by Kissmann et al. [START_REF] Kissmann | Instantiating general games using prolog or dependency graphs[END_REF] or a one step method equivalent to the one we are describing for the instantiation. The two methods perform similarly with a slight advantage for the second one that uses tabling.

Eliminating not, renaming true and does We distinguish the dynamic predicates depending on the state of the game from the remaining static ones, the instantiated values of which are independent from the state of the game and can be computed once and for all upon receiving the game description.

From each rule R, we construct a new rule g(R) by removing every (not T ) term where T is a dynamic term; renaming every (true T ) and (does T1 T2) term respectively with (base T ) and (input T1 T2).

Removing not in dynamic predicates allows us to compute any possible instantiation without risking an elimination by the negated term. It is a safe operation since GDL guarantees that any negated term always has to be fully instantiated. A drawback is that the process will produce useless grounded rules, since the not operator is never checked. This transformation allows us to ground all the rules in one pass.

Removing static terms Terms formed on static predicates do not need to appear in the instantiated rules since their truth is known regardless of the state of the game: if true they can be removed; if false the entire rule can be discarded; conversely if they appear within a not, they can be removed if false and the rule can be discarded if true. Consequently, we compute a rule s(R) from the initial rule R by removing any static term or its negation from R.

Adding the side effect and introducing a new symbol The rules g(R) and s(R) are combined to produce the two new rules that will be part of our final grounding program. Given a rule g(R) of the form (<= (p U1 . . . Up) T1 . . . Tn) we derive the two new rules:

(<= p# (U1 . . . Up) T1 . . . Tn (store s(R))) (<= (p U1 . . . Up) (p# U1 . . . Up))
where p is the original predicate symbol of the conclusion of g(R).

The store predicate has the side effect of storing the s(R) instanti-ated rule in a data structure shared between the Prolog interpreter and the driver program; it always evaluates as true. p# is a new unique symbol, different for each processed rule. It is necessary to prevent the tabling engine from tabling rules with side effects because it would lead to missed instantiations: the rule including the side effect is not tabled while the second is. These two rules are logically equivalent to the rule g(R).

Instantiation By querying all the solutions to the tabled predicates, the store predicate inserts instantiated rules into the shared data structure.

EXPERIMENTAL RESULTS

We tested the instantiation of the rules on all of the 246 games that were active in February 2014 on the Dresden server 3 . The experiments were run on one core of an Intel Xeon E5-4610 2.40GHz with 520GB of RAM. We measured that our method needs about 500MB to compute one million instantiated rules. We used YAP 6.2.2 Prolog interpreter as a library for our driver program written in C++.

In figure 1 we plotted the time performance of our grounding method. The time of the step computing input and base is not taken into account but we measured that it takes less than half the time of the whole instantiation process, demonstrating that our method does not need these predicates to be included in game descriptions. We represented the percentage of games that can be instantiated within the time budget represented in the x-axis. 24% of the games were instantiated in less than 100ms, 72% in less than one second and 94% in less than one minute. The remaining 6% that were grounded in more than one minute contained more than 10 7 rules and facts. The processing of three games was halted after 30 minutes of computation.

Figure 2 demonstrates that the time to ground increases almost linearly with the size of the grounded game description when it is greater than 104 . We also observed that a significant part of the time is used to translate the fluents from the Prolog internal representation into the GDL representation in the shared data structure.

We compared our grounder with the GGPBase flattener, a freely distributed GDL grounder 4 . The time needed by the flattener seemed to increase at least quadratically with the size of the grounded program whereas the time needed by our method has been established to increasing linearly.

We also compared our method with the two approaches presented by Kissmann and Edelkamp in [START_REF] Kissmann | Instantiating general games using prolog or dependency graphs[END_REF]. They were able to instantiate about 96 of 171 game descriptions in less than one minute with their Prolog-based approach, and 90 of 171 with their method using dependency graphs which are proportions that we attain in less than one second. Our method appears to be of two orders of magnitude faster when the instantiation time becomes significant.

CONCLUSION

We have demonstrated that it is possible to ground almost all the game descriptions found on the GGP servers in a time span compatible with the current GGP competition time settings. This relies on the use of tabling in a Prolog interpreter.

We have also established that the new predicates input and base can be considered as superfluous. The few game descriptions in which they can be helpful have a grounded size that is so large that building alternative representations such as Propositional Nets is problematic. Tweaking the Game Description Language for specific tasks is somewhat dubious since the description language should be as agnostic as possible in relation to methods that could be used by players.

Figure 1 .

 1 Figure 1. Percentage of instantiated game descriptions that were grounded within the time budget in the x-axis.

Figure 2 .

 2 Figure 2. The number of generated rules as a function of instantiation time for the 243 successfully instantiated games.

LIASD -University of Paris 8, France, email: jnv@ai.univ-paris8.fr

LIASD -University of Paris 8, France, email: jm@ai.univ-paris8.fr

The Dresden server is available at http://ggpserver.general-game-playing.de

The set of GGPBase Java libraries is distributed at http://www.ggp.org/developers/players.html


Jacques Curély
email: jacques.curely@u-bordeaux1.fr

Janis Kliava

Magnetism of 2d Heisenberg classical square lattices: theory vs experiments

Keywords: spin lattices, Heisenberg model, antiferromagnetism, renormalization group

In this article, we exclusively focus on the low-temperature behaviors of 2d square lattices composed of (2N+1) 2 classical spins with isotropic couplings between firstnearest neighbors (Heisenberg couplings), in the thermodynamic limit (N→+∞). After briefly recalling the theoretical closed-form expressions derived for (i) the zerofield partition function Z N (0), (ii) the spin-spin correlation between any two lattice sites, (iii) the correlation length and (iv) the static susceptibility, we build up a diagram characterized by three low-temperature magnetic phases. Moreover we show that the behaviors of these physical quantities as well as the diagram are in perfect agreement with the corresponding ones derived using a renormalization method. Finally we give criterions allowing to directly determine the magnetic phases characterizing 2d magnetic compounds described by our microscopic model. An experimental test is given for illustrating this theoretical study. It allows one to obtain a perfect fit of the susceptibility of [{Mn(DENA) 2 (N 3 ) 2 } n ] (DENA is the ligand diethylnicotinamide).

1 Introduction Thin magnetic layers play an important role in many areas of technology because they may show a great variety of very different physical properties. Namely, in the field of spintronics, these layers can appear at the interface between different semiconductors showing or not magnetic order. Two-dimensional (2d) antiferromagnets such as La 2 CuO 4 turn out to be high-temperature superconductors when properly doped [1]. Magnetic layers also constitute an intermediate step for building up 3d artificial magnets whose structure may be imposed (like for magnetic grains used in nanotechnologies) and are characterized by local spins of high quantum number.

As a result the theoretical modelling is of the highest interest, notably the 2d Heisenberg model [1]. The starting point of this model consists in considering an exchange Hamiltonian in which the spins are isotropically coupled. Thus the magnetic properties of La 2 CuO 4 may be described by this model. The lattice composed of spins ½ showing isotropic couplings between first-nearest neighbors is characterized by a square unit cell. Chakravarty et al. [1] have shown that the long-wave length excitations and low-energy properties are well-described by a mapping to a 2dclassical Heisenberg magnet because all the effects of quantum fluctuations can be resorbed by means of adapted renormalizations of the coupling constants. In other words the spins ½ may be replaced by classical spins i.e., spins characterized by a spin quantum number S≥5/2.

In previous papers [2] one of us gave a first treatment for the 2d square lattice composed of classical spins showing Heisenberg couplings (isotropic couplings between first-nearest neighbors). More recently, one of us published a couple of papers [START_REF] Curély | IEEE Proceedings of Optim 2010[END_REF] in which we rigorously established, for the first time, the closed-form expressions of the zerofield partition function and the spin-spin correlation, respectively. In this article, after recalling these important results we describe low-temperature behaviors of the correlation length and the static susceptibility derived from the spin-spin correlations for infinite lattices. This leads to the determination of a diagram including three magnetic phases. Moreover we show that all the behaviors are in perfect agreement with the corresponding ones derived by using the renormalization method. An experimental test is given for illustrating this theoretical study.

Theoretical background 2.1 Generalities

The starting point of the 2d classical Heisenberg model consists in expressing the general Hamiltonian describing a lattice characterized by a square unit cell and wrapped on a torus so that it contains (2N+1) 2 sites, each one being the carrier of a classical spin S i,j :

( )

, , N N ex mag i j i j i N j N H H H =- =- = + Â Â , (1) with
, 1 , 1 2 1 , , ( )
ex i j i j i j i j H J J + + = + S S S , (2)
, , ,

mag z i j i j i j H G S B = - , ( 3
) where G i,j = G if i+j is even or zero, G i,j = G' if i+j is odd. (4)
In ( 2) J 1 and J 2 refer to the exchange interaction between nearest neighbors belonging to the horizontal lines and vertical rows of the lattice, respectively. In addition J i >0 (respectively, J i <0), with i=1,2, denotes an antiferromagnetic (respectively, ferromagnetic) coupling. G i,j is the Landé factor characterizing each spin S i,j and expressed in μ B / unit, where μ B is Bohr's magneton and =h/2π, h being Planck's constant. Now one may wonder why we consider a lattice wrapped on a torus. At first sight, one can guess that it is due to the fact that a torus possesses more symmetry elements than a plane lattice. But, in the infinite lim-it (N→+∞), we previously showed [START_REF] Curély | IEEE Proceedings of Optim 2010[END_REF] that these two types of lattices are characterized by the same partition function Z N . As a result, all the thermodynamic functions derived from Z N are similar.

In the present article we exclusively focus on physical quantities expressed in the zero-magnetic field limit. In that case the zero-field partition function Z N (0) is given by , , , ,

... ... exp N N ex N NN i j N N i j i N j N Z d d d H -- =- =- Ê = - Á Ë Â Â Ú Ú Ú S S S β (0)
(5) where β=1/k B T. The magnetic susceptibility χ i,j per lattice site and the correlation length ξ are defined as , , , ' , ' '

i j i j i k j k k k k k G G χ β Γ + + = ÂÂ , ( 6
) 1/2 2 2 , ' ' , ' ' ( ' ) k k k k k k k k k k + Ê Á = Á Ë ÂÂ ÂÂ Γ ξ Γ , (7)
where Γ k,k' is the correlation function:

, '

.

k k i j i k j k i j i k j k Γ + + + + =< > -< >< > S S S S . (8)
In the previous equation, the bracket notation <…> means that we deal with a thermodynamic average. As a result we may define the spin-spin correlation between any two spins as:

, , ' , , , , ' , ' , , 1 .
... ... ... exp (0)

N N ex i j i k j k N N i j i j i k j k i k j k N N i j i N j N N d d d d H Z β + + -- + + + + =- =- Ê < > = - Á Ë Â Â Ú Ú Ú Ú S S S S S S S S . (9)
The spin correlation u < > S , with u=(i,j) or (i+k,j+k'), can be simply derived from (9) by replacing S i,j or S i+k,j+k' by unity.

Theoretical results

We shall not give here all the details allowing to determine closed-form expressions for the zero-field partition function, the spin-spin correlations, the static susceptibility as well as the correlation length. The reader may report himself to previous papers published by one of us [2,[START_REF] Curély | IEEE Proceedings of Optim 2010[END_REF]. However we just give below the results obtained. In the thermodynamic limit (N→+∞) the zero-field partition function can be written as:

( ) ( ) [ ] 2 2 4 8 , 1 2 0 (0) (4 ) N N N Z F J J π λ λ +• = = - - Â β β , ( 10
)
where the factor F , is (the factor 1/4π being omitted):

2 4 2 0 , 0 0 0 1 (2 1) 2 1 L L F C L = = + È Î + Â , 0,0 1 F = . ( 11
)
In ( 11) the current L-coefficient is a Clebsch-Gordan (CG) one. The temperature-dependent function λ (-βj) appearing in ( 10) is given by:

( ) ( ) 1/2 1/2 I 2 j j j π λ β β β + Ê - = - Á Ë ¯ , j = J 1 or J 2 , ( 12
)
where I +1/2 (-βj) is a modified Bessel function of the first kind.

As we are dealing with isotropic (Heisenberg) couplings, we have the following properties (with v = x, y or z):

, , ' , , ' 1 .
.

3 v v i j i k j k i j i k j k S S + + + + < >= < > S S , , , 1
3 v i j i j S < >= < > S , , , 1
3 v i j i j = Γ Γ , 2 , 1 ( ) 3 v i j S < >= , , , 3 i j v i j = χ χ , v = ξ ξ. ( 13
)
In addition we have found that

, z i j S < >=0, , ' z i k j k S + + < > =0,
for T>0 K, so that the correlation function , ' Γ z k k reduces to the spin-spin correlation , , '

.

z z i j i k j k S S + + < >.
Of course, when T = 0 K exactly, we have | ,

z i j S < >| = 1/3 (or equivalently | , i j < > S | = 1
). This result rigorously proves that the critical temperature is absolute zero, i.e.,

T C = 0 K [3].
The spin-spin correlation in the thermodynamic limit (N→+∞) is given by [START_REF] Curély | IEEE Proceedings of Optim 2010[END_REF]:

( ) ( ) [ ] [ ] 2 4 0,0 , ' , 1 2 1 ,0 1 0 1 . ( 1
) (0) N z z k k N S S F J J X X Z λ λ δ +• + - = < > = - - + - Â β β , k > 0, k' > 0, as N→+∞. ( 14
)
Z N (0) is the zero-field partition function given by ( 10), δ ,0 is the Dirac function and we have set:

( ) ( ) ( ) ' 2 , 1, 2, , k k F X C u u F ε ε ε ε ε + + + + + = , k ≥ 0, k' ≥ 0, 1 1 (2 1)(2 3) C + + = + + , 1 (2 1)(2 1) C -= + - , ( ) ( ) , ,
,

i i i F J u F J ε ε ε λ β λ β + + + - = - , ε = ±1, i = 1, 2. ( 15
)
F , is given by ( 11) and F , +ε can be expressed as the following CG-series (the factor 1/4π being omitted):

min(2 ,2 2 ) 2 0 0 , 0 0 0 0 0 1 (2 1)(2 2 1) 2 1 L L L F C C L ε ε ε ε ε + + + + = = + + + È Î + Â , 0, 1 F ε = , ε = ±1. ( 16
)
The correlation length may be derived from ( 7) and ( 14)-( 16):

( ) ( ) [ ] [ ] ( ) ( ) [ ] [ ] 2 2 1/2 4 , 1 2 1 , 0 1 0 4 , 1 2 1 , 0 1 0 (1 )
(1 )

N N F J J N N F J J D D λ λ δ ξ λ λ δ +• + - = +• + - = È - - + - Í Í = Í - - + - Í Î Â Â β β β β , as N→+∞, (17) with: ( ) ( )( ) 2 , , 1 , 2 , 1 1
1

F D C F u u + + + + + = - - ε ε ε ε ε , ( ) ( ) ( ) ( ) 1, 1, 2, 2, 2 2 1, 2, 1 1 1 1 u u u u N D u u + + + + + + + + È + + = + Í - - Í Î ˚ ε ε ε ε ε ε ε ε (18)
where C +ε , u i, +ε , F , +ε and F , are given by ( 15), ( 16) and ( 11), respectively. The static susceptibility per unit cell and averaged per site may be written as:

( )

0,0 0,1 1,0 1,1 1 4 z z z z χ χ χ χ χ = + + + ( 19
)
where the susceptibility per site is given by

( ) ( ) [ ] [ ] 2 4 , ' , 1 2 1 ,0 1 0 1 (1 ) (0) N k k N F J J Z χ λ λ χ δ χ +• + - = = - - + - Â β β , as N→+∞, ( 20
)
with k = 0 or 1 and k' = 0 or 1 so that owing to (19):

( )

2 2 2 , 1 , 2 , , 3 , ( ' ) 2
' 6 F G G W GG W C F W ε ε ε ε ε ε β χ + + + + + + + + = , ε = ±1, ( 21
) 2 2 1, 1, 2, 1, 2, [1 ( ) ][1 ( ) ] 4 W u u u u ε ε ε ε ε + + + + + = + + + , 2 2 2, 1, 2, 2, 1, 2[ (1 ( ) ) (1 ( ) )] W u u u u ε ε ε ε ε + + + + + = + + + , 2 2 3, 1, 2, [1 ( ) ][1 ( ) ] W u u ε ε ε + + + = - - , (22)
where C +ε , u i, +ε ,F , +ε and F , are given by ( 15), ( 16) and

(11), respectively, as noted after (18).

3 Study of the low-temperature behaviors We have seen that the quantity u i, ±1 given by ( 15) is of funda-mental importance because it appears in the closed-form expression of the spin-spin correlation. The latter is involved in the respective definitions of the susceptibility and the correlation length (( 6), ( 7)). Due to the fact that the critical temperature is absolute zero all the λ (-βj) given by ( 12) (in fact the eigenvalues of the problem) become equivalent including that one characterized by infinite-. We have previously shown that the ratio F , +ε /F , (ε=±1) tends to unity when →+∞ [START_REF] Curély | IEEE Proceedings of Optim 2012[END_REF]. As a result the lowtemperature study of the ratio u i, ±1 is reduced to that of λ ±1 (-βJ i )/λ (-βJ i ) or equivalently I ±1+1/2 (-βJ i )/ I +1/2 (-βJ i ) as →+∞ and β|J i |→+∞ (i=1,2). The behavior of Bessel function I +1/2 (-β|J i |)∼I (-β|J i |) for →+∞ and β|J i |→+∞ has been established by Olver [START_REF] Olver | [END_REF]. We have extended this work for a large-order (but not necessarily infinite) and for any real argument z i varying from a finite value to infinity [START_REF] Curély | IEEE Proceedings of Optim 2012[END_REF]. We have shown that it is necessary to introduce the dimensionless auxiliary variables:

2 2 1 l n 1 1 i i i i i i J z z J z ζ È Ê = - + + Í Á Ë Í + + Î ˚, i i J z β = . (23)
The numerical study of |ζ| is reported in Fig. 1. As expected we observe that there are two branches. |ζ| vanishes for a numerical value of |z| -1 = k B T/|J| very close to π/2 so that there are 3 domains which will be physically interpreted below. Let T 0 be the corresponding temperature; we set:

0 2 B i k T J π = , i = 1, 2. ( 24
)
In the formalism of renormalization group T 0 is called a fixed point. In the present 2d case we have →+∞. As a result we derive that T 0 →T C =0 K as →+∞ so that the critical temperature may be seen as a fixed point. This will allow one to expand all the thermodynamic functions as Taylor series of current term |T-T 0 | near T 0 ≈T C =0 K.

At this step we must recall that the square of the spin modulus S(S+1)∼S 2 , as S→+∞, is absorbed in the exchange energy Chakravarty et al. [1] as well as Chubukov et al. [1] have written the action S/ (which allows one to calculate the partition function) as proportional to J/2. In addition these authors have considered the spin density S/a where a is the lattice spacing. In our case the lattice spacing between two similar Landé factors G or G' is 2a. As a result, the left member of (24) can be written as k B T 0 /(|J|/2)(S/2a) 2 so that the right-hand side becomes 4π instead of π/2. We must keep in mind this remark because it will be very useful later.

Under these conditions and after a tedious calculus the key ratio u i, ±1 may be written as: The previous general low-temperature expansion may be finally expanded as Taylor series near T C =0 K if expressing the dimensionless auxiliary variables |ζ| and |z| in this limit.

( ) , 1 1 1 1 1 2 2 i i i i i i i i i J u z J z z z Ï Ê ˆÈ = - + + - Ì Á ˜Í Ë ¯Î Ó ∓ ± θ θ ( ) } 2 1 exp 2 ... 4 i i Ê ˆ- - - + Ë ¯˚ θ ζ , i = 1,
Noting that 0

1 i i X z z T -= , 0 0 i i B J T T X k T - = , 0 0 , , i i i T T X T T ρ Δ < Ï = Ì > Ó , (26)
the Taylor series giving |ζ i | becomes ln 1

2 i i X T π ζ Ê = + Ë ¯ , i = 1, 2, (27)
so that:

( )

2 exp 2 1 2 i i X T π ζ - Ê - = + Ë ¯ , i = 1, 2. ( 28
)
Noting that (1±u/ ) =exp(±u), as large (or infinite), we finally derive:

( ) exp 2 exp i i X T ζ π Ê - = - Á Ë ¯ , i = 1, 2, as →+∞. ( 29
)
The physical meaning of ρ i and Δ i will be given below.

At this step we define the coupling constants at temperature T and at the fixed point T 0 :

B k T g J = , 0 0 B k T g J = , 0 T g T = .
(30)

These ratios measure the strength of the quantum fluctuations because, if we recall that |J| must be read as |J|S |ζ| near T 0 we have the following equations:

( ) 31) so that the thermal study of |ζ| is reduced to two domains 1 g < i.e., g< g 0 (T<T 0 ) and 1 g > i.e., g> g 0 (T>T 0 ). Each of these domains may be itself divided in two subdomains according to as |ζ|>|ζ F | or |ζ|<|ζ F | . This is the reason for which we are led to introduce the |ζ

1 2 F g π ζ < = -, (T < T 0 ), ( )
1 2 F g π ζ > = -, (T > T 0 ), (
F |-variables 1 / i F x T < = ζ and 2 / i F x T > = ζ
. Then using the variables ρ i and Δ i defined in (26) we set

1 2 i i T x = πρ , 2 2 i i T x Δ = π . ( 32
)
Thus, as noted by Chubukov et al. [1], the parameters 1 i x and 2 i x control the scaling properties of the magnetic system. ρ i represents the spin stiffness of the ordered ground state (Néel state for an antiferromagnet) and Δ i is a spin gap between the ground state and a spin liquid phase. ρ i and Δ i (cf ( 26)) vanish at the fixed point T 0 . If we report to the definition of |ζ| (cf. ( 23)) |ζ| is dimensionless and homogeneous to T 0 /T i.e., 1/ g . If g <1 (T<T 0 ) we have two possibilities 1 i x >1 i.e., T>|ζ F | < and 1 i x <1 i.e., T<|ζ F | < . When T=T 0 , 1 i x and 2 i x become infinite due to the fact that ρ i and Δ i vanish. Finally, if g >1 (T>T 0 ) we have two possibilities 2 i x >1 i.e., T>|ζ F | > and 2 i x <1 i.e., T< |ζ F | > . As there is an analytical continuity between 1 i x and 2 i x while passing through T 0 , see (26), we derive that there are 3 domains:

(i) if T<|ζ F | < ( 1 i x <1) we deal with the Renormal- ized Classical Regime (RCR), (ii) if T<|ζ F | > ( 2 i
x <1) we deal with the Quantum Disordered Regime (QDR) and (iii) if

T>|ζ F | < ( 1 i x >1) or T>|ζ F | > ( 2 i
x >1) we have the Quantum Critical Regime (QCR). Along the line T=T 0 in this domain we directly reach the fixed point without a phase change. The magnetic diagram has been reported in Fig. 2. It is strictly similar to the one obtained by Chakravarty et al. and Chubukov et al. [1], separately. Finally, if T tends towards absolute zero when coming from the Renormalized Classical Regime (RCR) or from the Quantum Disordered Regime (QDR), or directly tends to the fixed point T 0 when coming from the Quantum Critical Regime (QCR), we directly reach the Néel line.

As noted before (25) we use the conventional notation of Chakravarty et al. and Chubukov et al. [1] for it allows one to make a reasoning with the simplest lattice unit cell of spacing a (instead of 2a in the more general case). Thus we reduce the physical discussion to the particular case G=G'. x and Δ i on the other hand, we have ρ i >>T and Δ i >>T, respectively. Expanding u i, ±1 in the variable k i

x we obtain after complicated calculations:

( ) { , 1 1 1 1 2 2 k i i i i i i i i i J J u z x J z z J Ê + =- - + È Î Á Ë ¯ ∓ ± ( ) ( ) } 8 1
.exp 1/2 ...

k k i i x x e + - - + ˙ π , k i x < 1, k = 1, 2. ( 33
)
For both regimes 1 i x <<1 and 2 i x <<1 a short numerical study shows that the exponentiel term is dominating over the term 2 k i x so that:

( ) { , 1 1 1 1 2 i i i i i i i i J J u z J z z J Ê + =- Á Ë ¯ ∓ ± ( ) ( ) } 8 1 .exp 1/ 2 ... i i k k x x e - - - + π , i = 1, 2, k = 1, 2. ( 34
)
In the Quantum Critical Regime (QCR, 1 i x >1 and 2 i

x >1), we now have 0 1 z z -<< . As a result, due to ( 26) and

(32), the expansion must be in 1/(4π k i x ). We have

( ) { , 1 1 1 1 2 2 i i i i i i i i J J u z J z z J Ê + =- Á Ë ¯ ∓ ± π 2 2 1 1 2 ... 4 4 i i k k x x Ȩ ˆÊ ˆÔ + - + Á ˜Á Ë ¯Ë ¯Ô ˛ π π π i = 1, 2, k = 1, 2. ( 35
)
Using again the relation between 1 i

x and ρ i on the one hand, 2 i

x and Δ i on the other hand, we now have ρ i <<T and Δ i <<1, respectively. Reporting these various expressions in the definitions of the correlation length and the susceptibility respectively given by ( 7) and ( 6) we can write:

2 J a ξ ª β , Néel line 1 x << 1, (QCR) 1 x > 1, ( 36a
) Ch Ch 2 2 1 e x p 2 8 2 4 e T a T Ê Ê â + Á ˜Á Ë ¯Ë ξ π ρ π π ρ , (RCR) 1 x < 1. (36b)
where ρ Ch represents the spin stiffness of the model proposed by Chakravarty et al. and Chubukov et al. [1]. The correspondence with our model is ρ=2ρ Ch (i.e., J=2J Ch ). It is well-know that, near T C =0 K, the product χT behaves as ξ x ξ y M 2 where M is the magnetic moment per unit cell. In our case, as we restrict our study to J=J 1 =J 2 , we have ξ=ξ x =ξ y where the correlation length is given by (36). Thus, for 2d non-compensated moments i.e., for ferromagnetic (F) or antiferromagnetic (AF) couplings, χT diverges as ξ 2 :

( ) 2

T J χ ª β , Néel line 1

x << 1, (QCR) 1

x > 1, M(T) ≠ 0, J > 0 (AF couplings), J < 0 (F couplings), (37a) x < 1, M(T) ≠ 0, J > 0 (AF couplings), J < 0 (F couplings). (37b)

In the compensated case the product χT may be written as: These results are in perfect agreement with the corresponding ones obtained by the renormalization group method [1].

In order to give an illustration of the theoretical model developed in this paper we have fitted the experimental susceptibility measured for a powder sample concerning the compound [{Mn(DENA) 2 (N 3 ) 2 } n ] because the corresponding Néel temperature T N =2.0 K is very low (the ligand DENA stands for the group diethylnicotinamide). This compound is characterized by 2d square lattices of classical spins (the manganese ions) isotropically coupled (i.e., showing Heisenberg couplings). That means that the 2d magnetic behavior occurs in the low-temperature region. The corresponding fit of the product χT is reported Fig. 3.

[{Mn(DENA) 2 (N 3 ) 2 } n ] is characterized by antiferromagnetic couplings (i.e., compensated magnetic moments). The experimental Landé factor G=1.98 μ B / is very close to the theoretical value 2. Similarly, the exchange energy J/k B =4.15 K is in agreement with tabulated experimental values previously obtained.

VFigure 1

 1 Figure 1 Variations of |ζ| for various values of k B T/|J|.

Figure 2

 2 Figure 2 Diagram of magnetic phases near T 0 = T C = 0 K.

  ρ i on the one hand, and between 2 i

  ) → 0, as T → 0 K J > 0 (AF couplings). (38)

Figure 3

 3 Figure 3 Theoretical fit of the experimental susceptibility for [{Mn(DENA) 2 (N 3 ) 2 } n ] (the ligand DENA stands for the group diethylnicotinamide).