text:
\section{Introduction}

The Sachdev-Ye-Kitaev (SYK) model \cite{Sachdev:1992fk,KitaevTalks,Polchinski:2016xgd,Maldacena:2016hyu},
\begin{equation}
  S_{\text{SYK}} = \int dt \left(\sum_j \frac{i}{2}\psi_j\partial_t \psi_j - i^{q/2}\sum_{i_1 \dots i_q} J_{i_1\dots i_q} \psi_{i_1}\dots \psi_{i_q} \right)\,,
\end{equation}
 is a quantum mechanical model describing $N$ Majorana fermions $\psi_i$ interacting via a non-linear potential involving gaussian random couplings $J_{i_1\dots i_q}$. It is solvable in the large $N$ limit, exhibits approximate conformal symmetry in the IR, and saturates the quantum chaos bound \cite{Shenker:2013pqa, Shenker:2013yza,Shenker:2014cwa,Maldacena:2015waa,Reynolds:2016pmi}. It has been proposed as the holographic dual of 2d dilaton gravity based on the observation that both sides of the correspondence share the same Goldstone mode effective action, written in terms of the Schwarzian derivative \cite{Almheiri:2014cka,Sachdev:2015efa,Maldacena:2016upp,Jensen:2016pah,Sekino:2008he,Engelsoy:2016xyb,Cvetic:2016eiv}.

In this paper, we consider a 1+1 dimensional generalisation of the SYK model, by considering the Thirring model with random couplings. i.e.
\begin{equation}
\label{IntroMod}
S ={1 \over 2  {\bf \pi} }  \int d^2z \ \left[ \sum_i \left({\bar\nu}^i \partial_z {\bar\nu}^i  + \nu^i  \partial_{\bar z}  \nu^i\right)   + \sum_{i<j,k<l}J_{ij;kl}\nu^i\nu^j{\bar\nu}^k{\bar\nu}^l\right]\,.
\end{equation}
with random couplings $J_{ij;kl}$ drawn from a gaussian ensemble with standard deviation $J$ (we also discuss generalisations of this model to higher powers of $\nu$ and $\bar\nu$). One can use much of the technology developed in the SYK context, though unlike previous generalisations \cite{Gu:2016oyy,Gross:2016kjj,Berkooz:2016cvq}, the couplings are not taken to be spatially disordered. Some further work on SYK can be seen in \cite{You:2016ldz,Anninos:2016szt,Jevicki:2016bwu,Bagrets:2016cdf,Jevicki:2016ito,Banerjee:2016ncu,Garcia-Garcia:2016mno,Fu:2016vas,Witten:2016iux,Cotler:2016fpe,Klebanov:2016xxf,Blake:2016jnn,Davison:2016ngz,Peng:2016mxj,Liu:2016rdi,Krishnan:2016bvg,Magan:2016ehs,Ferrari:2017ryl,Garcia-Garcia:2017pzl,Li:2017hdt,Gurau:2017xhf,Mandal:2017thl}.

The model is also an example of a conformal field theory -- in this case free fermions -- which is perturbed by a large set of marginal operators with random coefficients, some of them being relevant and some irrelevant. In the usual lore, couplings along marginally relevant directions increase, and couplings along marginally irrelevant directions decrease, as one flows towards the IR. In this case, however, RG mixes the two and the anomalous dimension of each operator depends on all of the other random couplings. If we fix the scale at which we define the theory, one expects it will eventually flow in the direction of some relevant operators, but the details and rates at which this happens are unclear.

It is also interesting to ask whether the theory is renormalizable, since we are necessarily turning on marginally irrelevant perturbations -- but on the other hand they are in the same statistical ensemble as the marginally relevant ones. For the model in \eqref{IntroMod}, the two point function cannot be solved completely, but we compute it at large distances (in the sense that we make precise below) for all values of $J$, without exhibiting a singularity, or the generation of a mass gap, in the physical domain\footnote{These statements are true in the large N limit, with a significant caveat that we have so far solved only for the two point function.}. From it we can extract an effective $\beta$ function
\begin{equation}
\beta(J)=4\pi^2 J^3
\label{intro-beta}
\end{equation}
and study the above questions. The $\beta$ function indicates that the interaction is marginally irrelevant so the theory is not conformal. It will also turn out not to be conformal in the limit of $J\to\infty$. Rather, the theory requires regularization for any value of $J$, and this regularization breaks conformal invariance already at the level of the 2-pt function.

Useful reference points are the Thirring or Gross-Neveu models \cite{Moshe:2003xn,ZinnJustin:2002ru,Gross:1974jv} (which are equivalent in our case), where the interaction term is non-random and diagonal between the left and right moving fermions. The $\beta$ function of these models is well understood -- in 2D, the leading term is quadratic in the 4-fermion coupling, and the interaction is marginally relevant for one sign of it. The theory then develops a mass gap (which is seen in \cite{Gross:1974jv} in perturbation theory via the appearance of a tachyonic pole at very low energies).

At finite $N$ we can expect a similar behavior in the extreme IR for each realisation, since each contains both relevant and irrelevant operators, and the former will eventually grow to be large, most likely triggering a mass gap. This, however, is the equivalent of the statement that in the 0+1 SYK the very late time behavior, or the behavior of states close enough to the ground state, differ from realisation to realisation \cite{Cotler:2016fpe}\footnote{See \cite{Balasubramanian:2014gla} for earlier work on black holes and random matrix theory.}.

The large $N$ ensemble average does not capture this expectation, but it captures the flow prior to chiral $\mathrm{SO}(N)_L\times \mathrm{SO}(N)_R$ symmetry breaking, where it is modified relative to the Thirring and Gross-Neveu models -- in our models we see no pole in the physical regime. Furthermore, the leading quadratic term in the $\beta$ function, which drove the RG in the Thirring and Gross-Neveu models, vanishes for the ensemble (in the large N limit). The $\beta$ function \eqref{intro-beta} is positive for any value of $J$, and it is driven by the cubic terms in the random couplings of the $\beta$ function. The latter is non-universal in the Thirring or Gross-Neveu models, but it is universal in our case for schemes preserving the statistical chiral symmetry. In fact, we reproduce the result \eqref{intro-beta} from suitable average considerations of the perturbative conformal field theory results reported in \cite{Gaberdiel:2008fn,Behr:2013vta}.

The 2-pt function therefore does not show signs of the chiral phase transition, at leading order in $1/N$, for any value of the coupling. It is known that in low dimensions quenched disorder can smooth out thermal phase transitions \cite{Imry:1975zz}. Our results generalize that statement to translationally invariant theories, with the chiral phase transition being smoothed out at this order. To our knowledge this is the first demonstration of smoothing out of a phase transition by translationally invariant ensemble average in a field theory.

One final note about the application of this model to black hole physics. The obvious analogy is to consider this model as related to a black hole in $AdS_3$. However, we do not want to consider this as the full dual to the entire $AdS_3$ space because the $\beta$ function is positive and the theory will not be asymptote to empty $AdS_3$. Perhaps a better way to think about this model is as describing the effective interactions of degrees of freedom inside a BTZ black hole \cite{Berkooz:2016cvq}. One then needs to 1) "prevent" these degrees of freedom from appearing at high momenta, and 2) couple additional "probe" fields to the model, which will mimic the degrees of freedom outside the black hole\footnote{A related statistical mechanical model appears in \cite{Banerjee:2016ncu}.}.

This paper is organised as follows. In section \ref{defmodel}, we define the Random Thirring model. In section \ref{SDsec}, we compute the fermion 2-pt function by solving the Schwinger-Dyson (SD) equations in the IR, for all values of $J$, and show how the 2-pt function flows. We also discuss the reparameterization invariance of the model. In section \ref{pCFT}, we compare our model to the flow of the usual Thirring model (or the Gross-Neveu model in our case). We compute the $\beta$ function and show which terms are washed out due to the randomness of the coupling, and which remain active. Section \ref{disc} contains some concluding remarks.

During the completion of this work, the preprint \cite{Turiaci:2017zwd} appeared which contains a 1+1 generalisation of the SYK model with the same interaction term. Although some of the intermediate steps are similar, the UV starting point of our model is different than the one described there, as is our solution and its implications for the RG flow of the model.

\vspace{20pt}
\section{The Random Thirring, or SYK, model in 1+1 dimensions}
\label{defmodel}

Consider a 2d non-chiral theory with N left moving $\nu^i$ and N right moving ${\bar\nu}^i$ ($i=1,\dots ,N$). Its free theory action and partition function in Euclidean space are given by
\begin{equation}
\label{FreeMaj2}
S_{\text{free}}(\nu,\,\bar{\nu}) = \frac{1}{2\pi} \int d^2z \ \sum_i\left[ {\bar\nu}^i \partial_z {\bar\nu}^i  + \nu^i  \partial_{\bar z}  \nu^i  \right],\quad Z=\int D\nu D{\bar \nu} \,e^{-S(\nu,\,\bar{\nu})}\,.
\end{equation}
The free propagators are
\begin{equation}
\begin{aligned}
\langle \nu_i(z_1,\bar z_1) \nu_j(z_2,\bar z_2) \rangle = \frac{\delta_{ij}}{z_1 - z_2} \equiv G_{\nu}^0(\vec x_1,\vec x_2) \delta_{ij}, \\
\langle \bar \nu_i(z_1,\bar z_1) \bar \nu_j(z_2,\bar z_2) \rangle = {\delta_{ij} \over \bar z_1 - \bar z_2}\equiv G_{\bar \nu}^0(\vec x_1,\vec x_2) \delta_{ij}
\end{aligned}
\end{equation}
where $\vec x$ denotes the pair $z,{\bar z}$. Their Fourier transforms equal
\begin{equation}
  G^{0}_\nu(p)=\frac{i\pi}{\bar p}, \quad G^{0}_{\bar\nu}(p)=\frac{i\pi}{p}
\end{equation}
For more details on our conventions, see appendix \ref{2dcft}.

The free theory is $\mathrm{SO}(N)_L \times \mathrm{SO}(N)_R$ invariant, with conserved currents
\begin{equation}
J^{[ij]}(z) = i \ \frac{\nu^{i}\nu^{j}(z) - \nu^{j}\nu^{i}(z)}{2} ,\quad {\bar J}^{[ij]}(\bar z) =i \ \frac{{\bar\nu}^{i}{\bar\nu}^{j}({\bar z}) - {\bar\nu}^{j}{\bar\nu}^{i}({\bar z}) }{2}  .
\end{equation}
defined for a pair $[ij]$ with $i<j$. Using these currents, we can add an interaction of the form
\begin{equation}
{\cal L}_{int}=- \sum_{i<j,k<l}J_{ij;kl} J^{[ij]}{\bar J}^{[kl]}={1\over 4}\sum_{i,j,k,l}J_{ij;kl}\nu^{i}\nu^{j}{\bar\nu}^{k}{\bar\nu}^{l},\ \ \ J_{ij;kl}=-J_{ji;kl}=-J_{ij;lk}
\label{model}
\end{equation}
The model we discuss in this work consists in treating the coupling constants $J_{ij;kl}$ as random variables drawn independently from a gaussian distribution defined by
\begin{equation}\label{NewJAv}
\langle \langle J_{ij;kl}^2 \rangle \rangle = \frac{2J^2}{ N^3}
\end{equation}
where $\langle \langle\ \  \rangle \rangle$ denotes ensemble average over $J_{ij;kl}$.

The resulting model can either be viewed as a translationally invariant continuum generalisation of the SYK model in 1+1 dimensions, or as a random cousin of the massless $\mathrm{SO}(N)$ Thirring model, where by the latter we mean the theory whose interaction Lagrangian involves a fixed diagonal coupling $J_{ij;kl} \sim g\,\delta_{ki}\delta_{lj}$
\begin{equation}
\label{ThirSec2}
  {\cal L}_{int} \propto \frac{g}{2} \sum_{i,j} J^{[ij]}\,{\bar J}^{[ij]}\,.
\end{equation}
We will refer to the model in equation \eqref{FreeMaj2} and \eqref{model} as the random Thirring model.

Though much of our discussion will be about \eqref{model}, there are two natural generalisations to consider. The first involves a higher order interaction term,
\begin{equation}
{\cal L}_{q,int}=\sum_{i_1,..i_q,k_1,..k_q}J_{i_1..i_q;k_1..k_q} \nu^{i_1}...\nu^{i_q}{\bar\nu}^{k_1}...{\bar\nu}^{k_q},\ \ q>2
\label{2qmodel}
\end{equation}
as already discussed in \cite{Turiaci:2017zwd}. The second involves the inclusion of a low-pass filter multiplying each fermion in Fourier space
\begin{multline}
{\cal L}_{q,int}=\int dp_1..dp_q du_1..du_q \delta(\sum p_i+\sum u_i) \sum_{i_1,..i_q,k_1,..k_q}J_{i_1..i_q;k_1..k_q} \\
(F(p_1)\nu^{i_1}(p_1))...(F(p_q)\nu^{i_q}(p_q))
(F(u_1){\bar\nu}^{k_1}(u_1))...(F(u_q){\bar\nu}^{k_q}(u_q)),\ \ q\ge 2,
\end{multline}
where the filter $F(k)$ decays at large momenta. The use of such filters, introduced in this context in \cite{Berkooz:2016cvq}, might be useful for holography in order to force a distinction between UV and IR degrees of freedom, but clearly it will be very restricted from a local field theory point of view.

Our model \eqref{model} has two types of symmetries : exact and statistical. The first ones hold for any realisation of the couplings, whereas the second ones only arise after carrying the coupling average. Poincar\'e symmetry and a $\mathbb{Z}_2$ symmetry $\nu\rightarrow -\nu$ and ${\bar\nu}\rightarrow{\bar \nu}$ for even q, or its composition for odd q, are exact symmetries of the Lagrangian. Once the ensemble average is carried out, and assuming self-averaging, i.e. that any realisation behaves similarly to the ensemble average, there is an emergent $\mathrm{SO}(N)_L \times \mathrm{SO}(N)_R$ symmetry, rotating left and right Majorana fermions independently, together with a parity exchanging $\nu^i\leftrightarrow {\bar\nu}^i$.

As in the case of the SYK model, we will sum over different realisations of the couplings $J_{ij;kl}$. If the model is self-averaging then this should reproduce, up to small corrections, the results of any specific realisation. However, for a specific realisation the random Thirring model and its $q>2$ generalisations are ordinary field theories. For $q>2$ the interaction term is irrelevant and we don't expect it to generate new interesting dynamics in the IR (although it can generate one in some intermediate scale). For $q=2$ the situation is more interesting, as we perturb the theory by a large number of operators, some of them relevant and some others irrelevant. In fact all the couplings mix together under RG flow but the qualitative picture that some are (marginally) relevant and some are (marginally) irrelevant is expected to hold. Further discussion on these points will appear in section 4.

\vspace{20pt}
\section{Two Point Function : Schwinger-Dyson Equations}
\label{SDsec}

To derive the Schwinger-Dyson (SD) equations for our models, we use the standard replica method \cite{Sachdev:2015efa} to perform the disorder average. If we further assume the replica symmetry is unbroken, the single replica action acquires the following bi-local interaction term
\begin{equation}\label{BilocalIntineta}
  S_{int} \propto J^2 N  \int d^2\!  \vec x_1 d^2\!  \vec x_2 \left( {\sum_i \bar{\nu}^i(\vec x_1) \, \bar\nu^i(\vec x_2 )\over N} \right)^2 \left( {\sum_j\nu^j (\vec x_1 ) \,\nu^j(\vec x_2 ) \over N} \right)^2\,.
\end{equation}

Next, we introduce two Lagrange multipliers $\Sigma_\nu(\vec x_1,\vec x_2), \Sigma_{\bar \nu}(\vec x_1,\vec x_2)$, whose equations of motion impose the constraints
\begin{equation}
G_\nu(\vec x_1,\vec x_2) =\frac{1}{N} \sum_i \nu^i(\vec x_1) \nu^i(\vec x_2)\,, \quad  G_{\bar \nu}(\vec x_1,\vec x_2) =\frac{1}{N}\sum_i \bar \nu^i(\vec x_1) \bar \nu^i(\vec x_2)\,.
\end{equation}
This is achieved by adding to the action the extra term
\begin{equation}
\delta S \propto N\int d^2\!  \vec x_1 \int d^2 \! \vec x_2 \left[ \Sigma_\nu( \vec x_1,\vec x_2) \left( G_\nu(\vec x_1,\vec x_2) - {\sum_i \nu^i(\vec x_1) \nu^i(\vec x_2) \over N}  \right) + (\nu^i \to \bar \nu^i)  \right]
\end{equation}
with $\Sigma(\vec x_1,\vec x_2)=-\Sigma(\vec{x}_2,\vec{x}_1)$. The resulting action is quadratic in the fermions. Performing their gaussian integral, we are left with the action:
\begin{equation}
\label{ColAction}
\begin{split}
- \frac{S}{N} & \propto   \log \text{Pf} (\partial_{\bar z} -\Sigma_\nu(\vec x_1, \vec x_2)) + \log \text{Pf} (\partial_{  z} -\Sigma_{\bar \nu}(\vec x_1, \vec x_2))  - \int d^2\! \vec x_1 d^2\! \vec x_2  \\
& \ \ \ \ \times \left[ \Sigma_\nu( \vec x_1,\vec x_2)   G_\nu(\vec x_1,\vec x_2) + \Sigma_{\bar \nu}( \vec x_1,\vec x_2)  G_{\bar \nu}(\vec x_1,\vec x_2) - \frac{J^2}{2} G_\nu(\vec{x}_1,\vec{x}_2)^2 G_{\bar \nu}(\vec{x}_1,\vec{x}_2)^2     \right]
\end{split}
\end{equation}
The equations of motion for $G_\nu(\vec{x}_1,\vec{x}_2)$ and $G_{\bar \nu}(\vec{x}_1,\vec{x}_2)$ are
\begin{equation}\label{DefG}
\begin{split}
\Sigma_\nu(\vec x_1,\vec x_2 )   &=  J^2 G_\nu(\vec x_1,\vec x_2 )\, \bar{G}_{\bar \nu}(\vec x_1,\vec x_2 )^2\,, \\
\Sigma_{\bar \nu}(\vec x_1,\vec x_2 )   &=   J^2  \bar{G}_{\bar \nu}(\vec x_1,\vec x_2 )  \, G_\nu(\vec x_1,\vec x_2 )^2\,,
\end{split}
\end{equation}
whereas the equations of motion for $\Sigma_\nu(\vec{x}_1,\vec{x}_2)$ and $\Sigma_{\bar \nu}(\vec{x}_1,\vec{x}_2)$ are then:
\begin{equation}\label{DefSigma}
\begin{split}
G_\nu(\vec p_1,\vec p_2 )^{-1} &=  G^{0}_{\nu}(\vec p_1,\vec p_2)^{-1} - \Sigma_\nu(\vec p_1,\vec p_2)\,,  \\
G_{\bar \nu}(\vec p_1,\vec p_2 )^{-1} &=  \bar{G}^{0}_{\bar \nu}(\vec p_1,\vec p_2)^{-1} - \bar{\Sigma}_{\bar \nu}(\vec p_1,\vec p_2)\,.
\end{split}
\end{equation}

The same procedure can be carried out for the 2q fermion interaction model \eqref{2qmodel}. The corresponding SD equations are
\begin{equation}\label{FullSDEqnq1}
\begin{aligned}
 G_\nu(\vec p_1,\vec p_2 )^{-1} &=  {G^{0}_\nu}(\vec p_1,\vec p_2 )^{-1} -    \Sigma_\nu(\vec p_1,\vec p_2 )\,,      \\
 G_{\bar \nu}(\vec p_1,\vec p_2 )^{-1} &=   G^{0}_{\bar \nu}(\vec p_1,\vec p_2 ){-1}   -    \Sigma_{\bar \nu}(\vec p_1,\vec p_2 )\,,\\
  \Sigma_\nu(\vec x_1,\vec x_2 )  &=  J^2\,G_\nu(\vec x_1,\vec x_2 )^{q-1}\, G_{\bar \nu}(\vec x_1,\vec x_2 )^{q}\,,  \\
  \Sigma_{\bar \nu}(\vec x_1,\vec x_2 ) & = J^2\, G_{\bar \nu}(\vec x_1,\vec x_2 )^{q-1}\, G_\nu(\vec x_1,\vec x_2 )^{q}\,.
\end{aligned}
\end{equation}

\vspace{10pt}
\subsection{A comment on reparametrization invariance}\label{sec:reparam}

Consider the $J\to \infty$ limit of the collective field action \eqref{ColAction} for the 2q fermion model \eqref{2qmodel}. This results in dropping the dependence on the free propagator in the equations of motion \eqref{FullSDEqnq1}. It is convenient to rewrite the first two equations in real space
\begin{equation}
\label{SDrspace}
\begin{aligned}
  \int d^2z\, G_\nu(z',z;\bar z',\bar z) \Sigma_\nu(z, z'';\bar z , \bar z'') &= -\delta(z'-z'') \delta(\bar z'-\bar z'')\,, \\
  \int d^2z\, G_{\bar \nu}(z',z;\bar z',\bar z) \Sigma_{\bar \nu}(z, z'';\bar z , \bar z'') &= -\delta(z'-z'') \delta(\bar z'-\bar z'')\,,
\end{aligned}
\end{equation}
to discuss their symmetries. As pointed out in \cite{Turiaci:2017zwd}, these equations of motion appear to be invariant under reparametrization.

For the case $q=2$, which we will argue below is the only physically interesting one for us, we can also keep the free kinetic term and reintroduce it into equations \eqref{SDrspace}. In this case the equations are invariant under conformal transformation $z\to f(z)$ and $\bar z \to \bar f(\bar z)$. To see this, consider the 2-pt function transformations
\begin{equation}\label{eq:TwoPointTransf}
\begin{aligned}
  G_\nu(z,z';\bar z,\bar z') &\to [f'(z)\,f'(z')]^{\Delta_L}  [\bar f'(\bar z)\bar f'(\bar z')]^{\bar\Delta_L}\, G_\nu(f(z),f(z');\bar f(\bar z),\bar f(\bar z'))\,, \\
  G_{\bar \nu}(z,z';\bar z,\bar z') &\to [f'(z) f'(z')]^{\Delta_R}  [\bar f' (\bar z)\bar f'(\bar z')]^{\bar\Delta_R} G_{\bar \nu}(f(z),f(z');\bar f(\bar z),\bar f(\bar z'))\,.
\end{aligned}
\end{equation}
The last two equations in \eqref{FullSDEqnq1} determine the transformation for the self-energies $\Sigma_\nu$ and $\Sigma_{\bar{\nu}}$, and the invariance of equations \eqref{SDrspace} (with the free kinetic term reinstated) requires that  $(\Delta_L,\Delta_R)=({1 \over 2},0)$ and $({\bar\Delta}_L,{\bar\Delta}_R)=(0,{1 \over 2})$
(for a general $q$, dropping the kinetic term, invariance under conformal transformations implies only that  $\Delta_L + \Delta_R = \bar{\Delta}_L + \bar{\Delta}_R = \frac{1}{q}$).

The conclusion from this analysis would suggest to solve the SD equations with a scale invariant ansatz for any $J$ for $q=2$ (and for large $J$ for $q>2$). Below, we will argue this is not necessarily the case. This quantum field theory requires regularization, which breaks conformal invariance. A-priori this symmetry may or may not be restored in the IR (or may be restored upon fine tuning some operators away). Actually, here it will not be restored. We will see this explicitly for the $q=2$ theory - the solution would not be scale invariant (although in a mild sense). It is important to emphasize that the regulator needs to be included already in the action, and the breaking of scale invariance is explicit (all the way to the IR) and not spontaneous.

As an aside we would also like to comment that the conformal symmetry of the action is also broken in the IR, beyond the effects of the regulator. The reason is that the action \eqref{ColAction} should be appended by boundary conditions, or, in the language of the equations of motions, the solutions are legitimate only when the functions decay fast enough at infinity. This is nothing but the usual argument why ordinary 2D CFTs can have only an $\mathrm{SL}(2)\times \mathrm{SL}(2)$ symmetry on the plane - all other generators of the Virasoro algebra introduce singularities at infinity, which change an n-point function into an n-point function in the presence of an operator at infinity.

This situation is different from reparametrization invariance in the SYK model in 0+1 dimensions. In that case, at the level of the 2-pt function, the action was consistently reparametrization invariant and a scale invariant solution could be found. Reparametrization was then broken spontaneously. This pattern of symmetry breaking was used heavily in computing the 4-pt function, although in order to make sense of it one had to eventually re-introduce the explicit breaking of reparametrization. Here the cut-off breaks reparametrization explicitly and the solution is not scale invariance. The net result of an explicit breaking of scale invariant (let alone reparametrization) is similar here and in the 0+1 SYK model, but the differences in the intermediate stage suggests that computing the 4-pt function would be considerably more involved.

\vspace{10pt}
\subsection{Solutions of the SD equations -- general q}

In this subsection we will look for solutions to the SD equations \eqref{FullSDEqnq1} for various values of $q$. We will show that for $q>2$ the interaction term is subleading in the IR, and for $q=2$ it introduces an unusual form of a logarithmic running into the 2-pt function. We will assume translation invariance and, to evaluate the importance of various terms, we will carry out a scaling argument in the IR. We will also assume that Lorentz symmetry (Euclidean rotation) is unbroken throughout the flow and hence the UV Lorentz numbers of the fermions remain fixed. We will also assume that parity is unbroken. Under these assumptions
\begin{equation}
G(z_1-z_2,\bar z_1-\bar z_2) \equiv G_{\nu}(z_1,\bar z_1, z_2, \bar z_2) = G_{\bar \nu}(\bar z_1,  z_1, \bar z_2,  z_2)
\end{equation}
and one can similarly define $\Sigma(z,\bar z)$. Then the SD equations \eqref{FullSDEqnq1} collapse to
\begin{eqnarray}\label{SimpSD}
G(p, \bar p)^{-1} = G^{(0)}(p, \bar p)^{-1} - \Sigma(p,\bar p)\,, \quad \Sigma(z,{\bar z}) = J^2 G(z,{\bar z})^{q-1} G({\bar z},z)^{q}
\end{eqnarray}

Consider a scaling solution
\begin{equation}
G(\lambda z,{\bar \lambda} {\bar z})=\lambda^{-2\Delta_L}{\bar\lambda}^{-2\Delta_R}G(z, {\bar z}),\quad G(\lambda^{-1}\,p, \bar{\lambda}^{-1}\bar p)=\lambda^{-2\Delta_L+1}{\bar\lambda}^{-2\Delta_R+1}G(p,\bar p)
\label{scaling}
\end{equation}
with $2(\Delta_L-\Delta_R)=1$, as dictated by invariance under Euclidean rotations. These solve the SD equations if
\begin{equation}
\Delta_L + \Delta_R = \frac{1}{q}  \implies \quad  2\Delta_{L} =  {1 \over q} + {1 \over 2} \quad , \quad 2\Delta_R  = {1 \over q} -{1 \over 2}
\end{equation}
We treat the $q>2$ and $q=2$ cases separately.

For $q>2$ we obtain $\Delta_R < 0$, which means the solutions are not physical. If the model is self-averaging (which we assumed before) then correlators should be close to being correlators in a unitary field theory. This necessitates that both $\Delta_L$ and $\Delta_R$ are non-negative. Or, related to it, the free propagator $G^{(0)}(p_z,p_{\bar z}) \sim {1 \over \bar p}$ already scales as $\bar \lambda$ and is leading at low momenta. Basically this means the theory is free in the IR\footnote{One might still be able to work in intermediate energy regimes, where the above approximation might be correct, but it will break down at sufficiently low energies.}. This is what one expects from field theory considerations as well. Deforming a free fermion theory by operators of the form $\nu^q{\bar \nu}^q$, which are operators of dimension $(q/2,q/2)$, the perturbation is irrelevant\footnote{This is also potentially where we differ from \cite{Turiaci:2017zwd} which add the interaction term to an altogether different theory.}. Thus, we will not consider the $q>2$ case further.

\vspace{10pt}
\subsection{Solutions of the SD equations for q=2}

The scaling argument \eqref{scaling} suggests that the solutions to the SD equation \eqref{SimpSD} for the $q=2$ model are
\begin{equation}
G(z,\bar z) = \frac{C}{z}, \ \ \ \Sigma(z,\bar z) = \frac{C^3 J^2}{z \bar z^2}\,,
\end{equation}
for some constant $C$. The difficulty with this ansatz originates in the Fourier transform of $\Sigma(z,\bar z)$. The latter is not well defined and requires the introduction of a scale. Following the conventions in appendix \ref{2dcft}, it is given by
\begin{equation}
\label{FT1byzbz2}
  \frac{1}{z \bar z^2} \propto \int {d^2\!p_z \over (2\pi)^2} e^{-i (pz + {\bar p} \bar z)}  {\bar p} \log(\Lambda^2/|p|^2)
\end{equation}
We would therefore like to see what the solution to the SD equations is. Our approach will be to modify our ansatz to include some dependence on logarithms, which will turn out to be essential to encode the RG flow of the model. We will present the large J solution in the IR before proceeding to the all-J solution (still at large distances).

\vspace{10pt}
\subsubsection{The Large J IR solution}

We first consider the large $J$ limit, where the free propagator terms in the SD equations can be neglected. If we interpret the presence of the logarithms above as inducing some soft RG flow on top of some leading power law term, as dictated by the scaling argument \eqref{scaling}, it may appear natural to consider the ansatz
 \begin{equation}\label{Jansatz}
G(z,\bar z) = \frac{A}{z} \left(\log(z{\bar z}\Lambda^2)\right)^\alpha\,, \quad \Sigma(z,\bar z)=\frac{A^3J^2}{z{\bar z}^2} \left(\log(z{\bar z}\Lambda^2)\right)^{3\alpha}\,.
\end{equation}
This depends on two parameters $A$ and $\alpha$, to be determined by solving the SD equations, and a cut-off scale $\Lambda$, whose role as a UV cut-off will become more apparent below. In this ansatz, we solve the "interaction" SD equation exactly, and then we analyse whether there is any regime of low momentum where our ansatz is self-consistent, i.e., the "propagator" SD equation is solved (at least in some approximation).

Working in momentum space, we observe that the Fourier transforms $G(p,{\bar p})$ and $\Sigma(p,{\bar p})$ of our ansatz \eqref{Jansatz} satisfy
\begin{equation}
\begin{aligned}
G(p,{\bar p}) &=\partial_{\bar p} {\hat G}(p,{\bar p})\,, \quad \text{with} \quad {\hat G}(p,{\bar p}) \equiv {A \over i} \int {d^2\!z \over 2 } {1 \over z{\bar z}} \log^\alpha(\Lambda^2z{\bar z})\,e^{ipz+i{\bar p}{\bar z}} \\
\partial_{\bar p}\Sigma(p,{\bar p}) &= A^3J^2i \int {d^2\!z \over 2} { \log^{3\alpha}(\Lambda^2z{\bar z}) \over z{\bar z}}\,  e^{ipz+i{\bar p}{\bar z}}\,.
\end{aligned}
\end{equation}
Hence, the integral we need to evaluate is
\begin{equation}
\label{original}
\begin{aligned}
F_\alpha(|p|/\Lambda)&\equiv \int {d^2\!z \over 2} {1\over z{\bar z}} \log(\Lambda^2z{\bar z})^\alpha e^{ipz+i{\bar p}{\bar z}}= \int { d^2\!\vec x   \over |\vec x|^2 } \log^\alpha(\Lambda^2 |\vec x|^2)\, e^{i \vec p . \vec x} \\
 &= 2^\alpha \int { dr \over r } \log^\alpha (\Lambda r) \int d\theta e^{i p r \cos \theta} =
(2 \pi) \ 2^\alpha   \int_0^\infty {dr \over r} \log^\alpha(\Lambda r) J_0(|p| r)\,,
\end{aligned}
\end{equation}
where we used the integral representation of the Bessel function $J_0(x)$ to perform the integration over the angular variable and $|p|^2=4p\,\bar p={\vec p}^2$ according to our conventions in appendix \ref{2dcft}.

To evaluate this integral, we start from the convergent formula\footnote{See formula 6.771 in \cite{gradshteyn}, for example.}
\begin{equation}
\begin{aligned}
G_\epsilon(|p|/\Lambda) &\equiv \int_0^\infty dr \ (\Lambda r)^\epsilon \  {\log(\Lambda r) \over r} \ J_0(|p| r) \\
&= {\Gamma({\epsilon \over 2}) |\Lambda/p|^\epsilon \over \Gamma(1-{\epsilon \over 2}) 2^{2 -\epsilon}  } \left[ \psi({\epsilon \over 2}) + \psi(1-{\epsilon \over 2}) +2 \log(2 |\Lambda/p| )\right]\,,
\end{aligned}
\end{equation}
valid for $0<\epsilon< \frac{3}{2}$ and where $\psi(x)$ is the Digamma function. Notice we can compute $F_\alpha(|p|/\Lambda)$ for $\alpha = n \in \mathbb Z$ from
\begin{equation}
  F_{n+1}(|p|/\Lambda)= 2^{n+2}  \pi \lim_{\epsilon \to 0} \partial_\epsilon^n G_\epsilon(|p|/\Lambda)\,.
\label{Fdev}
\end{equation}

To proceed, we can write $G_\epsilon(|p|/\Lambda)$ for small $\epsilon$ as
\begin{equation}
\begin{split}
G_\epsilon(|p|/\Lambda) =&   \left( {1\over\epsilon } + (\log 2- \gamma)+{\cal O}(\epsilon) \right)   e^{\epsilon \log(\Lambda/|p|)} \biggr(-{1\over \epsilon} -  \gamma +{\cal O}(\epsilon) + \log(2\Lambda/|p|) \biggr) \\
=& - {1 \over \epsilon^2} +  \sum_{n=0}^\infty  {\epsilon^n \log^{n+2}(\Lambda/|p|) \over n+2} \left[ 1 + {\cal O}({1 \over \log(\Lambda/|p|)}) \right] \\
\end{split}
\end{equation}
where $\gamma$ is the Euler's constan.
Plugging this leading logarithm into \eqref{Fdev}, and dropping the divergent momentum independent $\epsilon^{-2}$ term and the subleading momentum dependent logarithms, we get
\begin{equation}
\label{GenFormFn}
  F_n(|p|/\Lambda) = \frac{\pi}{n+1}  \log^{n+1}( \frac{\Lambda^2}{|p|^2})  \left( 1 +{\cal O}({1 \over \log(\Lambda/|p|)})\right) \,.
\end{equation}

Removing the $1/\epsilon^2$ divergences in the procedure above resembles dimensional regularization, but since we do not understand the renormalization of the model well enough we provide an explicit argument. In appendix \ref{alter}, we provide a specific regularisation of our original integral \eqref{original} leading to the same conclusion in the same IR regime. The gist of the argument is that since $\Lambda$ is our cut-off, we should not do the integral below a distance $\Lambda^{-1}$. Since the integral is convergent for $\epsilon > 0$, we can write
\begin{equation}
\int_{\Lambda^{-1}}^\infty dr\dots = \int_0^\infty dr \dots -  \int_0^{\Lambda^{-1}} dr \dots
\end{equation}
Changing variables to $y=\Lambda r$, working in the limit of small $|p|\Lambda$ and expanding $J_0$ (since the range of integration is finite in $y$), we obtain in the second integral a divergent $1/\epsilon^2$ piece which cancels the divergence from the first integral, justifying the derivation of equation \eqref{GenFormFn}.

We would like to extend our conclusion for integer $\alpha$ to a general one
\begin{equation}
F_\alpha(|p|/\Lambda)=\frac{\pi}{\alpha+1} \left(\log (\Lambda^2/|p|^2)\right)^{\alpha+1} \left(1+{\cal O}(1/\log(|p|/\Lambda)) \right)\,.
\label{Fresult}
\end{equation}
by the following qualitative argument (which becomes more precise for $\alpha$'s which are negative enough). Working with the dimensionless variable $y=\Lambda\,r$,
\begin{equation}
F_\alpha(|p|/\Lambda) = \pi\, 2^{\alpha+1}\,\int_1^\infty \frac{dy}{y} \left(\log y\right)^\alpha J_0(|p|\,y/\Lambda)\,.
\label{Fintegral}
\end{equation}
When $|p|/\Lambda$ is small, we can split the integral \eqref{Fintegral} into $[1,\, C\, \Lambda/|p|]$ and $[C\,\Lambda/|p|,\,\infty)$, where $C$ is finite and smaller than 1. In the first interval, the argument of the Bessel function is generically small and the function can be approximated by one. The resulting integral immediately reproduces \eqref{Fresult}. We are left to argue the contribution from the second interval is subleading. This is so for two reasons. First, when the argument $z$ of the Bessel function is large, the latter can be approximated by $z^{-1/2}\,\cos (z - \pi/4)$ (up to coefficients of order one). The integral involves a non-oscillating function multiplied by this oscillating decaying function. Second, for $\alpha$ negative and large enough in absolute value, the logarithm is small in this range.

In any case, the integral can also be evaluated numerically for the specific values of $\alpha$ that interest us in the same regime
$|p|/\Lambda \ll 1$. These confirm the expression \eqref{Fresult}. In the next subsection we present additional arguments for the validity of this large J solution.

Using \eqref{Fresult} determines the two Fourier transforms
\begin{equation}
\begin{aligned}
  G(p,{\bar p}) &=  i\frac{A\pi}{\bar p} \left(\log \frac{\Lambda^2}{|p|^2}\right)^{\alpha}\,, \\
  \partial_{\bar p}\Sigma(p,{\bar p}) &= i\frac{A^3J^2\pi}{3\alpha+1}\left(\log \frac{\Lambda^2}{|p|^2}\right)^{3\alpha+1} \quad \Rightarrow \quad \Sigma(p,{\bar p}) = i{\bar p} \frac{A^3J^2\pi}{3\alpha+1}\left(\log \frac{\Lambda^2}{|p|^2}\right)^{3\alpha+1}\,.
\end{aligned}
\end{equation}
The SD equation $G(p,{\bar p})\,\Sigma(p,{\bar p}) = -1$ requires
\begin{equation}
4\alpha+1=0,\ \ \ -\frac{A^4J^2 \pi^2}{3\alpha+1}=-1 \quad \Rightarrow \quad \alpha = -\frac{1}{4}\,, \,\,\, A^4=\frac{1}{4\pi^2\,J^2}\,.
\end{equation}

As a quick consistency check that this is a sensible result, note that $A$ is positive. This is a necessary condition because it has to correspond to a 2-pt function
\begin{equation}
\sum_i \langle \phi | \nu |i\rangle e^{-\tau E_i} \langle i | \nu | \phi \rangle
\end{equation}
(where $\nu$ is an Hermitian operator) which is positive.

\vspace{10pt}
\subsubsection{Solution for any J}

Another regime to consider in coupling space is small $J$, by carrying out perturbation theory in $J$. We can actually do better and find an all-$J$ solution, by remaining in the IR, i.e. large $\Lambda^2 z{\bar z}$ or $\Lambda^2/|p|^2$. We will assume an ansatz of the form
\begin{eqnarray}\label{perturbative anzatz}
G(z,\bar z) &=& \frac{1}{z}  \sum_{n=0}^\infty  a_n J^{2n} \log^n(\Lambda^2 z \bar z) \xrightarrow{|z| \gg \Lambda^{-1}, |p| \ll \Lambda} G(p , \bar p ) = \frac{i \pi}{\bar p}  \sum_{n=0}^\infty  a_n J^{2n} \log^n\frac{\Lambda^2}{|p|^2} \\
\nonumber
\Sigma(z,\bar z) &=& \frac{J^2}{z \bar z^2}  \sum_{n=0}^\infty  b_n J^{2n} \log^n(\Lambda^2 z \bar z) \xrightarrow{|z| \gg \Lambda^{-1}, |p| \ll \Lambda}  \Sigma(p , \bar p ) = i \pi  {\bar p}   J^2\sum_{n=0}^\infty  \frac{b_n}{n+1}\, J^{2n} \log^{n+1}\frac{\Lambda^2}{|p|^2}
\end{eqnarray}
with $a_0=1$ to match the free theory propagator. Note that this is not the most general expansion in $J$ and in logarithms that we could consider. In fact if we solve \eqref{DefG}-\eqref{DefSigma} perturbatively in $J$, the coefficient of $J^n$ in \eqref{perturbative anzatz} will have corrections which are lower powers of $\log(\Lambda^2 z \bar z)$ than indicated. However, at lengths much larger than the cutoff scale, these terms are subleading. We will show that the above ansatz is self-consistent, and also solves the SD equations with the right limits $J\to 0$ and large J, i.e. it re-sums the perturbative series allowing to interpolate between both expansions, as long as the IR criteria above is satisfied.

Instead of plugging our expansions in the SD equations to determine the coefficients $a_n,\,b_n$ recursively, we notice that the relevant Fourier transforms, summarized in eq. (\ref{Fourier2}-\ref{Fourier3}) (whose operation is denoted by ${\cal F}$ below) can be recast as
\begin{equation}
\begin{aligned}
{\cal F}\left[\frac{1}{z}\,F_G\bigl(\log(\Lambda^2z{\bar z}) \bigr)\right] &= \frac{i\pi}{\bar{p}}F_G\bigl(\log(\Lambda^2/|p|^2)\bigr)\,, \\
{\cal F}\left[\frac{1}{z\,\bar{z}^2}\, F_\Sigma \bigl(\log(\Lambda^2z{\bar z}) \bigr)\right] &= i\pi {\bar p}\,F_{\Sigma,I}\bigl(\log(\Lambda^2/|p|^2)\bigr)\,, \\
F_{\Sigma,I}(x)&\equiv\int^x dy F_\Sigma(y)
\end{aligned}
\end{equation}
Here $F_G,F_\Sigma$ should be thought of as a Taylor expansion in its arguments. This allows us to write our SD equations \eqref{DefG} and \eqref{DefSigma} as
\begin{equation}
F_\Sigma(x)=J^2F_G^3(x)\,, \quad \frac{1}{i\pi F_G(x)}=-i\pi \int_0^x dy F_\Sigma(y)+{1 \over i\pi} \,,
\end{equation}
where the integration constant is determined by the boundary condition at $J=0$, i.e. the absence of self energy ($\Sigma=0$) when the interaction is turned off ($J=0$). Taking derivatives, we derive the ODE equation
\begin{equation}
  F^\prime_G(x)=-\pi^2J^2 F^5_G(x)\,,\quad F_G(0)=1
\end{equation}
The ODE has a unique solution, giving rise to the 2-pt function
\begin{equation}\label{FullTwoPointFunc}
G(z,{\bar z})=\frac{1}{z}\,\frac{1}{\left(1+4\pi^2J^2\log(\Lambda^2\,z{\bar z})\right)^{1/4}}\,.
\end{equation}
The latter interpolates from small to large J, and has the correct limit at $J=0$ and large $J$.

\vspace{10pt}
\subsection{Beta Function}

Our analysis confirms the existence of an RG flow. We can extract the beta function $\beta (J)$ for the coupling $J$ that controls our random Thirring model from the Callan-Symanzik equation satisfied by the fermion propagator $G(z,{\bar z})$
\begin{equation}
\bigl( \Lambda \frac{\partial}{\partial\Lambda} +\beta(J) \partial_J +2\gamma(J)\bigr) {1\over z (1+ 4\pi^2 J^2\log(\Lambda^2 z{\bar z}))^{1/4}}=0\,,
\end{equation}
whose solution is given by
\begin{equation}\label{betagamma}
\beta(J)=4\pi^2 J^3\,, \quad \gamma(J)=\pi^2J^2\,.
\end{equation}

A positive $\beta$ function tells us the effective coupling becomes smaller in the IR. We are perturbing the action by a large number of marginal operators, which have non trivial 3-pt functions between them. This means that some of the coefficients of these operators will grow and some will decrease as we flow to the IR. In fact, which ones grow and which one decrease change as the couplings themselves evolve. If we think about $J$ as measuring the root mean square of the couplings, then some of them clearly decrease, generating a positive $\beta$ function. Still, the result is a bit peculiar if we want to think about $J\to\infty$, at least at finite $N$, which is not the case here. If we have a theory at finite cut-off and we turn on some relevant and irrelevant operators, then eventually some relevant directions will grow (and the irrelevant ones will decrease). This means that we expect that eventually the $\beta$ function will turn negative - we see no traces of this, presumably because of the large N scaling taken.

In the next section, we will gain a better understanding for the positivity of the beta function \eqref{betagamma} by using conformal perturbation theory around the free field fixed point.

\vspace{20pt}
\section{The $\beta$ function in conformal perturbation theory}
\label{pCFT}

To obtain a better understanding of the RG flow, we compare our theory to the Gross-Neveu \cite{Gross:1974jv} and Thirring models (for reviews, see \cite{ZinnJustin:2002ru, Moshe:2003xn}). These have been studied extensively, also in the context of the large N limit. Furthermore, since our model also deforms a free theory by 4-fermion marginal operators, conformal perturbation theory remains a very useful tool for any specific realisation of the couplings $J_{ij,kl}$, when these are small. The main difference is that our couplings are as random as possible (although they have a large statistical symmetry) and the key issue is the interplay between randomness and conformal perturbation theory.

In this section, we show that even though the $\beta$ function is quadratic in the couplings for a given realisation of the couplings $J_{ij,kl}$, this flow is suppressed at large N for the ensemble average in our models. We will argue that contrary to what happens to most known models, the next order contribution, i.e the cubic coupling term in the $\beta$ function, is universal\footnote{This statement holds for any scheme that preserves $\mathrm{SO}(N)_L\times \mathrm{SO}(N)_R$ invariance.}, drives the RG flow discovered in the previous section and in fact agrees with it for all values of $J$, and not just at weak coupling, in the $|p|/\Lambda \ll 1$ regime.

\vspace{10pt}
\subsection{RG in the Thirring and Gross-Neveu models}

As discussed in section \ref{defmodel}, our model is the "disordered" version of the Thirring model, given in equation \eqref{ThirSec2}. The latter is usually treated by introducing a gauge field $A_\mu^a$ and rewriting the interaction as
\begin{equation}
  \frac{1}{2} g J_\mu J_\mu\rightarrow \frac{A_\mu^2}{2g} + i\,A_\mu J_\mu\,,
\end{equation}
where we switched to a vector notation for the original currents and suppressed gauge indices \cite{ZinnJustin:2002ru, Moshe:2003xn}. The $A_\mu^2$ coupling is driven $g\to \infty$ in the IR. For the case of a single fermion in the fundamental representation, the theory is driven to a massive theory. If the starting UV field theory is a higher level WZW model, then it is expected\footnote{We are indebted to David Kutasov for a discussion of this  point.} to flow in the IR to a coset theory where we quotient by $\mathrm{SO}(N)$.

In our case, it is actually more useful to compare to the Gross-Neveu model \cite{Gross:1974jv}. For Majorana fermions, the Thirring model can be written as Gross-Neveu model by rewriting
\begin{equation}
\sum_{ab} \delta_{ab}J^a{\bar J}^b\sim \sum_{ij}\nu^i\nu^j{\bar\nu}^i{\bar\nu}^j\sim \sum_{ij}  \bigl(\nu^i{\bar\nu}^i\bigr) \bigl(\nu^j{\bar\nu}^j\bigr)
\end{equation}
Following the notation of Dirac fermions in Minkowski space used in \cite{Gross:1974jv}, this model is
\begin{equation}
  {\cal L} = i {\bar \psi}  {\slashed\partial}  \psi +  {g^2 \over 2 N } ({\bar\psi}\psi)^2\,.
\end{equation}
It is convenient to introduce an auxiliary bosonic field $\sigma$
\begin{equation}
{\cal L} = i {\bar \psi}  {\slashed\partial}  \psi    - \frac{\sigma^2}{2} - \frac{g}{\sqrt N} {\bar \psi} \psi\, \sigma\,,
\end{equation}
because at large $N$ the fermion propagator is not renormalized and the boson propagator becomes
\begin{equation}
D_R(p,\mu^2)= \frac{-i}{1+\frac{g^2}{2\pi} \log(-\frac{p^2}{\mu^2})}\,.
\end{equation}
One can use the Callan-Symanzik equation to extract the $\beta$ function at large $N$ to be
\begin{equation}
\beta(g) \equiv \mu \frac{\partial g}{\partial \mu} = - \frac{g^2}{2 \pi}
\label{betaGN}
\end{equation}
Hence, if $g>0$, the coupling is marginally relevant and the theory is asymptotically free in the UV. If $g<0$, the coupling is marginally irrelevant and the theory becomes free in the IR.

We expect a similar structure to exist in the random Thirring model - some of the couplings will be relevant and some irrelevant, their nature changing due to the non-linearity of the $\beta$ functions (as we will see below). In this situation it is more convenient to think about the theory as defined at some cut-off scale which is held fixed, in which case, at least for finite $N$, we expect the theory to emerge along some relevant directions and eventually develop a mass gap. However, this may not be accessible in the large $N$, in the phase where the statistical $\mathrm{SO}(N)_L\times \mathrm{SO}(N)_R$ is preserved.

For completeness, in the Gross-Neveu model the dynamical scale is given by
\begin{equation}
\Lambda(g) \propto \Lambda\, e^{-{2\pi \over g}}(1+{\cal O}(g)).
\end{equation}
where $\Lambda$ is a UV cutoff used to regularize the theory. In the IR, the theory breaks the chiral symmetry and flows to a massive theory. In fact, the theory is integrable and the complete spectrum of particles, labelled by $n$ (appearing in full multiplets of the symmetry) can be computed exactly and it is given by
\begin{equation}
m_n=\Lambda(g){N-2\over\pi}\sin	\biggl({n\pi\over (N-2)} \biggr)\,.
\end{equation}

\paragraph{$\beta$ function from conformal perturbation theory.}

The RG flows for the Thirring and Gross-Neveu models can also be understood using conformal perturbation theory. Consider a 2d CFT (the free fermion theory in our case) deformed by some marginal operators $\mathcal{O}_\alpha$ with conformal dimensions $(h,\bar{h})=(1,1)$
\begin{equation}
S_{\text{CFT}} - \sum_\alpha \lambda_\alpha \int \mathcal{O}_\alpha(z,\bar{z})\,d^2z\,.
\end{equation}
Conformal perturbation theory determines the first contribution to the $\beta$ function of these couplings to be\footnote{We are neglecting numerical factors in this subsection, but we will give precise formulas in subsection \ref{pform}.} \cite{cardy1988conformal}
\begin{equation}\label{BetaConf}
  \beta(\lambda_\alpha) \sim \sum_{\gamma,\sigma} C_{\alpha\gamma\sigma}\lambda_\gamma\lambda_\sigma + \dots
\end{equation}
where $C_{\alpha\gamma\sigma}$ is the 3-pt function $\langle \mathcal{O}_\alpha(z_\alpha,\bar{z}_\alpha) \mathcal{O}_\gamma(z_\gamma,\bar{z}_\gamma) \mathcal{O}_\sigma(z_\sigma,\bar{z}_\sigma)\rangle$ when we normalise the 2-pt functions canonically.

In our discussion, the deformation is of the form
\begin{equation}\label{SCFTDeform}
S_{\text{CFT}} -  J_{a;{\bar a}} \int J^{a}(z)\,\bar{J}^{{\bar a}}(\bar{z})\,d^2z\,.
\end{equation}
$J^a$ and $\bar J_{\bar a}$ are currents labelled by some index in the adjoint representation of $\mathrm{SO}(N)$ (see appendix \ref{currents} for more details on these conventions). (the same letter $J$ is used to denote both the currents and the coupling constants, but they will be distinguished by their index structure). The operators that we are perturbing by are $\mathcal{O}_{a{\bar a}}(z,\bar z)=J^a(z){\bar J}^{\bar a}(\bar z)$. Their three point functions are of the form
\begin{equation}\label{CintermsofF}
C_{a\bar a,b \bar b, c \bar c} \propto f_{abc}f_{{\bar a}{\bar b}{\bar c}}\,.
\end{equation}
Hence the $\beta$ function \eqref{BetaConf} is
\begin{equation}\label{BetaFunQuad}
  \beta_{aa'} \equiv \beta(J_{a;a'})  \propto \sum_{b,b',c,c'} f_{abc}f_{a'b'c'}\,J_{b;b'}\,J_{c;c'}\,.
\end{equation}
For the Gross-Neveu model,  the couplings equal $J_{a;\bar a} = \delta_{a\bar a} {g \over \sqrt N} $ and $f_{abc}$ are the structure constants of $\mathrm{SO}(N)$ given in appendix \ref{currents}. Using the identity \eqref{soNid}, the beta function \eqref{BetaFunQuad} reduces to
\begin{equation}
\beta(g) \sim -g^2\,,
\end{equation}
reproducing the behaviour in \eqref{betaGN}.

\vspace{10pt}
\subsection{$\beta$ function in the random Thirring model}

The standard use of conformal perturbation theory is for specific realisations of the set of couplings $J_{a;{\bar a}}$. In the following, we explore its extension to 2d CFT with disordered marginal deformations.

\paragraph{Flow of ensemble :} The 0+1 SYK model, the random Thirring model, or any of their cousins, have an infinite number of couplings in a given realisation. In principle we should track all of them. Once we declare that couplings are drawn from some class of distributions $f(\{J_{a;{\bar a}}\})$, e.g. Gaussian in our case, the number of parameters is greatly reduced. However, the distribution may flow, and its functional form may change with scale, i.e the ensemble may become a function of scale $f(\{J_{a;{\bar a}}\},\mu)$, in effect introducing more parameters.

There are two equivalent ways of thinking of the RG flow in disordered theories. The first is to think of the flow of ensemble of couplings. In this, the ensemble average is done with a scale dependent distribution denoted by $\langle \langle \rangle  \rangle_\mu$. For example
\begin{equation}\label{eq:flow ensemble}
\langle \langle \prod_{\alpha=1}^n  \lambda_\alpha \rangle  \rangle_\mu  = \int d\lambda_\alpha f(\{\lambda_\alpha\},\mu) \prod_{\alpha=1}^n\lambda_\alpha
\end{equation}
The second is to consider the flow of couplings for a specific realisation drawn from a fixed ensemble. This we denote by the usual $\langle \langle \rangle  \rangle$. For example
\begin{equation}\label{eq:flow couplings}
\langle \langle \prod_{\alpha=1}^n  \lambda_\alpha(\mu) \rangle  \rangle = \int d\lambda_\alpha f(\{\lambda_\alpha\}) \prod_{\alpha=1}^n\lambda_\alpha(\mu)
\end{equation}
Given the beta function of couplings in this way, one can deduce the corresponding flow of ensemble by demanding that \eqref{eq:flow ensemble} be the same as \eqref{eq:flow couplings} which is equivalent to
\begin{equation}
\mu \partial_\mu f(\{\lambda_\alpha\},\mu) = - \partial_\alpha \bigl (\beta^\alpha(\lambda) f(\{\lambda_\alpha\}) \bigr)
\end{equation}
where $\mu$ is the RG scale. We can characterize these distributions by their moments, and follow their RG flows by the infinite set of $\beta$ functions
\begin{equation}\label{n-moment}
\beta_n(\{J^{a;\bar a}\})=\mu\partial_\mu\ \langle\langle \Pi_{i=1}^n J^{a_i;{\bar a}_i}\rangle\rangle_\mu  =\langle\langle \mu \partial_\mu ( \Pi_{i=1}^n J^{a_i;{\bar a}_i}) \rangle\rangle\,.
\end{equation}
We will refer to them as the distribution $\beta$ functions. Working with the distribution $\beta$ functions might be a bit counter-intuitive at times. For example, the simplest case is when $n=1$. Using the leading $\beta$ function, quadratic in $J^{a ; \bar a}$, in \eqref{BetaFunQuad} and performing the ensemble average with $\langle\langle J_{b;b'}\,J_{c;c'} \rangle\rangle \sim \frac{J^2}{N^3} \delta_{bc}\delta_{b'c'}$, the average $\beta$ function equals
\begin{equation}\label{neq1}
\beta_1= \beta( \langle\langle J^{a;{\bar a}}  \rangle\rangle_\mu ) =\mu \partial_\mu \langle\langle J^{a ; \bar a} \rangle\rangle_\mu \propto   \sum_{b,\bar b,c,\bar c} f_{abc}f_{\bar a \bar b \bar c} \langle \langle J^{b;\bar b}\,J^{c;\bar c} \rangle \rangle = 0.
\end{equation}
However, this does not mean the theory does not flow.

In general, if we choose some functional form for the distribution, the RG flow can take us out of this subspace of functions. For example, working at finite $N$ in our model, the distribution does not remain Gaussian, or even symmetric under $J_{a;\bar a} \to -J_{a;\bar a}$, along the RG flow.  To see this, consider \eqref{n-moment} for $n=3$
\begin{equation}
\begin{aligned}
\beta_3=\mu\partial_\mu \langle\langle J^{a_1;{\bar a}_1} J^{a_2;{\bar a}_2} J^{a_3;{\bar a}_3}\rangle\rangle_\mu &\propto
\langle\langle J^{a_1;{\bar a}_1} J^{a_2;{\bar a}_2}f_{a_3bc}f_{{\bar a}_3{\bar b}{\bar c}} J^{b;{\bar b}}  J^{c;{\bar c}} \rangle\rangle\ + \text{permutations} \\
& \propto f_{a_1a_2a_3}{\bar f}_{\bar a_1 \bar a_2 \bar a_3} \frac{J^4}{N^6}\,.
\end{aligned}
\end{equation}
Hence, $\beta_3\neq 0$ at finite $N$, which is not compatible with a gaussian distribution. We observe, though, that this departure from gaussianity is subleading in $N$, compared to the nominal scaling in which $J^{a,{\bar a}}\sim J/N^{3/2}$.

\paragraph{Flow of the couplings:}
The various $\beta_n$ are statistical averages of various couplings in the theory. In principle, all of them appear in different Callan-Symanzik equations and one needs to track the RG flow for all of them. At large $N$, and as long as the statistical symmetry is unbroken, some simplifications occur. Here, we will focus on some additional aspects of the flow generated by the leading term in the $\beta$ function given in \eqref{BetaFunQuad}. We will see this term gives vanishing contributions (in the large N limit) to some of the interesting $\beta_n$'s.
\begin{itemize}
\item  First, the ensemble average of the $\beta$ function was already computed in \eqref{neq1}
\begin{equation}
\beta_1=\langle\langle \beta_{aa'} \rangle\rangle =  \langle\langle \mu \partial_\mu( J^{a;\bar a} ) \rangle\rangle = 0.
\end{equation}
Note this result also follows from symmetry --  the $\beta$ function transforms in the $\text{ad}(\mathrm{SO}(N)_L) \times \text{ad}(\mathrm{SO}(N)_R)$ representation and $\mathrm{SO}(N)_L\times \mathrm{SO}(N)_R$ is restored after the ensemble average. Hence the averaged $\beta$ function must vanish.

\item Although we find that the averaged $\beta$ function vanishes, it could be that there are large fluctuations around this mean value. To gain some intuition on what the physics may be for a given realisation, we compute the standard deviation of this averaged beta function. This is given by $\beta_{a,a'}^2$ (not summed over $a,a'$)
\begin{equation}
\begin{split}
\langle \langle \beta_{a,a'}^2 \rangle \rangle = & \sum_{bb'cc'{\hat b}{\hat b}'{\hat c}{\hat c}'} f_{abc}f_{a'b'c'}f_{a{\hat b}{\hat c}}f_{a'{\hat b}'{\hat c}'}  \langle \langle J_{b;b'}J_{c;c'}J_{{\hat b};{\hat b}'}J_{{\hat c};{\hat c}'}\rangle \rangle \\
= & {J^4\over N^6} \sum_{bb'cc'{\hat b}{\hat b}'{\hat c}{\hat c}'} f_{abc}f_{a'b'c'}f_{a{\hat b}{\hat c}}f_{a'{\hat b}'{\hat c}'}
\bigl( \delta_{b{\hat b}}  \delta_{b'{\hat b}'} \delta_{c{\hat c}}  \delta_{c'{\hat c}'} + \delta_{b{\hat c}}  \delta_{b'{\hat c}} \delta_{c{\hat b}}  \delta_{c'{\hat b}'} \bigr) \\
\sim & {J^4\over N^6} \sum_{bcb'c'} f_{abc}f_{abc}f_{a'b'c'}f_{a'b'c'}\sim {J^4\over N^4}
\end{split}
\end{equation}
The associated nominal scaling is
\begin{equation}
{\partial {J\over N^{3/2}} \over \partial \log \mu } \propto {J^2\over N^2},\ \implies {\partial J\over \partial \log \mu}\sim {J^2\over \sqrt{N}}
\end{equation}
and hence to leading order in the large N expansion (with J held fixed), the leading term in the $\beta$ function should be taken to be zero.

\item If the ensemble remains gaussian to leading order in $N$, the only parameter characterising the ensemble is $J$ (and in any case, the standard deviation is the leading statistical moment). Hence it is interesting to study its flow by studying
\begin{equation}
J^2(\mu) ={1\over N} \langle \langle \sum_{a{\bar a}} {J_{a;{\bar a}}}^2 \rangle\rangle_\mu\ .
\end{equation}
At this order of the $\beta$ function, i.e. quadratic order in the couplings, $\beta_2$ also vanishes
 \begin{equation}
\beta_2=\mu \partial_\mu \langle\langle J^{a_1 ; \bar a_1} J^{a_2 ; \bar a_2} \rangle\rangle_\mu \propto f_{a_1bc}f_{\bar a_1 \bar b \bar c}  \langle\langle J^{b ; \bar b} J^{c ; \bar c} J^{a_2 ; \bar a_2} \rangle\rangle + 1 \leftrightarrow 2 = 0\ ,
\end{equation}
i.e.,
\begin{equation}
 \partial_{\log \mu} J^2 = \partial_{\log \mu } \frac{1}{N} \langle\langle J_{a{\bar a}}J^{a{\bar a}} \rangle\rangle
 =0\ ,
\end{equation}

\end{itemize}
These are actually different measures why the leading quadratic term in the $\beta$ function \eqref{BetaFunQuad} does not contribute to the flow, due to cancellation between the different couplings. These arguments are correct in perturbation theory, or as long as the $\mathrm{SO}(N)_L\times \mathrm{SO}(N)_R$ is unbroken. They may be limited in teaching us about global issues of the RG flow because the latter is non-linear in the parameters. For example, if we fix the couplings at some high scale $\Lambda$, then since relevant couplings are turned on, then eventually an IR dynamical scale would be generated (at least at finite N). Computing this dynamical scale for a specific realisation, as a function of $J_{a;\bar a}$, and then averaging over the $J_{a;\bar a}$ is different from first averaging the $\beta$ function and then trying to deduce the flow's end.

\vspace{10pt}
\subsection{$\beta$ function from conformal perturbation theory }
\label{pform}

Conformal perturbation theory provides an expansion of the $\beta$ function in higher powers of the couplings $J_{a;\bar a}$.
We saw the quadratic terms are effectively zero at large $N$ in our models. Yet the distribution parameter $J^2$ still flows as we see explicitly in our computations \eqref{betagamma}. In this section, we argue this RG flow originates from the cubic terms in the $\beta$ function.

Before starting this discussion, let us comment on the status of these cubic terms, which are usually scheme dependent and hence non-universal. For example, if we redefine the couplings appearing in \eqref{BetaConf} by
\begin{equation}\label{schemedep}
{\tilde \lambda}_\alpha=\lambda_\alpha+ \sum_{\beta \gamma} A_{\alpha\beta\gamma}\lambda_\beta\lambda_\gamma + \dots
\end{equation}
the higher order terms in the $\beta$ function expansion will change. However,  we can argue that such terms are universal in schemes preserving the $\mathrm{SO}(N)_L \times \mathrm{SO}(N)_R$ statistical symmetry. To see this, notice that the redefinition \eqref{schemedep} would change the cubic term in the beta function as follows\footnote{Here $\dot \lambda \equiv \mu \partial_\mu \lambda$.}
\begin{equation}
\begin{split}
{\dot{\tilde\lambda}_\alpha} &= {\dot\lambda}_\alpha + 2A_{\alpha\beta\gamma}{\dot\lambda}_\beta\lambda_\gamma =C_{\alpha\beta\gamma}\lambda_\beta\lambda_\gamma+{\cal O}(\lambda^3)+2A_{\alpha\beta\gamma}C_{\beta\sigma\delta}{\lambda}_\sigma\lambda_\delta \lambda_\gamma\\
&= C_{\alpha\beta\gamma}{\tilde\lambda}_\beta{\tilde\lambda}_\gamma +{\cal O}(\lambda^3) - 2C_{\alpha\beta\gamma}A_{\beta\sigma\delta}{\lambda}_\sigma\lambda_\delta \lambda_\gamma + 2A_{\alpha\beta\gamma}C_{\beta\sigma\delta}{\lambda}_\sigma\lambda_\delta \lambda_\gamma\,,
\end{split}
\end{equation}
where $C_{\alpha \beta \gamma}$ are given in \eqref{CintermsofF}. If the redefinition preserves the $\mathrm{SO}(N)_L\times \mathrm{SO}(N)_R$, the matrix $A$ has to intertwine $\mathrm{SO}(N)_L \times \mathrm{SO}(N)_R$ to $\mathrm{SO}(N)_L \times \mathrm{SO}(N)_R \times \mathrm{SO}(N)_L \times \mathrm{SO}(N)_R$.  The only option is to take $A_{\alpha \beta \gamma} \propto C_{\alpha \beta \gamma}$ but then the cubic term in the $\beta$ function does not change.

In the following, we work with such a set of schemes, dictated by symmetry and having a universal cubic term contribution to the $\beta$ function. We use the results in \cite{Gaberdiel:2008fn,Behr:2013vta}. The formula for the $\beta$ function evaluated to this order is given in equations (5.12) and (5.13) of \cite{Behr:2013vta} which when adapted to our notation gives \footnote{Our convention for the $\beta$ function has an extra sign compared to that of \cite{Behr:2013vta}. In what follows, we will raise and lower adjoint indices $a,b$ etc by $\delta^{ab}, \delta_{ab}$ respectively.}
\begin{equation}
\beta_{a{\bar a}}=-\pi\,f_{bca}\,{ f}_{\bar b\bar c\bar a}\,J^{b;{\bar b}} J^{c;{\bar c}} -\beta_{a{\bar a} b{\bar b}c{\bar c}d{\bar d}} J^{b;{\bar b}} J^{c;{\bar c}} J^{d;{\bar d}}
\label{betak}
\end{equation}
where
\begin{equation}
\begin{aligned}
\beta_{a{\bar a}   b{\bar b}c{\bar c}d{\bar d}} &= \frac{\pi^2}{3!} \biggl(E_{abcd,{\bar a}{\bar b}{\bar c}{\bar d}}+{\bar E}_{abcd,{\bar a}{\bar b}{\bar c}{\bar d}}\biggr) \\
E_{abcd,{\bar a}{\bar b}{\bar c}{\bar d}}&=
(\delta_{ad}\delta_{bc}-\delta_{ac}\delta_{bd}){f}_{{\bar a}{\bar b}}^{\ \ \bar r}{f}_{{\bar r}{\bar c}{\bar d}}+
 (\delta_{ab}\delta_{cd}-\delta_{ad}\delta_{bc}){f}_{{\bar a}{\bar c}}^{\ \ \bar r}{f}_{{\bar r}{\bar d}{\bar b}}  \\
 & \quad + (\delta_{ac}\delta_{bd}-\delta_{ab}\delta_{cd}){f}_{{\bar a}{\bar d}}^{\ \ \bar r}{f}_{{\bar r}{\bar b}{\bar c}}\,, \\
{\bar E}_{abcd,{\bar a}{\bar b}{\bar c}{\bar d}}&= (\delta_{\bar a\bar d}\delta_{\bar b\bar c}-\delta_{\bar a\bar c}\delta_{\bar b\bar d}) {  f}_{{a}{b}}^{\ \ r}f_{{ r}{c}{d}} + (\delta_{\bar a\bar b}\delta_{\bar c\bar d}-\delta_{\bar a\bar d}\delta_{\bar b\bar c}) {  f}_{{a}{c}}^{\ \ r}f_{{ r}{d}{b}} \\
& \quad + (\delta_{\bar a\bar c}\delta_{\bar b\bar d}-\delta_{\bar a\bar b}\delta_{\bar c\bar d}) {  f}_{{a}{d}}^{\ \ r}f_{{ r}{b}{c}}\,. \\
&= E_{{\bar a}{\bar b}{\bar c}{\bar d}, a bc d}
\end{aligned}
\label{betak2}
\end{equation}

We are interested in evaluating the ensemble average of the $\beta$ function of the mean squared couplings, i.e $\tilde \beta \equiv \langle \langle \beta (J^{a;\bar a}  J_{a;\bar a}) \rangle \rangle$. Assuming the ensemble to be gaussian at all scales gives
\begin{equation}\label{betafun}
\tilde \beta  = \sum_{a,\bar a}\mu\partial_\mu \langle \langle J_{a;{\bar a}} J_{a;{\bar a}} \rangle \rangle_\mu = \sum_{a,\bar a} \mu\partial_\mu {2J^2\over N^3} \delta^a_a \delta^{\bar a}_{\bar a} =  J  N (1 - N^{-1})^2 \mu\partial_\mu J\,.
\end{equation}
Whereas the usual definition of $\beta$ function of ensemble gives
\begin{eqnarray}\label{RHS}
\tilde \beta &=& 2 \sum_{a,\bar a} \langle \langle J_{a; \bar a} \mu\partial_\mu J_{a;\bar a} \rangle \rangle = 2 \sum_{a,\bar a} \langle \langle J_{a; \bar a} \beta^{a \bar a}\rangle \rangle \\
&=& - 2 \beta_{a{\bar a} b_1{\bar b}_1 b_2{\bar b}_2 b_3{\bar b}_3} \langle \langle J^{a;{\bar a}} J^{b_1;{\bar b}_1} J^{b_2;{\bar b_2}} J^{b_3;{\bar b_3}}\rangle \rangle\\
\nonumber
&=& -{2\pi^2 \over 3!} \left( E_{ab_1b_2b_3,\bar a \bar b_1 \bar b_2 \bar b_3} + E_{\bar a \bar b_1 \bar b_2 \bar b_3,ab_1b_2b_3} \right) \times   {4 J^4 \over N^6}[  \delta^{ab_1}\delta^{b_2b_3}\delta^{{\bar a}{\bar b}_1}\delta^{{\bar b}_2{\bar b}_3}+ 1,2,3 \ \text{cyclic}] \\
&=& -{16 \pi^2 J^4 \over 3! N^6} \quad   E_{ab_1b_2b_3,\bar a \bar b_1 \bar b_2 \bar b_3}  \quad  [  \delta^{ab_1}\delta^{b_2b_3}\delta^{{\bar a}{\bar b}_1}\delta^{{\bar b}_2{\bar b}_3}+ 1,2,3 \ \text{cyclic}] \\
\nonumber
&=& 4 \pi^2 J^4 N (1 + {\cal O}(N^{-1}))\,,
\end{eqnarray}
where we used relevant identities satisfied by the structure constants described in appendix \ref{currents}. Notice that the above has the correct nominal scaling ${1 \over N^3}\times N^4$.  Comparing with \eqref{betafun}, we get
\begin{equation}
\beta(J) \equiv \mu \partial_\mu  J = 4 \pi^2 J^3
\end{equation}
This matches the $\beta$ function in \eqref{betagamma} obtained from direct computation using the Callan-Symanzik equation.

\paragraph{Flow of ensemble :} Finally, it is worth noting that this cubic term preserves the Gaussianity of the distribution of the couplings at large $N$.  To prove this we can show that the RG flow of any statistical average of $2n$ $J$'s is given by taking into account, at leading order in N, by the RG flow of each pair. i.e., Gaussianity implies that
\begin{equation}\label{Gaussian Claim}
\begin{split}
\mu \partial_\mu \langle\langle J^{a_1;{\bar a}_1}....J^{a_{2n};{\bar a}_{2n}} \rangle\rangle_\mu = \sum_{\sigma} \langle\langle \mu \partial_\mu ( J^{a_{\sigma(1)};{\bar a}_{\sigma(1)}}  J^{a_{\sigma(2)};{\bar a}_{\sigma(2)}}) \rangle\rangle
\prod_{i=2}^{n}  \langle\langle   J^{a_{\sigma(2i-1)};{\bar a}_{\sigma(2i-1)}}  J^{a_{\sigma(2i)};{\bar a}_{\sigma(2i)}}) \rangle\rangle
\end{split}
\end{equation}
where the permutation $\sigma \subset S_{2n}$ has $n$ 2-cycles - basically all possible Wick contractions. To see this, we evaluate the left hand side first
\begin{equation}
\mu \partial_\mu \langle\langle J^{a_1;{\bar a}_1}....J^{a_{2n};{\bar a}_{2n}} \rangle\rangle_\mu  = - \sum_{k=1}^{2n}\beta_{a_k \bar a_k b_1 \bar b_1 b_2 \bar b_2 b_3 \bar b_3} \langle \langle  J^{b_1;\bar b_1} J^{b_2;\bar b_2} J^{b_3;\bar b_3}   \prod_{i=1,i\ne k}^{2n} J^{a_i; \bar a_i}   \rangle\rangle\,.
\end{equation}
The evaluation of the right hand side has two kinds of terms. One in which two of the $J^{b_i;\bar b_i}$ contract among themselves. Notice these are exactly the terms captured by \eqref{Gaussian Claim}. Their nominal scaling is $N^{-3n}$. The second kind involves no contractions among the $J^{b_i;\bar b_i}$. Consider one such term below (where say $J^{b_i;\bar b_i}$ contracts with $J^{a_i;\bar a_i}$ for $i=1,2,3$ and $k>3$). We show it has subleading nominal scaling. Hence, it is negligible.
\begin{equation}
\begin{aligned}
&  \beta_{a_k \bar a_k b_1 \bar b_1 b_2 \bar b_2 b_3 \bar b_3} \prod_{i=1}^3 \langle \langle  J^{b_i;\bar b_i}  J^{a_i;\bar a_i} \rangle\rangle
 \langle \langle  \prod_{i=4,i\ne k}^{2n} J^{a_i; \bar a_i}   \rangle\rangle \\
 & \sim  N^{-{9  }} E_{a_k b_1b_2b_3,\bar a_k \bar b_1 \bar b_2\bar b_3} \prod_{i=1}^3 \delta^{a_i b_i} \delta^{\bar a_i \bar b_i} \langle \langle  \prod_{i=4,i\ne k}^{2n} J^{a_i; \bar a_i}   \rangle\rangle \\
  & \sim  N^{-9}\left[ \left( \delta_{a_k a_3} \delta_{a_1a_2} - \delta_{a_k a_2} \delta_{a_1 a_3} \right)  f_{\bar a_k \bar a_1}^{\ \ \ \ \bar r} f_{\bar r \bar a_2 \bar a_3 }  + 1,2,3 \ \text{cyclic} \right] \langle \langle  \prod_{i=4,i\ne k}^{2n} J^{a_i; \bar a_i}   \rangle\rangle
\end{aligned}
\end{equation}
Since $\sum_{\bar r}f_{\bar a_k \bar a_1}^{\ \ \ \ \bar r} f_{\bar r \bar a_2 \bar a_3 }$ does not scale with $N$, we conclude the nominal scaling of this term is $N^{-3n -3}$, which is subleading compared to the $N^{-3n}$ captured by the gaussianity preserving term in \eqref{Gaussian Claim}.

\section{Discussion and Future Directions}
\label{disc}

We computed the 2-pt function for a set of $N$ 1+1 dimensional Majorana fermions with a random Thirring interaction. We obtained that the $\beta$ function is positive, which means that the theory is not renormalizable. The running is, however, weaker than the usual logarithmic running of marginal operators.

As an effective theory below some scale, it is perfectly valid, and its RG exhibits some new features. This is the context in which the theory might be useful for black holes in AdS -- the model above might serve as a heuristic model for the degrees of freedom inside a black hole at some finite energy, and if we want to understand its surrounding weakly curved AdS outside the horizon, we need to couple it to additional "probe" degrees of freedom \cite{Berkooz:2016cvq}. The energy scale is therefore set by the temperature and the degrees of freedom of the black hole "never make it" to high energies.

There are some natural future directions
\begin{enumerate}
\item[1.] A better understanding of the solution to the Schwinger-Dyson equation: We treated the Schwinger-Dyson equations in an approximation in which the we solve the spatial "interaction" equation exactly, and the momentum "propagator" equation only approximately. This was convenient since the "interaction" equation is more non-linear. The fact that we were able to approximately solve the equations consistently for all values of $J$ for sufficiently low energies lends support to this approximation scheme. However, it would be worthwhile to see if the approximation can be justified, or tested on other examples.

\item[2.] Thermal partition function: One natural extension is to compute the thermal partition function.  We expect that we will be able to do it at low temperature, where we can use a long distance approximations of the type that we used above.

\item[3.] 4-pt function: In the 0+1 dimensional SYK there was a truncation of the action, in terms of 2-pt functions, to a reparametrization invariant action, and there was no need to regulate the action in order to make sense of the 2-pt function. This enabled partial computation of the 4-pt function, although the final expression required a scale (or the re-introduction of the RG flow). Here we need to include the RG flow already at the level of the 2-pt function, which means the computation of the 4-pt function might be considerably more difficult. It is, however, the main indication whether the theory can be related to a bulk theory of any sort.

\item[4.] Renormalizability and stability: We would like to see whether the theory is renormalizable, and at what scales it develops a mass gap in the IR. The former might be easier in the bi-local description of the mode. To check for a mass gap one can look for poles in futher correlation functions. In the Gross-Neveu model, for example, one indication for the mass scale is a tachyonic pole in the 2-pt function for $\sigma$, which feeds into the a pole in the 4-pt function of the fermions (there is no pole in the 2-pt function of the fermion which is not renormalized when writing the theory with the $\sigma$ field).  To see this effect in our model, we again need to compute the 4-pt function of the fermions in the model.

\item[5.] The Thirring model is dual to Sine-Gordon models via bosonization dualities. This suggests that one can obtain an SYK like models with bosonic fields too, atleast in 2 dimensions. It is interesting to then explore if there are other interesting bosonic SYK models.

\item[6.] Finally, it would be interesting to couple this model, at finite temperature, to additional "probe" degrees of freedom and examine whether it resembles a black hole in $AdS_3$.

\end{enumerate}

We hope to return to these issues in future work.

\vspace{20pt}
\subsection*{Acknowledgements}
We would like to thank Ofer Aharony and David Kutasov for illuminating discussions. The work of MB is supported by an ISF center of excellence grant (1989/14). PN gratefully acknowledges the support from International Centre for Theoretical Sciences (ICTS), India. The work of MR is supported by a Discovery grant from NSERC. The work of JS is supported by the Science and Technology Facilities Council (STFC) [grant number ST/L000458/1]. MB holds the Charles and David Wolfson Professorial chair of Theoretical Physics. JS's research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research, Innovation and Science.

\vspace{20pt}
\begin{appendix}

\section{2d Euclidean CFT conventions}
\label{2dcft}

Given the Euclidean action $S[\nu,\bar \nu]$ for the Majorana fermions $\nu,\bar \nu$, the euclidean partition function is given by
\begin{equation}
Z = \int [D\nu] [D \bar \nu] e^{-S[\nu,\bar \nu]}
\end{equation}
Euclidean coordinates are labelled by $\vec x= (x_1,x_2)$ and momentum by $\vec p = (p_1,p_2)$. The definition of Fourier transform
and its inverse are
\begin{equation}
{\cal F}(\vec p) = \int {d^2\vec x}\, e^{i \vec p . \vec x} f(\vec x)\,, \quad
f(\vec x) = \frac{1}{(2\pi)^2}\int {d^2\vec p}\,  e^{-i \vec p . \vec x} {\cal F}(\vec p)\,.
\end{equation}
Complex coordinates are defined as
\begin{eqnarray}
z = x_1+i x_2\,, \quad \bar z = x_1 - i x_2 \quad \implies \quad \underbrace{dz d\bar z}_{d^2\!z} = 2 d^2\! \vec x
\end{eqnarray}
and similarly for momenta
\begin{equation}
p = {p_1 - i p_2 \over 2}\,, \quad \bar p = {p_1 + i p_2 \over 2} \quad \implies \quad \underbrace{dp d\bar{p}}_{d^2\!p} = {d^2\vec p \over 2}
\end{equation}
One can check that  $z \bar z = |\vec x|^2$ and $|p|^2 \equiv 4 p \bar p = |\vec p|^2$
\begin{equation}
\vec p .\vec x = p z + \bar p \bar z\,, \quad \delta^2(\vec x) = 2 \delta(z) \delta(\bar z) \,, \quad \delta^2(\vec p) = {  \delta(p) \delta(\bar p)\over 2}
\end{equation}
The definition of Fourier transform in complex coordinates translates to
\begin{equation}
\begin{split}
{\cal F}(p, {\bar p})[ f(z,\bar z)] &\equiv \int {d^2\!z \over 2} e^{i (p z + {\bar p} \bar z)} f(z,\bar z) \\
f(z, \bar z) &= {\cal F}^{-1}[{\cal F}(p, {\bar p})] \equiv \int {2d^2\!p \over (2\pi)^2} e^{-i (p z + {\bar p} \bar z)} {\cal F}(p, {\bar p})\,. \\
\end{split}
\end{equation}

\paragraph{Summary of Fourier transforms.} The main Fourier transforms used in the main text is given below.

\begin{eqnarray}\label{Fourier1}
	\int {d^2\!z \over 2} e^{i (p  z + \bar p  \bar z)} { \log^\alpha(\Lambda^2 z \bar z) \over z \bar z   } &=& { \pi  \over \alpha +1 }   \log^{\alpha+1} (\Lambda^2/|p|^2) \left( 1 +{\cal O}({ 1 \over \log(\Lambda^2 / |p|^2)  })\right)   \\
	\label{Fourier2}
	\int {d^2\!z \over 2} e^{i (p  z + \bar p  \bar z)} { \log^\alpha(\Lambda^2 z \bar z) \over z   } &=& { i \pi  \over \bar p }   \log^{\alpha} (\Lambda^2/|p|^2)\left( 1 +{\cal O}({ 1 \over \log(\Lambda^2 /|p|^2)  })\right)  \\
	\label{Fourier3}
	\int {d^2\!z \over 2} e^{i (p  z + \bar p  \bar z)} { \log^\alpha(\Lambda^2 z \bar z) \over z \bar z^2   } &=& { i \pi \bar p \over \alpha +1 }   \log^{\alpha+1} (\Lambda^2/|p|^2)\left( 1 +{\cal O}({ 1 \over \log(\Lambda^2 / |p|^2)  })\right)
\end{eqnarray}

\vspace{20pt}
\section{A different regularisation of the main Fourier transform}
\label{alter}

In this appendix we provide an alternative regularisation for the integral \eqref{original}
\begin{equation}
  F_\alpha(|p|/\Lambda) = (2 \pi) \ 2^\alpha   \int_0^\infty {dr \over r} \log^\alpha(\Lambda r) J_0(|p| r)\,.
\end{equation}

Since the integral diverges at small $r$, we need to cut it off. It is natural to do so at $r=1/\Lambda$ (or larger values so that the logarithm does not become negative). This is another indication that the scale $\Lambda$ behaves like a UV cut-off. The resulting integral reduces to
\begin{equation}
F_\alpha(|p|/\Lambda) = \pi\, 2^{\alpha+1}\,\int_1^\infty \frac{dy}{y} \left(\log y\right)^\alpha J_0(|p|\,y/\Lambda)\,,
\label{Fintegral1}
\end{equation}
where we introduced the dimensionless variable $y=\Lambda\,r$.

Let us consider the $\alpha= n \in \mathbb{Z}$ to get some intuition. We are interested in studying the momentum dependence of this integral. For this purpose, define $\tau \equiv |p|/\Lambda$ and $t=\tau\,y$, so that \eqref{Fintegral1} equals
\begin{equation*}
I_\alpha(\tau) \equiv \frac{F_\alpha(|p|/\Lambda)}{\pi\, 2^{\alpha+1}}= \int^\infty_\tau \frac{dt}{t}(\log z/\tau)^\alpha\,J_0(t)\,.
\end{equation*}
Its derivative with respect to $\tau$ satisfies
\begin{equation}
\frac{dI_\alpha(\tau)}{d\tau} = -\frac{\alpha}{\tau} I_{\alpha-1}(\tau)\,.
\label{recurrence}
\end{equation}
Notice the contribution from the lower limit of integration does not contribute since the integrand vanishes at $t=\tau$ due to the logarithm.

To proceed, we use an integral appearing in \cite{abramowitz+stegun}
\begin{equation}
I_0(\tau) = \int^\infty_\tau \frac{J_0(t)}{t}\,dt = -\gamma - \log\frac{\tau}{2} - \sum_{k=1}^\infty (-1)^k \frac{(\tau/2)^{2k}}{2k\,(k!)^2}\,.
\end{equation}
This result is valid for any $\tau$. In the following, we will explore the physical regime corresponding to the IR, $\tau\equiv |p|/\Lambda \ll 1$, where the dominant contribution to $I_0(\tau)$ is captured by the logarithm and all the analytic terms are dropped.

It is easy to solve the recurrence relation \eqref{recurrence} when focusing on this dominant logarithmic contribution. Indeed, by induction, we can prove that the ansatz
\begin{equation}
I_\alpha(\tau) \sim \frac{(-1)^{\alpha+1}}{\alpha+1} \left(\log \frac{\tau}{2}\right)^{\alpha + 1}\,,
\end{equation}
solves \eqref{recurrence}. Notice that the integer nature of $\alpha$ was used in the induction step, so that $I_0(\tau)$ belongs to our series.

The above derivation for $\alpha\in \mathbb{Z}$ also determines the logarithmic nature of the subleading contributions
\begin{equation}
F_\alpha(|p|/\Lambda)=\frac{\pi}{\alpha+1} \left(\log (\Lambda^2/|p|^2)\right)^{\alpha+1} \left(1+{\cal O}(1/\log(|p|/\Lambda)) \right)\,,
\end{equation}
in agreement with the result \eqref{GenFormFn} in the main text.

\vspace{20pt}
\section{Normalisation of currents}
\label{currents}

When matching the free fermionic notation with the $\mathrm{SO}(N)$ at level one, the currents are written in terms of adjoint indices $a$
\begin{equation}
  J^a(z) = \frac{1}{2}\sum_{i,j} \left(\nu^i\,t^a_{ij}\,\nu^j\right)(z)\,,
\end{equation}
with a similar expression for $\bar{J}^b(\bar{z})$ in terms of $\bar{\nu}^k$, where $a$ stands for a pair of integers $(kl)$ satisfying $1\leq k < l \leq N$ and the matrices $t^a_{ij}$ satisfy\footnote{Following the conventions in \cite{DiFrancesco:1997nk}.}
\begin{equation}
\begin{aligned}
 t^a_{ij} &\equiv i\left(\delta^k_i\delta^l_j - \delta^k_j\delta^l_i\right) \\
 \text{Tr}\left(t^at^b\right) &= 2\delta^{ab} \equiv 2\left(\delta^{ki}\delta^{lj}-\delta^{kj}\delta^{li}\right) \\
 \sum_a t^a_{ij}t^a_{kl} &= -\delta_{ik}\delta_{jl}+ \delta_{jk}\delta_{il}\,,
\end{aligned}
\end{equation}
where in the second line $a=(ij)$ and $b=(kl)$. Notice $\sum_a \delta^{aa} = \frac{N^2}{2}(1-N^{-1})$ due to the range of the indices $(kl)$. These matrices have a commutator $\left[t^a,\,t^b\right]= \sum_c if_{abc}\,t^c$ whose structure constants are explicitly given by
\begin{equation}
 f_{abc}  \equiv  f_{(ij)(kl)(mn)}=\delta_{mi}\left(\delta_{nl}\delta_{jk}-\delta_{nk}\delta_{jl}\right) + \delta_{mj}\left(\delta_{il}\delta_{nk}-\delta_{nl}\delta_{ik}\right)
\end{equation}
It is easy to show these structure constants satisfy the identity
\begin{equation}
  \sum_{a,b} f_{abc}f_{abd} = 2(N-2)\delta_{cd}\,.
\label{soNid}
\end{equation}

Finally, the OPE between currents is
\begin{equation*}
  J^a(z)\,J^b(\omega) \sim \sum_c \frac{i\,f_{abc}\,J^c(\omega)}{(z-\omega)} + \frac{1}{2}\frac{\text{Tr}(t^at^b)}{(z-\omega)^2}\,,
\end{equation*}
with an analogous expression for the opposite chirality. These determine the 3-pt functions \eqref{CintermsofF}.

\end{appendix}

\vspace{10pt}
\bibliographystyle{JHEP}

redpajama_set_name: RedPajamaArXiv


text:
Home » NEWS » David44's "I Feel Better On My Own" Is Your New Happy Song
David44's "I Feel Better On My Own" Is Your New Happy Song
Posted on March 1, 2019 in NEWS
Photo: Courtesy of Copenhagen Records
With rippling shapes of expansive atmosphere, David44 showcases a voice that is way more kinetic than any artificial beats in his latest single "I Feel Better On My Own." There is a malleability in David44's voice that sways between fluttering lightness to emotional heftiness. Through the glowing layers, David44 explores finding happiness within himself – enjoy below:
"The only one who can make you happy is yourself. This song is about when I realized that, and that's a really empowering feeling that comes with it." explained Davd44.
David44 will be supporting Alex Vargas in March and playing a headline show in April. Go see him:
3/14 – Vega (Copenhagen, Denmark)

redpajama_set_name: RedPajamaCommonCrawl


text:
Classic test for lightly raced Miroku
Caulfield Guineas hopes for Miroku failed to materialise but the colt gets the chance to make a spring impression in the Group Three Caulfield Classic.
The Phillip Stokes-trained Miroku impressed on debut to win a maiden on the Ballarat synthetic track in early September before finishing unplaced in his next two starts at Geelong and Flemington.
Miroku notched the second win of his campaign as a short-priced favourite over 1500m at Mornington at his most recent start and will now be tried at stakes level for the first time in Saturday's Caulfield Classic for three-year-olds.
It will also be the colt's first try over 2000m and he has barrier one in the field of nine.
"He gives the indication he will run it," Stokes said.
"He has got a good barrier and he's a last-start winner, so we'll see. But I think he'll be OK.
"We thought he was going to be a Guineas horse but that didn't work out. We went to a softer option and got a win back on the board.
"He's by Shooting To Win and he's a big striding colt. I think if he relaxes he can run it out."
Miroku holds a nomination for the Group One Victoria Derby (2500m) at Flemington on November 2 which Stokes said they would consider if he ran well on Saturday.
The only filly in the race, the Danny O'Brien-trained Miami Bound, was the $2.70 favourite on Friday with Miroku at $17.
Racehorse breeders demand action after 'appalling' ABC abattoir investigation
Melbourne's letter to Sydney: Why we don't care about 'The Everest'


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text:
from django.http import HttpResponseNotFound,HttpResponseForbidden
from webinars_web.webinars import models
from django.shortcuts import render_to_response
from marketplace.decorators import marketplace
from django.conf import settings
from django.views.decorators.http import require_GET
from sanetime import sanetime
from webinars_web.webinars import utils
from django.template import RequestContext
#from pprint import pprint

redpajama_set_name: RedPajamaGithub


text:
Your Daily Horoscope for Tuesday, April 17
After 7 years in Pisces, Chiron enters Aries, which can make all signs feel a little brighter, a little bigger, and a little more ready to take on the world. This aspect lasts for several months, and it's a great time to begin to heal yourself and move forward. Today, Saturn turns retrograde, which may lead to intensity, second-guessing, and some projects from the past landing back on your plate. Look at this aspect as a way to learn and grow. Here, daily horoscopes for all signs for Tuesday, April 17.
See the future. Know the future. Download your premium astrology report today.
Aries Daily Horoscope
Today is a day to hang back a bit and try to let life unfold. Things may not make sense right now, but trust the process and trust that things will fall into place soon enough.
Taurus Daily Horoscope
You may be tempted to try a different tack, but don't — trust that staying the course is the best course of action, as changing direction could be problematic.
Gemini Daily Horoscope
You may want to socialize, but the stars say it's best to buckle down and stick to your to-do list. You have plenty of time to spend with friends, but for the next few days, it's all about getting things done.
Cancer Daily Horoscope
The moon in Taurus suggests this can be a day where slow and steady wins the race. If you've felt frustrated with how slowly things have been flowing, take heart, as you may finally see results.
Leo Daily Horoscope
You may feel overwhelmed with possibility, Leo. Instead of trying to do everything at once, stick to one or two things on your to do list, and focus on getting those done well.
Virgo Daily Horoscope
Today is all about the details. You may feel like you're the only one who cares about dotting i's or crossing t's, but trust that your diligence will pay off sooner rather than later.
Libra Daily Horoscope
Finally, things feel like they're going in the right direction, and it's a relief after a few weeks of uphill battles. Feel confident you're on the right track.
Scorpio Daily Horoscope
Relationships take center stage, and after months of feeling like things weren't quite clicking, you may feel like things are finally settling into place, and that any struggles have made your bond even tighter.
Sagittarius Daily Horoscope
You may feel a lot of nervous energy. It's time to channel that into action. Figure out what you want for the future, and go after it. Today is a day where a small action, like an email to someone, leads to big rewards.
Capricorn Daily Horoscope
You may feel like your mind and heart are at war, but it's smart to follow your heart where it leads — some things may not make sense on paper, but make so much sense for your well being.
Aquarius Daily Horoscope
You may be swayed or tempted by someone else's advice, but try to figure out what your mind and heart really want. You have the best instinct for the right path forward, and while some advice may be helpful, too much may make it even tougher to sort out your next steps.
Pisces Daily Horoscope
Socializing has implications that radiate through your career, and networking is incredibly important this week, Pisces. Prioritize spending time with family and friends — everything will get done, and too much solo time may be counterproductive.
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Sorted out the offending corner brace on the chassis that was blocking the fitting of the roll bar bolts by cutting it out, bit of a delicate job with the angle grinder as i did not want to scratch anything and not a lot off room to work on, all done now with a sigh of relief.
Looked at getting a few more bits from the loft and came across the inner wheel arches, noticed there was a cut out on one of them, wondering if it was cut out at some stage as if it was rubbing on the wheel, I certainly can't leave a unsightly hole there some have started on glassing it in, a bit tricky getting the curve right.
Rear lights are now refitted and sit even between the rear number plate and also the front indicator lights are now in place after repairing the fibreglass that they bolt onto.
Recently I met with a upholsterer whose work I have been impressed with and talked to him about some ideas for my interior, I know it would be easy enough to hand over the car and cheque book to Marcos Heritage but have a few little tweaks I want to include, nothing drastic but feel that I just want to add a few little touches that I have not seen before, all rather exciting but will keep this under wraps for the present time.


text:
James Guy seals 200m Fly gold at British Champs
James Guy claimed his first title of the British Swimming Championships 2018, claiming gold in the men's 200m Butterfly final.
The 23-year-old, who is on Team England's swimming squad for the Gold Coast, was over a second faster than the rest of the field.
He was the only English swimmer to make the podium, touching in 1:58.05 for the title.
Though this was a way off his PB time set at the British Swimming Champs 2017, he looks in good form for his Commonwealth campaign next month.
Guy said: "It's nice to come here and race as part of my preparations for the Commonwealth Games.
"It's great to win but I'm just thinking of long term goals – Commonwealth Games, the Europeans and in the long term, the Olympics in Tokyo.
"The weather in the Gold Coast is better than in Edinburgh today so I can't wait to get out there and represent my country."
Tom Derbyshire takes bronze in 1500m Free
Bath University's Tom Derbyshire claimed silver in the 1500m Freestyle after a tight fought battle against Spain's Miguel Duran.
The 20-year-old was just four seconds behind his Spanish rival, touching in 15:19.21.
Wales' Daniel Jervis took the gold, emerging victorious with a time of 15:01.87. Duran touched in 15:15.48.
Aimee Willmott secures silver in 400m IM
Middlesbrough-born Aimee Willmott took silver behind Aberdeen's Hannah Miley in the 400m Individual Medley final.
The University of Stirling swimmer put in a 4:41.00 effort to take her place on the podium.
Pretty pleased with this shiny 🥈 medal tonight! 400IM is a tough on the best of days, never mind in the middle of a hard block of training 👊🏻
200IM up tomorrow! #EISM18pic.twitter.com/Yhyrz22Jin
— Aimee Willmott (@aimee_willmott) March 1, 2018
Loughborough's Abbie Wood was less than a second and a half behind, claiming bronze in 4:42.38.
Miley, who was the oldest athlete in the field and is going for her third straight Commonwealth title in this event next month, took the title with 4:38.91.
View full results from day one of the British Swimming Championships 2018.
British Swimming Championships 2018
National Event Camp successfully delivered as part of talent pathway
'Existing talent and promising potential' in National Junior Squad for 2020-21 season
British Swimming Championships to be held at the London Aquatics Centre in 2021
Willmott and Tai named as members of Team England's Athlete Advisory Panel


Conveniently located near I 77, I 480, and the Ohio Turnpike, The Estates II Apartments offers luxurious one bedroom with a den and two bedroom suites. Located in beautiful Broadview Heights just minutes from Cleveland, Hopkins Airport, the Metro Parks, golf courses, and shopping malls. Also convenient to Akron/Canton.
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In recent years, advancements and nonsurgical cosmetic treatments have opened the door to eliminating unwanted fat, even the most stubborn, diet-resistant areas.
At MDCS: Medical Dermatology & Cosmetic Surgery, Dr. Snehal Amin and Dr. Noah Gratch teamed up to offer men and women in Midtown Manhattan, New York, the most effective and advanced nonsurgical approaches to skin tightening and body contouring available. Under the care of their experienced team, you can not only melt away cellulite, but you can also improve the strength and muscle tone of your abdomen and buttocks.
When you're looking to banish unwanted body fat and cellulite, the MDCS team turns to the most innovative technology available, including EMSCULPT and Excilis. These two treatments offer proven results, but work in markedly different ways.
EMSCULPT focuses on improving your muscle tone in two primary areas: the abdomen and buttocks. When using this technique to contour your body, you can expect stronger, more toned muscles, which results in a firmer abdomen or rear. This technology is the only treatment available that contours your body by building and toning your muscles.
The team's Exilis aesthetic system is an ideal approach for wrinkles and cellulite reduction. And, because Exilis uses heat to temporarily firm and tone your skin, it can also increase circulation in the treatment area, while also providing temporary pain relief.
The proven results you can expect with EMSCULPT and Excilis are different from each other. EMSCULPT delivers HIFEM®, or High-Intensity Focused Electromagnetic, technology directly to your abdominal or buttocks muscles to trigger muscle contractions. This targeted approach causes approximately 20,000 powerful muscle contractions in 30 minutes, results that aren't possible through normal exercise alone.
As your muscles react to this noninvasive energy, they release chemicals in the area causing fat cells to break down and release free fatty acids. This large number of fatty acids makes your fat cells malfunction and die, and your body absorbs them and flushes them from your system. At the same time, your muscles go through cellular changes, which leads them to strengthen and thicken.
Unlike EMSCULPT, Excilis body contouring treatments combine radiofrequency and ultrasound waves. This innovative technology delivers heat into the deepest layers of your skin to damage your fat cells, while keeping your surface skin temperature cool and comfortable. When exposed to heat, your fat cells die, and your body flushes them from your system. The precise amount of heat used with Excilis also triggers collagen production in the treatment site. Collagen helps tighten and tone skin in the area, which creates a firmer and more youthful appearance.
There's no downtime following EMSCULPT or Excilis treatments.
You can also expect fast results, especially when compared to other body contouring technologies and regular diet and exercise. Instead of waiting months, EMSCULPT offers noticeable changes within a day or two after a single visit with continued improvement within two to four weeks of your final appointment.
Similar to EMSCULPT body contouring treatments, the MDCS team usually suggests a series of Excilis sessions to reach maximum results. Because your body triggers new collagen production after each session, it's common to have the most dramatic results within the three to six months of your last treatment.
To learn more about noninvasive skin tightening and body contouring with EMSCULPT and Exilis, call MDCS: Medical Dermatology & Cosmetic Surgery, or schedule an appointment online today.
Hair loss, wrinkles, and age spots are common as you get older, but you don't have to live with them. Platelet-rich plasma (PRP) therapy is a minimally invasive treatment option that provides results.
As a chronic inflammatory skin condition, psoriasis can't be cured — but it can be kept under control with the right approach. Find out what the most common psoriasis triggers are and what you can do to manage or avoid them.
While your skin can benefit from laser resurfacing any time of the year, there are advantages to doing it in the wintertime. Find out why the months between December and March are sometimes referred to as laser skin resurfacing season.
Double chins are pesky—and no one wants to deal with them. The chin area is super difficult to tone and eliminate fat cells from. We follow three women as they undergo Liposuction, Kybella, and Coolsculpt in their journey to fix their double chin.
MDCS's own Snehal Amin, MD writes about Cynosure Elite MPX, a multi-modality laser that can treat skin imperfections such as extra hair, superficial varicose veins and brown spots on the skin simultaneously, for Cosmetic Town Journal.