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We perform a Monte Carlo study of the SU(3) non-Abelian Higgs model. We discuss phase structure and non-Abelian vortices by gauge-invariant operators. External magnetic fields induce non-Abelian vortices in the color–flavor-locked phase. The spatial distribution of non-Abelian vortices suggests the repulsive vortex–vortex interaction.

A quantum vortex is a visible topological phenomenon in the quantum world. It is observed in superfluid helium, type-II superconductors, and atomic Bose–Einstein condensates under magnetic fields or rotation [

We can study the semi-classical properties of vortices by solving classical equations of motion or mean-field equations. However, this is insufficient to reveal the full quantum properties of vortices. We need first-principle calculation of quantum field theory, such as a lattice Monte Carlo simulation. Abelian vortices have been generated by Monte Carlo simulation in relativistic theories [

In this work we perform a Monte Carlo study of non-Abelian vortices. Although the most interesting theory is high-density QCD, it suffers from the fermion sign problem. Alternatively, we adopt the non-Abelian Higgs model without the sign problem. This model is a good starting point because it is known as an effective theory of dense quark matters [

Let us consider the

The

The potential term is

Vortices are defined by the winding number

We focus on the

Before analyzing the topological property of this model, let us understand the basic property at vanishing external U(1) gauge fields

The tachyonic mass

Condensate

The scalar-coupling constant

When

Ratio of the off-diagonal component

We introduce external U(1) magnetic fields to generate non-Abelian vortices. We consider a homogeneous magnetic field

A vortex is defined by the counter integral of a phase. One would naively calculate the phase of the scalar field

The expectation value

Vortex number

In

Coordinate dependence of the vortex density

Although the vortex number is

Spatial distribution of the vortex density

Multi-vortex distributions can be obtained in the same manner. At

There are two remarks: The first one is about volume scaling. The obtained results are the superpositions of all vortex states with finite action. Some vortices, known as global vortices, have logarithmically divergent action in the infinite-volume limit [

We studied non-Abelian vortices by Monte Carlo simulation of the non-Abelian Higgs model. We applied external magnetic fields to excite non-Abelian vortices. Similar analysis will be possible by rotating the lattice [

The author was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 15K17624. The numerical simulations were carried out on SX-ACE in Osaka University.

Open Access funding: SCOAP