بولتزمان العظيم
https://www.youtube.com/watch?v=HTBuIciRan4&feature=youtu.be
بولتزمان هو واحد من اعظم الفيزيائيين و الفيزيائيين النظريين الذين انتجهم الغرب و انتجتهم الانسانية..
بل هو برأيى ثالث اعظم فيزيائى و فيزيائى نظرى بعد نيوتن و اينشتاين...
فهو مؤسس اساسى من مؤسسى الميكانيك الاحصائى الذى يضع اسس الترموديناميك..
وهو صاحب النظرية الحركية للغازات التى تعطى تفسير لترموديناميك الغاز انطلاقا من الميكانيك الكلاسيكى لنيوتن وهذا هو تعريف الميكانيك الاحصائى..
فالميكانيك الاحصائى يعطى تفسيرا للتصرف الماكروسكوبى او العيانى للاجسام (مثلا ترموديناميك الغاز من حجم و ضغط و درجة حررة...) بدلالة الخواص الميكروسكوبية او المجهرية للاجسام (مثلا الحالات الممكنة للذرات المشكلة للغاز)..
وبولتزمان هو مكتشف الانطروبى الذى يقيس بالضبط عدد الحالات الميكروسكوبية المتاحة او الممكنة او المفتوحة امام جملة ما التى تعطى نفس الحالة الماكروسكوبية اى تقابل مثلا نفس الججم و درجة الحرارة و الضغط......
ولهذا نقول ان الانطروبى يقيس اللانظام او العشوائية...
فالغاز مثلا له انظروبى اكبر من السائل و السائل له انطروبى اكبر من الصلب لان عدد الحالات الذرية المتاحة لذرات الغاز اكبر من عدد الحالات الذرية المتاحة لذرات الغاز و عدد الحالات المتاحة امام ذرات السائل اكبر من عدد الحالات المتاحة امام ذرات الصلب..
وبولتزمان هو من ادخل مقياس الانطروبى للفيزياء و ربطه بالاحتمال او عدد الحالات الميكروسكوبية عبر المعادلة الشهيرة الموضوعة على شاهد قبره...
هذه المعادلة هى
S=k*log W
حيث S هو الانطروبى.
و W هو الاحتمال او عدد الحالات الميكروسكوبية المتاحة المقابلة لنفس الحالة الماكروسكوبية..
و k هو ثابت بولتزمان احد اشهر الثوابت الطبيعية الذى يتحكم فى كل الظواهر الحرارية..
و الانطروبى هو الكمية الفيزيائية الوحيدة التى لا تحترم تناظر العكس فى الزمن فهو يزيد بشكل مضطرد وهذا هو المبدأ الثانى للترموديناميك الذى يعبر عن معضلة السهم فى الزمن فى ابسط صورها التى كان لبولتزمان حلول عبقرية لها مازالت هى افضل الحلول الى غاية يومنا هذا رغم عدم معرفتهم فى وقته بالكوسمولوجيا المتوفرة لدينا اليوم...
و من اكتشافات بولتزمان ايضا معادلة بولتزمان الشهيرة (المعادلة 16.13)التى تقيس كيف يقترب غاز معين (وبصورة عامة جملة ما) من حالة التوزازن..
وحل هذه المعادلة يؤدى مباشرة الى توزيع بولتزمان-ماكسويل الشهير للسرعات الذى يتحكم فى التصرف الميكروسكوبى للذرات عند التوزان الحرارى...
وقد بدأ بولتزمان حياته الاستاذية كأستاذ الفيزياء الرياضية بجامعة غراز ثم انتقل الى كرسى الفيزياء التجريبية بنفس الجامعة ثم الى كرسى الفيزياء النظرية بجامعة ميونيخ ثم اخيرا استاذ الفيزياء النظرية بجامعة فينا.
وهو يسمى فترته فى غراز اسعد سنوات عمره وهناك تعرف على زوجته التى كانت اول امرأة طالبة فيزياء و رياضيات بجامعة غراز (رغم محاولة المنع التى تعرضت لها) و انجبت له ثلاث بنات و ولدان.
وفى فينا قدم محاضرات فى فلسفة الفيزياء كانت من اشهر المحاضرات على الاطلاق ورغم هذا فان الفلاسفة الالمان هاجموا و بشراسة بولتزمان و نظريته الذرية للميكانيك الاحصائى لأن كل الفلاسفة كانوا و مازالوا يرفضون وجود الذرة و كل الميكانيك الاحصائى (النظرى) الذى وضعه بولتزمان يعتمد و بشكل محورى على فكرة الذرة.
اذن دافع بولتزمان عن نظرية الميكانيك الاحصائى الذرية امام الفلاسفة و تعلم كثيرا من الفلسفة حتى يستطيع دحر اللغو الفلسفى الذى كان يقرأه لكنه يأس فى الاخير و ترك هذا الامر الذى زاد على ما يبدو من تدهور حالته النفسية التى كانت اصلا سيئه لاسباب اخرى ذاتية.
وفى الاخير قرر بولتزمان وضع حد لحياته فانتحر بالقرب من ترياسته بايطاليا عام 1906 و اوصى ان تكتب معادلته الخالدة اعلاه على شاهد قبره فهو كان يعرف انها اكبر انجازاته الفيزيائية و الفلسفية.
فمعادلة الانظروبى هى اكبر برهان فيزيائى و كذا فلسفى على وجود الذرة و على سهم الزمن و غيرها من الامور الفيزيائية و الفلسفية العميقة.
بل هو برأيى ثالث اعظم فيزيائى و فيزيائى نظرى بعد نيوتن و اينشتاين...
فهو مؤسس اساسى من مؤسسى الميكانيك الاحصائى الذى يضع اسس الترموديناميك..
وهو صاحب النظرية الحركية للغازات التى تعطى تفسير لترموديناميك الغاز انطلاقا من الميكانيك الكلاسيكى لنيوتن وهذا هو تعريف الميكانيك الاحصائى..
فالميكانيك الاحصائى يعطى تفسيرا للتصرف الماكروسكوبى او العيانى للاجسام (مثلا ترموديناميك الغاز من حجم و ضغط و درجة حررة...) بدلالة الخواص الميكروسكوبية او المجهرية للاجسام (مثلا الحالات الممكنة للذرات المشكلة للغاز)..
وبولتزمان هو مكتشف الانطروبى الذى يقيس بالضبط عدد الحالات الميكروسكوبية المتاحة او الممكنة او المفتوحة امام جملة ما التى تعطى نفس الحالة الماكروسكوبية اى تقابل مثلا نفس الججم و درجة الحرارة و الضغط......
ولهذا نقول ان الانطروبى يقيس اللانظام او العشوائية...
فالغاز مثلا له انظروبى اكبر من السائل و السائل له انطروبى اكبر من الصلب لان عدد الحالات الذرية المتاحة لذرات الغاز اكبر من عدد الحالات الذرية المتاحة لذرات الغاز و عدد الحالات المتاحة امام ذرات السائل اكبر من عدد الحالات المتاحة امام ذرات الصلب..
وبولتزمان هو من ادخل مقياس الانطروبى للفيزياء و ربطه بالاحتمال او عدد الحالات الميكروسكوبية عبر المعادلة الشهيرة الموضوعة على شاهد قبره...
هذه المعادلة هى
S=k*log W
حيث S هو الانطروبى.
و W هو الاحتمال او عدد الحالات الميكروسكوبية المتاحة المقابلة لنفس الحالة الماكروسكوبية..
و k هو ثابت بولتزمان احد اشهر الثوابت الطبيعية الذى يتحكم فى كل الظواهر الحرارية..
و الانطروبى هو الكمية الفيزيائية الوحيدة التى لا تحترم تناظر العكس فى الزمن فهو يزيد بشكل مضطرد وهذا هو المبدأ الثانى للترموديناميك الذى يعبر عن معضلة السهم فى الزمن فى ابسط صورها التى كان لبولتزمان حلول عبقرية لها مازالت هى افضل الحلول الى غاية يومنا هذا رغم عدم معرفتهم فى وقته بالكوسمولوجيا المتوفرة لدينا اليوم...
و من اكتشافات بولتزمان ايضا معادلة بولتزمان الشهيرة (المعادلة 16.13)التى تقيس كيف يقترب غاز معين (وبصورة عامة جملة ما) من حالة التوزازن..
وحل هذه المعادلة يؤدى مباشرة الى توزيع بولتزمان-ماكسويل الشهير للسرعات الذى يتحكم فى التصرف الميكروسكوبى للذرات عند التوزان الحرارى...
وقد بدأ بولتزمان حياته الاستاذية كأستاذ الفيزياء الرياضية بجامعة غراز ثم انتقل الى كرسى الفيزياء التجريبية بنفس الجامعة ثم الى كرسى الفيزياء النظرية بجامعة ميونيخ ثم اخيرا استاذ الفيزياء النظرية بجامعة فينا.
وهو يسمى فترته فى غراز اسعد سنوات عمره وهناك تعرف على زوجته التى كانت اول امرأة طالبة فيزياء و رياضيات بجامعة غراز (رغم محاولة المنع التى تعرضت لها) و انجبت له ثلاث بنات و ولدان.
وفى فينا قدم محاضرات فى فلسفة الفيزياء كانت من اشهر المحاضرات على الاطلاق ورغم هذا فان الفلاسفة الالمان هاجموا و بشراسة بولتزمان و نظريته الذرية للميكانيك الاحصائى لأن كل الفلاسفة كانوا و مازالوا يرفضون وجود الذرة و كل الميكانيك الاحصائى (النظرى) الذى وضعه بولتزمان يعتمد و بشكل محورى على فكرة الذرة.
اذن دافع بولتزمان عن نظرية الميكانيك الاحصائى الذرية امام الفلاسفة و تعلم كثيرا من الفلسفة حتى يستطيع دحر اللغو الفلسفى الذى كان يقرأه لكنه يأس فى الاخير و ترك هذا الامر الذى زاد على ما يبدو من تدهور حالته النفسية التى كانت اصلا سيئه لاسباب اخرى ذاتية.
وفى الاخير قرر بولتزمان وضع حد لحياته فانتحر بالقرب من ترياسته بايطاليا عام 1906 و اوصى ان تكتب معادلته الخالدة اعلاه على شاهد قبره فهو كان يعرف انها اكبر انجازاته الفيزيائية و الفلسفية.
فمعادلة الانظروبى هى اكبر برهان فيزيائى و كذا فلسفى على وجود الذرة و على سهم الزمن و غيرها من الامور الفيزيائية و الفلسفية العميقة.
Lattice Quantum Field Theory (of Matrix Models)- Final Update Annaba University
Abstract
We attempt to systematically develop the matrix-model/quantum-theory
correspondence by working out explicitly various non-trivial examples.
Semester
Fall 2019-2020
Level
Master II, Theoretical Physics, Badji-Mokhtar Annaba University.
Phi-four in two dimensions, the disordered/uniform 2nd order phase transition, Metropolis algorithm, critical behavior in 2 and 4 dimensions, continuum limit and renormalization group equation.
-Model II: Matrix Models (Real Quartic Matrix Model)
The quartic matrix model, the disordered/non-uniform 3rd order phase transition, Metropolis algorithm, multitrace matrix models and noncommutative phi-four, Wigner semi-circle law and emergent geometry.
-Model III: Gauge Models (M-(atrix) Theory)
M-(atrix) theory in large dimension limit, Metropolis algorithm for the coordinates and the holonomy angles, Hagedorn/deconfinement 2nd order phase transition, Gross-Witten-Wadia 3rd order phase transition and the emergence of a gap, relation to the black-string/black-hole first order phase transition via gauge/gravity duality.
Contents
-Introductory Remarks:
Path integrals and partition functions, thermal field theory, phase transitions, Monte Carlo methods (only Metropolis algorithm).
-Model I: Vector Models (Ising Model)Phi-four in two dimensions, the disordered/uniform 2nd order phase transition, Metropolis algorithm, critical behavior in 2 and 4 dimensions, continuum limit and renormalization group equation.
-Model II: Matrix Models (Real Quartic Matrix Model)
The quartic matrix model, the disordered/non-uniform 3rd order phase transition, Metropolis algorithm, multitrace matrix models and noncommutative phi-four, Wigner semi-circle law and emergent geometry.
-Model III: Gauge Models (M-(atrix) Theory)
M-(atrix) theory in large dimension limit, Metropolis algorithm for the coordinates and the holonomy angles, Hagedorn/deconfinement 2nd order phase transition, Gross-Witten-Wadia 3rd order phase transition and the emergence of a gap, relation to the black-string/black-hole first order phase transition via gauge/gravity duality.
Simulations
Simulation I- The Lattice $\Phi_2^4$.
Monte Carlo results: tests_Ising.pdf
statistical errors: error.f
random number generators: random.f
gnuplot script: script
main code: ising.f
Newton-Raphson method: newton-raphson-phi**4.f
Monte Carlo results: tests_Ising.pdf
statistical errors: error.f
random number generators: random.f
gnuplot script: script
main code: ising.f
Newton-Raphson method: newton-raphson-phi**4.f
Simulation II- Quartic Matrix Model:
Monte Carlo results: tests_multitrace.pdf
main code: matrix_four.f
eigenvalues sorting: eigen-sorting.f
eigenvalues histogram: eigen-histogram.f
Note: Results are obtained by one of our students.
Monte Carlo results: tests_multitrace.pdf
main code: matrix_four.f
eigenvalues sorting: eigen-sorting.f
eigenvalues histogram: eigen-histogram.f
Note: Results are obtained by one of our students.
Simulation III- Large number of dimensions of the BFSS Matrix Model:
Monte Carlo results: tests_BFSS.pdfcode without gauge field: matrixQM_v1.f
code with gauge field: matrixQM_v2.f
Background Material
Ising Model
Lattice QFT (of Matrix Models)
Phases of gauge theory in lower dimensions and the black-hole/black-string transition
The multitrace matrix models of noncommutative geometry and quantum gravity
The Ising model
The model
The Euclidean phi-four theory in $d$ dimensions with $O(N)$ symmetry is given by the action
\begin{eqnarray}
S[\phi]=\int d^dx \bigg(\frac{1}{2}(\partial_{\mu}\phi^i)^2+\frac{\mu_0^2}{2}\phi^i\phi^i+\frac{\lambda}{4}(\phi^i\phi^i)^2\bigg).\label{contaction}
\end{eqnarray}
We will employ lattice regularization in which $x=an$, $\int d^dx =a^d\sum_n$, $\phi^i(x)=\phi_n^i$ and $\partial_{\mu}\phi^i=(\phi_{n+\hat{\mu}}^i-\phi_n^i)/a$. The lattice action reads then
\begin{eqnarray}
S[\phi]
&=&\sum_n\bigg(-2\kappa \sum_{\mu}\Phi_n^i\Phi_{n+\hat{\mu}}^i+{\Phi}_n^i\Phi_n^i+g(\Phi_n^i\Phi_n^i-1)^2 \bigg).
\end{eqnarray}
The mass parameter $\mu_0^2$ is replaced by the so-called hopping parameter $\kappa$ and the coupling constant $\lambda$ is replaced by the coupling constant $g$ where
\begin{eqnarray}
{\mu}_{0L}^2\equiv \mu_0^2a^2=\frac{1-2g}{\kappa}-2d~,~\lambda_{L}\equiv \frac{\lambda}{a^{d-4}}=\frac{g}{\kappa^2}.
\end{eqnarray}
The fields $\phi_n^i$ and $\Phi_n^i$ are related by an irrelevant rescaling.
In the simulations we will start by fixing the lattice quartic coupling $\lambda_L$ and the lattice mass parameter $\mu_{0L}^2$ which then allows us to fix $\kappa$ and $g$ as
\begin{eqnarray}
\kappa=\frac{\sqrt{8\lambda_L+(\mu_{0L}^2+4)^2}-(\mu_{0L}^2+4)}{4\lambda_L}.
\end{eqnarray}
\begin{eqnarray}
g=\kappa^2\lambda_L.
\end{eqnarray}
The phase diagram will be drawn originally in the $\mu_{0L}^2-\lambda_L$ plane which is the lattice phase diagram.
The limit $g\longrightarrow \infty $ of the $O(1)$ model is precisely the Ising model in $d$ dimensions.
The mean field approximation
There are two phases in this model. A disordered (paramagnetic) phase characterized by $\langle \Phi_n^i\rangle=0$ and an ordered (ferromagnetic) phase characterized by $\langle \Phi_n^i\rangle=v_i\neq 0$.
This can be seen, for example, by using the mean field approximation which is an exact prescription in dimension $d=4$. In this method, we approximate the spins $\Phi_n^i$ at the $2d$ nearest neighbors of each spin $\Phi_n^i$ by the average $v^i=\langle \Phi_n^i\rangle$, viz
\begin{eqnarray}
\frac{\sum_{\mu}(\Phi_{n+\hat{\mu}}^i+\Phi_{n-\hat{\mu}}^i)}{2d}=v^i.
\end{eqnarray}
However, from the definition of $\langle \Phi_n^i\rangle$ we have
\begin{eqnarray}
\langle \Phi_n^i\rangle
&=&\frac{\int d\mu(\Phi)~\Phi_n^i e^{2\kappa\sum_n\sum_{\mu}\Phi_n^i\Phi_{n+\hat{\mu}}^i}}{\int d\mu(\Phi)~e^{2\kappa\sum_n\sum_{\mu}\Phi_n^i\Phi_{n+\hat{\mu}}^i}}
\end{eqnarray}
In the mean field approximation this definition becomes a condition on $v^i$ given explicitly by
\begin{eqnarray}
v^i
&=&\frac{\int d\mu(\Phi)~\Phi_n^i e^{4\kappa d \sum_n \Phi_n^i v^i}}{\int d\mu(\Phi)~e^{4\kappa d \sum_n\Phi_n^i v^i}}.
\end{eqnarray}
From the one hand, we get in the limit $g\longrightarrow 0$ the solution
\begin{eqnarray}
v^i=2\kappa d v^i+...\Rightarrow \kappa_c=\frac{1}{2d}.
\end{eqnarray}
On the other hand, in the limit $g\longrightarrow \infty$ we get the solution
\begin{eqnarray}
v^i=\frac{4\kappa d v^i}{N}+...\Rightarrow \kappa_c=\frac{N}{4d}.
\end{eqnarray}
The critical line $\kappa_c=\kappa_c(g)$ interpolates in the $\kappa-g$ plane between these two lines.
For example, for the $O(1)$ model in the limit $g\longrightarrow \infty$ the solution must satisfy the condition
\begin{eqnarray}
v=\tanh 4\kappa d v.
\end{eqnarray}
For $\kappa<\kappa_c$ (where $\kappa_c=1/4d$) there is only one solution at $v=0$ whereas for $\kappa>\kappa_c$ there are two solutions away from the zero, i.e. $v\neq 0$. Clearly for $\kappa$ near $\kappa_c$ the solution $v$ is near $0$ and thus we can expand the above equation as
\begin{eqnarray}
\frac{1}{3}(4d)^2\kappa^3 v^2=\kappa-\kappa_c.
\end{eqnarray}
Thus only for $\kappa>\kappa_c$ there is a non-zero solution.
In summary we have the two phases
\begin{eqnarray}
\kappa>\kappa_c:~{\rm broken, ordered, ferromagnetic}
\end{eqnarray}
\begin{eqnarray}
\kappa<\kappa_c:~{\rm symmetric, disordered, paramagnetic}.
\end{eqnarray}
The one-loop result in four dimensions
As we have already mentioned the mean field approximation becomes exact in dimension four. This translates into the fact that in four dimensions the critical behavior is fully controlled by the Gaussian fixed point (since the mean field approximation effectively reduces the theory to free fields).
Indeed, for $d=4$ the critical value at $g=0$ is $\kappa_c=1/8$ for all $N$ which can be derived from the one-loop order of the continuum phi-four theory with $O(N)$ symmetry as follows. The renormalized mass at one-loop order in this theory is given by the equation (with $a=1/\Lambda$)
\begin{eqnarray}
a^2m_R^2
&=&am^2+\frac{(N+2)\lambda}{16\pi^2}+\frac{(N+2)\lambda}{16\pi^2}a^2m^2\ln a^2m^2\nonumber\\
&+&\frac{(N+2)\lambda}{16\pi^2}a^2m^2{\bf C}+a^2\times {\rm finite}~{\rm terms}.
\end{eqnarray}
Thus we obtain in the continuum limit $a\longrightarrow 0$ the result
\begin{eqnarray}
a^2m^2&\longrightarrow &-\frac{(N+2)\lambda}{16\pi^2}-\frac{(N+2)\lambda}{16\pi^2}a^2m^2\ln a^2m^2\nonumber\\
&-&\frac{(N+2)\lambda}{16\pi^2}a^2m^2{\bf C}-a^2\times {\rm finite}~{\rm terms}.
\end{eqnarray}
In other words, we have (with $r_0=(N+2)/8\pi^2$)
\begin{eqnarray}
a^2m^2\longrightarrow a^2m_c^2=-\frac{r_0}{2}\lambda+O(\lambda^2).
\end{eqnarray}
This is the critical line for small values of the coupling constant which when expressed in terms of $\kappa$ and $g$ we obtain
\begin{eqnarray}
\kappa \longrightarrow \kappa_c=\frac{1}{8}+(\frac{r_0}{2}-\frac{1}{4})g+O(g^2).
\end{eqnarray}
The continuum limit $a\longrightarrow 0$ corresponds precisely to the limit in which the mass approaches its critical value. This happens for every value of the coupling constant and hence the continuum limit $a\longrightarrow 0$ is the limit in which we approach the critical line. The continuum limit is therefore a second order phase transition.
The continuum limit in two dimensions: Monte Carlo and renormalization group equation methods
We are interested in $d=2$ and $N=1$ which is the Ising model in two dimensions. This shares with the more realistic case in $d=3$ and $N=1$ (which describes all second order phase transitions in nature) the same critical behavior. However, in two dimensions the model admits a quasi-exact solution in terms of quantum field theory (super-renormalizability) and the renormalization group equation (Wilson-Fisher fixed point) which in the limit $g\longrightarrow\infty$ should reduce to the remarkable Onsager's exact solution.
Monte Carlo method
In two dimensions the lattice points are labelled by $n=(n_1,n_2)$ where $n_1=1,...,L_1$ and $n_2=1,...,L_2$. We will also impose periodic boundary conditions, i.e. $\Phi_{n_1+L_1,n_2}\equiv \Phi_{n_1,n_2}$ and $\Phi_{n_1,n_2+L_2}\equiv \Phi_{n_1,n_2}$.
The lattice action for $\Phi^4$ theory in two dimensions with $O(N)$ symmetry reads explicitly
\begin{eqnarray}
S[\phi]
&=&\sum_{n_1=1}^{L_1}\sum_{n_2=1}^{L_2}\bigg(-2\kappa \Phi_{n_1,n_2}^i(\Phi_{n_1+1,n_2}^i+\Phi_{n_1,n_2+1}^i)+{\Phi}_{n_1,n_2}^i\Phi_{n_1,n_2}^i+g(\Phi_{n_1,n_2}^i\Phi_{n_1,n_2}^i-1)^2 \bigg).\label{act}
\end{eqnarray}
This Euclidean quantum field theory is a statistical dynamical system with a Boltzmann weight given by
\begin{eqnarray}
{\cal P}[\Phi]=\frac{\exp(-S[\Phi])}{Z}.
\end{eqnarray}
$Z$ is the partition function of the system which is defined in an obvious way.
In Monte Carlo simulations we can use the Metropolis algorithm to generate a collection of thermalized configurations $\Phi_n^i$ according to this probability distribution.
Indeed, in the Metropolis algorithm we make small changes to the field configuration, for example $\Phi_{p_1,p_2}^j\longrightarrow \Phi_{p_1,p_2}^j+h$, then accept/reject these changes using precisely the Boltzmann probability distribution
\begin{eqnarray}
W(\Phi_{p_1,p_2}^j\longrightarrow \Phi_{p_1,p_2}^j+h) ={\rm min}(1,\exp(-\Delta S_{p_1,p_2}^j(h)).\label{met}
\end{eqnarray}
The variation $\Delta S_{p_1,p_2}^j(h)$ due to the change $\Phi_{p_1,p_2}^j\longrightarrow \Phi_{p_1,p_2}^j+h$ is given explicitly by
\begin{eqnarray}
\Delta S_{p_1,p_2}^j(h)&=&-2\kappa h\big(\Phi_{p_1+1,p_2}^j+\Phi_{p_1-1,p_2}^j+\Phi_{p_1,p_2+1}^j+\Phi_{p_1,p_2-1}^j\big)+(1-2g)\big(h^2+2h\Phi_{p_1,p_2}^j\big)\nonumber\\
&+&g\big(h^2+2h\Phi_{p_1,p_2}^j\big)\big(h^2+2h\Phi_{p_1,p_2}^j+2\Phi_{p_1,p_2}^i\Phi_{p_1,p_2}^i\big).\label{var}
\end{eqnarray}
In the actual code (written in Fortran which is our preferred programming language) we will go through the following steps:
- We write a subroutine which computes the energy given by the action (\ref{act}). We can also include other primary observables in this subroutine for example the magnetization
\begin{eqnarray}
M=\frac{1}{NL_1L_2}\langle m\rangle~,~m=|\sum_{i,n_1,n_2}\Phi_{n_1,n_2}^i|.
\end{eqnarray}
-We write a subroutine which computes the variation (\ref{var}). This depends on $j$, $p_1$, $p_2$ and $h$.
- In preparation for writing the Metropolis process we include a random number generator. We prefer Mersenne Twistor or ran2 of numerical recipes.
- We write a subroutine which implements the Metropolis algorithm (\ref{met}). So we change all the fields at all lattice points successively and accept/reject according to the probability (\ref{met}). The value of the variation $h$ is drawn randomly and its range is tuned so that the acceptance rate is fixed around $1/3$.
- In preparation for writing thermalization and measurement processes we write a subroutine which computes the average, the error and the auto-correlation time for arbitrary observables.
- We write down the main program in which the random number generator, the physical parameters of the model and the field configuration are initialized followed by a thermalization process which changes the system until thermalization (or equilibrium) is reached then followed by a measurement process in which a set of thermalized configurations $\Phi_n^i$ is collected and primary observables are measured.
- We measure the secondary observables such as second moments like the specific heat (associated with the energy) and susceptibility (associated with the magnetization). These are given by
\begin{eqnarray}
C_v=\langle S^2\rangle -\langle S\rangle^2.
\end{eqnarray}
\begin{eqnarray}
\chi=\langle m^2\rangle -\langle m\rangle^2.
\end{eqnarray}
Renormalization group equation
In the simulations we will start by fixing the lattice quartic coupling $\lambda_L$ and the lattice mass parameter $\mu_{0L}^2$ which then allows us to fix $\kappa$ and $g$. The phase diagram will be drawn originally in the $\mu_{0L}^2-\lambda_L$ plane which is the lattice phase diagram. This should be extrapolated to the infinite volume limit defined by $l_1=L_1a\longrightarrow \infty$ and $l_2=L_2a\longrightarrow \infty$ with $a$ fixed.
However, the Euclidean quantum field theory phase diagram should be drawn in terms of the renormalized parameters and is obtained from the lattice phase diagram by taking the limit $a\longrightarrow 0$.
In two dimensions the $\Phi^4$ theory requires only mass renormalization while the quartic coupling constant is finite (recall that in four dimensions three renormalizations are required: the mass, the quartic coupling and the wave function renormalizations).
Indeed, in two dimensions the bare mass $\mu_0^2$ diverges logarithmically when we remove the cutoff, i.e. in the limit $\Lambda\longrightarrow \infty$ where $\Lambda=1/a$ while $\lambda$ is independent of $a$. As a consequence, the lattice parameters will go to zero in the continuum limit $a\longrightarrow 0$.
We know that mass renormalization is due to the tadpole diagram which is the only divergent Feynman diagram in the theory and takes the form of a simple reparametrization given by
\begin{eqnarray}
\mu_0^2=\mu^2-\delta\mu^2.\label{ren}
\end{eqnarray}
$\mu^2$ is the renormalized mass parameter and $\delta\mu^2$ is the counter term which is fixed via an appropriate renormalization condition. The unltraviolet divergence $\ln\Lambda$ of $\mu_0^2$ is contained in $\delta\mu^2$ while the renormalization condition will split the finite part of $\mu_0^2$ between $\mu^2$ and $\delta\mu^2$. The choice of the renormalization condition can be quite arbitrary. A convenient choice suitable for Monte Carlo measurements and which distinguishes between the two phases of the theory is given by the usual normal ordering prescription \cite{Loinaz:1997az} .
Quantization at one-loop (together with a self-consistent Hartree treatement) gives in the symmetric phase where $\mu^2>0$ the $2-$point function
\begin{eqnarray}
\Gamma^{(2)}(p)
&=&p^2+\mu^2+\Sigma(p)~,~\Sigma(p)=3\lambda A_{\mu^2}-\delta\mu^2+....
\end{eqnarray}
The dots $...$ denote two-loop effect. $A_{\mu^2}$ is precisely the value of the tadpole diagram given by
\begin{eqnarray}
A_{\mu^2}=\int\frac{d^2k}{(2\pi)^2}\frac{1}{k^2+\mu^2}.
\end{eqnarray}
The renormalization condition which is equivalent to normal ordering the interaction is equivalent to the choice
\begin{eqnarray}
\delta\mu^2=3\lambda A_{\mu^2}.\label{ren1}
\end{eqnarray}
A dimensionless coupling constant can the be defined by
\begin{eqnarray}
f=\frac{\lambda}{\mu^2}.
\end{eqnarray}
The action becomes
\begin{eqnarray}
S[\phi]=\int d^2x \bigg(\frac{1}{2}(\partial_{\mu}\phi)^2+\frac{1}{2}\mu^2(1-3fA_{\mu^2})\phi^2+\frac{f\mu^2}{4}\phi^4\bigg).
\end{eqnarray}
For sufficiently small $f$ the exact effective potential is well approximated by the classical potential with a single minimum at $\phi_{\rm cl}=0$. For larger $f$, the coefficient of the mass term in the above action can become negative and as a consequence a transition to the broken symmetry phase is possible, although in this regime the effective potential is no longer well approximated by the classical potential. Indeed, a transition to the broken symmetry phase was shown to be present in \cite{Chang:1976ek}, where a duality between the strong coupling regime of the above action and a weakly coupled theory normal-ordered with respect to the broken phase was explicitly constructed.
On a finite volume lattice with periodic boundary conditions equation (\ref{ren}), together with equation (\ref{ren1}), becomes
\begin{eqnarray}
\mu_L^2-3\lambda_L A_{\mu_L^2}-\mu_{0L}^2=0.\label{reno}
\end{eqnarray}
The Feynman diagram $A_{\mu^2}$ takes explicitly the form
\begin{eqnarray}
A_{\mu^2}
&=&\frac{1}{N^2}\sum_{n_1=1}^N\sum_{n_2=1}^N\frac{1}{4\sin^2 {\pi n_1}/{N}+4\sin^2{\pi n_2}/{N}+\mu_L^2}.
\end{eqnarray}
Given the critical value of $\mu_{0L}^2$ for every value of $\lambda_L$ we need then to determine the corresponding critical value of $\mu_{L}^2$. This can be done numerically using the Newton-Raphson algorithm. The continuum limit $a\longrightarrow 0$ is then given by extrapolating the results into the origin, i.e. taking $\lambda_L=a^2\lambda\longrightarrow 0$, $\mu_L^2=a^2\mu^2\longrightarrow 0$ in order to determine the critical value
\begin{eqnarray}
f_*={\rm lim}_{\lambda_L,\mu_L^2\longrightarrow 0}\frac{\lambda_L}{\mu_{L*}^2}.
\end{eqnarray}
References
%\cite{Loinaz:1997az}
\bibitem{Loinaz:1997az}
W.~Loinaz and R.~S.~Willey,
'Monte Carlo simulation calculation of critical coupling constant for continuum phi**4 in two-dimensions,'
Phys.\ Rev.\ D {\bf 58}, 076003 (1998)
[hep-lat/9712008].
%%CITATION = HEP-LAT/9712008;%%
%31 citations counted in INSPIRE as of 23 May 2015
%\cite{Chang:1976ek}
\bibitem{Chang:1976ek}
S.~J.~Chang,
'The existence of a second order phase transition in the two-dimensional phi**4 field theory,'
Phys.\ Rev.\ D {\bf 13}, 2778 (1976)
[Phys.\ Rev.\ D {\bf 16}, 1979 (1977)].
%%CITATION = PHRVA,D13,2778;%%
%113 citations counted in INSPIRE as of 25 May 2015
فرضية بولتزمان-شوتز
و
أول مرة قرأت عن "فرضية بولتزمان-شوتز" عند فيلسوف الزمن هيو برايس فى
معرض انتقاده الشديد لها وجدتنى اقول لنفسى: هذا هو الامر الذى كنت ابحث
عنه منذ مدة طويلة جدا. فهو نموذج رياضى من ثالث العمالقة فى الفيزياء
-بولتزمان- (وهى ليست فكرته بل هى فكرة مساعده شوتز كما يذكر بولتزمان
نفسه).
اذن هو نموذج رياضى يجمع الكثير من الامور التى تبدو متناقضة للوهلة الاولى.
و عندما قرأت عن هذه الفكرة عند برايش شدنى جدا شرحه المقتضب لها و المنحنى الوحيد الذى وضعه لتوضيحها ثم لما تابعت القراءة وجدته يعطى انتقادا طويلا لها شدنى هو الآخر لها و ليس ضدها.
اذن انتقادات برايس ل "فرضية بولتزمان-شوتز" شدتنى ليس للموقف المعارض لها المتحمس ضدها كما كان يهدف يرايس ان يفعل قُراءه بل شدنى للموقف المناصر لها و المتحمس لها.
على كل حال كنت اعرف ان برايس هو من دعاة نظرية ارخميدس فى الزمن (التى هى نظرة الفيزياء الحديثة لنيوتن و اينشتاين) فى انه لا يوجد هناك سهم للزمن.
و كنت ارى ان هذه النظرة نظرة قاصرة جامدة تختزل جزءا اساسيا من القضية كان قد تكلم عنه القديس اوغستين وهو ان سهم الزمن هو جزء موضوعى من العقل و الذاكرة و الخيال فهناك جزء نفسى مهم فى ماهية الزمن.
اذن اننى من المتحمسين لفرضية "بولتزمان-شوتز" فهى تحتوى من بين ما تحتوى عليه على:
-فرضية عديد-الاكوان (الكوسمولوجية) و ان هذا الكون الذى نعيش فيه هو جزء ضئيل ربما مهمل من عديد-الاكوان.
-عديد-الاكوان هو جملة ميتة حراريا وهذا يعنى بكل بساطة انها جملة ثابتة لا تتغير و هى تقابل الزمن الاول الذى هو الدهر و هو زمن لانهائى.
-هذا الكون هو تقلب حرارى واحد من عدد لانهائى من التقلبات الحرارية الممكنة و كل تقلب حرارى هو كون مختلف.
وهذا يقابل الزمن الذى نعرفه الذى له سهم (الذى يتبع القانون الثانى للترموديناميك فى ان الانطروبى دائما فى حالة زيادة مضطردة).
-اذن الزمن له سهم فى كوننا و قد يكون له سهم معاكس فى كون آخر لكن عديد-الاكوان ليس له سهم (وهذا اول تطبيق لفكرة المنظورية).
-الزمن الذى له سهم هو زمن نهائى (لان التقلب الحرارى محدود زمنيا) لكن عديد-الاكوان له دهر لانهائى.
-رغم كل هذه التقلبات (وعددها لانهائى) فان عديد-الاكوان يبقى ثابت غير متغير (ونقول ميت حراريا بلغة الفيزياء). فعديد-الاكوان هو لانهائى و هو اكبر من كل اللانهايات الكونية بمرات لانهائية (فهناك عدد لانهائى من اللانهايات اكبر من بعضها البعض بمرات لانهائية كما نعرف من فلسفة الرياضيات).
-فرضية المحاكاة (وقد يكون ليس كل شيء فى الواقع هو محاكاة لكن جزء معتبر منه محاكاة). فالمحاكاة تكلف ظاقويا اقل بكثير من التقلب الحرارى الحقيقى.
-والمحاكاة ليست بالضرورة غير-حقيقية اذا كان كل الواقع (او جزء معتبر منه) محاكاة. لانه اذا كان كل شيء محاكاة فتلك هى الحقيقة لا اقل و لا اكثر.
-المبدأ الانظروبيكى (او المبدأ البشرى). اذن الارض (والانسان) رغم انها ليست هى مركز الكون (كما تقول الطريقة العلمية) الا انها مكان مميز فى الكون. وهذا اول تطبيق لفكرة تميز الانسان رغم عدم مركزيته.
-المونيزم. لا يوجد شيء آخر فعلا الا عديد-الاكوان و هو ثابت وهو الموجود الحقيقى بل هو الوجود نفسه.
-ثم عندما يدخل الميكانيك الكمومى الى الفرضية فيمكن عندها تفسير سهم الترموديناميك و سهم الزمن بالرصد و انهيار دالة الموجة وهذا من اقوى تطبيقات المنظورية. وهذا يشير ايضا الى نفسية السهم فى الزمن فى الكون الذى نعيش فيه.
-اذن الانسان يتميز بعقل و وعى و علم وهو انعكاس للزمن ليس شيئا آخر.
والوعى يقابل درجات حرية (وهذا مصطلح فيزيائى نقصد به متغيرات) يجب ان تكون ناجمة هى الاخرى عن تقلب حرارى انطلاقا من عديد-الاكوان.
اذن عديد-الاكوان نفسه يجب ان يتميز بعقل و وعى و علم و عدد درجات الحرية المرفقة بكل ذلك هو لانهائى (وهذا لا يتطلب ان يكون الكون الذى نعيش فيه واعى و عاقل و عالم فقط عديد-الاكوان يجب ان يكون كذلك).
-والكمومى يؤدى ايضا الى فكرة ان الكون الذى نعيش فيه هو نفسه تركيب خطى لفروع مختلفة. هذه الفروع مرتبطة ببعضها البعض عبر برزخ. وهذه الفروع وجودها كمونى غير-متحقق اما البرزخ فوجوده اكثر تحققا.
فالعقل و المادة نفسهما هما تركيب خطى لنفس الحالة (هنا المونيزم مرة اخرى) مرتبطان ببعضها البعض ببرزخ و هو الخيال الذى يذكره ابن عربى فهو اكثر تحققا-وجوديا من كل من العقل و المادة (بالنسبة لبولتزمان اما بالنسبة لابن عربى فالخيال اكثر تحققا وجوديا من المادة فقط و ليس من العقل).
-لكن كون الانسان هو تقلب حرارى (او تجلى) من عديد-الاكوان يعنى ان ارادته خاضعة للقوانين الترموديناميكية التى تحكم كل شيء وهذه هى التواؤمية التى هى بالاساس جبر محض على كل موجود الا المبدأ الانطروبيكى (الذى يتحول عنده الجبر الى تواؤمية).
-و كما ان المبدأ الانطروبيكى هو مركز الكون فان الحقيقة المحمدية هى مركز المبدأ الانطروبيكى.
-اما معضلة الشر فتتطلب مبدأ انطروبيكى ليس مركزه الانسان بل مركزه الشيطان. فهناك شر حقيقى لا يمكن تفسيره الا بحرية حقيقية لا يمتلكها الانسان لكن يمتلكها الشيطان (فحظ الانسان من الحرية هو التواؤمية لا اقل لا اكثر) .
وهذه بعض الافكار التى تنطوى عليها فرضية "بولتزمان-شوتز" التى انتقدها كثير من فلاسفة الفيزياء فى هذا العصر لان اجندتهم تختلف عن اجندتنا و هذا دون ان يقدموا بديل حقيقى لتفسير الزمن و سهمه و تدفقه و نسبيته.
لاحظوا اننى لم اشرح بالضبط ماهية فرضية بولتزمان و شوتز التى نتركها اذن لفرصة اخرى ان شاء الله.
-
اذن هو نموذج رياضى يجمع الكثير من الامور التى تبدو متناقضة للوهلة الاولى.
و عندما قرأت عن هذه الفكرة عند برايش شدنى جدا شرحه المقتضب لها و المنحنى الوحيد الذى وضعه لتوضيحها ثم لما تابعت القراءة وجدته يعطى انتقادا طويلا لها شدنى هو الآخر لها و ليس ضدها.
اذن انتقادات برايس ل "فرضية بولتزمان-شوتز" شدتنى ليس للموقف المعارض لها المتحمس ضدها كما كان يهدف يرايس ان يفعل قُراءه بل شدنى للموقف المناصر لها و المتحمس لها.
على كل حال كنت اعرف ان برايس هو من دعاة نظرية ارخميدس فى الزمن (التى هى نظرة الفيزياء الحديثة لنيوتن و اينشتاين) فى انه لا يوجد هناك سهم للزمن.
و كنت ارى ان هذه النظرة نظرة قاصرة جامدة تختزل جزءا اساسيا من القضية كان قد تكلم عنه القديس اوغستين وهو ان سهم الزمن هو جزء موضوعى من العقل و الذاكرة و الخيال فهناك جزء نفسى مهم فى ماهية الزمن.
اذن اننى من المتحمسين لفرضية "بولتزمان-شوتز" فهى تحتوى من بين ما تحتوى عليه على:
-فرضية عديد-الاكوان (الكوسمولوجية) و ان هذا الكون الذى نعيش فيه هو جزء ضئيل ربما مهمل من عديد-الاكوان.
-عديد-الاكوان هو جملة ميتة حراريا وهذا يعنى بكل بساطة انها جملة ثابتة لا تتغير و هى تقابل الزمن الاول الذى هو الدهر و هو زمن لانهائى.
-هذا الكون هو تقلب حرارى واحد من عدد لانهائى من التقلبات الحرارية الممكنة و كل تقلب حرارى هو كون مختلف.
وهذا يقابل الزمن الذى نعرفه الذى له سهم (الذى يتبع القانون الثانى للترموديناميك فى ان الانطروبى دائما فى حالة زيادة مضطردة).
-اذن الزمن له سهم فى كوننا و قد يكون له سهم معاكس فى كون آخر لكن عديد-الاكوان ليس له سهم (وهذا اول تطبيق لفكرة المنظورية).
-الزمن الذى له سهم هو زمن نهائى (لان التقلب الحرارى محدود زمنيا) لكن عديد-الاكوان له دهر لانهائى.
-رغم كل هذه التقلبات (وعددها لانهائى) فان عديد-الاكوان يبقى ثابت غير متغير (ونقول ميت حراريا بلغة الفيزياء). فعديد-الاكوان هو لانهائى و هو اكبر من كل اللانهايات الكونية بمرات لانهائية (فهناك عدد لانهائى من اللانهايات اكبر من بعضها البعض بمرات لانهائية كما نعرف من فلسفة الرياضيات).
-فرضية المحاكاة (وقد يكون ليس كل شيء فى الواقع هو محاكاة لكن جزء معتبر منه محاكاة). فالمحاكاة تكلف ظاقويا اقل بكثير من التقلب الحرارى الحقيقى.
-والمحاكاة ليست بالضرورة غير-حقيقية اذا كان كل الواقع (او جزء معتبر منه) محاكاة. لانه اذا كان كل شيء محاكاة فتلك هى الحقيقة لا اقل و لا اكثر.
-المبدأ الانظروبيكى (او المبدأ البشرى). اذن الارض (والانسان) رغم انها ليست هى مركز الكون (كما تقول الطريقة العلمية) الا انها مكان مميز فى الكون. وهذا اول تطبيق لفكرة تميز الانسان رغم عدم مركزيته.
-المونيزم. لا يوجد شيء آخر فعلا الا عديد-الاكوان و هو ثابت وهو الموجود الحقيقى بل هو الوجود نفسه.
-ثم عندما يدخل الميكانيك الكمومى الى الفرضية فيمكن عندها تفسير سهم الترموديناميك و سهم الزمن بالرصد و انهيار دالة الموجة وهذا من اقوى تطبيقات المنظورية. وهذا يشير ايضا الى نفسية السهم فى الزمن فى الكون الذى نعيش فيه.
-اذن الانسان يتميز بعقل و وعى و علم وهو انعكاس للزمن ليس شيئا آخر.
والوعى يقابل درجات حرية (وهذا مصطلح فيزيائى نقصد به متغيرات) يجب ان تكون ناجمة هى الاخرى عن تقلب حرارى انطلاقا من عديد-الاكوان.
اذن عديد-الاكوان نفسه يجب ان يتميز بعقل و وعى و علم و عدد درجات الحرية المرفقة بكل ذلك هو لانهائى (وهذا لا يتطلب ان يكون الكون الذى نعيش فيه واعى و عاقل و عالم فقط عديد-الاكوان يجب ان يكون كذلك).
-والكمومى يؤدى ايضا الى فكرة ان الكون الذى نعيش فيه هو نفسه تركيب خطى لفروع مختلفة. هذه الفروع مرتبطة ببعضها البعض عبر برزخ. وهذه الفروع وجودها كمونى غير-متحقق اما البرزخ فوجوده اكثر تحققا.
فالعقل و المادة نفسهما هما تركيب خطى لنفس الحالة (هنا المونيزم مرة اخرى) مرتبطان ببعضها البعض ببرزخ و هو الخيال الذى يذكره ابن عربى فهو اكثر تحققا-وجوديا من كل من العقل و المادة (بالنسبة لبولتزمان اما بالنسبة لابن عربى فالخيال اكثر تحققا وجوديا من المادة فقط و ليس من العقل).
-لكن كون الانسان هو تقلب حرارى (او تجلى) من عديد-الاكوان يعنى ان ارادته خاضعة للقوانين الترموديناميكية التى تحكم كل شيء وهذه هى التواؤمية التى هى بالاساس جبر محض على كل موجود الا المبدأ الانطروبيكى (الذى يتحول عنده الجبر الى تواؤمية).
-و كما ان المبدأ الانطروبيكى هو مركز الكون فان الحقيقة المحمدية هى مركز المبدأ الانطروبيكى.
-اما معضلة الشر فتتطلب مبدأ انطروبيكى ليس مركزه الانسان بل مركزه الشيطان. فهناك شر حقيقى لا يمكن تفسيره الا بحرية حقيقية لا يمتلكها الانسان لكن يمتلكها الشيطان (فحظ الانسان من الحرية هو التواؤمية لا اقل لا اكثر) .
وهذه بعض الافكار التى تنطوى عليها فرضية "بولتزمان-شوتز" التى انتقدها كثير من فلاسفة الفيزياء فى هذا العصر لان اجندتهم تختلف عن اجندتنا و هذا دون ان يقدموا بديل حقيقى لتفسير الزمن و سهمه و تدفقه و نسبيته.
لاحظوا اننى لم اشرح بالضبط ماهية فرضية بولتزمان و شوتز التى نتركها اذن لفرصة اخرى ان شاء الله.
-
The multitrace matrix models of noncommutative geometry and quantum gravity
In here we are interested in a noncommutative scalar $\Phi^4$ theory. As an example, we will take the underlying noncommutative space to be fuzzy ${\bf CP}^n_l$ which is obtained by Berezin quantization \cite{Berezin:1974du} of the projective space ${\bf CP}^n=SU(n+1)/U(n)$.
Fuzzy ${\bf CP}^n_l$ is given by the spectral triple \cite{Frohlich:1993es,Connes:1994yd} (see also \cite{Dolan:2001gn})
\begin{eqnarray}
{\bf CP}^n_l=({\bf H}_{n,l}, {\rm Mat}_{N_{n,l}},\Delta_{n,l}).
\end{eqnarray}
${\bf H}_{n,l}$ is the Hilbert space associated with the irreducible representation of $su(n+1)$ which is given by the totally symmetrized tensor product of $l$ fundamental representations, viz $(l,0,0,...,0)$. The dimension of this representation is given by
\begin{eqnarray}
N_{n,l}=\frac{(n+l)!}{n!l!}.
\end{eqnarray}
The Hilbert space ${\bf H}_{n,l}$ is thus acted on by the complete matrix algebra ${\rm Mat}_{N_{n,l}}$ of finite dimension $N_{n,l}$ with inner product defined by
\begin{eqnarray}
(f,g)=\frac{1}{N_{n,l}}Tr f^{\dagger}g.
\end{eqnarray}
The $\Delta_{n,l}$ is the Laplace operator on fuzzy ${\bf CP}^n_l$ defined, in terms of the generators $L_i$ of $su(n+1)$ in the representation $(l,0,0,...,0)$, by the quadratic Casimir
\begin{eqnarray}
\Delta_{n,l}(f)=[L_i,[L_i,f]].
\end{eqnarray}
Derivation on fuzzy ${\bf CP}^n_l$ are given precisely by the adjoint action of the generators $L_i$, i.e.
\begin{eqnarray}
{\rm ad}L_i(f)=[L_i,f]=(L_i^L-L_i^R)(f).
\end{eqnarray}
This shows explicitly that the space of functions on fuzzy ${\bf CP}^n_l$, which is the matrix algebra ${\rm Mat}_{d_{n,l}}$, decomposes under the action of $SU(n+1)$ as the direct sum of the irreducible representations $(m,...,m)$ of $SU(n + 1)$,viz
\begin{eqnarray}
(l,...,0)\otimes \overline{(l,...,0)}=\oplus_{m=0}^l(m,0,...,0,m).
\end{eqnarray}
Hence, the matrix algebra ${\rm Mat}_{N_{n,l}}$ is the endomorphism ${\bf Hom}(V_{n,l})=V_{n,l}\otimes V_{n,l}^*$ where $V_{n,l}$ is the vector space associated with the representation $(l,0,...,0)$.
The polarization tensors $T_{m \sigma}$ which transform in the irreducible representation $(m,...,m)$ are the eigenmatrices of the Laplace $\Delta_{n,l}$ with eigenvalue $\lambda_m$ and degeneracy $d_{n,m}$ given by
\begin{eqnarray}
\lambda_m=2m(m+n).
\end{eqnarray}
And
\begin{eqnarray}
d_{n,m}=\frac{n(2m+n)((m+n-1)!)^2}{(n!)^2(m!)^2}.
\end{eqnarray}
Indeed, the index $\sigma$ in $T_{m \sigma}$ denotes the other quantum numbers required to specify the representation $(m,...,m)$.
The above spectrum is precisely the spectrum of the Laplace operator on commutative ${\bf CP}^n$ only cutoff at $m=l$. The commutative limit is therefore $l\longrightarrow\infty$ in which the eigenmatrices $T_{m \sigma}$ go over to the correct eigenfunctions $Y_{m \sigma}$ (spherical harmonics) on ${\bf CP}^n$.
The coordinate functions $X_i$ on fuzzy ${\bf CP}^n_l$ are obtained from suitably
rescaling the generators $L_i$ of $su(n + 1)$ in the representation $(l,0,...,0)$, namely $X_i=a_l L_i$. We have then the commutation relations
\begin{eqnarray}
[X_i,X_j]=ia_l f_{ijk}X_k.
\end{eqnarray}
The data contained in the spectral triple $({\bf H}_{n,l}, {\rm Mat}_{N_{n,l}},\Delta_{n,l})$ is sufficient to write down an Euclidean action for a scalar field theory on fuzzy ${\bf CP}^n_l$. A scalar field $\Phi$ is a hermitian $N\times N$ matrix in ${\rm Mat}_{N_{n,l}}$, i.e. $N\equiv N_{n,l}$. An action on fuzzy ${\bf CP}^n_l$ is then given by
\begin{eqnarray}
S=Tr(\Phi C_2\Phi+r\Phi^2+g\Phi^4).
\end{eqnarray}
The partition function of the model is given by
\begin{eqnarray}
Z=\int d\mu_D(\Phi)\exp(-\beta S[\Phi]).
\end{eqnarray}
$d\mu_D(\Phi)$ is Dyson measure on ${\rm Mat}_{N}$.
We can diagonalize the matrix $\Phi$ as $\Phi=U\Lambda U^{\dagger}$. We also write $\Phi=\Phi_{\mu}T_{\mu}$ or $\Phi_{\mu}=Tr\Phi T_{\mu}$ where $T_{\mu}$, $\mu=1,...,N^2$ are the generators of $u(N)$. The partition function then becomes
\begin{eqnarray}
Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\nonumber\\
&\times &\int d\mu_H(U)\exp\big(-\beta K_{\mu\nu}Tr U\Lambda U^{\dagger}T_{\mu}Tr U\Lambda U^{\dagger}T_{\nu}\big).
\end{eqnarray}
The Vandermonde $\Delta^2(\Lambda)$ is given by
\begin{eqnarray}
\Delta^2(\Lambda)=\prod_{i=1}^N(\lambda_i-\lambda_j)^2.
\end{eqnarray}
And $d\mu_H(U)$ is the Haar measure on $U(N)$. The kinetic matrix $K_{\mu\nu}$ is given on the other hand by
\begin{eqnarray}
K_{\mu\nu}=-Tr[L_i,T_{\mu}][L_i,T_{\nu}].
\end{eqnarray}
We can now perform a hopping parameter expansion to perform the integral over $U$ (by using the properties $(A\otimes B)(C\otimes D)=(AC\otimes BD)$ and $Tr(A\otimes B)=TrA TrB$ and the orthogonality relation of the Haar measure) as follows \cite{Saemann:2010bw,OConnor:2007ibg}
\begin{eqnarray}
Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\nonumber\\
&\times &\sum_k\frac{(-\beta)^k}{k!}\int d\mu_H(U)\prod_{i=1}^kK_{\mu_i\nu_i}Tr U\Lambda U^{\dagger}T_{\mu_i}Tr U\Lambda U^{\dagger}T_{\nu_i}\nonumber\\
&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\sum_k\frac{(-\beta)^k}{k!}K_{\mu_1\nu_1}...K_{\mu_k\nu_k}\nonumber\\
&\times & \int d\mu_H(U)Tr \bigg((U\otimes...\otimes U)(\Lambda\otimes...\otimes\Lambda)(U^{\dagger}\otimes...\otimes U^{\dagger})(T_{\mu_1}\otimes T_{\nu_1}\otimes...\otimes T_{\mu_k}\otimes T_{\nu_k})\bigg)\nonumber\\
&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\sum_k\frac{(-\beta)^k}{k!}K_{\mu_1\nu_1}...K_{\mu_k\nu_k}\nonumber\\
&\times &\sum_{\rho}\frac{1}{{\rm dim}(\rho)} Tr_{\rho}(\Lambda\otimes...\otimes\Lambda)Tr (T_{\mu_1}\otimes T_{\nu_1}\otimes...\otimes T_{\mu_k}\otimes T_{\nu_k}).
\end{eqnarray}
After a very long and involved calculation we find upto the order $O(\beta^2)$ the multitrace action \cite{Saemann:2010bw,OConnor:2007ibg}
\begin{eqnarray}
Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\exp(-\beta (S_1+S_2)),
\end{eqnarray}
where
\begin{eqnarray}
S_1={\cal J}_1=a_{2,0}Tr\Lambda^2+a_{1,1}(Tr\Lambda)^2.
\end{eqnarray}
And
\begin{eqnarray}
S_2&=&\frac{\beta}{2}({\cal J}_1^2-{\cal J}_2)\nonumber\\
{\cal J}_2&=&a_{4,0}Tr\Lambda^4+a_{3,1}Tr\Lambda^3 Tr\Lambda+a_{2,2}(Tr\Lambda^2)^2+a_{2,1,1}Tr\Lambda^2(Tr\Lambda)^2+a_{1,1,1,1}(Tr\Lambda)^4.
\end{eqnarray}
Fuzzy ${\bf CP}^n_l$ is given by the spectral triple \cite{Frohlich:1993es,Connes:1994yd} (see also \cite{Dolan:2001gn})
\begin{eqnarray}
{\bf CP}^n_l=({\bf H}_{n,l}, {\rm Mat}_{N_{n,l}},\Delta_{n,l}).
\end{eqnarray}
${\bf H}_{n,l}$ is the Hilbert space associated with the irreducible representation of $su(n+1)$ which is given by the totally symmetrized tensor product of $l$ fundamental representations, viz $(l,0,0,...,0)$. The dimension of this representation is given by
\begin{eqnarray}
N_{n,l}=\frac{(n+l)!}{n!l!}.
\end{eqnarray}
The Hilbert space ${\bf H}_{n,l}$ is thus acted on by the complete matrix algebra ${\rm Mat}_{N_{n,l}}$ of finite dimension $N_{n,l}$ with inner product defined by
\begin{eqnarray}
(f,g)=\frac{1}{N_{n,l}}Tr f^{\dagger}g.
\end{eqnarray}
The $\Delta_{n,l}$ is the Laplace operator on fuzzy ${\bf CP}^n_l$ defined, in terms of the generators $L_i$ of $su(n+1)$ in the representation $(l,0,0,...,0)$, by the quadratic Casimir
\begin{eqnarray}
\Delta_{n,l}(f)=[L_i,[L_i,f]].
\end{eqnarray}
Derivation on fuzzy ${\bf CP}^n_l$ are given precisely by the adjoint action of the generators $L_i$, i.e.
\begin{eqnarray}
{\rm ad}L_i(f)=[L_i,f]=(L_i^L-L_i^R)(f).
\end{eqnarray}
This shows explicitly that the space of functions on fuzzy ${\bf CP}^n_l$, which is the matrix algebra ${\rm Mat}_{d_{n,l}}$, decomposes under the action of $SU(n+1)$ as the direct sum of the irreducible representations $(m,...,m)$ of $SU(n + 1)$,viz
\begin{eqnarray}
(l,...,0)\otimes \overline{(l,...,0)}=\oplus_{m=0}^l(m,0,...,0,m).
\end{eqnarray}
Hence, the matrix algebra ${\rm Mat}_{N_{n,l}}$ is the endomorphism ${\bf Hom}(V_{n,l})=V_{n,l}\otimes V_{n,l}^*$ where $V_{n,l}$ is the vector space associated with the representation $(l,0,...,0)$.
The polarization tensors $T_{m \sigma}$ which transform in the irreducible representation $(m,...,m)$ are the eigenmatrices of the Laplace $\Delta_{n,l}$ with eigenvalue $\lambda_m$ and degeneracy $d_{n,m}$ given by
\begin{eqnarray}
\lambda_m=2m(m+n).
\end{eqnarray}
And
\begin{eqnarray}
d_{n,m}=\frac{n(2m+n)((m+n-1)!)^2}{(n!)^2(m!)^2}.
\end{eqnarray}
Indeed, the index $\sigma$ in $T_{m \sigma}$ denotes the other quantum numbers required to specify the representation $(m,...,m)$.
The above spectrum is precisely the spectrum of the Laplace operator on commutative ${\bf CP}^n$ only cutoff at $m=l$. The commutative limit is therefore $l\longrightarrow\infty$ in which the eigenmatrices $T_{m \sigma}$ go over to the correct eigenfunctions $Y_{m \sigma}$ (spherical harmonics) on ${\bf CP}^n$.
The coordinate functions $X_i$ on fuzzy ${\bf CP}^n_l$ are obtained from suitably
rescaling the generators $L_i$ of $su(n + 1)$ in the representation $(l,0,...,0)$, namely $X_i=a_l L_i$. We have then the commutation relations
\begin{eqnarray}
[X_i,X_j]=ia_l f_{ijk}X_k.
\end{eqnarray}
The data contained in the spectral triple $({\bf H}_{n,l}, {\rm Mat}_{N_{n,l}},\Delta_{n,l})$ is sufficient to write down an Euclidean action for a scalar field theory on fuzzy ${\bf CP}^n_l$. A scalar field $\Phi$ is a hermitian $N\times N$ matrix in ${\rm Mat}_{N_{n,l}}$, i.e. $N\equiv N_{n,l}$. An action on fuzzy ${\bf CP}^n_l$ is then given by
\begin{eqnarray}
S=Tr(\Phi C_2\Phi+r\Phi^2+g\Phi^4).
\end{eqnarray}
The partition function of the model is given by
\begin{eqnarray}
Z=\int d\mu_D(\Phi)\exp(-\beta S[\Phi]).
\end{eqnarray}
$d\mu_D(\Phi)$ is Dyson measure on ${\rm Mat}_{N}$.
We can diagonalize the matrix $\Phi$ as $\Phi=U\Lambda U^{\dagger}$. We also write $\Phi=\Phi_{\mu}T_{\mu}$ or $\Phi_{\mu}=Tr\Phi T_{\mu}$ where $T_{\mu}$, $\mu=1,...,N^2$ are the generators of $u(N)$. The partition function then becomes
\begin{eqnarray}
Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\nonumber\\
&\times &\int d\mu_H(U)\exp\big(-\beta K_{\mu\nu}Tr U\Lambda U^{\dagger}T_{\mu}Tr U\Lambda U^{\dagger}T_{\nu}\big).
\end{eqnarray}
The Vandermonde $\Delta^2(\Lambda)$ is given by
\begin{eqnarray}
\Delta^2(\Lambda)=\prod_{i=1}^N(\lambda_i-\lambda_j)^2.
\end{eqnarray}
And $d\mu_H(U)$ is the Haar measure on $U(N)$. The kinetic matrix $K_{\mu\nu}$ is given on the other hand by
\begin{eqnarray}
K_{\mu\nu}=-Tr[L_i,T_{\mu}][L_i,T_{\nu}].
\end{eqnarray}
We can now perform a hopping parameter expansion to perform the integral over $U$ (by using the properties $(A\otimes B)(C\otimes D)=(AC\otimes BD)$ and $Tr(A\otimes B)=TrA TrB$ and the orthogonality relation of the Haar measure) as follows \cite{Saemann:2010bw,OConnor:2007ibg}
\begin{eqnarray}
Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\nonumber\\
&\times &\sum_k\frac{(-\beta)^k}{k!}\int d\mu_H(U)\prod_{i=1}^kK_{\mu_i\nu_i}Tr U\Lambda U^{\dagger}T_{\mu_i}Tr U\Lambda U^{\dagger}T_{\nu_i}\nonumber\\
&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\sum_k\frac{(-\beta)^k}{k!}K_{\mu_1\nu_1}...K_{\mu_k\nu_k}\nonumber\\
&\times & \int d\mu_H(U)Tr \bigg((U\otimes...\otimes U)(\Lambda\otimes...\otimes\Lambda)(U^{\dagger}\otimes...\otimes U^{\dagger})(T_{\mu_1}\otimes T_{\nu_1}\otimes...\otimes T_{\mu_k}\otimes T_{\nu_k})\bigg)\nonumber\\
&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\sum_k\frac{(-\beta)^k}{k!}K_{\mu_1\nu_1}...K_{\mu_k\nu_k}\nonumber\\
&\times &\sum_{\rho}\frac{1}{{\rm dim}(\rho)} Tr_{\rho}(\Lambda\otimes...\otimes\Lambda)Tr (T_{\mu_1}\otimes T_{\nu_1}\otimes...\otimes T_{\mu_k}\otimes T_{\nu_k}).
\end{eqnarray}
After a very long and involved calculation we find upto the order $O(\beta^2)$ the multitrace action \cite{Saemann:2010bw,OConnor:2007ibg}
\begin{eqnarray}
Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\exp(-\beta (S_1+S_2)),
\end{eqnarray}
where
\begin{eqnarray}
S_1={\cal J}_1=a_{2,0}Tr\Lambda^2+a_{1,1}(Tr\Lambda)^2.
\end{eqnarray}
And
\begin{eqnarray}
S_2&=&\frac{\beta}{2}({\cal J}_1^2-{\cal J}_2)\nonumber\\
{\cal J}_2&=&a_{4,0}Tr\Lambda^4+a_{3,1}Tr\Lambda^3 Tr\Lambda+a_{2,2}(Tr\Lambda^2)^2+a_{2,1,1}Tr\Lambda^2(Tr\Lambda)^2+a_{1,1,1,1}(Tr\Lambda)^4.
\end{eqnarray}
The coefficients $a$ can be found in \cite{Saemann:2010bw,OConnor:2007ibg}.
For example, we compute on ${\bf CP}^1_l$, where the highest weight $l$ is related to the ordinary spin $s$ by $s=l/2$, the following quantities $N=l+1$, $Tr K\sim N^4$, $Tr K^2\sim 4N^6/3$, $Tr K^3\sim 2N^8$ and $a_{3,1}\sim 8/3$.
Whereas we compute in four dimensions on ${\bf CP}^2_l$ the following quantities $N=(l+1)(l+2)/2$, i.e. $l^2\sim 2N$ and as a consequence $Tr K\sim 8 N^3/3$, $Tr K^2\sim 8N^4$, $Tr K^3\sim 128 N^5/5$ and $a_{3,1}\sim -256/9N^2$.
It was argued in \cite{Ydri:2017riq} that the non-vanishing of the multitrace term proportional to $a_{3,1}$ is a necessary and a sufficient condition for a stable Ising phase and a stable emergent background geometry. It was also argued there that quantum gravity in (or more precisely quantized spaces of) dimensions higher than two can thus be obtained from pure multitrace matrix models along these same lines.
By neglecting all multitrace terms we find that the quartic matrix model $N Tr(\tilde{r}\Phi^2+\tilde{g}\Phi^4)$ (where $\tilde{r}=r/N^{3/2}$ and $\tilde{g}=g/N^2$) in the limit of very weak coupling $\tilde{g}\longrightarrow 0$ is dominated by the Wigner semi-circle law given by
\begin{eqnarray}
\rho(\lambda)=\frac{\tilde{r}}{\pi}\sqrt{\frac{2}{\tilde{r}}-\lambda^2}.
\end{eqnarray}
However, by including all multitrace terms we get noncommutative $\Phi^4$ on fuzzy ${\bf CP}^n_l$ which is also dominated (or more precisely its eigenvalue sector) in the limit of weak coupling $\tilde{g}\longrightarrow 0$ by a Wigner semi-circle corresponding to the following simple model \cite{Steinacker:2005wj}
\begin{eqnarray}
S=\frac{2N}{\alpha_0^2(m)}Tr\Phi^2.
\end{eqnarray}
In other words, we must make in the above Wigner semi-circle law the following replacement
\begin{eqnarray}
\tilde{r}\longrightarrow\frac{2}{\alpha_0^2(m)}.
\end{eqnarray}
$\alpha_0$ is the maximum eigenvalue and it is given in terms of the cutoff $\Lambda\sim l$ on fuzzy ${\bf CP}^n_l$ and the mass $m^2\sim r$ by the relation \cite{Steinacker:2005wj}
\begin{eqnarray}
\alpha_0^2(m)=4c(m,\Lambda)\Lambda^{d-2}.
\end{eqnarray}
We have explicitly
\begin{eqnarray}
c(m,\Lambda)=\frac{1}{16\pi^2}\bigg(1-\frac{m^2}{\Lambda^2}\ln(1+\frac{\Lambda^2}{m^2})\bigg)~,~d=4,n=2.
\end{eqnarray}
\begin{eqnarray}
c(m,\Lambda)=\frac{1}{4\pi}\ln(1+\frac{\Lambda^2}{m^2})~,~d=2,n=1.
\end{eqnarray}
Thus, by studying the multitrace approximation at weak coupling we can determine when the Wigner behavior goes from the behavior of the pure matrix model $d=0$ to the behavior of noncommutative matrix models at $d=2$ and $d=4$ as we vary for example the parameter $a_{3,1}$. The emergence of the correct behavior is an indication of a stable Ising phase and as a consequence a stable emergent background geometry.
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F.~A.~Berezin, ``General Concept of Quantization,'' Commun.\ Math.\ Phys.\ {\bf 40}, 153 (1975). doi:10.1007/BF01609397
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B.~P.~Dolan, D.~O'Connor and P.~Presnajder, ``Matrix phi**4 models on the fuzzy sphere and their continuum limits,''
JHEP {\bf 0203}, 013 (2002) doi:10.1088/1126-6708/2002/03/013 [hep-th/0109084].
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\bibitem{Saemann:2010bw}
C.~Saemann,
``The Multitrace Matrix Model of Scalar Field Theory on Fuzzy CP^n,''
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D.~O'Connor and C.~Saemann,
``Fuzzy Scalar Field Theory as a Multitrace Matrix Model,''
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H.~Steinacker,
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%2 citations counted in INSPIRE as of 19 Jan 2020
For example, we compute on ${\bf CP}^1_l$, where the highest weight $l$ is related to the ordinary spin $s$ by $s=l/2$, the following quantities $N=l+1$, $Tr K\sim N^4$, $Tr K^2\sim 4N^6/3$, $Tr K^3\sim 2N^8$ and $a_{3,1}\sim 8/3$.
Whereas we compute in four dimensions on ${\bf CP}^2_l$ the following quantities $N=(l+1)(l+2)/2$, i.e. $l^2\sim 2N$ and as a consequence $Tr K\sim 8 N^3/3$, $Tr K^2\sim 8N^4$, $Tr K^3\sim 128 N^5/5$ and $a_{3,1}\sim -256/9N^2$.
It was argued in \cite{Ydri:2017riq} that the non-vanishing of the multitrace term proportional to $a_{3,1}$ is a necessary and a sufficient condition for a stable Ising phase and a stable emergent background geometry. It was also argued there that quantum gravity in (or more precisely quantized spaces of) dimensions higher than two can thus be obtained from pure multitrace matrix models along these same lines.
By neglecting all multitrace terms we find that the quartic matrix model $N Tr(\tilde{r}\Phi^2+\tilde{g}\Phi^4)$ (where $\tilde{r}=r/N^{3/2}$ and $\tilde{g}=g/N^2$) in the limit of very weak coupling $\tilde{g}\longrightarrow 0$ is dominated by the Wigner semi-circle law given by
\begin{eqnarray}
\rho(\lambda)=\frac{\tilde{r}}{\pi}\sqrt{\frac{2}{\tilde{r}}-\lambda^2}.
\end{eqnarray}
However, by including all multitrace terms we get noncommutative $\Phi^4$ on fuzzy ${\bf CP}^n_l$ which is also dominated (or more precisely its eigenvalue sector) in the limit of weak coupling $\tilde{g}\longrightarrow 0$ by a Wigner semi-circle corresponding to the following simple model \cite{Steinacker:2005wj}
\begin{eqnarray}
S=\frac{2N}{\alpha_0^2(m)}Tr\Phi^2.
\end{eqnarray}
In other words, we must make in the above Wigner semi-circle law the following replacement
\begin{eqnarray}
\tilde{r}\longrightarrow\frac{2}{\alpha_0^2(m)}.
\end{eqnarray}
$\alpha_0$ is the maximum eigenvalue and it is given in terms of the cutoff $\Lambda\sim l$ on fuzzy ${\bf CP}^n_l$ and the mass $m^2\sim r$ by the relation \cite{Steinacker:2005wj}
\begin{eqnarray}
\alpha_0^2(m)=4c(m,\Lambda)\Lambda^{d-2}.
\end{eqnarray}
We have explicitly
\begin{eqnarray}
c(m,\Lambda)=\frac{1}{16\pi^2}\bigg(1-\frac{m^2}{\Lambda^2}\ln(1+\frac{\Lambda^2}{m^2})\bigg)~,~d=4,n=2.
\end{eqnarray}
\begin{eqnarray}
c(m,\Lambda)=\frac{1}{4\pi}\ln(1+\frac{\Lambda^2}{m^2})~,~d=2,n=1.
\end{eqnarray}
Thus, by studying the multitrace approximation at weak coupling we can determine when the Wigner behavior goes from the behavior of the pure matrix model $d=0$ to the behavior of noncommutative matrix models at $d=2$ and $d=4$ as we vary for example the parameter $a_{3,1}$. The emergence of the correct behavior is an indication of a stable Ising phase and as a consequence a stable emergent background geometry.
References
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%2 citations counted in INSPIRE as of 19 Jan 2020
Phases of gauge theory in lower dimensions and the black-hole/black-string transition
The model
Gauge theory in one dimension with $d$ adjoint scalar fields is intimately related to string theory (for example D branes on tori) and black hole physics (for example the Gregory-Laflamme instability). The action can be obtained from the dimensional reduction of $(d+1)-$dimensional Yang-Mills theory to $1$ dimension. The action is of the form
\begin{eqnarray}
S=\frac{1}{g^2}\int_0^{\beta}dt Tr\bigg[\frac{1}{2}(D_t\Phi_i)^2-\frac{1}{4}[\Phi_i,\Phi_j]^2\bigg].\label{bfssB}
\end{eqnarray}
This matrix quantum mechanics is in fact the bosonic part of the BFSS M-(atrix) theory which corresponds to DLCQ of M-theory and describes D0-branes (which is perhaps the most important connection to string theory and black holes).
The only free parameter in this model is the Hawking temperature $T=1/\beta$. On the lattice with the time direction given by a circle the inverse temperature $\beta$ is precisely equal the circumference of the circle, viz $\beta=a.\Lambda$ where $a$ is the lattice spacing and $\Lambda$ is the number of links.
The Polyakov line (which acts as our macroscopic order parameter) is defined in terms of the holonomy matrix $U$
\begin{eqnarray}
P=\frac{1}{N}Tr U~,~U={\cal P}\exp(-i\int_0^{\beta}dt A(t)).
\end{eqnarray}
After gauge-fixing on the lattice (we choose the static gauge $A(t)=-(\theta_1,\theta_2,...,\theta_N)/\beta$) we write the Polyakov line $P$ in terms of the holonomy angles $\theta_a$ as
\begin{eqnarray}
P=\frac{1}{N}\sum_a\exp(i\theta_a).
\end{eqnarray}
We actually measure in Monte Carlo simulation the expectation value
\begin{eqnarray}
\langle |P|\rangle=\int d\theta \rho(\theta)\exp(i\theta).
\end{eqnarray}
The eigenvalue distribution $\rho(\theta)$ of the holonomy angles is our microscopic order parameter used to characterize precisely the various phases of this model. This eigenvalue distribution is given formally by
\begin{eqnarray}
\rho(\theta)=\frac{1}{N}\sum_{a=1}^N \langle\delta(\theta-\theta_a)\rangle.
\end{eqnarray}
The energy $E/N^2$ of the bosonic truncation of the BFSS matrix model and its corresponding extent of space $R^2$ are the other very important observables in this model. They are given explicitly by
\begin{eqnarray}
\frac{E}{N^2}=\frac{3T}{N^2}\langle {\rm commu}\rangle~,~{\rm commu}=-\frac{1}{4g^2}\int_{0}^{\beta}dt{Tr}[{\Phi}_i^{},{\Phi}_j^{}]^2.
\end{eqnarray}
\begin{eqnarray}
R^2=\frac{a}{\Lambda N^2}\langle {\rm radius}\rangle~,~{\rm radius}=\frac{N}{a}\sum_{n=1}^{\Lambda}{Tr}{\Phi}_i^{2}(n).
\end{eqnarray}
The phase structure
The phase diagram of this model was determined numerically by means of the Monte Carlo method in \cite{Kawahara:2007fn} to be consisting of two phase transitions and three stable phases. In more detail, we haveThe confinement/deconfinement phase transition: This is a second order phase transition associated with the spontaneous breakdown of the $U(1)$ symmetry
\begin{eqnarray}
A(t)\longrightarrow A(t)+C.{\bf 1}.
\end{eqnarray}
At low temperatures this symmetry is unbroken and as a consequence we have a confining phase characterized by a uniform eigenvalue distribution, i.e.
\begin{eqnarray}
\rho(\theta)=\frac{1}{2\pi}~,~T\longrightarrow 0.
\end{eqnarray}
As the temperature increases the above $U(1)$ symmetry gets spontaneously broken at some temperature $T_{c 2}$ and we enter the deconfining phase which is characterized by a non-uniform eigenvalue distribution.
We can easily check that the Polyakov line $\langle |P|\rangle$ vanishes identically in a uniform eigenvalue distribution. But numerically it is observed that $\langle |P|\rangle$ vanishes only as $1/N$ at low temperatures \cite{Kawahara:2007fn}.
The energy $E/N^2$ and the extent of space $R^2$ in the confining uniform phase are both constant which is consistent with the so-called Eguchi-Kawai equivalence \cite{Eguchi:1982nm} (The expectation values of single-trace operators in $d-$dimensional large $N$ gauge theories are independent of the volume if the $U(1)^d$ symmetry is not spontaneoulsy broken. In our case $d=1$ and independence of the volume is precisely independence of the temperature which is the inverse Euclidean time).
The constant value of the energy in the confining uniform phase is identified with the ground state energy. The energy in the deconfining non-uniform phase ($T\gt T_{c2}$) deviates from this constant value quadratically, i.e. as $(T-T_{c 2})^2$. This is confirmed from the behavior of the extent of space $R^2$ which is constant in the confining uniform phase then deviates from this constant value quadratically as well \cite{Kawahara:2007fn}.
The Gross-Witten-Wadia phase transition:
This is a third order phase transition occurring at a temperature $T_{c1}\gt T_{c2}$ dividing therefore the non-uniform phase into two distinct phases: The gapless phase in the intermediate region $T_{c 2}\le T\le T_{c 1}$ and the gapped phase at high temperatures $T\gt T_{c1}$.
It is observed in numerical simulations \cite{Kawahara:2007fn} that this phase transition is well described by the Gross-Witten-Wadia one-plaquette model given explicitly by \cite{Gross:1980he,Wadia:1980cp}
\begin{eqnarray}
Z_{GWW}=\int dU \exp(\frac{N}{\kappa}Tr U+{\rm h.c}).
\end{eqnarray}
The deconfined non-uniform gapless phase is described by a gapless eigenvalue distribution (and hence the name: gapless phase) of the form
\begin{eqnarray}
\rho_{\rm gapless}=\frac{1}{2\pi}(1+\frac{2}{\kappa}\cos\theta)~,~-\pi\lt\theta\le+\pi~,~\kappa\ge 2.
\end{eqnarray}
The fact that the angle $\theta$ takes values in the full range $]-\pi,+\pi]$ is precisely what is meant by the word "gapless", i.e. there are no gaps on the circle. This solution is valid only for $\kappa\ge 2$ where $\kappa$ is a function of the temperature.
At $\kappa=2$ (corresponding to $T=T_{c1}$) a third order phase transition occurs to a gapped eigenvalue distribution given explicitly by
\begin{eqnarray}
\rho_{\rm gapped}=\frac{1}{\pi\sin^2\frac{\theta_0}{2}}\cos\frac{\theta}{2}\sqrt{\sin^2\frac{\theta_0}{2}-\sin^2\frac{\theta}{2}}~,~-\theta_0\le\theta\le+\theta_0~,~\kappa\lt 2.
\end{eqnarray}
The eigenvalue distribution is non-zero only in the range $[-\theta_0,\theta_0]$ (arbitrarily chosen to be centered around $0$ for simplicity) where the angle $\theta_0$ is given explicitly by
\begin{eqnarray}
\sin^2\frac{\theta_0}{2}=\frac{\kappa}{2}.
\end{eqnarray}
This is a gapped distribution since only the interval $[-\theta_0,\theta_0]$ is filled. At high temperatures corresponding to $\kappa\longrightarrow 0$ the above distribution approaches a delta function \cite{Aharony:2003sx}.
This third order phase transition is associated therefore with the appearance of a gap in the eigenvalue distribution. We notice that the deconfining non-uniform phase is dominated by the gapped phase since the region of the gapless phase is extremely narrow.
Thus, the Polyakov line suffers another phase transition in the non-uniform phase where it rises from $0$ to the value $1/2$ at $\kappa=2$ ($T=T_{c1}$) in the gapless phase then rises further from $1/2$ to the value $1$ in the gapped phase. We can calculate explicitly the behavior
\begin{eqnarray}
P=1-\frac{\kappa}{4}~,~\kappa\lt 2.
\end{eqnarray}
\begin{eqnarray}
P=\frac{1}{\kappa}~,~\kappa\ge 2.
\end{eqnarray}
Hagedorn transition
It has been argued that the deconfinement phase transition in gauge theory such as the above discussed phase trnasition is precisely the Hagedorn phase in string theory \cite{Aharony:2003sx}. It has also been argued there that the Hagedorn transition could be a single first order transition and not a deconfinement second order transition followed by a gapped third order transition, i.e. the gapless phase may not be there (recall that its range is very narrow).
$1/d$ expansion
The above phase structure was derived in \cite{Mandal:2009vz} in the limit $d\longrightarrow\infty$ (a $1/d$ expansion is considered in lattice models originally in \cite{Drouffe:1979dh,Drouffe:1983fv}). See the related treatment in \cite{Kabat:2000zv,Kabat:2001ve}.
By using a $1/d$ expansion around the $d=\infty$ (here $d=9$ which is reasonably large) saddle point of the model (\ref{bfssB}) which is characterized by a non-zero value of the condensate $\langle Tr\Phi_i\Phi_i\rangle$ we can show explicitly that the phase structure of the model (\ref{bfssB}) consists of two phases: $1)$ a confinement/deconfinement second order phase transition marking the onset of non-uniformity in the eigenvalue distribution closely followed by $2)$ a GWW third order phase transition marking the onset of a gap in the eigenvalue distribution.
The large $d$ limit at finite $N$ is defined, in analogy with the large $N$ limit, by $d\longrightarrow\infty$ and $g^2\longrightarrow\infty$ keeping $g^2d={\rm fixed}$. Thus, the large $N$ and large $d$ limit is defined by $\tilde{\lambda}=g^2Nd= \lambda d$ where $\lambda=g^2N$ is 't Hooft limit which can be chosen to be $1$, i.e. $g^2=1/N$. It is found that fluctuations around the saddle point in this limit of large $d$ and large $N$ are suppressed by powers of $1/d$.
In this large $d$ and large $N$ limit, and after the introduction of an $SO(d)-$invariant field $B_{ij}$ to get ride of the Yang-Mills term, the dynamically massive adjoint scalars $\Phi^i$ can be integrated out yielding an effective action depending only on the gauge field $A$. As it turns out this effective action is gauge-invariant since it depends only on the eigenvalues $e^{i\theta_a}$ of the gauge-covariant holonomy matrix or Wilson loop $U$. Explicitly, the effective action is a Landau-Ginzburg energy functional of the form (see \cite{Aharony:2003sx} for the original derivation)
\begin{eqnarray}
\frac{S}{N^2 d}=\frac{3}{8}\beta\tilde{\lambda}^{1/3}+a_1|u_1|^2+b_1|u_1|^4+\sum_{n=2}a_n|u_n|^2+...
\end{eqnarray}
The $u_n$ are the moments of the Wilson loop $U$, viz
\begin{eqnarray}
u_n=\frac{1}{N}\sum_a\exp(in\theta_a)=\int d\theta \rho(\theta)\exp(in\theta). \label{moments}
\end{eqnarray}
The constants $a_n$ and $b_1$ are given by
\begin{eqnarray}
a_n=\frac{1}{n}(\frac{1}{d}-\bar{x}^n)~,~b_1=\frac{1}{3}\beta\tilde{\lambda}^{1/3}\bar{x}^2~,~\bar{x}=\exp(-\beta\tilde{\lambda}^{1/3}).
\end{eqnarray}
Obviously, for $\bar{x}\lt 1/d$ all the $a_n$ are positive and thus $u_n=0$ for all $n$ is a global minimum of the potential. The vanishing of all the $u_n$ corresponds, from (\ref{moments}), to a uniform eigenvalue distribution and thus this phase is the confinement phase of gauge theory where $TrU=0$.
As the temperature increases $\bar{x}$ crosses the value $1/d$ and as a consequence the quadratic coefficient $a_1$ becomes negative while the rest of the $a_n$ and the $b_n$ remain positive. The $|u_n|$ for $n\gt 1$ remain zero while $|u_1|$ takes the value $|u_1|=\sqrt{-a_1/2b_1}$ for $T\geq T_{c2}$ where the critical temperature $T_{c2}$ is determined from the condition $a_1=0$ to be given by \cite{Mandal:2009vz}
\begin{eqnarray}
T_{c2}=\frac{\lambda^{1/3}d^{1/3}}{\ln d}.
\end{eqnarray}
This is a second order phase transition (the second derivative of the free energy is found to be discontinuous) which marks the onset of non-uniformity in the eigenvalue distribution.
Above $T_{c2}$ (where all the $|u_n|$ for $n\gt 1$ vanishes) the eigenvalue distribution can be given by
\begin{eqnarray}
\rho(\theta)=\frac{1}{2\pi}(1+2|u_1|\cos\theta).
\end{eqnarray}
For $|u_1|=0$ this is a uniform distribution whereas for small $|u_1|$ in the region $|u_1|\leq 1/2$ we have a GWW gapless eigenvalue distribution, i.e. a distribution which does not vanish in the whole interval $]-\pi,+\pi]$. As $|u_1|$ increases from $0$ to $1/2$ the above eigenvalue dsitribution vanishes at $\theta=\pi$. Here we enter a new phase characterized by a GWW gapped eigenvalue distribution. The critical temperature $T_{c1}$ can be determined from the condition where the saddle point value $\langle |u_1|\rangle=\sqrt{-a_1/2b_1}$ reaches the value $1/2$. We get \cite{Mandal:2009vz}
\begin{eqnarray}
T_{c1}=T_{c2}+\frac{\tilde{\lambda}^{1/3}}{6d\ln d}.
\end{eqnarray}
This GWW transition is associated with the onset of a gap in the eigenvalue distribution and it is of a third order (the third derivative of the free energy is discontinuous) as shown originally in \cite{Aharony:2003sx, AlvarezGaume:2005fv}.
The comparison between the Monte Carlo data of \cite{Kawahara:2007fn} which is performed for $d=9$ adjoint scalars (the dimension of the reduced space is $d=9$) and the $1/d$ analysis around the $d=\infty$ saddle point of \cite{Mandal:2009vz} is very favourable. We get
\begin{eqnarray}
T_{c2}=0.895(d=\infty)~,~0.8761(d=9).
\end{eqnarray}
\begin{eqnarray}
T_{c1}=0.917(d=\infty)~,~0.905(d=9).
\end{eqnarray}
The $1/d$ expansion can also be applied to gauge theory in $0$ dimension (the infinite temperature limit of the model (\ref{bfssB}) which is essentially the celebarted IKKT model \cite{Ishibashi:1996xs}) where it can be shown that the $1/N$ expansion can be performed exactly \cite{Mandal:2009vz}. This expansion in $0$ dimension was actually done originally in \cite{Hotta:1998en}.
Connection to two-dimensional Yang-Mills theory and D-branes description
The connection between the phase structure of low dimensional gauge theories and D-branes (especially the $D0-$branes and $D1-$branes in the current case) is discussed in great detail in \cite{Aharony:2004ig,Aharony:2005ew}. See also {Azeyanagi:2009zf,Kawahara:2007fn,Mandal:2009vz}.
In summary, the confinement/deconfinement phase transition observed in the model (\ref{bfssB}), which is the analogue of the confinement/deconfinement phase transition of ${\cal N}=4$ supersymmetric Yang-Mills theory on ${\bf S}^3$, is the weak coupling limit of the black-string/black-hole phase transition observed in the dual garvity theory of two-dimensional Yang-Mills theory.
Thus, the phase structure of gauge theory in low dimensions can also be obtained from considerations of holography and the dual gravitational theory (beside and supplementing the numerical Monte Carlo and the analytical $1/d$ and $1/N$ expansions considered in previous sections). In particular, the thermodynamics of a given phase of Yang-Mills gauge theory can be deduced from the Bekenstein-Hawking thermodynamics of the corresponding charged black (string or hole) solution \cite{Aharony:2004ig}.
Conversely, the black-string/black-hole phase transition (which is an example of the Gregory-Laflamme instability) can be successfully studied beyond the supergravity approximation using gauge theory methods thanks to the gauge/gravity duality and the AdS/CFT correspondence \cite{Itzhaki:1998dd}.
We start from two-dimensional super Yang-Mills gauge theory on the torus ${\bf T}^2$, i.e. (with $\mu,\nu=0,1$ and $I,J=1,...,8$)
\begin{eqnarray}
S=\frac{1}{g^2_{\rm YM}}\int d^{2} x\big[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}(D_{\mu}X_I)(D^{\mu}X_I)+\frac{1}{4}[X_I,X_J]^2+\frac{i}{2}\bar{\psi}\gamma^{\mu}D_{\mu}\psi+\frac{1}{2}\bar{\psi}\gamma^{I}[X_{I},\psi]\big].
\end{eqnarray}
This is equivalent to type IIB superstring theory around the black 1-brane background spacetime. Thus, we are dealing with a system of $N$ coincident D1-branes forming a black 1-brane solution. This black 1-brane solution can be mapped via S-duality to a black string solution winding around the circle in ${\bf R}^8\times{\bf S}^1$ \cite{Itzhaki:1998dd}.
The dimensionless parameters of this model are $\tilde{T}\tilde{L}$ and $\tilde{\lambda}\tilde{L}^2$ where $\tilde{\beta}=1/\tilde{T}$ and $\tilde{L}$ are the circumferences of the two cycles of ${\bf T}^2$.
At strong 't Hooft coupling $\tilde{\lambda}\longrightarrow\infty$ and small temperature $\tilde{T}$ it was shown in \cite{Aharony:2004ig} that the above $2-$dimensional Yang-Mills theory exhibits a first order phase transition at the value
\begin{eqnarray}
\tilde{T}\tilde{L}=\frac{2.29}{\sqrt{\tilde{\lambda}\tilde{L}^2}}.
\end{eqnarray}
This corresponds in the dual gravity theory side to a transition between the black hole phase (gapped phase) and the black string phase (the uniform and gapless phases) \cite{Susskind:1997dr}. This black-hole/black-string first order phase transition is the Gregory-Laflamme instability in this case \cite{Gregory:1993vy}.
Indeed, for very large compactification circumference $\tilde{L}$ the two-dimensional super Yang-Mills gauge theory describes $N$ coincident $D1-$branes in type IIB string theory which are winding on the circle. When $\tilde{L}\longrightarrow 0$ the appropriate description becomes given by $T-$duality \cite{Taylor:1996ik} in terms of $N$ coincident $D0-$branes in type IIA string theory which are winding on a circle of circumference $\alpha^{\prime}/\tilde{L}$.
The positions of these $D-$particles on the $T-$dual circle are given by the eigenvalues of the Wilson loop winding on the circle, i.e. of the holonomy matrix
\begin{eqnarray}
W={\cal P}\exp(i\oint dx A_x).\label{Wl}
\end{eqnarray}
By an appropriate gauge transformation the Wilson line can be diagonalized as
\begin{eqnarray}
W={\rm diag}(\exp(i\theta_1),...,\exp(i\theta_N)).
\end{eqnarray}
The phase $\theta_a$ is precisely the position of the $a$th $D0-$brane on the $T-$dual circle. If all the angles $\theta_a$ accumulate at the same point then we obtain a black hole at that location whereas if they are distributed uniformly on the circle we obtain a uniform black string. We can also obtain a non-uniform black string phase or a phase with several black holes depending on the distribution of the eigenvalues $\theta_a$.
In the high temperature limit $\tilde{T}\longrightarrow 0$ and weak 't Hooft coupling $\tilde{\lambda}\longrightarrow 0$ the above two-dimensional super Yang-Mills gauge theory on the torus ${\bf T}^2$ reduces to the one-dimensional bosonic gauge theory (\ref{bfssB}).
The Wilson loop (\ref{Wl}) winding around the spatial circle in the $2-$dimensional theory becomes in the $1-$dimensional theory the Polyakov line $P=TrU/N$ since the time direction of this $1-$dimensional model is the spatial direction of the $2-$dimensional model. More precisely, we have
\begin{eqnarray}
\lambda=\tilde{\lambda}\tilde{T}~,~
\tilde{L}=\beta.
\end{eqnarray}
The high temperature $2-$dimensional Yang-Mills theory on a circle was studied in \cite{Aharony:2004ig,Harmark:2004ws} where a phase transition around $\lambda_{\rm eff}=1.4$ was observed. This result was made more precise by studying the $1-$dimensional matrix quantum mechanics in \cite{Kawahara:2007fn} where two phase transitions were identified of second and third order respectively at the values (see next section for detailed discussion)
\begin{eqnarray}
\lambda_{\rm eff}=1.35(1)\Rightarrow \tilde{T}\tilde{L}=\frac{1.35(1)}{\tilde{\lambda}\tilde{L}^2}.
\end{eqnarray}
\begin{eqnarray}
\lambda_{\rm eff}=1.487(2)\Rightarrow \tilde{T}\tilde{L}=\frac{1.487(2)}{\tilde{\lambda}\tilde{L}^2}.
\end{eqnarray}
The second order transition separates between the gapped phase and the non-uniform phase whereas the third order separates between the non-uniform phase and the uniform phase. These phases, in the $2-$dimensional phase diagram with axes given by the dimensionless parameters $\tilde{T}\tilde{L}$ and $\tilde{\lambda}\tilde{L}^2$, occur at high temperatures in the region where the $2-$dimensional Yang-Mills theory reduces to the bosonic part of the $1-$dimensional BFSS quantum mechanics. It is conjectured in \cite{Kawahara:2007fn} that by continuing the above two lines to low temperatures we will reach a triple point where the two lines intersect and as a consequence the non-uniform phase ceases to exist below this tri-critical point.
Gaussianity
It has been argued in \cite{Filev:2015hia} that the dynamics of the bosonic BFSS model (\ref{bfssB}) is fully dominated by the large $d$ behavior encoded in the quadratic action (with $m=d^{1/3}$)\begin{eqnarray}
S_{}=\frac{1}{g^2}\int_0^{\beta}dt{\rm Tr}\bigg[\frac{1}{2}(D_t\Phi_i)^2+\frac{1}{2}m^2\Phi_i^2\bigg].\label{gauss}
\end{eqnarray}
This has been checked in Monte Carlo simulations where a Hagedorn/deconfinement transition is observed consisting of a second order confinement/deconfinement phase transition closely followed by a GWW third order transition which marks the emergence of a gap in the eigenvalue distribution.
In this approximation it is observed that the eigenvalues of the adjoint scalar fields $\Phi_i$ are distributed according to the Wigner semi-circle law with a radius $r$ following the temperature behavior of the extent of space $R^2$ since $r^2=4R^2/d$. Thus, only the radius of the eigenvalue distribution undergoes a phase transition not in its shape (which is always a Wigner semi-circle law). At low temperature this radius becomes constant given by $r=\sqrt{2/m}$.
An analytic study of the model (\ref{gauss}) is given in the beautiful paper \cite{Furuuchi:2003sy} where its relevance to the plane wave matrix model and string theory is discussed at length.
The BMN plane wave gauge theory
The action (\ref{bfssB}) is the bosonic part of the BFSS matrix model which is given by the supersymmetric one-dimensional gauge theory \cite{Banks:1996vh}
\begin{eqnarray}
S_{\rm BFSS}=\frac{1}{g^2}\int_0^{\beta}dt{\rm Tr}\bigg[\frac{1}{2}(D_t\Phi_i)^2-\frac{1}{4}[\Phi_i,\Phi_j]^2+\frac{1}{2}\psi_{\alpha}D_t\psi_{\alpha}-\frac{1}{2}\psi_{\alpha}(\gamma_i)_{\alpha\beta}[\Phi_i,\psi_{\beta}]\bigg].\label{BFSS}
\end{eqnarray}
The only mass deformation that can be added to this action so that maximal supersymmetry is preserved is the plane wave deformation given by the following quadratic mass terms and Chern-Simons term:
\begin{eqnarray}
\Delta S_{\rm defor}&=&\frac{1}{g^2}\int_0^{\beta}dt{Tr}\bigg[\frac{\mu^2}{2}\sum_{a=1}^3\Phi_a^2+\frac{\mu^2}{8}\sum_{i=4}^9\Phi_i^2+i\mu\sum_{a,b,c=1}^3\epsilon_{abc}\Phi_a\Phi_b\Phi_c\bigg]\nonumber\\
&+&\frac{3i\mu}{8}\psi_{\alpha}(\gamma^{123})_{\alpha\beta}\psi_{\beta}\bigg].
\end{eqnarray}
Indeed, the celebrated BMN matrix model is given simply by the sum of the above two actions, viz \cite{Berenstein:2002jq}
\begin{eqnarray}
S_{\rm BMN}=S_{\rm BFSS}+\Delta S_{\rm defor}.\label{BMN}
\end{eqnarray}
The bosonic truncation of this model gives us then the one-dimensional gauge theory (with $\mu_1=\mu_2=2\alpha/3\equiv \mu$)
\begin{eqnarray}
S_{}=\frac{1}{g^2}\int_0^{\beta}dt{\rm Tr}\bigg[\frac{1}{2}(D_t\Phi_i)^2-\frac{1}{4}[\Phi_i,\Phi_j]^2+\frac{\mu_1^2}{2}\sum_{a=1}^3\Phi_a^2+\frac{\mu_2^2}{8}\sum_{i=4}^9\Phi_i^2+\frac{2i\alpha}{3}\sum_{a,b,c=1}^3\epsilon_{abc}\Phi_a\Phi_b\Phi_c\bigg].
\end{eqnarray}
The phase structure of the model $\mu_1=\mu_2=0$ was studied using the Monte Carlo method in \cite{Kawahara:2007nw}. In particular, it was shown that a fuzzy sphere phase exists above some value $\tilde{\alpha}_c$ of the coupling constant $\tilde{\alpha}=\alpha_cN^{1/3}$ where the Hagedorn/deconfinement transition is found to be absent. This critical value $\tilde{\alpha}_c$ is a function of the scaled temperature $\tilde{T}=T/N^{2/3}$. Below $\tilde{\alpha}_c$ the Yang-Mills phase is observed to be divided by the Hagedorn/deconfinement temperature $T_H$ of the original bosonic BFSS model (the model with $\alpha=0$).
The supersymmetric BMN model (\ref{BMN}) seems however to support the Hagedorn/deconfinement transition (a single phase of of first order) in the fuzzy sphere background \cite{Furuuchi:2003sy, Kawahara:2006hs}. A first order Hagedorn/deconfinement transition is also expected to appear in the supersymmetric BFSS model (\ref{BFSS}) \cite{Barbon:1998cr, Aharony:2005ew}
It is very interesting to consider also the phase structure of the Gaussian approximation of the above model obtained in the limit $d\longrightarrow \infty$ which is given by (with $m=d^{1/3}$)
\begin{eqnarray}
S_{}=\frac{1}{g^2}\int_0^{\beta}dt{\rm Tr}\bigg[\frac{1}{2}(D_t\Phi_i)^2+\frac{1}{2}m^2\Phi_i^2+\frac{\mu_1^2}{2}\sum_{a=1}^3\Phi_a^2+\frac{\mu_2^2}{8}\sum_{i=4}^9\Phi_i^2+i\alpha\sum_{a,b,c=1}^3\epsilon_{abc}\Phi_a\Phi_b\Phi_c\bigg].
\end{eqnarray}
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