Thermal Phase Transition in the Bosonic BFSS Matrix Model

I recently got involved with numerically simulating the BFSS and BMN matrix models as a means to study quantum gravity via holographic dualities. These models are quantum mechanical models with hermitian N x N matrices as degrees of freedom, i.e. 0+1-dimensional quantum field theories.

In its field theory description, the BFSS model arises from dimensionally reducing maximally supersymmetric SU(N) Yang-Mills theory in 9+1 dimensions down to 0+1 dimensions. From string theory, it can be obtained as a description of N D0-branes on flat space, which can form a bound state with the properties of certain black hole solutions in type IIa supergravity in the large N and strong coupling limit.

The BMN model is a massive deformation of the BFSS model that features a more complicated vacuum structure on the field theory side, but better stability of the numerical simulations. On the gravity side, the massive deformation corresponds to changing the flat background into a certain plane wave.

Despite being quantum mechanical models, both the BFSS and BMN models are quite hard to simulate in interesting regimes, as they feature fermions and require the matrix size N to be large enough to accurately probe the large N limit which connects to the description via classical gravity.

An interesting starting point is to consider only the bosonic part of the BFSS model, which arises as the high-temperature limit of 1+1-dimensional supersymmetric Yang-Mills theory. This theory again has a gravity description and is of particular interest to study the transition from black strings to localised black holes, see e.g. this recent paper.  Using this truncation, numerical simulations are much faster and access to N=64 was feasible using the ATHENE cluster in Regensburg.

In our first joint paper, we investigated the thermal confinement / deconfinement phase transition in the bosonic BFSS model. Conflicting results on the order of the transition were reported in the literature and we set out to clarify the situation with a careful study at sufficiently large N.

It turned out that N=64 was required to obtain a clear signal and previous investigation reported contradictory results as a consequence of accessing only N=32 due to limited computational resources.

Our simulations revealed that the deconfinement transition is of first order. The order parameter of the transition is the (absolute value of the) Polyakov loop P, the trace of the closed loop parallel transport of the gauge field in the time direction (periodic boundary conditions in the time-like direction result in a finite temperature). The first oder nature of the transition can be seen in the following plot.

Histogram of the order parameter |P|. At the transition temperature T=0.885, two peaks are visible. In the large N, these peaks become sharply localised at |P|=0 (confined) and |P|=0.5 (dedonfined). 

The first order nature of the transition becomes apparent due to the system spending most time in either the small |P| or |P|=0.5 phase at the critical temperature (green). In the Monte Carlo history, repeated tunnelling between those phases is visible. For slightly lower or higher temperature, the system remains in either phase.

Future work will address similar issues in the full supersymmetric models. In an upcoming post, I will also discuss interesting results on "partial deconfinement" that can be found in the above paper.