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.
Highlights
The modelThe field Φ being N × N hermitian matrix, Pα momenta and R the curvature of htr space projected onto Z = 0 section
Values at different points in space and where periodic non-uniform magnetisation patterns appear [18,19,20]
We present the phase diagram for matrices of size N = 24 and results for infinite matrix size limit of disordered to ordered phase transition line when the interaction with curvature is turned off
Summary
The field Φ being N × N hermitian matrix, Pα momenta and R the curvature of htr space projected onto Z = 0 section. We modeled peaks with triangular distribution of width w and took w/(2 6) as a measure of uncertainty of their position, which gives 65% confidence interval. In the absence of kinetic and curvature terms, it is possible to simplify the integration over hermitian matrices in (2.10), leaving only computationally much cheaper integration over eigenvalues. The pure potential (PP) model, with only mass and quartic term, exhibits the phase for c2 < 0 and a 3rd order phase transition between and ↑↓ phases for large enough c2 > 0. In the limit of negligible kinetic term, a diagonal solution exists that combines the effects of the curvature and the potential provided that. We here concentrate mostly on the model without curvature, while the detailed investigation of curvature effects is pending
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