Geometrical and topological study of the Kosterlitz–Thouless phase transition in the XY model in two dimensions

Bel-Hadj-Aissa, Ghofrane; Gori, Matteo; Franzosi, Roberto; Pettini, Marco

doi:10.1088/1742-5468/abda27

Cited by 9 publications

(12 citation statements)

References 50 publications

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“…To the contrary, the set of uncoupled harmonic oscillators is integrable, however, each single harmonic oscillator is ergodic in its own two-dimensional phase space, and, since all the oscillators have the same frequency, so that they are interchangeable, and the initial conditions are random, also this systems behaves as if it was ergodic, as the stability of the results of the computation of the geometric observables has been checked by changing the initial conditions. In fact, given an observable, Φ, defined on the phase space, the microcanonical averages can be measured along the dynamics as follows [23,34]:…”

Section: Numerical Resultsmentioning

confidence: 99%

“…On the other hand, the Landau theory, which relates phase transitions with the symmetry-breaking phenomenon, is not an all-encompassing theory because there are many systems undergoing phase transitions in the absence of an order parameter, and thus in the absence of symmetry-breaking [22,23]. Therefore, looking for generalizations of the existing theories is a well motivated and timely purpose.…”

Section: Introductionmentioning

confidence: 99%

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Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions

Cairano

Gori

Pettini

2021

Entropy

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Different arguments led to supposing that the deep origin of phase transitions has to be identified with suitable topological changes of potential related submanifolds of configuration space of a physical system. An important step forward for this approach was achieved with two theorems stating that, for a wide class of physical systems, phase transitions should necessarily stem from topological changes of energy level submanifolds of the phase space. However, the sufficiency conditions are still a wide open question. In this study, a first important step forward was performed in this direction; in fact, a differential equation was worked out which describes how entropy varies as a function of total energy, and this variation is driven by the total energy dependence of a topology-related quantity of the relevant submanifolds of the phase space. Hence, general conditions can be in principle defined for topology-driven loss of differentiability of the entropy.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions

Cairano

Gori

Pettini

2021

Entropy

Self Cite

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“…That is, a continuous change in the configuration space geometry brought about a divergence in the mixing time, and while such events are often accompanied by topological changes they are not always. The significance of the configuration space geometry is supported by a recent study of the topological and geometric properties of the two-dimensional XY model by Bel-Hadj-Aissa et al [13]; they computed the mean geometric curvature of the equipotential energy level sets and observed that the location of the phase transition could be inferred from the level set curvatures.…”

Section: Introductionmentioning

confidence: 93%

A geometric conjecture about phase transitions

Ericok¹,

Mason²

2022

Preprint

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As phenomena that necessarily emerge from the collective behavior of interacting particles, phase transitions continue to be difficult to predict using statistical thermodynamics. A recent proposal called the topological hypothesis suggests that the existence of a phase transition could perhaps be inferred from changes to the topology of the accessible part of the configuration space. This paper instead suggests that such a topological change is often associated with a dramatic change in the configuration space geometry, and that the geometric change is the actual driver of the phase transition. More precisely, a geometric change that brings about a discontinuity in the mixing time required for an initial probability distribution on the configuration space to reach steady-state is conjectured to be related to the onset of a phase transition in the thermodynamic limit. This conjecture is tested by evaluating the diffusion diameter and -mixing time of the configuration spaces of hard disk and hard sphere systems of increasing size. Explicit geometries are constructed for the configuration spaces of these systems, and numerical evidence suggests that a discontinuity in the -mixing time coincides with the solid-fluid phase transition in the thermodynamic limit.

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“…We evaluated the geometric observables in Eq. ( 9) exploiting the ergodic hypothesis, that is, we converted the microcanonical averages into time averages [18,20]. More precisely, given a generic phase-space-valued function, f : Λ → R, we have…”

Section: Expression Of the Geometric Observablesmentioning

confidence: 99%

The geometric theory of phase transitions

Cairano¹

2022

J. Phys. A: Math. Theor.

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We develop a geometric theory of phase transitions (PTs) for Hamiltonian systems in the microcanonical ensemble. Such a theory allows to rephrase the Bachmann's classification of PTs for finite-size systems in terms of geometric properties of the energy level sets (ELSs) associated to the Hamiltonian function. Specifically, by defining the microcanonical entropy as the logarithm of the ELS’s volume equipped with a suitable metric tensor, we obtain an exact equivalence between thermodynamics and geometry. In fact, we show that any energy-derivative of the entropy can be associated to a specific combination of geometric curvature structures of the ELSs which, in turn, are well-precise combinations of the potential function derivatives. In so doing, we establish a direct connection between the microscopic description provided by the Hamiltonian and the collective behavior which emerges in a PT. Finally, we also analyze the behavior of the ELSs' geometry in the thermodynamic limit showing that nonanalyticities of the energy-derivatives of the entropy are caused by nonanalyticities of certain geometric properties of the ELSs around the transition point. We validate the theory studying the PTs that occur in the $\phi^4$ and Ginzburg-Landau-like models.

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Geometrical and topological study of the Kosterlitz–Thouless phase transition in the XY model in two dimensions

Cited by 9 publications

References 50 publications

Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions

Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions

A geometric conjecture about phase transitions

The geometric theory of phase transitions

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