We study O(N ) models with power-law interactions by renormalization group (RG) methods: when the wave function renormalization is not present or not field dependent, their critical exponents can be computed from the ones of the corresponding short-range O(N ) models at an effective fractional dimension. Explicit results in 2 and 3 dimensions are given for the exponent ν. We propose an improved RG to describe the full theory space of the models where both short-range and long-range interactions are present and competing, and no a priori choice among the two in the RG flow is done: the eigenvalue spectrum of the full theory for all possible fixed points is drawn and the effective dimension shown to be only approximate. A full description of the fixed points structure is given, including multicritical long-range universality classes. PACS numbers: 11.10.Hi, 05.70.Fh, 11.10.Kk Preprint: CP 3 -Origins-2014-22 DNRF90 and DIAS-2014-22 O(N ) models are celebrated and tireless workhorses of statistical mechanics and play a key role in the field of critical phenomena: from one side the interest for their properties motivated the developments of numerous -analytical and numerical -techniques, from the other side they are concretely used as a test ground to benchmark the validity of new techniques for critical phenomena and lattice models.Among the interactions studied in the context of O(N ) models an important and paradigmatic role is played by long-range (LR) interactions, having the form of power-law decaying couplings.A first reason is that the results can be contrasted with the findings obtained for short-range (SR) interactions, to explore how universal and non-universal quantities change increasing the range where S i denote a unit vector with N components in the site i of a lattice in dimension d, J is a coupling energy and d + σ is the exponent of the power-law decay (we refer in the following to cubic lattices). When σ ≤ 0 a diverging energy density is obtained and to well define the thermodynamic limit it is necessary to rescale the coupling constant J [5]. When σ > 0 the model may have a phase transition of the second order, in particular as a function of the parameter σ three different regimes occur [6,7]: (i) for σ ≤ d/2 the mean-field approximation is valid even at the critical point; (ii) for σ greater than a critical value, σ * , the model has the same critical behaviour of the SR model (formally, the SR model is obtained in the limit σ → ∞); (iii) for d/2 < σ ≤ σ * the system exhibits peculiar LR critical exponents. For the Ising model in d = 1[2-4] the value σ * = 1 is found, and for σ = σ * a phase transition of the Berezinskii-KosterlitzThouless universality class occur [8][9][10] (see more references in [11]). Many efforts have been devoted to the determination of σ * and to the characterization of the universality classes in the region d/2 < σ ≤ σ * for general N in dimension d ≥ 2, which is the case we are going to consider in this paper. In the classical paper [6] the expression η = 2 − σ...

Several recent experiments in atomic, molecular and optical systems motivated a huge interest in the study of quantum long-range systems. Our goal in this paper is to present a general description of their critical behavior and phases, devising a treatment valid in d dimensions, with an exponent d+σ for the power-law decay of the couplings in the presence of an O(N ) symmetry. By introducing a convenient ansatz for the effective action, we determine the phase diagram for the N -component quantum rotor model with long-range interactions, with N = 1 corresponding to the Ising model. The phase diagram in the σ − d plane shows a non trivial dependence on σ. As a consequence of the fact that the model is quantum, the correlation functions are anisotropic in the spatial and time coordinates for σ smaller than a critical value and in this region the isotropy is not restored even at criticality. Results for the correlation length exponent ν, the dynamical critical exponent z and a comparison with numerical findings for them are presented. arXiv:1704.00528v2 [cond-mat.quant-gas]

We compute critical exponents of O(N )-models in fractional dimensions between two and four, and for continuos values of the number of field components N , in this way completing the RG classification of universality classes for these models. In d = 2 the N -dependence of the correlation length critical exponent gives us the last piece of information needed to establish a RG derivation of the Mermin-Wagner theorem. We also report critical exponents for multi-critical universality classes in the cases N ≥ 2 and N = 0. Finally, in the large-N limit our critical exponents correctly approach those of the spherical model, allowing us to set N ∼ 100 as threshold for the quantitative validity of leading order large-N estimates.

In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/r 1+σ , where r is the distance length between distinct sites. We introduce and test an order N Monte Carlo algorithm and we determine as a function of σ the critical value Cc at which percolation occurs. The critical exponents in the range 0 < σ < 1 are reported and compared with mean-field and ε-expansion results. Our analysis is in agreement, up to a numerical precision ≈ 10 −3 , with the mean field result for the anomalous dimension η = 2 − σ, showing that there is no correction to η due to correlation effects.