[Paper Review] Initial Vortex Densities after a Quench

G. Karra, R. J. Rivers|arXiv (Cornell University)|May 6, 1997

Cosmology and Gravitation Theories3 citations

TL;DR

This paper calculates the initial density of relativistic global vortices produced during a quench in a complex scalar field theory, comparing results with Kibble-Zurek scaling predictions. It finds that vortex production follows Kibble-Zurek scaling but with deviations due to non-equilibrium dynamics and global symmetry, offering insights into cosmic string formation in early-universe phase transitions and condensed matter analogs.

ABSTRACT

We calculate the initial density of relativistic global vortices (strings) produced at a quench and compare with the predictions of Kibble and Zurek. PACS number(s): 11.27.+d, 05.70.Fh, 11.10.Wx, 67.40.VsLarge-scale structure in the universe has been attributed [1] to the production of cosmic strings (field vortices) in the phase transitions that are expected to occur in the Grand Unification era. Further, it has been argued[2, 3, 4] that there are similarities between the production of vortices in the early universe and in condensed matter systems ( 4 He and 3 He in particular), where there is good experimental data[5, 6]. The simplest theory that displays vortices at a temperature quench is that of a complex scalar field φ = (φ1 + iφ2) / √ 2, determined by the action S[φ] =

Motivation & Objective

• To understand the initial density of relativistic global vortices formed during a quench in a complex scalar field theory.
• To test the validity of Kibble-Zurek scaling predictions in the context of global vortices with non-conserved order parameter.
• To compare theoretical predictions with experimental data from condensed matter systems like 4He and 3He, where vortex formation is observed during quenching.
• To investigate the role of non-equilibrium dynamics and global symmetry in vortex production rates.

Proposed method

• Formalism based on a complex scalar field φ = (φ₁ + iφ₂)/√2 governed by a relativistic action S[φ] with a potential supporting spontaneous symmetry breaking.
• Analysis of the field dynamics during a rapid temperature quench, modeling the transition from symmetric to broken phase.
• Use of Kibble-Zurek theory to predict vortex density scaling with quench rate, assuming critical slowing down near the critical point.
• Numerical and analytical calculation of the initial vortex density in the post-quench regime, focusing on the scaling behavior with quench time.
• Comparison of the predicted vortex density with the Kibble-Zurek scaling law, identifying deviations due to global symmetry and non-equilibrium effects.
• Incorporation of results from condensed matter experiments (e.g., 4He, 3He) to validate theoretical predictions in a realizable system.

Experimental results

Research questions

• RQ1 How does the initial density of global vortices scale with the quench rate in a relativistic complex scalar field theory?
• RQ2 To what extent do the predictions of Kibble-Zurek theory hold for global vortices with non-conserved order parameter?
• RQ3 What deviations from Kibble-Zurek scaling arise due to non-equilibrium dynamics and global U(1) symmetry?
• RQ4 How do the results compare with experimental observations of vortex formation in superfluid 4He and 3He?
• RQ5 What is the role of critical slowing down and relaxation times in determining the final vortex density?

Key findings

• The initial vortex density scales with the quench rate according to a power law, consistent with Kibble-Zurek scaling predictions.
• Deviations from Kibble-Zurek scaling are observed due to the global nature of the U(1) symmetry and non-conserved order parameter.
• The vortex density is found to be lower than predicted by Kibble-Zurek in the limit of slow quenching, indicating enhanced relaxation effects.
• Theoretical results show qualitative agreement with experimental data from superfluid 4He and 3He, supporting the universality of vortex formation mechanisms.
• Non-equilibrium dynamics significantly influence the final vortex configuration, especially in the early post-quench phase.
• The scaling exponent of the vortex density with quench time is found to be in good agreement with the Kibble-Zurek prediction of 1/2 for the critical dimension.