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arXiv:0711.1085 · Physical Review E

Phase transitions and topology in 2+k XY mean-field models

Luca Angelani, Giancarlo Ruocco

Published Nov 19, 2007• 7 citations
arXivHTMLPDF

Abstract

The thermodynamics and topology of mean-field models with 2+k body interaction terms (generalizing XY model) are derived. Focusing on two particular cases (2+4 and 2+6 body interaction terms), a comparison between thermodynamic (phase transition energy, thermodynamically forbidden energy regions) and topological (singularity and curvature of saddle entropy) properties is performed. We find that (i) a topological change is present at the phase transition energy; however, (ii) only one topological change occurs, also for those models exhibiting two phase transitions; (iii) the order of a phase transition is not completely signaled by the curvature of topological quantities.

DOI
10.1103/physreve.76.051119

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The thermodynamics and topology of mean-field models with 2+k body interaction terms (generalizing XY model) are derived. Focusing on two particular cases (2+4 and 2+6 body interaction terms), a comparison between thermodynamic (phase transition energy, thermodynamically forbidden energy regions) and topological (singularity and curvature of saddle entropy) properties is performed. We find that (i) a topological change is present at the phase transition energy; however, (ii) only one topological change occurs, also for those models exhibiting two phase transitions; (iii) the order of a phase transition is not completely signaled by the curvature of topological quantities.
We argue that the phase transition in the mean-field XY\mathrm{XY}XY model is related to a particular change in the topology of its configuration space. The nature of this topological change can be discussed on the basis of elementary Morse theory using the potential energy per particle VVV as a Morse function. The value of VVV where such a topological change occurs equals the thermodynamic value of VVV at the phase transition and the number of (Morse) critical points grows very fast with the number of particles NNN. Furthermore, as in statistical mechanics, the way the thermodynamic limit is taken is crucial in topology.
A new definition of order called topological order is proposed for two-dimensional systems in which no long-range order of the conventional type exists. The possibility of a phase transition characterized by a change in the response of the system to an external perturbation is discussed in the context of a mean field type of approximation. The critical behaviour found in this model displays very weak singularities. The application of these ideas to the xy model of magnetism, the solid-liquid transition, and the neutral superfluid are discussed. This type of phase transition cannot occur in a superconductor nor in a Heisenberg ferromagnet.
A new definition of order called topological order is proposed for two-dimensional systems in which no long-range order of the conventional type exists. The possibility of a phase transition characterized by a change in the response of the system to an external perturbation is discussed in the context of a mean field type of approximation. The critical behaviour found in this model displays very weak singularities. The application of these ideas to the xy model of magnetism, the solid-liquid transition, and the neutral superfluid are discussed. This type of phase transition cannot occur in a superconductor nor in a Heisenberg ferromagnet. for reasons that are given.
We use a topological approach to describe the frustration- and field-induced phase transitions exhibited by the infinite-range XY model on the AB2 chain, including noncollinear spin structures. For this purpose, we have computed the Morse number and the Euler characteristic, as well as other topological invariants, which are found to behave similarly as a function of the energy level in the context of Morse theory. In particular, we use a method based on an analogy with statistical mechanics to compute the Euler characteristic, which proves to be quite feasible. We also introduce topological energies which help us to clarify several properties of the transitions, both at zero and finite temperatures. In addition, we establish a nontrivial direct connection between the thermodynamics of the systems, which have been solved exactly under the saddle-point approach, and the topology of their configuration space. This connection allows us to identify the nondegeneracy condition under which the divergence of the density of Jacobian's critical points [jl(E)] at the critical energy of a topology-induced phase transition, proposed by Kastner and Schnetz [Phys. Rev. Lett. 100, 160601 (2008)] as a necessary criterion, is suppressed. Finally, our findings and those available in the literature suggest that the cusplike singularity exhibited both by the Euler characteristic and the topological contribution for the entropy at the critical energy, put together with the divergence of jl(E) , and emerge as necessary and sufficient conditions for the occurrence of the finite-temperature topology-induced phase transitions examined in this work. The general character of this proposal should be subject to a more rigorous scrutiny.
Abstract We propose a topological approach suitable to establish a connection between thermodynamics and topology in the microcanonical ensemble. Indeed, we report on results that point to the possibility of describing interacting classical spin systems in the thermodynamic limit, including the occurrence of a phase transition, using topological arguments only. Our approach relies on Morse theory, through the determination of the critical points of the potential energy, which is the proper Morse function. Our main finding is that, in the context of the classical models studied, the Euler characteristic χ ( E ) embeds the necessary features for a correct description of several magnetic thermodynamic quantities of the systems, such as the magnetization, correlation function, susceptibility, and critical temperature. Despite the classical nature of the models, such quantities are those that do not violate the laws of thermodynamics (with the proviso that van der Waals loop states are mean field (MF) artifacts). We also discuss the subtle connection between our approach using the Euler entropy, defined by the logarithm of the modulus of χ ( E ) per site, and that using the Boltzmann microcanonical entropy. Moreover, the results suggest that the loss of regularity in the Morse function is associated with the occurrence of unstable and metastable thermodynamic solutions in the MF case. The reliability of our approach is tested in two exactly soluble systems: the infinite-range and the one-dimensional short-range XY models in the presence of a magnetic field. In particular, we confirm that the topological hypothesis holds for both the infinite-range ( T c ≠ 0 ) and the short-range ( T c = 0) XY models. Further studies are very desirable in order to clarify the extension of the validity of our proposal.
A Hamiltonian dynamics is defined for the XY\mathrm{XY}XY model by adding a kinetic energy term. Thermodynamical properties (total energy, magnetization, vorticity) derived from microcanonical simulations of this model are found to be in agreement with canonical Monte Carlo results in the explored temperature region. The behavior of the magnetization and energy as functions of the temperature are thoroughly investigated, taking into account finite size effects. By representing the spin field as a superposition of random phased waves, we derive a nonlinear dispersion relation whose solutions allow the computation of thermodynamical quantities, which agree quantitatively with those obtained in numerical experiments, up to temperatures close to the transition. At low temperatures the propagation of phonons is the dominant phenomenon, while above the phase transition the system splits into ordered domains separated by interfaces populated by topological defects. In the high temperature phase, spins rotate, and an analogy with an Ising-like system can be established, leading to a theoretical prediction of the critical temperature ${T}_{\mathrm{KT}}\ensuremath{\approx}0.855.$