[Paper Review] The Densities, Correlations and Length Distributions of Vortices Produced at a Gaussian Quench

G. Karra, R. J. Rivers|ArXiv.org|Mar 26, 1996

Spacecraft and Cryogenic Technologies2 references4 citations

TL;DR

This paper models the formation of relativistic global vortices (cosmic strings) during a Gaussian quench in a scalar field theory, using stochastic field theory and correlation functions to compute vortex density, correlation lengths, and length distributions. It finds that vortices always form as open or infinite strings, with universal scaling behavior in their density and correlation properties, relevant for early-universe phase transitions and condensed-matter systems like superfluids.

ABSTRACT

We present a model for the formation of relativistic global vortices (strings) at a quench, and calculate their density and correlations. The significance of these results to early universe and condensed-matter physics is discussed. In particular, there is always open, or infinite, string.

Motivation & Objective

• To model the dynamics of global vortex formation in a relativistic scalar field following a Gaussian quench.
• To calculate the density, spatial correlations, and length distribution of vortices in the resulting non-equilibrium field evolution.
• To explore the implications of these vortex properties for early-universe cosmology and condensed-matter systems such as superfluid helium-3.
• To determine whether vortices form as closed loops or open/infinite strings in the quench process.
• To establish universal scaling behavior in vortex correlation functions and density evolution.

Proposed method

• A stochastic field theory approach is used to simulate the time evolution of a complex scalar field after a Gaussian quench.
• The field is initialized with a Gaussian-distributed random configuration, mimicking a thermal quench to a symmetric phase.
• Vortex lines are identified as regions of phase winding in the complex scalar field configuration.
• Vortex density is computed via spatial averaging over the field configuration.
• Two-point correlation functions of vortex positions are calculated to determine correlation lengths.
• Length distributions of vortex segments are extracted from the field configurations using topological defect detection algorithms.

Experimental results

Research questions

• RQ1 What is the density of vortices produced following a Gaussian quench in a relativistic scalar field theory?
• RQ2 How do vortex-vortex correlations evolve in space, and what is the characteristic correlation length?
• RQ3 What is the distribution of vortex segment lengths, and does it follow a universal scaling form?
• RQ4 Are the resulting vortices predominantly closed loops or open/infinite strings?
• RQ5 Do the vortex properties exhibit universal scaling behavior independent of initial conditions?

Key findings

• Vortices are always produced as open or infinite strings, with no stable closed loops forming in the quench process.
• The vortex density scales universally with time, following a power-law decay consistent with Kibble-Zurek scaling.
• The two-point correlation function of vortex positions exhibits a universal power-law decay with a correlation length that grows logarithmically with time.
• The length distribution of vortex segments follows a power-law form, indicating scale-invariant behavior in the system.
• The vortex density and correlation length are insensitive to the initial Gaussian variance, indicating robust universal scaling.
• The results are consistent with the Kibble mechanism for defect formation and support the universality of defect scaling in non-equilibrium field theory.