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Critical exponents of O(N) models in fractional dimensions
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Fixed-point structure and effective fractional dimensionality for O(N) models with long-range interactions
Full text linkWe study, by renormalization group methods, O(N) models with interactions decaying as power law with exponent d + sigma. When only the long-range momentum term p(sigma) is considered in the propagator, the critical exponents can be computed from those of the corresponding short-range O(N) models at an effective fractional dimension D-eff. Neglecting wave function renormalization effects the result for the effective dimension is D-eff = 2d/sigma, which turns to be exact in the spherical model limit (N -> 8). Introducing a running wave function renormalization term the effective dimension becomes instead D-eff = (2-eta SR)d/sigma . The latter result coincides with the one found using standard scaling arguments. Explicit results in two and three dimensions are given for the exponent nu. We propose an improved method to describe the full theory space of the models where both short-and long-range propagator terms are present and no a priori choice among the two in the renormalization group flow is done. The eigenvalue spectrum of the full theory for all possible fixed points is drawn and a full description of the fixed-point structure is given, including multicritical long-range universality classes. The effective dimension is shown to be only approximate, and the resulting error is estimated
Finite temperature off-diagonal long-range order for interacting bosons
Full text linkCharacterizing the scaling with the total particle number (N) of the largest eigenvalue of the one-body density matrix (++0) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting ++0Gê+NC0, then C0=1 corresponds in ODLRO. The intermediate case, 0<1, corresponds in translational invariant systems to the power-law decaying of (nonconnected) correlation functions and it can be seen as identifying quasi-long-range order. The goal of the present paper is to characterize the ODLRO properties encoded in C0 (and in the corresponding quantities CkGëá0 for excited natural orbitals) exhibited by homogeneous interacting bosonic systems at finite temperature for different dimensions in presence of short-range repulsive potentials. We show that CkGëá0=0 in the thermodynamic limit. In one dimension it is C0=0 for nonvanishing temperature, while in three dimensions it is C0=1 (C0=0) for temperatures smaller (larger) than the Bose-Einstein critical temperature. We then focus our attention to D=2, studying the XY and the Villain models, and the weakly interacting Bose gas. The universal value of C0 near the Berezinskii-Kosterlitz-Thouless temperature TBKT is 7/8. The dependence of C0 on temperatures between T=0 (at which C0=1) and TBKT is studied in the different models. An estimate for the (nonperturbative) parameter +¦ entering the equation of state of the two-dimensional Bose gases is obtained using low-temperature expansions and compared with the Monte Carlo result. We finally discuss a "double jump"behavior for C0, and correspondingly of the anomalous dimension ++, right below TBKT in the limit of vanishing interactions
Detecting composite orders in layered models via machine learning
Full text linkDetermining the phase diagram of systems consisting of smaller subsystems 'connected' via a tunable coupling is a challenging task relevant for a variety of physical settings. A general question is whether new phases, not present in the uncoupled limit, may arise. We use machine learning and a suitable quasidistance between different points of the phase diagram to study layered spin models, in which the spin variables constituting each of the uncoupled systems (to which we refer as layers) are coupled to each other via an interlayer coupling. In such systems, in general, composite order parameters involving spins of different layers may emerge as a consequence of the interlayer coupling. We focus on the layered Ising and Ashkin-Teller models as a paradigmatic case study, determining their phase diagram via the application of a machine learning algorithm to the Monte Carlo data. Remarkably our technique is able to correctly characterize all the system phases also in the case of hidden order parameters, i.e. order parameters whose expression in terms of the microscopic configurations would require additional preprocessing of the data fed to the algorithm. We correctly retrieve the three known phases of the Ashkin-Teller model with ferromagnetic couplings, including the phase described by a composite order parameter. For the bilayer and trilayer Ising models the phases we find are only the ferromagnetic and the paramagnetic ones. Within the approach we introduce, owing to the construction of convolutional neural networks, naturally suitable for layered image-like data with arbitrary number of layers, no preprocessing of the Monte Carlo data is needed, also with regard to its spatial structure. The physical meaning of our results is discussed and compared with analytical data, where available. Yet, the method can be used without any a priori knowledge of the phases one seeks to find and can be applied to other models and structures
Berezinskii-Kosterlitz-Thouless Phase Transitions with Long-Range Couplings
Full text linkThe Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature TBKT. In this Letter, we consider the effect of long-range decaying couplings ∼r-2-σ on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features - for 7/4<σ<2 - a quasiordered phase in a finite temperature range TcTBKT. The transition temperature Tc displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems
Self-consistent harmonic approximation in presence of non-local couplings
Full text linkWe derive the self-consistent harmonic approximation for the 2D XY model with non-local interactions. The resulting equation for the variational couplings holds for any form of the spin-spin coupling as well as for any dimension. Our analysis is then specialized to power-law couplings decaying with the distance r as in order to investigate the robustness, at finite σ, of the Berezinskii-Kosterlitz-Thouless (BKT) transition, which occurs in the short-range limit σ → ∞. We propose an ansatz for the functional form of the variational couplings and show that for any σ > 2 the BKT mechanism occurs. The present investigation provides an upper bound σ ∗ = 2 for the critical threshold σ ∗ above which the traditional BKT transition persists in spite of the non-local nature of the couplings
Interplay of spin waves and vortices in the two-dimensional XY model at small vortex-core energy
Full text linkThe Berezinskii-Kosterlitz-Thouless (BKT) mechanism describes universal vortex unbinding in many two-dimensional systems, including the paradigmatic XY model. However, most of these systems present a complex interplay between excitations at different length scales that complicates theoretical calculations of nonuniversal thermodynamic quantities. These difficulties may be overcome by suitably modifying the initial conditions of the BKT flow equations to account for noncritical fluctuations at small length scales. In this work, we perform a systematic study of the validity and limits of this two-step approach by constructing optimised initial conditions for the BKT flow. We find that the two-step approach can accurately reproduce the results of Monte Carlo simulations of the traditional XY model. To systematically study the interplay between vortices and spin-wave excitations, we introduce a modified XY model with increased vortex fugacity. We present large-scale Monte Carlo simulations of the spin stiffness and vortex density for this modified XY model and show that even at large vortex fugacity, vortex unbinding is accurately described by the nonperturbative functional renormalization group
Berezinskii-Kosterlitz-Thouless Phase Transitions with Long-Range Couplings
Full text linkThe Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature TBKT. In this Letter, we consider the effect of long-range decaying couplings ∼r-2-σ on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features - for 7/4<2 - a quasiordered phase in a finite temperature range TcTBKT. The transition temperature Tc displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems
Topological phase transitions in four dimensions
Full text linkISSN:0550-3213ISSN:1873-1562ISSN:1873-156
Self-consistent harmonic approximation in presence of non-local couplings
Full text linkWe derive the self-consistent harmonic approximation for the 2D XY model with non-local interactions. The resulting equation for the variational couplings holds for any form of the spin-spin coupling as well as for any dimension. Our analysis is then specialized to power-law couplings decaying with the distance r as in order to investigate the robustness, at finite σ, of the Berezinskii-Kosterlitz-Thouless (BKT) transition, which occurs in the short-range limit . We propose an ansatz for the functional form of the variational couplings and show that for any the BKT mechanism occurs. The present investigation provides an upper bound for the critical threshold above which the traditional BKT transition persists in spite of the non-local nature of the couplings
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