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.

Dynamical quantum phase transitions hold a deep connection to the underlying equilibrium physics of the quench Hamiltonian. In a recent study [J. C. Halimeh et al., arXiv:1810.07187], it has been numerically demonstrated that the appearance of anomalous cusps in the Loschmidt return rate coincides with the presence of bound domain walls in the spectrum of the quench Hamiltonian. Here, we consider transverse-field Ising chains with power-law and exponentially decaying interactions, and show that by removing domain-wall coupling via a truncated Jordan-Wigner transformation onto a Kitaev chain with long-range hopping and pairing, anomalous dynamical criticality is no longer present. This indicates that bound domain walls are necessary for anomalous cusps to appear in the Loschmidt return rate. We also calculate the dynamical phase diagram of the Kitaev chain with long-range hopping and pairing, which in the case of power-law couplings is shown to exhibit rich dynamical criticality including a doubly critical dynamical phase. arXiv:1902.08621v1 [cond-mat.stat-mech]

Slow variations (quenches) of the magnetic field across the paramagnetic-ferromagnetic phase transition of spin systems produce heat. In systems with short-range interactions the heat exhibits universal power-law scaling as a function of the quench rate, known as Kibble-Zurek scaling. In this work we analyze slow quenches of the magnetic field in the Lipkin-Meshkov-Glick (LMG) model, which describes fully connected quantum spins. We analytically determine the quantum contribution to the residual heat as a function of the quench rate δ by means of a Holstein-Primakoff expansion about the mean-field value. Unlike in the case of short-range interactions, scaling laws in the LMG model are only found for a ramp ending at the critical point. If instead the ramp is symmetric, as in the typical Kibble-Zurek scenario, after crossing the critical point the system tends to reabsorb the defects formed during the first part of the ramp: the number of excitations exhibits a crossover behavior as a function of δ and tends to a constant in the thermodynamic limit. Previous, and seemingly contradictory, theoretical studies are identified as specific limits of this dynamics. Our results can be tested on several experimental platforms, including quantum gases and trapped ions.

We consider anisotropic long-range interacting spin systems in d dimensions. The interaction between the spins decays with the distance as a power law with different exponents in different directions: we consider an exponent d1 + σ1 in d1 directions and another exponent d2 + σ2 in the remaining d2 ≡ d − d1 ones. We introduce a low energy effective action with non analytic power of the momenta. As a function of the two exponents σ1 and σ2 we show the system to have three different regimes, two where it is actually anisotropic and one where the isotropy is finally restored. We determine the phase diagram and provide estimates of the critical exponents as a function of the parameters of the system, in particular considering the case of one of the two σ's fixed and the other varying. A discussion of the physical relevance of our results is also presented.

The study of critical properties of systems with long-range interactions has attracted in the last decades a continuing interest and motivated the development of several analytical and numerical techniques, in particular in connection with spin models. From the point of view of the investigation of their criticality, a special role is played by systems in which the interactions are long-range enough that their universality class is different from the short-range case and, nevertheless, they maintain the extensivity of thermodynamical quantities. Such interactions are often called weak long-range. In this paper we focus on the study of the critical behaviour of spin systems with weak-long range couplings using renormalization group, and we review their remarkable properties. For the sake of clarity and self-consistency, we start from the classical O(N ) spin models and we then move to quantum spin systems. I. INTRODUCTIONIn this Special Issue several aspects of the equilibrium and dynamical properties of systems with long-range (LR) interactions are discussed. In the paper we apply concepts and tools developed for LR systems, including the modern renormalization group approach, to the study of spin models with weak LR interactions. Such interactions are able to modify the universal properties of the systems in which they act, but anyway preserve the extensivity of the thermodynamic quantities. Our main goal is to present and apply the formalism of the functional renormalization group (FRG) to weak LR systems and show the capability of the FRG to clarify their critical properties, which are in many cases still unknown and sometimes diffucult to obtain with other approaches.Given the wide framework already presented in the other contributions to this Special Issue, it is not necessary to extensively emphasize that LR interactions play an important and paradigmatic role in the study of many body interacting systems, such as O(N ) symmetric models. Indeed, the importance of LR interactions is motivated by their presence in several systems ranging from plasma physics to astrophysics and cosmology [1,2].In order to understand the typical phenomena occurring in spin models with power-law LR couplings and define the weak LR regime, let us first consider the classical O(N ) symmetric models, whose Hamiltonian reads(1)The spin variables S i are unit vectors with N components, placed at the sites, labeled by the index i, of a d dimensional lattice. The coupling constant J is constant for a decay exponent σ > 0, conversely for σ ≤ 0 the coupling constant J needs to be rescaled by an appropriate power of the system size to absorb the divergence of the interaction energy density the thermodynamic limit [1,2]. When σ > 0 the model may have a second order phase transition. The main result is that three different regimes can occur as a function of the parameter σ [3, 4]:• for σ ≤ d/2 the mean-field approximation correctly describes the universal behavior;• for σ greater than a threshold value, σ * , the model has the same critical...