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Trasição de fase topológica no modelo de Heisenberg anisotrópico em duas dimensões

Tatiana Pena Figueiredo

Published Sep 28, 2017• 0 citations
OpenAlex

Abstract

The importance of topological excitations in physics, mainly in the description of phase transitions, has a long history beginning with the superconductivity phenomenon in the rst years of the last century. However, the ideas behind this new concept has been more accurately described by Kosterlitz and Thouless, about 40 years ago. Conductor-superconductor, uidsuperuid, rough transitions and many others are characterized as being due the appearance of topological excitations in the system. Another concept of great importance is that the universality in phase transitions . Close to a phase transition the correlations of the system become innity, so that, the details of the model become irrelevant. In this situation, systems originaly so dierent as superconductors and ferromagnets, can be described by the same theory. The critical expoents, that caracterize the transition, depend only on the system dimension, the range of the potential and of their symmetries. This work is dedicated to study of the anisotropic Heisenberg model in two dimensions. This model has a non-usual phase transition, with quasi-long range order, characterized by a change in the behavior of the spin-spin correlation function C(r). In low temperature, T < TBKT, C(r) decays as a power law with distance, r. For temperatures greater than TBKT the correlation function falls o exponencially. The free energy has all the continuous derivatives, for this reason the transition is known as innity order transition. The BKT teory assume that the transition is driven by a vortice-antivortice unbiding mechanism in the system. In our study we did extensive numerical simulations with the tecnique known as Replica Exchange Wang-Landau. This method allow us to estimate the density of states g(E) of the model. With g(E) we can calculated the relevant termodynamic functions of the system (energy, magnetization, suceptibility and correlations for instance). Using Finite Size Scaling tecniques we have determinade the transition temperature TBKT and the behavior of the correlation function for a large range of temperatures. The calculations were done for various dilutions of non-magnetic sites, p, with p = 0,0;0,20;0,30 e 0,35, that allowed us to describe with great precision the critical behavior of the model in particular the correlation lentgh

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In this paper, we use the Replica-Exchange Wang–Landau (REWL) technique to study the behavior of the two-point correlation function of the site diluted classical anisotropic Heisenberg (AH) model in two dimensions. The model has a topological Berezinskii–Kosterlitz–Thouless (BKT) transition for any dilution down to the percolation threshold, [Formula: see text]. Contrary to earlier results that predict the exponent of the correlation function, [Formula: see text], to decay linearly to zero as a function of temperature, we found that it decreases with temperature to a finite value [Formula: see text] at [Formula: see text] for any dilution [Formula: see text].

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In this thesis we study spin systems in low dimension at zero temperature, analyzing their quantum phase transitions. In particular, we used a variational method and a linear spin wave theory for obtaining the approximate ground state. Firstly, we studied the order-disorder transitions of the quasi-one-dimensional isotropic antiferromagneticHeisenberg model of spin-1=2, dened on a square lattice with exchange interactions between nearest neighbors, J and J0. The latter interaction is responsible for onedimensional or two-dimensional behavior of the model namely, when it is zero and when it has a nonzero value, respectively. We obtained the staggered magnetizations of the antiferromagnetic and collinear antiferromagnetic phases. In the absence of exchange interactions between second neighbors, the collinear antiferromagnetic phase is obtained for J0 < 0. This model in the one-dimensional limit has a critical value for the ratio J0=J, where we have a phase transition of the order-disorder type. We founda critical value J0=J = 0, which corresponds to the value physically expected, and compare our results with the linear spin wave theory, that predicts a nonzero critical value. The variational method is also used in the study of the Heisenberg anisotropic model with frustration induced by the inclusion of exchange interactions between second neighbors J2. We renamed the exchange interactions of the rst neighbors J1 e J0 1 , so this model is known as J1 J0 1 J2. Furthermore, we include in this modelthe exchange anisotropy which follows the x and y components of the spin operators. Thus, the more general Hamiltonian presents the isotropic Heisenberg and Ising model limits for certain values of the anisotropy. With the goal to analyze the real role played by this anisotropy, we distinguish between rst and seconds neighbors by the labels 1and 2, respectively. We analyzed the staggered magnetization and the energies for the antiferromagnetic, collinear antiferromagnetic and quantum paramagnetic phase. From these analyzes, we observed quantum phase transitions for certain values of the exchange anisotropy and frustration parameter, the latter one given by the ratio J2=J1. The global phase diagram was also obtained for the cases where the rst and second neighbor exchange anisotropies are equal and where they dier from each other. In the rst case, we can observe the ordered phases, Néel and collinear antiferromagnetic, and between them a magnetically disordered phase. We checked the existence of a critical endpoint separating the three phases. In the second case, we note that the exchange anisotropy inuences the region of the disordered phase, aecting even the existence of the critical endpoint. Besides presenting, at certain intervals of the anisotropy, a behavior that was not expected a priori. We also analyze the quasi-one-dimensional behavior J01 ! 0, where the shape of the phase diagram is unchanged. We explore the degree of dimer formation via the dimerization parameter for some values of exchange anisotropy and obtain a higher degree of dimerization for the disordered phase in comparison with the ordered phases, as expected. We end this work with a generalization of the rst study without frustration in function only of the exchange anisotropy. As a challenge to the variational method, we study the more general J1 J0 1 J2 modelfor integer spin-1 without the exchange anisotropy and avoiding the unidimensional case. Dierent from what happens in the semi-integer spins case, we did not nd the disordered phase for integer spin S= 1. We note only a rst-order phase transition between the ordered phases. Finally, we return to the frustrated model with exchange anisotropies 1 and 2 to study the model from a perspective of linear spin wavetheory, in order to compare the results with those obtained via the variational method. A detailed analysis was performed based on the comparison between the two methods, which agreed qualitatively in all explored boundaries and even those behaviors that were not expected from the variational method were conrmed by the spin wave theory.This consistency of the results allowed us to get a dierent view of the role played by the exchange anisotropy when dierentiated for rst and second neighbors. If the anisotropies are considered equal they work only as a parameter that carries the model from the isotopic Heisenberg limit to the Ising model. However, when dierent, they promote disorder in the system.

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The importance of topological excitations in physics, mainly in the description of phase transitions, has a long history beginning with the superconductivity phenomenon in the rst years of the last century. However, the ideas behind this new concept has been more accurately described by Kosterlitz and Thouless, about 40 years ago. Conductor-superconductor, uidsuperuid, rough transitions and many others are characterized as being due the appearance of topological excitations in the system. Another concept of great importance is that the universality in phase transitions . Close to a phase transition the correlations of the system become innity, so that, the details of the model become irrelevant. In this situation, systems originaly so dierent as superconductors and ferromagnets, can be described by the same theory. The critical expoents, that caracterize the transition, depend only on the system dimension, the range of the potential and of their symmetries. This work is dedicated to study of the anisotropic Heisenberg model in two dimensions. This model has a non-usual phase transition, with quasi-long range order, characterized by a change in the behavior of the spin-spin correlation function C(r). In low temperature, T < TBKT, C(r) decays as a power law with distance, r. For temperatures greater than TBKT the correlation function falls o exponencially. The free energy has all the continuous derivatives, for this reason the transition is known as innity order transition. The BKT teory assume that the transition is driven by a vortice-antivortice unbiding mechanism in the system. In our study we did extensive numerical simulations with the tecnique known as Replica Exchange Wang-Landau. This method allow us to estimate the density of states g(E) of the model. With g(E) we can calculated the relevant termodynamic functions of the system (energy, magnetization, suceptibility and correlations for instance). Using Finite Size Scaling tecniques we have determinade the transition temperature TBKT and the behavior of the correlation function for a large range of temperatures. The calculations were done for various dilutions of non-magnetic sites, p, with p = 0,0;0,20;0,30 e 0,35, that allowed us to describe with great precision the critical behavior of the model in particular the correlation lentgh
In this paper, we use the Replica-Exchange Wang–Landau (REWL) technique to study the behavior of the two-point correlation function of the site diluted classical anisotropic Heisenberg (AH) model in two dimensions. The model has a topological Berezinskii–Kosterlitz–Thouless (BKT) transition for any dilution down to the percolation threshold, [Formula: see text]. Contrary to earlier results that predict the exponent of the correlation function, [Formula: see text], to decay linearly to zero as a function of temperature, we found that it decreases with temperature to a finite value [Formula: see text] at [Formula: see text] for any dilution [Formula: see text].
The classical Heisenberg model is one of the most fundamental models in statistical and condensed matter physics. Extensive theoretical and numerical studies suggest that, in two dimensions, this model does not exhibit a finite-temperature phase transition but instead manifests asymptotic freedom. However, some research has also proposed the possibility of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition over the years. In this study, we revisit the classical two-dimensional (2D) Heisenberg model through large-scale simulations with linear system sizes up to L=16384L=16384L=16384. Our Monte Carlo data, without any extrapolation, clearly reveal an exponential divergence of the correlation length $\ensuremath{\xi}$ as a function of the inverse temperature $\ensuremath{\beta}$, a hallmark of asymptotic freedom. Moreover, extrapolating $\ensuremath{\xi}$ to the thermodynamic limit in the low-temperature regime achieves close agreement with the three-loop perturbative calculations. We further propose a finite-size scaling (FSS) ansatz for $\ensuremath{\xi}$, demonstrating that the pseudocritical point ${\ensuremath{\beta}}_{L}$ diverges logarithmically with LLL. The thermodynamic and finite-size scaling behaviors of the magnetic susceptibility $\ensuremath{\chi}$ are also investigated and corroborate the predictions of asymptotic freedom. Our paper provides solid evidence for asymptotic freedom in the 2D Heisenberg model and advances understanding of finite-size scaling in such systems.
In this thesis we study spin systems in low dimension at zero temperature, analyzing their quantum phase transitions. In particular, we used a variational method and a linear spin wave theory for obtaining the approximate ground state. Firstly, we studied the order-disorder transitions of the quasi-one-dimensional isotropic antiferromagneticHeisenberg model of spin-1=2, dened on a square lattice with exchange interactions between nearest neighbors, J and J0. The latter interaction is responsible for onedimensional or two-dimensional behavior of the model namely, when it is zero and when it has a nonzero value, respectively. We obtained the staggered magnetizations of the antiferromagnetic and collinear antiferromagnetic phases. In the absence of exchange interactions between second neighbors, the collinear antiferromagnetic phase is obtained for J0 < 0. This model in the one-dimensional limit has a critical value for the ratio J0=J, where we have a phase transition of the order-disorder type. We founda critical value J0=J = 0, which corresponds to the value physically expected, and compare our results with the linear spin wave theory, that predicts a nonzero critical value. The variational method is also used in the study of the Heisenberg anisotropic model with frustration induced by the inclusion of exchange interactions between second neighbors J2. We renamed the exchange interactions of the rst neighbors J1 e J0 1 , so this model is known as J1 J0 1 J2. Furthermore, we include in this modelthe exchange anisotropy which follows the x and y components of the spin operators. Thus, the more general Hamiltonian presents the isotropic Heisenberg and Ising model limits for certain values of the anisotropy. With the goal to analyze the real role played by this anisotropy, we distinguish between rst and seconds neighbors by the labels 1and 2, respectively. We analyzed the staggered magnetization and the energies for the antiferromagnetic, collinear antiferromagnetic and quantum paramagnetic phase. From these analyzes, we observed quantum phase transitions for certain values of the exchange anisotropy and frustration parameter, the latter one given by the ratio J2=J1. The global phase diagram was also obtained for the cases where the rst and second neighbor exchange anisotropies are equal and where they dier from each other. In the rst case, we can observe the ordered phases, Néel and collinear antiferromagnetic, and between them a magnetically disordered phase. We checked the existence of a critical endpoint separating the three phases. In the second case, we note that the exchange anisotropy inuences the region of the disordered phase, aecting even the existence of the critical endpoint. Besides presenting, at certain intervals of the anisotropy, a behavior that was not expected a priori. We also analyze the quasi-one-dimensional behavior J01 ! 0, where the shape of the phase diagram is unchanged. We explore the degree of dimer formation via the dimerization parameter for some values of exchange anisotropy and obtain a higher degree of dimerization for the disordered phase in comparison with the ordered phases, as expected. We end this work with a generalization of the rst study without frustration in function only of the exchange anisotropy. As a challenge to the variational method, we study the more general J1 J0 1 J2 modelfor integer spin-1 without the exchange anisotropy and avoiding the unidimensional case. Dierent from what happens in the semi-integer spins case, we did not nd the disordered phase for integer spin S= 1. We note only a rst-order phase transition between the ordered phases. Finally, we return to the frustrated model with exchange anisotropies 1 and 2 to study the model from a perspective of linear spin wavetheory, in order to compare the results with those obtained via the variational method. A detailed analysis was performed based on the comparison between the two methods, which agreed qualitatively in all explored boundaries and even those behaviors that were not expected from the variational method were conrmed by the spin wave theory.This consistency of the results allowed us to get a dierent view of the role played by the exchange anisotropy when dierentiated for rst and second neighbors. If the anisotropies are considered equal they work only as a parameter that carries the model from the isotopic Heisenberg limit to the Ising model. However, when dierent, they promote disorder in the system.
A noncritical scaling model is reported to describe the behavior of low-dimensional magnetic systems. Unlike the classical phase transition approach, nonsingular solutions are deduced that are worthwhile when correlations exist, but which are not large enough to trigger a long range order at $T\ensuremath{\ne}0$. The notion of ``universality class'' is extended to these systems that stand at or below a lower critical dimensionality, and illustrated for the quantum Heisenberg ferromagnetic chain. In this system, the TTT dependence of $\ensuremath{\chi}T$ exhibits a power law with a negative critical exponent, $\ensuremath{-}1.25S$, and a negative critical temperature.
Frustrated magnetism is a very active research topic in condensed matter physics. The competition between different ground states often leads to a suppression of the ordering temperature and opens up the possibility that novel states emerge. One state of particular interest are the spin liquids, for which we find no broken symmetry down to T —› 0, where T is the temperature. Moreover, theses states show topological order and are notable for their peculiar features, such as long-range entanglement and the emergence of fractionalized excitations. The Kitaev model is a fascinating example of an exactly solvable model displaying a spin-liquid ground state in two dimensions. In real materials, however, deviations from the original Kitaev model are expected to appear, and we discuss relevant perturbations here. In this work, we investigate the fate of Kitaevs spin-liquid in the presence of disorder bond defects or vacancies for an extended version of the model. Considering static flux backgrounds, we observe a power-law divergence in the low-energy limit of the density of states, with a non-universal exponent. We link this power-law distribution of energy scales to weakly coupled droplets inside the bulk, in an uncanny similarity to the Griffiths phase often present in the vicinity of disordered quantum phase transitions. If time-reversal symmetry is broken, we find that power-law singularities are tied to the destruction of the topological phase of the Kitaev model in the presence of bond disorder alone. For weak to moderate site dilution, there is a transition from this topologically trivial phase with power-law singularities to a topologically non-trivial one. Therefore, diluted Kitaev materials are potential candidates to host Kitaevs chiral spin-liquid phase.
Spontaneous symmetry breaking of 2D isotropic Heisenberg magnet is restricted by Mermin-Wagner theorem at any finite temperature in presence of short-range exchange interaction.Kosterlitz and Thouless using XY spin model showed that how an order state could developed in 2D spin system in presence of short range isotropic interaction.Very recent discovery of several van der waals magnet revised and redefined our understanding on 2D Heisenberg magnet and its ground state properties.After a rigorous and careful study of several 2D magnetic material we have realized from both experimentally and numerically that the finite size of a 2D system has great impact on the ground state symmetry breaking.Because of that finite size effect more often an anisotropic residual magnetic moment is generated and trigger the spontaneous symmetry breaking at finite temperature(T) and even only presence of short-range interaction we observed the phase transition of that Heisenberg spin system.In this present work we have shown the basic role of finite size,anisotropy during the symmetry breaking of 2D Heisenberg XY magnet.Here we have simulated Kosterlitz-Thouless transition using classical Monte-carlo simulation and study the effect of anisotropy during the phase transition.We presented the behaviour of different thermodynamic properties of 2D XY spin model system during the Kosterlitz-Thouless(KT) transition.The generic characteristic of KT transition which make it distinct from other critical phenomena is that the peak of heat capacity is not diverging with increase of system size rather peak is decreasing with the increasing of system size near at transition temperature.Here we are observing that behaviour in our present simulation and that specific behaviour help us for classifying the present transition as Kosterlitz-Thouless(KT) transition.