Detecting scaling in phase transitions on the truncated Heisenberg algebra

Abstract

We construct and analyze a phase diagram of a self-interacting matrix field coupled to curvature of the non-commutative truncated Heisenberg space. The model reduces to the renormalizable Grosse-Wulkenhaar model in an infinite matrix size limit and exhibits a purely non-commutative non-uniformly ordered phase. Particular attention is given to scaling of model’s parameters. We additionally provide the infinite matrix size limit for the disordered to ordered phase transition line.

Second moment fuzzy-field-theory-like matrix models

“…The numerical analysis of these models led to most of the results for the phase structure of the fuzzy field theories. Moreover, recent simulations of the matrix model describing a version of noncommutative field theory that is free of the UV/IR mixing showed signs of retreat of the non-uniform order phase [22,23], further confirming the connection between the existence of this phase and the UV/IR mixing.…”

Beyond second-moment approximation in fuzzy-field-theory-like matrix models

“…The numerical analysis of these models led to most of the results for the phase structure of the fuzzy field theories. Moreover, recent simulations of the matrix model describing a version of noncommutative field theory that is free of the UV/IR mixing showed signs of retreat of the non-uniform order phase [24,25], further confirming the connection between the existence of this phase and the UV/IR mixing.…”

Renormalization footprints in the phase diagram of the Grosse-Wulkenhaar model

“…We here continue inspection of the matrix regularization S N of (3) started in [44]. Let us reintroduce the model and walk through its main features.…”

“…For comparison, a pure potential model would show a →↑↓ transition line c 2 = 2 √ c 4 in the infinite N limit. In [44] we presented a simple argument that, due to diagonality, curvature acts as a quasi-mass term and should shift transition lines proportionally to the c r by δ c 2 relative to the R-off case:…”

“…In [44], we started a numerical comparison of the twodimensional GW-model matrix regularization with (Ron) and without (R-off) the curvature term, focusing mainly on the latter. Here, we wish to present more details on the former and to see how it bares under the oscillator term switching off procedure that ensures the φ 4 -model's renormalizability [12].…”