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Topological defect formation in quantum phase transitions

Edward Gillman

Published May 1, 2018• 0 citations
Publisher

Abstract

The Kibble Zurek mechanism describes the universal formation of topological defects during continuous symmetry breaking phase transitions. It has been confirmed in a wide variety of systems and is of interest within theoretical high energy physics and cosmology. While during high temperature phase transitions classical methods can be used to study topological defect formation, in zero temperature quantum phase transitions, quantum effects can dominate dynamics such that classical approximations fail. This is problematic in quantum field theory because topological defect formation constitutes a non-perturbative non-equilibrium phenomenon, yet there are at present no well-developed non-perturbative non-equilibrium methods available for calculations. Nonetheless, due to its generality and confirmation in other cases, the Kibble-Zurek mechanism is expected to hold. This means that in addition to being of physical interest in its own right, the Kibble-Zurek mechanism is also an excellent test for any potential non-perturbative non-equilibrium techniques. In this thesis, tensor network techniques are applied to the problem of confirming the Kibble Zurek mechanism in the phi^4 quantum field theory in D = (1 + 1) spacetime dimensions. Such techniques have already been highly successful in condensed matter and to some extent in quantum field theory. The kink defects of the theory are studied both in equilibrium and in the non-equilibrium scenario of their formation. Results consistent with the Kibble-Zurek mechanism are found, which provides evidence that the mechanism holds in this case and confirms tensor networks as a promising non-perturbative non-equilibrium method for quantum field theory. As tensor network methods are developed further to higher dimensions and more sophisticated theories, they could one day provide a powerful method for the non-perturbative study of non-equilibrium high energy physics and cosmology, an area of physics which remains essentially unexplored yet important for our understanding of nature.

DOI
10.25560/62633

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The Kibble Zurek mechanism describes the universal formation of topological defects during continuous symmetry breaking phase transitions. It has been confirmed in a wide variety of systems and is of interest within theoretical high energy physics and cosmology. While during high temperature phase transitions classical methods can be used to study topological defect formation, in zero temperature quantum phase transitions, quantum effects can dominate dynamics such that classical approximations fail. This is problematic in quantum field theory because topological defect formation constitutes a non-perturbative non-equilibrium phenomenon, yet there are at present no well-developed non-perturbative non-equilibrium methods available for calculations. Nonetheless, due to its generality and confirmation in other cases, the Kibble-Zurek mechanism is expected to hold. This means that in addition to being of physical interest in its own right, the Kibble-Zurek mechanism is also an excellent test for any potential non-perturbative non-equilibrium techniques. In this thesis, tensor network techniques are applied to the problem of confirming the Kibble Zurek mechanism in the phi^4 quantum field theory in D = (1 + 1) spacetime dimensions. Such techniques have already been highly successful in condensed matter and to some extent in quantum field theory. The kink defects of the theory are studied both in equilibrium and in the non-equilibrium scenario of their formation. Results consistent with the Kibble-Zurek mechanism are found, which provides evidence that the mechanism holds in this case and confirms tensor networks as a promising non-perturbative non-equilibrium method for quantum field theory. As tensor network methods are developed further to higher dimensions and more sophisticated theories, they could one day provide a powerful method for the non-perturbative study of non-equilibrium high energy physics and cosmology, an area of physics which remains essentially unexplored yet important for our understanding of nature.
The Kibble Zurek mechanism in a relativistic ${\ensuremath{\phi}}^{4}$ scalar field theory in D=(1+1)D=(1+1)D=(1+1) is studied using uniform matrix product states. The equal time two point function in momentum space G2(k){G}_{2}(k)G2​(k) is approximated as the system is driven through a quantum phase transition at a variety of different quench rates ${\ensuremath{\tau}}_{Q}$. We focus on looking for signatures of topological defect formation in the system and demonstrate the consistency of the picture that the two point function G2(k){G}_{2}(k)G2​(k) displays two characteristic scales, the defect density nnn and the kink width dK{d}_{K}dK​. Consequently, G2(k){G}_{2}(k)G2​(k) provides a clear signature for the formation of defects and a well defined measure of the defect density in the system. These results provide a benchmark for the use of tensor networks as powerful nonperturbative nonequilibrium methods for relativistic quantum field theory, providing a promising technique for the future study of high energy physics and cosmology.
Systems passing through quantum critical points at finite rates have a finite probability of undergoing transitions between different eigenstates of the instantaneous Hamiltonian. This mechanism was proposed by Kibble as the underlying mechanism for the formation of topological defects in the early universe and by Zurek for condensed matter systems. Here, we use a system of nuclear spins as an experimental quantum simulator undergoing a non-equilibrium quantum phase transition. The experimental data confirm the validity of the Kibble–Zurek mechanism of defect formation.
The Kibble-Zurek mechanism (KZM) describes the nonequilibrium dynamics and topological defect formation in systems undergoing second-order phase transitions. KZM has found applications in fields such as cosmology and condensed matter physics. However, it is generally not suitable for describing first-order phase transitions. It has been demonstrated that transitions in systems like superconductors or charged superfluids, typically classified as second order, can exhibit weakly first-order characteristics when the influence of fluctuations is taken into account. Moreover, the order of the phase transition (i.e., the extent to which it becomes first rather than second order) can be tuned. We explore quench-induced formation of topological defects in such tunable phase transitions and propose that their density can be predicted by combining KZM with nucleation theory.
Abstract We study the dynamics of second-order phase transitions that may have taken place in the early universe and analyze the Kibble-Zurek (KZ) scaling as well as the formation of topological defects emerging from a quench in the one-dimensional Bose-Hubbard model. We describe the universal dynamics of the KZ mechanism (KZM) and the topological defect formation by simulating the Bose–Hubbard model at zero temperature. We study the effects of thermalization on the ground state by analyzing the transition between Mott and superfluid phases. The out-of-equilibrium dynamics induced by a quench of the hopping parameter is presented. We find a KZ scaling behaviour at zero temperature and analyze the KZ dynamics.
In the course of a nonequilibrium continuous phase transition, the dynamics ceases to be adiabatic in the vicinity of the critical point as a result of the critical slowing down (the divergence of the relaxation time in the neighborhood of the critical point). This enforces a local choice of the broken symmetry and can lead to the formation of topological defects. The Kibble–Zurek mechanism (KZM) was developed to describe the associated nonequilibrium dynamics and to estimate the density of defects as a function of the quench rate through the transition. During recent years, several new experiments investigated the formation of defects in phase transitions induced by a quench both in classical and quantum mechanical systems. At the same time, some established results were called into question. We review and analyze the Kibble–Zurek mechanism focusing in particular on this surge of activity, and suggest possible directions for further progress.