Triple point of a scalar field theory on a fuzzy sphere

Samuel Kováčik,Denjoe O’Connor

doi:10.1007/jhep10(2018)010

Abstract

The model of a scalar field with quartic self-interaction on the fuzzy sphere has three known phases: a uniformly ordered phase, a disordered phase and a non-uniformly ordered phase, the last of which has no classical counterpart. These three phases are expected to meet at a triple point. By studying the infinite matrix size limit, we locate the position of this triple point to within a small triangle in terms of the parameters of the model. We find the triple point is closer to the coordinate origin of the phase diagram than previous estimates but broadly consistent with recent analytic predictions.

Highlights

A real scalar field theory with φ4 potential on a sphere of radius R is described by the action
We find the triple point is closer to the coordinate origin of the phase diagram than previous estimates but broadly consistent with recent analytic predictions
We have refined the position of the triple point of a scalar field theory on a fuzzy sphere by using hybrid Monte Carlo simulations and extrapolating to the N → ∞ limit using matrices of size up to N = 127

Summary

The fuzzy φ4 model

A real scalar field theory with φ4 potential on a (commutative) sphere of radius R is described by the action. In th√e large N limit the pure potential model has a third order phase transition, at bc = −2 N c, to a phase where the eigenvalues are distributed between both minima of the potential and do not jump the barrier. Due to the repulsion between the eigenvalues discussed above, whenbc2 < ̃b < ̃bc and cis sufficiently large, there is an additional matrix phase, which lies between the disordered and the uniform phase. It has no classical counterpart in the commutative model and is characterised by some eigenvalues in one well and the rest of them in the other, λi = ±iλ0. It is the location of this triple point we seek

Methods and results

Findings

Conclusion