Exact spectra of periodic samples are computed up to N = 36. Evidence of an extensive set of low-lying levels, lower than the softest magnons, is exhibited. These low-lying quantum states are degenerated in the thermodynamic limit; their symmetries and dynamics as well as their finite-size scaling are strong arguments in favor of Neel order. It is shown that the Neel order parameter agrees with first-order spin-wave calculations. A simple explanation of the low-energy dynamics is given as well as the numerical determinations of the energies, order parameter, and spin susceptibilities of the studied samples. It is shown how suitable boundary conditions, which do not frustrate Neel order, allow the study of samples with N = 3@+1 spins. A thorough study of these situations is done in parallel with the more conventional case N = 3p.

We show how the broken symmetries of the Neel state are embodied in the exact spectrum of the triangular Heisenberg antiferromagnet on finite lattices as small as N = 21 (spectra up to N = 36 have been computed). We present the first numerical evidence of an extensive set of low-lying levels that are below the softest magnons and collapse to the ground state in the thermodynamic limit, This set of quantum states represents the quantum counterpart of the classical Neel ground state.We develop an approach relying on the symmetry analysis and finite-size scaling and we provide new arguments in favor of an ordered ground state for the S = 2 triangular Heisenberg model. PACS numbers: 75.10.Jm, 75.30.Kz, 75.40. Mg In the last decades, a large amount of work has been devoted to the understanding of the quantum ground state of two-dimensional antiferromagnets.In the early seventies, Anderson launched a debate on the possible existence of a "resonating valence bond" (RVB) state which could represent an alternative to the Neel antiferromagnetic state [1]. The first candidate to be considered was the spin-2 Heisenberg antiferromagnet on the triangular lattice:where the sum runs over Erst neighbor pairs. A variational RVB state was proposed to be more stable than the Neel state [2]. From the spin-wave analysis, it was later concluded that quantum fiuctuations were insufficient to destabilize Neel's classical state [3]; perturbative approaches led to the same conclusion and variational ones did not weaken it [4,5]. However, exact results of diagonalization on small periodic samples up to N = 27 were extrapolated and gave the opposite result [6,7]. But in the above numerical studies, the spin-liquid hypothesis was not really explored, nor was the Neel long-range order (NLRO) assumption convincingly discarded. Usually NLRO is checked on the finite-size scaling of the ground-state energy and magnetization [8,9]. The magnon dispersion relation being linear in k, the leading finite-size correction to the ground-state energy per particle F is O(N~) and that for the magnetization modulus per particle M is O(N t 2) (M~i s defined in[10]). Figure 1 shows the values of E~and M~for small periodic samples. We present the results for the erst calculation of the N = 36 sample, a calculation made possible by using all the symmetries of the Hamiltonian and the lattice. We find (2S, S~)ss = -0.3735823(1) and Mss = 0.400575(1). Prom the values, it is clear that the magnetization modulus does not extrapolate to zero in the N~oo limit; but, it is difficult to assert that the finite-size sealing of the ground-state energy behaves as N~. Therefore, no de6nite conclusion can be drawn from the ground-state evaluations on small samples. In this Letter we show how the hypothesis of NLRO implies a list of drastic conditions on the symmetries, dynamics, and the finite-size scaling of an extensive [O(Ns~~)] set of low-lying levels of the spectrum. Some of these conditions are new, others go back to Anderson's seminal paper on antiferromagnets [11] or ...

A group symmetry analysis of the low lying levels of the spin-1/2 kagomé Heisenberg antiferromagnet is performed for small samples up to N = 27.This new approach allows to follow the effect of quantum fluctuations when the sample size increases. The results contradict the scenario of "order by disorder" which has been advanced on the basis of large S calculations. A large enough second neighbor ferromagnetic exchange coupling is needed to stabilize the √ 3 × √ 3 pattern: the finite size analysis indicates a quantum critical transition at a non zero coupling.

We study the exact low energy spectra of the spin 1/2 Heisenberg antiferromagnet on small samples of the kagomé lattice of up to N = 36 sites. In agreement with the conclusions of previous authors, we find that these low energy spectra contradict the hypothesis of Néel type long range order. Certainly, the ground state of this system is a spin liquid, but its properties are rather unusual. The magnetic (∆S = 1) excitations are separated from the ground state by a gap. However, this gap is filled with nonmagnetic (∆S = 0) excitations. In the thermodynamic limit the spectrum of these nonmagnetic excitations will presumably develop into a gapless continuum adjacent to the ground state. Surprisingly, the eigenstates of samples with an odd number of sites, i.e. samples with an unsaturated spin, exhibit symmetries which could support long range chiral order. We do not know if these states will be true thermodynamic states or only metastable ones. In any case, the low energy properties of the spin 1/2 Heisenberg antiferromagnet on the kagomé lattice clearly distinguish this system from either a short range RVB spin liquid or a standard chiral spin liquid. Presumably they are facets of a generically new state of frustrated two-dimensional quantum antiferromagnets.

AMS 2000 subject classifications: Primary 62G05, 62G20; secondary 62E17, 62H30.The stochastic block model (SBM) is a probabilistic model designed to describe heterogeneous directed and undirected graphs. In this paper, we address the asymptotic inference in SBM by use of maximum-likelihood and variational approaches. The identifiability of SBM is proved while asymptotic properties of maximum-likelihood and variational estimators are derived. In particular, the consistency of these estimators is settled for the probability of an edge between two vertices (and for the group proportions at the price of an additional assumption), which is to the best of our knowledge the first result of this type for variational estimators in random graphs

In this paper, we use a new hybrid method to compute the thermodynamic behavior of the spin- 1 / 2 Kagome antiferromagnet under the influence of a large external magnetic field. We find a T2 low-temperature behavior and a very low sensitivity of the specific heat to a strong external magnetic field. We display clear evidence that this low-temperature magnetothermal effect is associated with the existence of low-lying fluctuating singlets, but also that the whole picture ( T2 behavior of C(v) and the thermally activated spin susceptibility) implies contribution of both nonmagnetic and magnetic excitations. Comparison with experiments is made.

Motivated by recent experiments on an S = 1/2 antiferromagnet on the kagomé lattice, we investigate the Heisenberg J1 − J2 model with ferromagnetic J1 and antiferromagnetic J2. Classically the ground state displays Néel long-range order with 12 noncoplanar sublattices. The order parameter has the symmetry of a cuboctahedron, it fully breaks SO(3) as well as the spin flip symmetry, and we expect from the latter a Z2 symmetry breaking pattern. As might be expected from the Mermin-Wagner theorem in two dimensions, the SO(3) symmetry is restored by thermal fluctuations while the Z2 symmetry breaking persists up to a finite temperature. A complete study of S = 1/2 exact spectra reveals that the classical order subsists for quantum spins in a finite range of parameters. First-order spin wave calculations give the range of existence of this phase and the renormalizations at T = 0 of the order parameters associated to both symmetry breakings. This phase is destroyed by quantum fluctuations for a small but finite J2/|J1| ≃ 3, consistently with exact spectra studies, which indicate a gapped phase. I. THEORETICAL AND EXPERIMENTAL ISSUESWhatever the nature of the spin, classical or quantum, the first neighbor Heisenberg antiferromagnet on the kagomé lattice fails to display Néel-like long-range order. Classically, it is characterized by an extensive entropy 1,2 at T = 0. Quantum mechanically the spin-1/2 system has an exceptionally large density of low lying excitations 3,4 reminiscent of the classical extensive entropy. It is still debated whether and eventually how this degeneracy is lifted in the quantum limit 5,6 . An essential issue concerns the influence of perturbations: classically the effect of a second neighbor coupling J 2 has been very early studied by Harris and co-workers 7 . They showed that an infinitesimal J 2 is sufficient to drive the system toward an ordered phase with the three spins around a triangle pointing 120 • from each other. Antiferromagnetic second-neighbor coupling (J 2 > 0) favors the q = 0 Néel order of this pattern on the Bravais lattice, whereas there are nine spins per unit cell for J 2 < 0 (q = √ 3 × √ 3 order). The effect of Dzyaloshinsky-Moriya interactions has also been analyzed 8 . To our knowledge the reduction of the order parameter by quantum fluctuations has only been studied through exact diagonalizations 9 . This approach points to an immediate transition from the "disordered phase" at the pure J 1 > 0 point, to the semiclassical Néel phases.Up until now the J 1 − J 2 model on the kagomé lattice has only been studied for antiferromagnetic J 1 . Many magnetic compounds 10-13 with this geometry have been studied so far, but most of them have spin S = 3/2. A few compounds with S = 1/2 Cu ions have recently been synthetized [14][15][16] . None of them can be described by a pure isotropic first neighbor antiferromagnetic Heisenberg model. Recent experimental work on an organic compound with copper ions on a kagomé lattice 17 gives indication of competing ferromagnetic and antiferrom...

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