text:
\section{Introduction}

The effect of many-body interaction on the topological properties of materials has been a fascinating subject in the research of topological insulators (TIs). From the symmetry perspective, the noninteracting TIs have been well understood within the context of symmetry classification\cite{Schnyder08,Kitaev09,Chiu16}, and a significant amount of effort has been made to generalize the symmetry classification scheme to incorporate many-body interactions\cite{Morimoto15,Queiroz16,Song17,Wang17}. On the other hand, through investigating concrete models, interactions have been shown to lead to novel and intriguing phenomena, such as the notion of topological Mott insulators\cite{Raghu08}, fractional topological insulators\cite{Levin09,Maciejko10,Maciejko15}, topological Kondo insulators\cite{Dzero10}, dynamical axion field\cite{Li10}, fractional Chern insulators (FCIs)\cite{Neupert11,Regnault11}, and first-order topological phase transitions\cite{Amaricci15,Amaricci16,Imriska16}, among many others.

Limiting our discussion to weakly interacting TIs, a suitable framework to address the effect of interactions is the Green's function formalism\cite{So85,Niu85,Ishikawa86,Volovik88,Volovik03,Qi08,Wang10,Gurarie11,
Essin11,Wang12,Wang12_2,Wang13,Grandi15}.
Within this formalism, the quantity that characterizes the topology of the system, namely the topological invariant, can be expressed in terms of the single-particle Green's function. In particular, in certain symmetry classes, the topological invariant of a $D$-dimensional TI takes the form of a $D$-dimensional momentum space integral or a $(D+1)$-dimensional momentum-frequency space integral over a certain combination of the single-particle Green's function and its derivatives. For the rest of the article, the integrand in this $D$- or $(D+1)$-dimensional integral is referred to as the curvature function, synonymous to the integration of the local curvature of a closed string counts the number of knots it contains. Depending on the symmetry class, the curvature function takes different forms. The advantage of this Green's function formalism is that the many-body effects, such as disorder or correlations, can be conveniently incorporated in a perturbative manner, provided the system remains in the weak coupling regime and is continuously connected to a noninteracting topological phase\cite{He16_2}.

To investigate the many-body effect for these particular symmetry classes using the Green's function formalism, however, appears to be a tedious task in practice. This is because within the framework of perturbation theory, the Green's function itself already contains at least a $D$-dimensional momentum space integration for the self-energy, making it totally a $2D$- or $(2D+1)$-dimensional integration problem. This is obviously a very costly calculation, especially if one's interest is to search for the topological phase transitions in a large parameter space for TIs defined at some high spatial dimension.

A generic feature near topological phase transitions seems to indicate an algorithm that can circumvent these integrations. In noninteracting systems, in which the curvature function is calculated from Berry connection, Berry curvature\cite{Chen16}, or Pfaffian of the time-reversal operator\cite{Chen16_2}, one observes that the curvature function generally diverges at a certain high symmetry point (HSP) in the momentum space as the system approaches the critical point, and the divergence changes sign across the critical point. This divergence is essentially due to the closing of the bulk gap at the HSP. The topological phase transition that features this divergence is said to be of second-order. Recently, it is revealed that FCIs near interaction-driven topological phase transitions also manifest such a divergence, in which the curvature function is a many-body Berry curvature calculated within the twisted boundary condition\cite{Kourtis17}. Based on this divergence, a renormalization group approach (RG) has been proposed\cite{Chen16,Chen16_2,Kourtis17}, which is a iterative procedure to find the trajectory (RG flow) in the parameter space along which the divergence is reduced, from which the topological phase transitions can be identified. In this article, we will demonstrate that, at least for the Dirac models and the interactions we examined, the curvature function features such a divergence and hence the RG approach is applicable. A significance of this RG approach is that it only requires to calculate the curvature function at few points in the momentum or momentum-frequency space, and consequently the $D$- or $(D+1)$-dimensional integration of the curvature function is avoided.

Besides demonstrating the RG approach as a numerically convenient tool, another purpose of this article is to introduce the following statistical aspects into weakly interacting TIs based on the divergence of the curvature function\cite{Chen17}, using 1D class BDI and 2D class A Dirac models in the presence of electron-electron and electron-phonon interaction as examples. The first is a real space or spacetime correlation function as the Fourier transform of the curvature function, which takes the form of a product of real space or spacetime Green's function propagated over several segments. Remarkably, at the critical point, the correlation length or correlation time diverges due to the divergence of the curvature function, signifying the scale invariance. The critical exponents of the correlation length and correlation time, as well as that of the curvature function at the HSP, are shown to be constrained by a scaling law due to conservation of the topological invariant. In addition, through investigating concrete models, we unveils that interaction may affect the critical exponents of the system.

The article is structured in the following manner. In Sec.~II, we introduce the general formalism of Green's function and the perturbative approach to Dirac models under weak electron-electron and electron-phonon interactions. The divergence of curvature function and the RG approach are then introduced on a phenomenological level. In Sec.~III, we discuss 1D class BDI models with interactions, including the specific form of topological invariant and the accompanied correlation function. A concrete model is then used to demonstrate the RG approach and to extract the critical exponents. In Sec.~IV, we discuss 2D class A models, for both of frequency-dependent and frequency-independent self-energies. We introduce the correlation function for both cases, and discuss how the interactions influence the critical exponents. Sec.~V summarizes the results.

\section{Generic critical behavior investigated by perturbative treatment\label{sec:generic_critical_perturbative}}

\subsection{Green's function formalism for Dirac models with interactions\label{sec:Green_fn_Dirac_interaction}}

Our goal is to use Green's function formalism to study the $2\times 2$ Dirac models under the influence of many-body interactions, and also to use the Green's function to calculate the many-body curvature function introduced in the next section, so here we briefly outline the plan of attack. Starting from a noninteracting spinless $2\times 2$ Dirac Hamiltonian
\begin{eqnarray}
&&{\cal H}_{0}=\sum_{\bf k}
\left(
\begin{array}{cc}
c_{A{\bf k}}^{\dag} & c_{B{\bf k}}^{\dag} \\
\end{array}
\right)H_{0}({\bf k})
\left(
\begin{array}{c}
c_{A{\bf k}} \\
c_{B{\bf k}}
\end{array}
\right)\;,
\nonumber \\
&&H_{0}({\bf k})=d_{0}({\bf k})\sigma_{0}+d_{1}({\bf k})\sigma_{1}+d_{2}({\bf k})\sigma_{2}+d_{3}({\bf k})\sigma_{3}\;.
\label{general_2by2_Dirac_Hamiltonian}
\end{eqnarray}
which is the suitable minimal model for 1D class BDI and 2D class A systems. Depending on the symmetry class, not every $d_{i}$ is present, and also the eveness and oddness of each $d_{i}$ depends on the symmetry class. The two degrees of freedoms will be referred to as $A$ and $B$ sublattices, although in general it may represent some other pseudospin degrees of freedom. The $d_{0}$ component may be set to zero at noninteracting level, but interactions can induce $d_{0}$, as explain later. Here $c_{I{\bf k}}$ denotes the spinless fermion operator that satisfies the anticommutation relation $\left\{c_{I{\bf k}},c_{I^{\prime}{\bf k}^{\prime}}^{\dag}\right\}=\delta_{II^{\prime}}\delta_{{\bf k}{\bf k}^{\prime}}$ with sublattice index $I=\left\{A,B\right\}$.

In the presence of a weak interaction written in the second-quantized form ${\cal H}_{int}$, our strategy is to calculate the effect of interaction perturbatively using Matsubara Green's function
\begin{eqnarray}
G({\bf k},\tau)=\left(
\begin{array}{ll}
G_{AA}({\bf k},\tau) & G_{AB}({\bf k},\tau) \\
G_{BA}({\bf k},\tau) & G_{BB}({\bf k},\tau)
\end{array}
\right)\;,
\end{eqnarray}
with the matrix elements defined by
\begin{eqnarray}
G_{IJ}({\bf k},\tau)=-\langle T_{\tau}c_{I{\bf k}}(\tau)c_{J{\bf k}}^{\dag}(0)\rangle\;.
\end{eqnarray}
where $T_{\tau}$ is the time ordering. The full Green's function with discrete frequency $i\omega_{n}$ can be obtained from Dyson's equation
\begin{eqnarray}
G&=&G_{0}+G_{0}\Sigma G=G_{0}+G_{0}\Sigma G_{0}+G_{0}\Sigma G_{0}\Sigma G_{0}+...
\nonumber \\
&=&\left(G_{0}^{-1}-\Sigma\right)^{-1}=\left(i\omega_{n}-H_{0}-\Sigma\right)^{-1},
\label{Dyson_equation}
\end{eqnarray}
where $\Sigma$ is the self-energy. The interacting part of Dyson's equation in Eq.~(\ref{Dyson_equation}) is
\begin{eqnarray}
&&\left(G_{0}\Sigma G\right)_{IJ}=-\sum_{n=1}^{\infty}(-1)^{n}\int_{0}^{\beta}d\tau_{1}\int_{0}^{\beta}d\tau_{2}...
\int_{0}^{\beta}d\tau_{n}
\nonumber \\
&&\times\langle T_{\tau} c_{I{\bf k}}(\tau){\cal H}_{int}(\tau_{1}){\cal H}_{int}(\tau_{2})...{\cal H}_{int}(\tau_{n})c_{J{\bf k}}^{\dag}(0)\rangle,
\end{eqnarray}
that evaluates different, connected diagrams. In this article we restrict our calculation to one-loop level.

To be specific, we examine two kinds of interactions. The first is the short range interaction between spinless fermions\cite{Wen10,Grushin13,Jia13,Daghofer14,Guo14,Luo15}. In particular, we consider the density-density interaction between two sublattices that takes the form
\begin{eqnarray}
{\cal H}_{e-e}=\sum_{\bf kk^{\prime}q}V_{\bf q}c_{A{\bf k+q}}^{\dag}c_{B{\bf k^{\prime}-q}}^{\dag}c_{B{\bf k}^{\prime}}c_{A{\bf k}}\;.
\label{general_density_density_interaction}
\end{eqnarray}
The precise form of $V_{\bf q}$ will be discussed in later sections. The second is the electron-phonon interaction\cite{Zhu12,Li12,Kim12,Huang12,Parente13,DasSarma13,Li13,Luo13,Sobota14,
Howard14,Gupta14,Glinka15,Glinka15_2,Zhao15,Sharafeev17,Tamtogl17,Heid17}, which has been a realistic issue, for instance on the transport of surface states. To examine the perturbative approach, we consider a polar coupling between spinless fermions and a longitudinal optical phonon mode, described by the Fr\"{o}hlich Hamiltonian\cite{Mahan00}
\begin{eqnarray}
{\cal H}_{e-ph}&=&\sum_{\bf kq}M_{\bf q}\left(c_{A{\bf k+q}}^{\dag}c_{A{\bf k}}+c_{B{\bf k+q}}^{\dag}c_{B{\bf k}}\right)A_{\bf q}\;,
\nonumber \\
A_{\bf q}&=&a_{\bf q}+a_{\bf -q}^{\dag}\;,
\nonumber \\
M_{\bf q}&=&\frac{u\,\omega_{0}}{q}\;.
\label{general_electron_phonon_interaction}
\end{eqnarray}
Here $a_{\bf q}$ is the phonon annihilation operator, $\omega_{0}$ is the optical phonon frequency, and $u$ is a phenomenological prefactor that takes care of the dielectric constant. We will treat all the variables as dimensionless for simplicity.

The calculation of self-energy for these two types of interactions is detailed in Appendix \ref{appendix:self_energy}. After the Dyson's equation is solved, we then treat the discrete Matsubara frequency continuously $i\omega_{n}\rightarrow i\omega$ to obtain the Green's function with continuous frequency variable for the sake of calculating the topological invariant, as detailed in the following sections. Taking into account how the self-energy enters the Green's function, the full Green's function in momentum-frequency space takes the form
\begin{eqnarray}
G({\bf k},i\omega)
&=&\frac{1}{(i\omega+d_{0}^{\prime})^{2}-d^{\prime 2}}
\nonumber \\
&&\times\left(
\begin{array}{cc}
i\omega+d_{0}^{\prime}+d_{3}^{\prime} & d_{1}^{\prime}-id_{2}^{\prime} \\
d_{1}^{\prime}+id_{2}^{\prime} & i\omega+d_{0}^{\prime}-d_{3}^{\prime}
\end{array}
\right)\;,
\nonumber \\
G^{-1}({\bf k},i\omega)&=&\left(
\begin{array}{cc}
i\omega+d_{0}^{\prime}-d_{3}^{\prime} & -d_{1}^{\prime}+id_{2}^{\prime} \\
-d_{1}^{\prime}-id_{2}^{\prime} & i\omega+d_{0}^{\prime}+d_{3}^{\prime}
\end{array}
\right)\;,
\label{2D_G_dprime}
\end{eqnarray}
where $d^{\prime}=\sqrt{d_{1}^{\prime 2}+d_{2}^{\prime 2}+d_{3}^{\prime 2}}$, and the self-energy-renormalized ${\bf d}^{\prime}$-vector is
\begin{eqnarray}
&&d_{1}^{\prime}=d_{1}+{\rm Re}\Sigma_{AB}\;,\;\;\;d_{2}^{\prime}=d_{2}-{\rm Im}\Sigma_{AB}\;,
\nonumber \\
&&d_{3}^{\prime}=d_{3}+\frac{\Sigma_{AA}-\Sigma_{BB}}{2}\;,
\;\;\;d_{0}^{\prime}=\frac{-\Sigma_{AA}-\Sigma_{BB}}{2}\;.
\label{renormalized_d_vector_mu}
\end{eqnarray}
Notice that because of the self-energy $\Sigma_{IJ}=\Sigma_{IJ}({\bf k},i\omega)$, each $d_{i}^{\prime}=d_{i}^{\prime}({\bf k},i\omega)$ is generally a function of both momentum and frequency. In 1D class BDI models, the $i\omega$ is eventually analytically continued to $i\omega\rightarrow E$ and then taken as a real number, whereas in 2D class A models the $i\omega$ remains a imaginary number that is to be integrated out.

\subsection{Divergence of curvature function\label{sec:criticality}}

In noninteracting TIs, the topological phase transition can be identified from the closing of the bulk gap. However, in interacting systems, the single-particle spectral function is in general not a $\delta$-function but spreads out over a range of frequency, so the bulk gap is strictly speaking a rather ambiguous notion. In this case, the Green's function formalism offers a way to identify the topological phase transition without explicitly invoking the notion of bulk gap. In this article, we examine the Green's function formalism and discuss the quantum criticality near the topological phase transitions within the context of the divergence of the curvature function, as introduced below.

In the Green's function formalism, the integer-valued topological invariant ${\cal C}={\cal C}({\bf M})$ is formulated in terms of single-particle Green's functions, and depends on a set of tuning parameters ${\bf M}=(M_{1},M_{2}...M_{D_{M}})$ that form a $D_{M}$-dimensional parameter space. Each $M_{i}$ is either a noninteracting or interacting parameter in the Hamiltonian. The topological invariant for the two classes under consideration takes the form of a $\tilde{D}$-dimensional integration over momentum space or momentum-frequency space
\begin{eqnarray}
{\cal C}&=&\int d^{\tilde D}{\bf K}\,F({\bf K},{\bf M})\;,
\nonumber \\
\tilde{D}&=&1\;,\;{\bf K}=k\;{\rm for\;1D\;class\;BDI}\;,
\nonumber \\
\tilde{D}&=&3\;,\;{\bf K}=(\omega,k_{x},k_{y})\;{\rm for\;2D\;class\;A}.
\label{C_FkM_class_BDI_class_A}
\end{eqnarray}
The function $F({\bf K},{\bf M})$ is referred to as the curvature function, and is expressed in terms of the single-particle Green's function calculated perturbatively in Sec.~\ref{sec:Green_fn_Dirac_interaction}. We assume that there is a certain HSP in the momentum or momentum-frequency space, denoted by ${\bf K}_{0}$ and satisfying ${\bf K}_{0}=-{\bf K}_{0}$, around which the curvature function is an even function
\begin{eqnarray}
F({\bf K}+\delta{\bf K},{\bf M})=F({\bf K}_{0}-\delta{\bf K},{\bf M})\;
\end{eqnarray}
due to certain symmetries, such as inversion symmetry, where $\delta{\bf K}$ is a small displacement.

At least for the models investigated in this article, we observe the following critical behavior for the curvature function. Firstly, the curvature function around the HSPs takes the Ornstein-Zernike form in all $\tilde{D}$-directions in the momentum or momentum-frequency space
\begin{eqnarray}
F({\bf K}_{0}+\delta{\bf K},{\bf M})=\frac{F({\bf K}_{0},{\bf M})}{\prod_{i=1}^{\tilde{D}}\left(1+\xi_{i}^{2}\delta K_{i}^{2}\right)}\;,
\label{Ornstein_Zernike}
\end{eqnarray}
where $\xi_{i}$ is a length or time scale, as indicated schematically in Fig.~\ref{fig:Fk0_xi_RG_schematics}. Secondly, as approaching the critical point ${\bf M}\rightarrow{\bf M}_{c}$, the extremum $F({\bf K}_{0},{\bf M})$ diverges, and the time or length scale $\xi_{i}$ diverges or saturates to a constant
\begin{eqnarray}
&&\lim_{{\bf M}\rightarrow{\bf M}_{c}^{+}}F({\bf K}_{0},{\bf M})=-\lim_{{\bf M}\rightarrow{\bf M}_{c}^{-}}F({\bf K}_{0},{\bf M})=\pm\infty\;,
\nonumber \\
&&\lim_{{\bf M}\rightarrow{\bf M}_{c}}\xi_{i}=\infty\;{\rm or}\;{\rm const.},
\label{Fk0_xi_divergence}
\end{eqnarray}
where ${\bf M}_{c}^{+}$ and ${\bf M}_{c}^{-}$ denote the two sides of the phase boundary in the parameter space. In short terms, the Lorentzian shape of Eq.~(\ref{Ornstein_Zernike}) gradually narrows as ${\bf M}\rightarrow{\bf M}_{c}$, and flips sign abruptly as ${\bf M}$ crosses ${\bf M}_{c}$. The critical exponents then follows
\begin{eqnarray}
F({\bf K}_{0},{\bf M})\propto |{\bf M}-{\bf M}_{c}|^{-\gamma}\;,\;\;\;\xi_{i}\propto |{\bf M}-{\bf M}_{c}|^{-\nu_{i}}\;.
\label{Fk0_xi_critical_exponent}
\end{eqnarray}
The numerator $F({\bf K}_{0},{\bf M})$ in the Ornstein-Zernike form of Eq.~(\ref{Ornstein_Zernike}) is assigned to the exponent of the susceptibility $\gamma$, and the length or time scale $\xi_{i}$ is assigned to the exponent of the correlation length $\nu_{i}$, for the reason that will become clear later.

\begin{figure}[ht]
\begin{center}
\includegraphics[clip=true,width=0.99\columnwidth]{Fk0_xi_RG_schematics}
\caption{ (a) Schematics of the Ornstein-Zernike form of the curvature function in the vicinity of a HSP ${\bf K}_{0}$ along certain direction $K_{i}$ in the momentum or momentum-frequency space. The constant volume underneath the Lorentzian peak leads to the scaling law in Eq.~(\ref{scaling_law}) that constrains the critical exponent of $F({\bf K}_{0},{\bf M})$ and that of $\xi_{i}$. (b) Schematics of the RG procedure, which demands $F({\bf K}_{0}+\delta{\bf K},{\bf M})$ (red dot) to be equal to $F({\bf K}_{0},{\bf M}^{\prime})$ (orange dot), as indicated by the dashed line. The divergence of the curvature function at ${\bf K}_{0}$ is gradually reduced under this procedure whereas the topological invariant remains unchanged, hence the system gradually flows away from the critical point. }
\label{fig:Fk0_xi_RG_schematics}
\end{center}
\end{figure}

The critical exponents $\gamma$ and $\nu_{i}$ are not independent, but constrained by a scaling law due to the conservation of topological invariant as ${\bf M}\rightarrow {\bf M}_{c}$. This can be seen by considering the integration of the Ornstein-Zernike form of Eq.~(\ref{Ornstein_Zernike}) over the width of the Lorentzian peak, which contributes to a fraction of the topological invariant that remains constant
\begin{eqnarray}
{\cal C}_{div}&=&F({\bf K}_{0},{\bf M})\prod_{i=1}^{\tilde D}\left(\int_{-\xi_{i}^{-1}}^{\xi_{i}^{-1}}\frac{dK_{i}}{1+\xi_{i}^{2}K_{i}^{2}}\right)
\nonumber \\
&=&\frac{F({\bf K}_{0},{\bf M})}{\prod_{i=1}^{\tilde D}\xi_{i}}\times{\cal O}(1)={\rm Const}.
\end{eqnarray}
A scaling law then follows after applying Eq.~(\ref{Fk0_xi_critical_exponent}),
\begin{eqnarray}
\gamma=\sum_{i=1}^{\tilde D}\nu_{i}\;,
\label{scaling_law}
\end{eqnarray}
which constraints the critical exponents\cite{Chen17}.

Previous investigations\cite{Chen17,Kourtis17} seem to suggest a general principle of introducing the correlation function that characterizes the topological phase transitions, which is found to also apply to weakly interacting systems: if the topological invariant is an integral of a certain curvature function, then the Fourier transform of the curvature function represents a correlation function
\begin{eqnarray}
\lambda_{\bf R}&=&\int d^{\tilde D}{\bf K}\,e^{i{\bf K}\cdot{\bf R}}F({\bf K},{\bf M})\;,
\nonumber \\
{\bf R}&=&
\left\{
\begin{array}{l}
r\;{\rm for\;1D\;class\;BDI}\;,\\
(-t,r_{x},r_{y})\;{\rm for\;2D\;class\;A}\;.
\end{array}
\right.
\label{general_Fourier_trans_lambdaR}
\end{eqnarray}
Here for 2D class A one has $e^{i{\bf K}\cdot{\bf R}}=e^{ik_{x}r_{x}+ik_{y}r_{y}-i\omega t}$ (defining $R_{0}=t$ or $-t$ does not make a difference since the curvature function is even in $\omega$, but we follow the usual convention of Fourier transform). Together with the Ornstein-Zernike form of the curvature function in Eq.~(\ref{Ornstein_Zernike}), this implies the correlation function decays exponentially in all ${\tilde D}$-directions with a correlation length or correlation time $\xi_{i}$. In contrast to previous investigations that consider certain kinds of two-point correlation of the field operator\cite{Ringel11,You14,Toldin15}, the $\lambda_{\bf R}$ in Eq.~(\ref{general_Fourier_trans_lambdaR}) is generally a product of real space or spacetime Green's functions propagated over several segments, as we shall see in the following sections. This Fourier transform principle is motivated by the progress in the theory of charge polarization and theory of orbital magnetization, which are real space representations of the topological order in noninteracting systems, and the correlation function therein measures a certain overlap of Wannier functions\cite{Chen17}. We should also emphasize that the correlation functions are nonzero in both topologically trivial and nontrivial phases, hence a more general measure for the quantum criticality than the edge state, as the later only manifests in the topologically nontrivial phase.

\subsection{Renormalization group approach\label{sec:RG}}

The discussion in Sec.~\ref{sec:criticality} motivates us to apply a recently developed RG scheme\cite{Chen16,Chen16_2} to identify the topological phase transitions in the $D_{M}$-dimensional parameter space, which renders a very convenient tool that significantly reduces the numerical effort to solve the many-body problem at hand. The RG procedure is based on the divergence of the curvature function described by Eqs.~(\ref{Ornstein_Zernike}) and (\ref{Fk0_xi_divergence}). The procedure demands that at a given ${\bf M}$, one solves for the new ${\bf M}^{\prime}$ that satisfies
\begin{eqnarray}
F({\bf K}_{0}+\delta{\bf K},{\bf M})=F({\bf K}_{0},{\bf M}^{\prime})\;,
\label{RG_procedure_general}
\end{eqnarray}
where ${\bf K}_{0}$ is an HSP, and $\delta{\bf K}$ is a small deviation away from it along a scaling direction, as indicated schematically in Fig.~\ref{fig:Fk0_xi_RG_schematics}. Iteratively solving for the mapping ${\bf M}\rightarrow{\bf M}^{\prime}$ yields an RG flow in the parameter space along which the divergence of $F({\bf K}_{0},{\bf M})$ is reduced but the topological invariant remains unchanged, in a way analogous to stretching a messy string until the number of knots becomes obvious, from which the topological phase transitions can be identified. To obtain the RG flow for each tuning parameter $M_{i}$, we use Eq.~(\ref{RG_procedure_general}) but fix all other tuning parameters unchanged
\begin{eqnarray}
&&F({\bf K}_{0}+\delta{\bf K},(M_{1},M_{2},...M_{i}...M_{d_{M}}))
\nonumber \\
&&=F({\bf K}_{0},(M_{1},M_{2},...M_{i}^{\prime}...M_{d_{M}}))\;.
\label{scaling_procedure_each_Mi}
\end{eqnarray}
Denoting $M_{i}^{\prime}-M_{i}=dM_{i}$ and $dl=|\delta{\bf K}|^{2}$, expanding Eq.~(\ref{scaling_procedure_each_Mi}) to leading order yields the generic RG equation
\begin{eqnarray}
\frac{dM_{i}}{dl}=\frac{1}{2}\frac{(\delta {\bf K}\cdot{\boldsymbol\nabla}_{\bf K})^{2}F({\bf K},{\bf M})|_{{\bf K}={\bf K}_{0}}}{\partial_{M_{i}}F({\bf K}_{0},{\bf M})}\;.
\label{generic_RG_eq}
\end{eqnarray}
For interacting systems, often times the numerical calculation is performed on discrete mash points, in which case Eq.~(\ref{generic_RG_eq}) may be approximated by
\begin{eqnarray}
\frac{dM_{i}}{dl}=\frac{\Delta M_{i}}{|\Delta{\bf K}|^{2}}\frac{F({\bf K}_{0}+\Delta{\bf K},{\bf M})-F({\bf K}_{0},{\bf M})}{F({\bf K}_{0},{\bf M}+\Delta{\bf M}_{i})-F({\bf K}_{0},{\bf M})}\;,
\label{generic_RG_eq_discrete}
\end{eqnarray}
where $\Delta {\bf K}$ is the grid spacing along the scaling direction in the momentum or momentum-frequency space, $\Delta M_{i}$ is the grid spacing along ${\hat{\bf M}}_{i}$ direction in the parameter space, and $\Delta {\bf M}_{i}=\Delta M_{i}{\hat{\bf M}}_{i}$. Equation (\ref{generic_RG_eq_discrete}) demonstrates the advantage of this RG scheme, namely at a given ${\bf M}$, one only needs to calculate the curvature function at three mesh points $F({\bf K}_{0},{\bf M})$, $F({\bf K}_{0}+\Delta{\bf K},{\bf M})$, and $F({\bf K}_{0},{\bf M}+\Delta{\bf M}_{i})$ to calculate the RG flow $dM_{i}/dl$ without explicitly performing the $\tilde{D}$-dimensional integration in Eq.~(\ref{C_FkM_class_BDI_class_A}). In addition, to extract the critical exponents in Eq.~(\ref{Fk0_xi_critical_exponent}) also requires no more than these mesh points, provided $\Delta{\bf K}$ is along the direction reciprocal to $\xi_{i}$. For an arbitrarily given model, one should apply RG to all HSPs since the curvature function may diverge at any of them, but in this article we focus on the HSP where the noninteracting model is known to close the bulk gap. We remark that this RG scheme is applicable even if the Green's function (and subsequently the curvature function) is calculated by means other than the perturbative treatment in Sec.~\ref{sec:Green_fn_Dirac_interaction}, for instance quantum Monte Carlo (QMC) or exact diagonalization. Because it renormalizes the curvature function, the RG scheme is hereafter referred to as the curvature renormalization group (CRG) approach.

In this CRG scheme, the critical point and fixed point are identified from the RG flow from the flow rate and the direction of the flow, summarized below:
\begin{eqnarray}
{\rm Critical\;point}:&&\left|\frac{d{\bf M}}{dl}\right|\rightarrow\infty,\;{\rm flow\;directs\;away},
\nonumber \\
{\rm Stable\;fixed\;point}:&&\left|\frac{d{\bf M}}{dl}\right|\rightarrow 0,\;{\rm flow\;directs\;into},
\nonumber \\
{\rm Unstable\;fixed\;point}:&&\left|\frac{d{\bf M}}{dl}\right|\rightarrow 0,\;{\rm flow\;directs\;away}.
\nonumber \\
\label{identifying_Mc_Mf}
\end{eqnarray}
The critical point ${\bf M}_{c}$ form a $(D_{M}-1)$-dimensional surface in the $D_{M}$-dimensional parameter space. The flow rate diverges at ${\bf M}_{c}$ because the numerator in Eq.~(\ref{generic_RG_eq}) diverges, as can be seen from the Ornstein-Zernike form in Eq.~(\ref{Ornstein_Zernike}) and the critical behavior in Eq.~(\ref{Fk0_xi_divergence}). The flow rate vanishes at the fixed point ${\bf M}_{f}$ because the numerator in Eq.~(\ref{generic_RG_eq}) vanishes, which corresponds to a vanishing correlation length or correlation time $\xi_{i}$. The divergence of $\xi_{i}$ at ${\bf M}_{c}$ and vanish of it at ${\bf M}_{f}$ is the interpretation of scale invariance in this CRG scheme,  since the amplitude of the correlation does not depend on the distance ${\bf R}$.

\section{1D class BDI with interactions\label{sec:1D_class_BDI}}

\subsection{Topological invariant in the presence of interactions\label{sec:Green_function_1D}}

We proceed to use 1D class BDI spinless fermionic systems to demonstrate the principles in Sec.~\ref{sec:generic_critical_perturbative}. The appropriate minimal model is that in Eq.~(\ref{general_2by2_Dirac_Hamiltonian}) with
\begin{eqnarray}
d_{0,1}(k)=d_{0,1}(-k)\;,\;\;\;d_{2}(k)=-d_{2}(-k)\;.
\label{BDI_1D_d_vector}
\end{eqnarray}
That is, the noninteracting Dirac Hamiltonian only contains the $d_{0,1}$ components that are even in $k$, and the $d_{2}$ component that is odd in $k$. The $d_{3}$ component is absent. In the presence of interactions, the ${\bf d}$-vector is changed to the ${\bf d}^{\prime}$-vector according to Eq.~(\ref{renormalized_d_vector_mu}).

We adopt the topological invariant expressed in terms of the full Green's function\cite{Essin11}
\begin{eqnarray}
{\cal C}=\int_{0}^{2\pi} \frac{dk}{4\pi i} \;{\rm Tr}\left(\sigma_{3}G^{-1}\partial_{k}G\right)|_{i\omega=-d_{0}^{\prime}}\;,
\label{topo_invariant}
\end{eqnarray}
evaluated at $i\omega=-d_{0}^{\prime}$, where $-d_{0}^{\prime}$ can be viewed as the interaction-induced chemical potential. In the case of density-density interaction in which the self-energy is frequency-independent, the requirement of $i\omega=-d_{0}^{\prime}$ physically means that the entire spectrum is shifted by $-d_{0}^{\prime}$ and hence the spectrum is particle-hole symmetric with respect to $-d_{0}^{\prime}$. Consequently, in the Green's function formalism, the reference energy shall be shifted to $i\omega=-d_{0}^{\prime}$ such that the diagonal element of Green's function vanishes and Eq.~(\ref{topo_invariant}) is applicable (see the calculation below). In the noninteracting model described by Eq.~(\ref{BDI_1D_d_vector}), the $d_{0}$ component may be taken to be zero $d_{0}=0$ following the usual convention, and hence $i\omega=0$ in Eq.~(\ref{topo_invariant})\cite{Essin11,Gurarie11}.

From Eq.~(\ref{Dyson_equation}) one sees that the full Green's function at $i\omega=-d_{0}^{\prime}$ has the same property and the unperturbed Green's function, namely it only has off-diagonal elements
\begin{eqnarray}
G(k,i\omega=-d_{0}^{\prime})=
\left(
\begin{array}{cc}
0 & \frac{-Q_{k}-\Sigma_{AB}}{|Q_{k}+\Sigma_{AB}|^{2}} \\
\frac{-Q_{k}^{\ast}-\Sigma_{AB}^{\ast}}{|Q_{k}+\Sigma_{AB}|^{2}} & 0
\end{array}
\right)\;,
\label{G_full_E0}
\end{eqnarray}
where $\Sigma_{AB}\equiv\Sigma_{AB}(k,i\omega=-d_{0}^{\prime})=\Sigma_{BA}(k,i\omega=-d_{0}^{\prime})^{\ast}$ is the upper-right off-diagonal element of the $2\times 2$ self-energy matrix at $i\omega=-d_{0}^{\prime}$. Putting Eq.~(\ref{G_full_E0}) into Eq.~(\ref{topo_invariant}), the topological invariant is
\begin{eqnarray}
{\cal C}&=&\frac{1}{4\pi i}\int_{0}^{2\pi} dk \left\{\frac{-1}{Q_{k}+\Sigma_{AB}}\partial_{k}\left(Q_{k}+\Sigma_{AB}\right)
-h.c.\right\}
\nonumber \\
&=&\frac{1}{2\pi}\int_{0}^{2\pi} dk \;\partial_{k}\varphi\;,
\label{topo_invariant_dphi}
\end{eqnarray}
where $\varphi_{k}$ is the argument of $Q_{k}+\Sigma_{AB}=|Q_{k}+\Sigma_{AB}|e^{-i\varphi_{k}}$. Thus the topology of the system simply counts the number of times that the phase of $Q_{k}+\Sigma_{AB}$ winds as $k$ changes from $0$ to $2\pi$, which is determined by both the off-diagonal element of the noninteracting Hamiltonian $Q_{k}=d_{1}-id_{2}$ and that of the self-energy $\Sigma_{AB}$. Obviously, the winding number only takes integer values.

Equation (\ref{topo_invariant_dphi}) indicates that the topological invariant is calculated from the integration of a curvature function, in this case simply the gradient of the phase
\begin{eqnarray}
F(k,{\bf M})\equiv\;\frac{1}{2i}{\rm Tr}\left(\sigma_{3}G^{-1}\partial_{k}G\right)|_{i\omega=-d_{0}^{\prime}}=\partial_{k}\varphi_{k}\;.
\label{curvature_function}
\end{eqnarray}
Assuming that the curvature function displays the divergence discussed in Sec.~\ref{sec:criticality}, which is indeed the case for the 1D class BDI Dirac model with the density-density interaction, a correlation function that characterizes the topological phase transition can be introduced. Consider the Fourier transform between real space and momentum space Green's function
\begin{eqnarray}
G(r,i\omega)&=&\frac{1}{N}\sum_{k}e^{ikr}G(k,i\omega)=\int\frac{dk}{2\pi}e^{ikr}G(k,i\omega)\;,
\nonumber \\
G(k,i\omega)&=&\sum_{r}e^{-ikr}G(r,i\omega)=\int dr\,e^{-ikr}G(r,i\omega)\;,
\nonumber
\label{Gr_Gk_Fourier_1D}
\end{eqnarray}
and likewisely for $G^{-1}$, where we have set the lattice constant $a=1$ to be unity and dimensionless, and hence the momentum and position are also dimensionless $\left[k\right]=\left[r\right]=1$, and the volume of the system is $v=Na=N$. The topological invariant can then be expressed in terms of the real space Green's function
\begin{eqnarray}
{\cal C}=-\frac{1}{2}\int dr\,{\rm Tr}\left[\sigma_{3}G^{-1}(-r,i\omega)\,r\,G(r,i\omega)\right]_{i\omega=-d_{0}}\;.
\end{eqnarray}
In addition, the Fourier transform of the curvature function yields a correlation function
\begin{eqnarray}
&&\lambda_{r}=\int_{0}^{2\pi} \frac{dk}{2\pi} \,e^{ikr}\,F(k,{\bf M})
\nonumber \\
&&=-\frac{1}{2}\int dr_{1}\,{\rm Tr}\left[\sigma_{3}G^{-1}(r-r_{1},i\omega)\,r_{1}\,G(r_{1},i\omega)\right]_{i\omega=-d_{0}}\;.
\nonumber \\
\label{lambdaR_1D}
\end{eqnarray}
In this representation, both ${\cal C}$ and $\lambda_{r}$ take the form of a product of the Green's function, its inverse, and the distance it propagates, as depicted
graphically in Fig.~\ref{fig:1D_class_BDI_topo_inv_corr_fn_graphics}. Putting the Ornstein-Zernike form of Eq.~(\ref{Ornstein_Zernike}) into the Fourier transform in Eq.~(\ref{lambdaR_1D}), we see that the correlation function exponentially decays at large distance $\lambda_{r}\propto e^{-r/\xi}$ in either topologically trivial or nontrivial phase, with $\xi$ playing the role of the correlation length. Moreover, the correlation length diverges at the topological phase transition ${\bf M}_{c}$, signifying scale invariance.

\begin{figure}[ht]
\begin{center}
\includegraphics[clip=true,width=0.85\columnwidth]{1D_class_BDI_topo_inv_corr_fn_graphics}
\caption{ (a) Graphic presentation of the Green's function times the polarization $r\,G(r,i\omega=-d_{0}^{\prime})$ as a thick line and the inverse of the Green's function $G^{-1}(r,i\omega=-d_{0}^{\prime})$ as a thin line. (b) Topological invariant for the interacting 1D class BDI models expressed in terms of the spacetime Green's function. The blue point denotes the origin, and the black points denote the position $r$ that is to be integrated. (c) The correlation function $\lambda_{R}$ presented graphically, which decays with a correlation length that diverges at the topological phase transition. }
\label{fig:1D_class_BDI_topo_inv_corr_fn_graphics}
\end{center}
\end{figure}

\subsection{Su-Schrieffer-Heeger model with nearest-neighbor interaction\label{sec:SSHnn}}

As a concrete example for 1D class BDI, we consider the spinless Su-Schrieffer-Heeger (SSH) model in the presence of a density-density interaction. The noninteracting part of the Hamiltonian is
\begin{eqnarray}
{\cal H}_{0}&=&\sum_{i}(t+\delta t)c_{Ai}^{\dag}c_{Bi}+(t-\delta t)c_{Ai+1}^{\dag}c_{Bi}+h.c.
\nonumber \\
&=&\sum_{k}Q_{k}c_{Ak}^{\dag}c_{Bk}+Q_{k}^{\ast}c_{Bk}^{\dag}c_{Ak}\;,
\label{SSH_unperturbed}
\end{eqnarray}
where $Q_{k}=(t+\delta t)+(t-\delta t)e^{-ik}$. We examine the nearest-neighbor density-density interaction
\begin{eqnarray}
{\cal H}_{e-e}=V\sum_{i}\left(n_{Ai}n_{Bi}+n_{Bi}n_{Ai+1}\right)\;,
\label{SSH_ee_interaction}
\end{eqnarray}
where $n_{Ii}\equiv c_{Ii}^{\dag}c_{Ii}$, whose Fourier transform takes the form of Eq.~(\ref{general_density_density_interaction}) with $V_{q}=V(1+\cos q)$. The model remains spinless in the presence of the interaction in Eq.~(\ref{SSH_ee_interaction}), in contrast to the spinful version of this model under the influence of Hubbard interactions, which has been investigated previously\cite{Manmana14,Yoshida14}.

Following Appendix \ref{appendix:self_energy}, the one-loop calculation gives the self-energies
\begin{eqnarray}
&&\Sigma_{AA}(k)=\Sigma_{BB}(k)=V\;,
\nonumber \\
&&\Sigma_{AB}(k)=\frac{1}{2}\sum_{q}V_{q}e^{-i\alpha_{k+q}}
=\left[\Sigma_{BA}(k)\right]^{\ast}\;,
\label{SSHnn_self-energy}
\end{eqnarray}
where the phase $\alpha$ is defined from the diagonal element of the noninteracting Hamiltonian $Q_{k}\equiv|Q_{k}|e^{-i\alpha_{k}}$, and $n_{F}(x)=\theta(-x)$ is the Fermi distribution function that takes the form of a step function at zero temperature. In deriving Eq.~(\ref{SSHnn_self-energy}), we have used Eqs.~(\ref{BDI_1D_d_vector}), (\ref{Hartree_Fock_self_energy}), and the result that follow $G_{0AA}(p,\tau=0)=1/2$ and $G_{0AB}(p+q,\tau=0)=-e^{-i\alpha_{p+q}}/2$. The full Green's function is therefore
\begin{eqnarray}
&&G(k,i\omega)=\frac{1}{(i\omega-V)^{2}+|Q_{k}+\Sigma_{AB}(k)|^{2}}
\nonumber \\
&&\times\left(
\begin{array}{cc}
i\omega-V & Q_{k}+\Sigma_{AB}(k) \\
\left[Q_{k}+\Sigma_{AB}(k)\right]^{\ast} & i\omega-V
\end{array}
\right)\;.
\end{eqnarray}
As explained after Eq.~(\ref{G_full_E0}), at $i\omega=V$ the full Green's function becomes off-diagonal, and the topology simply counts the winding of the phase $\varphi_{k}$ of the off-diagonal element across the BZ
\begin{eqnarray}
&&\varphi(k,{\bf M})=-\arg\left(Q_{k}+\Sigma_{AB}\right)
\nonumber \\
&&=-\arg\left(Q_{k}+\frac{1}{2}\sum_{q}V_{q}e^{-i\alpha_{k+q}}\right)\;.
\label{SSHnn_phik_analytical_result}
\end{eqnarray}
It is then clear that the calculation of topological invariant ${\cal C}$ in Eq.~(\ref{topo_invariant_dphi}) involves two momentum integrations: one integrates over $k$ in Eq.~(\ref{topo_invariant_dphi}) for the topological invariant itself, and the other integrates over $q$ in Eq.~(\ref{SSHnn_phik_analytical_result}) for the one-loop self-energy.

As explain in Sec.~\ref{sec:RG}, we now apply the CRG approach to study the topological phase transitions in the parameter space ${\bf M}=(\delta t,V)$ without explicitly calculating the integration over $k$ in Eq.~(\ref{topo_invariant_dphi}). Since $\partial_{k}\varphi_{k}$ plays the role of the curvature function, the RG equation is that in Eqs.~(\ref{generic_RG_eq}) and (\ref{generic_RG_eq_discrete}) with $F({\bf K},{\bf M})=F(k,{\bf M})=\partial_{k}\varphi_{k}$. Going one step further, since often times the phase $\varphi_{k}=\varphi(k,{\bf M})$ itself is the quantity that is easiest to be solved numerically, one can express the RG equation in terms of $\varphi(k,{\bf M})$ on several mesh points
\begin{eqnarray}
&&\frac{dM_{i}}{dl}=\left(\frac{\Delta M_{i}}{\Delta k^{2}}\right)\left\{\varphi(k_{0}+2\Delta k,{\bf M})-2\varphi(k_{0}+\Delta k,{\bf M})\right.
\nonumber \\
&&\left.+\varphi(k_{0},{\bf M})\right\}
\times\left\{\varphi(k_{0}+\Delta k,{\bf M}+\Delta{\bf M}_{i})\right.
\nonumber \\
&&\left.-\varphi(k_{0},{\bf M}+\Delta{\bf M}_{i})-\varphi(k_{0}+\Delta k,{\bf M})+\varphi(k_{0},{\bf M})\right\}^{-1}\;,
\nonumber \\
\label{1D_generic_RG_eq_phi}
\end{eqnarray}
from which one sees that only five mesh points of $\varphi(k,{\bf M})$ are required to obtain the RG flow at a given ${\bf M}$, which is numerically much more economical compared to a brute force integration of Eq.~(\ref{topo_invariant_dphi}).

The resulting RG flow in the ${\bf M}=(\delta t,V)$ parameter space by using $k_{0}=\pi$ is shown in Fig.~\ref{fig:SSHnn_RGflow}. The RG flow identifies the phase boundary between two topologically distinct phases according to the rule in Eq.~(\ref{identifying_Mc_Mf}). The critical point in the noninteracting limit located at $(\delta t_{c},V)=(0,0)$ separates the topologically trivial ${\cal C}=0$ and nontrivial ${\cal C}=1$ phases, a well-known result for the noninteracting SSH model, and the phase boundary in the interacting model is a continuous line that passes through this critical point. The fixed point for the nontrivial phase seems to locate at ${\bf M}_{f}=(\delta t_{f},V_{f})=(-t,-\infty)$, i.e., with an infinite attractive nearest-neighbor interaction, and for the trivial phase seems to locate at ${\bf M}_{f}=(\delta t_{f},V_{f})=(t,\infty)$, i.e., an infinite repulsive nearest-neighbor interaction. Certainly our Green's function approach is limited within weak coupling $V\apprle t$ so these fixed points are out of the reach of this approach, whose investigation may require other types of numerical approach, such as exact diagonalization.

Figure \ref{fig:SSHnn_RGflow} also shows the critical behavior of $F(k_{0},{\bf M})$ and $\xi$ as approaching the critical point from either $\delta t\rightarrow\delta t_{c}$ or $V\rightarrow V_{c}$ direction. Both $F(k_{0},{\bf M})^{-1}$ and $\xi^{-1}$ clearly demonstrate a linear behavior as approaching the phase boundary from either direction, indicating the critical exponents defined in Eq.~(\ref{Fk0_xi_critical_exponent}) are $\gamma\approx \nu\approx 1$. Besides satisfying the scaling law introduced in Eq.~(\ref{scaling_law}), these exponents coincide with those in the noninteracting limit\cite{Chen17}.

\begin{figure}[ht]
\begin{center}
\includegraphics[clip=true,width=0.8\columnwidth]{SSHnn_RGflow}
\includegraphics[clip=true,width=0.99\columnwidth]{SSHnn_exponent_fixV}
\includegraphics[clip=true,width=0.99\columnwidth]{SSHnn_exponent_fixdt}
\caption{ (a) RG flow $d{\bf M}/dl=(d\delta t/dl,dV/dl)$ (black arrows) of SSH model with nearest-neighbor interaction. The $dV/dl$ component is manually reduced by a factor of $0.1$ for the sake of presentation. The log of the flow rate $\ln(|d{\bf M}/dl|)$ is indicated by the color scale, with brown the low flow rate and green the high flow rate. The flow identifies a phase boundary (green line in the middle from which the arrows direct away) between the topologically nontrivial ${\cal C}=1$ and the trivial ${\cal C}=0$ phase. The fixed points seem to locate at ${\bf M}_{f}=(\delta t_{f},V_{f})=(-t,-\infty)$ and $(t,\infty)$ for the two phases, respectively. (b) and (c) show the $F(k_{0}=\pi,{\bf M})^{-1}$ and $\xi^{-1}$ as functions of $\delta t$ at several values of $V$, and (d) and (e) show them as functions of $V$ at several values of $\delta t$. The linear behavior of them as approaching the critical points (where they vanish) indicates the critical exponents in Eq.~(\ref{Fk0_xi_critical_exponent}) are $\gamma\approx\nu\approx 1$.  }
\label{fig:SSHnn_RGflow}
\end{center}
\end{figure}

\section{2D class A with interactions}

\subsection{Frequency-dependent self-energy\label{sec:Green_function_2D}}

We proceed to consider the 2D spinless fermionic systems in class A that do not preserve any of the three nonspatial symmetries. The ${\bf d}$-vector in the noninteracting model of Eq.~(\ref{general_2by2_Dirac_Hamiltonian}) in this symmetry class satisfies
\begin{eqnarray}
d_{1,2}({\bf k})=-d_{1,2}(-{\bf k})\;,\;\;\;d_{0,3}({\bf k})=d_{0,3}(-{\bf k})\;.
\end{eqnarray}
That is, the noninteracting Dirac Hamiltonian has $d_{1,2}$ component that are odd in ${\bf k}$, and the $d_{0,3}$ component that are even in ${\bf k}$. The $d_{0}$ component is conventionally set to zero, but can be generated by interactions.

 An expression for the topological invariant in terms of full Green's function has been proposed\cite{Niu85,Gurarie11}
\begin{eqnarray}
{\cal C}=&&\frac{\pi}{3}\int_{BZ} \frac{d^{2}{\bf k}}{(2\pi)^{2}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}
\nonumber \\
&&\times\epsilon^{abc}{\rm Tr}\left[(G^{-1}\partial_{a}G)(G^{-1}\partial_{b}G)(G^{-1}\partial_{c}G)\right]
\label{2D_C_interacting}
\end{eqnarray}
where $\epsilon^{abc}$ is antisymmetric in the three indices $\left\{a,b,c\right\}=\left\{\omega,k_{x},k_{y}\right\}$, and $G\equiv G({\bf k},i\omega)$ is the interaction-dressed single-particle Green's function. We now examine the integrand in this expression using the renormalized ${\bf d}^{\prime}$-vector in Eqs.~(\ref{2D_G_dprime}) and (\ref{renormalized_d_vector_mu}). Keeping in mind that $d_{i}^{\prime}$ is a function of both momentum and frequency, and hence $\partial_{\omega}d_{i}^{\prime}$ and $\partial_{k_{j}}d_{i}^{\prime}$ are both nonzero, a straight forward expansion of the integrand in Eq.~(\ref{2D_C_interacting}) gives
\begin{eqnarray}
&&\frac{\pi}{3}\epsilon^{abc}{\rm Tr}\left[(G^{-1}\partial_{a}G)(G^{-1}\partial_{b}G)(G^{-1}\partial_{c}G)\right]_{\left\{a,b,c\right\}=\left\{\omega,k_{x},k_{y}\right\}}
\nonumber \\
&&=\frac{4\pi i}{\left[(i\omega+d_{0}^{\prime})^{2}-d^{\prime 2}\right]^{2}}\left\{
-i\epsilon^{abc}d_{a}^{\prime}\partial_{x}d_{b}^{\prime}\partial_{y}d_{c}^{\prime}
|_{\left\{a,b,c\right\}=\left\{1,2,3\right\}}
\right.
\nonumber \\
&&+\epsilon^{abcd}d_{a}^{\prime}\partial_{\omega}d_{b}^{\prime}\partial_{x}d_{c}^{\prime}\partial_{y}d_{d}^{\prime}
|_{\left\{a,b,c,d\right\}=\left\{0,1,2,3\right\}}
\nonumber \\
&&\left.+i\omega\epsilon^{abc}\partial_{\omega}d_{a}^{\prime}\partial_{x}d_{b}^{\prime}\partial_{y}d_{c}^{\prime}
|_{\left\{a,b,c\right\}=\left\{1,2,3\right\}}\right\},
\label{TrGGGGGG_to_dprime_to_FkM}
\nonumber \\
&&=F({\bf K},{\bf M})\;,
\end{eqnarray}
where $\epsilon^{abc}$ and $\epsilon^{abcd}$ are antisymmetric in permutations of two neighboring indices. In the spirit of Eq.~(\ref{topo_invariant}), this integrand is treated as a curvature function $F({\bf K},{\bf M})$ defined in the momentum-frequency space ${\bf K}=(\omega,k_{x},k_{y})$, and for the models examined in this article, we find that it exhibits the critical behavior described in Sec.~\ref{sec:criticality}.

To demonstrate the critical behavior, we first examine the following continuous limit of the noninteracting lattice model $d_{i}^{\prime}\rightarrow d_{i}$\cite{Bernevig13}
\begin{eqnarray}
&&d_{0}=0\;,\;\;\;d_{1}=k_{x},\;\;\;d_{2}=k_{y},\;\;\;d_{3}=M,
\nonumber \\
&&\epsilon^{abc}d_{a}\partial_{x}d_{b}\partial_{y}d_{c}=M.
\end{eqnarray}
In this limit, the second and third term in Eq.~(\ref{TrGGGGGG_to_dprime_to_FkM}) drops out because $\partial_{\omega}d_{i}=0$, and hence
\begin{eqnarray}
&&\frac{\pi}{3}\epsilon^{abc}{\rm Tr}\left[(G_{0}^{-1}\partial_{a}G_{0})(G_{0}^{-1}\partial_{b}G_{0})(G_{0}^{-1}\partial_{c}G_{0})\right]
\nonumber \\
&&=\frac{4\pi M}{(\omega^{2}+k^{2}+M^{2})^{2}}
\approx\frac{\tilde{F}({\bf K}_{0},M)}{1+\xi^{2}K^{2}}\;,
\label{FkM_2D_class_A_noninteracting_limit}
\end{eqnarray}
where in the last line we have expanded with $K\ll 1$ and used $d^{2}=k^{2}+M^{2}$. The result suggests that the curvature function takes the Ornstein-Zernike form around the HSP ${\bf K}_{0}=(0,0,0)$ in all three directions in the momentum-frequency space, with the height of the Lorentzian $\tilde{F}({\bf K}_{0},M)=4\pi/M^{3}$ that diverges with critical exponent $\gamma=3$ and changes sign at the critical point $M_{c}=0$, and the inverse of the Lorentzian width in all three directions $\xi_{i}=\sqrt{2}/|M|$ diverges with critical exponent $\nu_{i}=1$. Thus the noninteracting Dirac model treated within the Green's function formalism satisfies the critical behavior and the scaling law $\gamma=\nu_{t}+\nu_{x}+\nu_{y}$ introduced in Sec.~\ref{sec:criticality}.

Consider the Fourier transform of the Green's function
\begin{eqnarray}
G({\bf R})&=&\int\frac{d^{3}{\bf K}}{(2\pi)^{3}}\,e^{i{\bf K\cdot R}}\,G({\bf K})\;,
\nonumber \\
G({\bf K})&=&\int d^{3}{\bf R}\,e^{-i{\bf K\cdot R}}\,G({\bf R})\;,
\end{eqnarray}
and likewisely for $G^{-1}$, where $e^{i{\bf K}\cdot{\bf R}}=e^{ik_{x}r_{x}+ik_{y}r_{y}-i\omega t}$, $\int d^{3}{\bf K}\equiv\int_{BZ} d^{2}{\bf k}\int_{-\infty}^{\infty} d\omega$, and ${\bf R}=(-t,R_{x},R_{y})$ is the spacetime coordinate. A direct Fourier transform of  Eq.~(\ref{2D_C_interacting}) gives the topological invariant expressed in terms of spacetime Green's functions
\begin{eqnarray}
{\cal C}&=&\frac{i\pi}{3}\left(\prod_{i=1}^{5}\int d^{3}{\bf R}_{i}\right)\epsilon^{abc}
\nonumber \\
&&\times{\rm Tr}\left[\right.G^{-1}(-{\bf R}_{1}-{\bf R}_{2}-{\bf R}_{3}-{\bf R}_{4}-{\bf R}_{5})R_{1a}G({\bf R}_{1})
\nonumber \\
&&\times\left.G^{-1}({\bf R}_{2})R_{3b}G({\bf R}_{3})
G^{-1}({\bf R}_{4})R_{5c}G({\bf R}_{5})\right].
\end{eqnarray}
Following Eq.~(\ref{general_Fourier_trans_lambdaR}), the correlation function is introduced from the Fourier transform of the curvature function
\begin{eqnarray}
&&\lambda_{\bf R}=\int_{BZ} \frac{d^{2}{\bf k}}{(2\pi)^{2}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,e^{i{\bf K}\cdot{\bf R}}F({\bf K},{\bf M})
\nonumber \\
&&=\frac{i\pi}{3}\left(\prod_{i=1}^{5}\int d^{3}{\bf R}_{i}\right)\epsilon^{abc}
\nonumber \\
&&\times{\rm Tr}\left[\right.G^{-1}({\bf R}-{\bf R}_{1}-{\bf R}_{2}-{\bf R}_{3}-{\bf R}_{4}-{\bf R}_{5})R_{1a}G({\bf R}_{1})
\nonumber \\
&&\times\left.G^{-1}({\bf R}_{2})R_{3b}G({\bf R}_{3})
G^{-1}({\bf R}_{4})R_{5c}G({\bf R}_{5})\right]\;.
\end{eqnarray}
Similar to those for 1D class BDI models discussed in Sec.~\ref{sec:1D_class_BDI}, the topological invariant and the correlation function in this representation are a product of the spacetime Green's functions propagated over several segments, as graphically presented in Fig.~\ref{fig:2D_class_A_topo_inv_corr_fn_graphics}.

\begin{figure}[ht]
\begin{center}
\includegraphics[clip=true,width=0.8\columnwidth]{2D_class_A_topo_inv_corr_fn_graphics}
\caption{ (a) Graphic presentation of the spacetime Green's function times a distance $R_{a}G({\bf R})$ as a thick line and the inverse of the spacetime Green's function $G^{-1}({\bf R})$ as a thin line. (b) Topological invariant for the interacting 2D class A models expressed in terms of the spacetime Green's function. The blue point denotes the origin, and the 5 black points denote the positions ${\bf R}_{1}\sim{\bf R}_{5}$ that are to be integrated. (c) The correlation function $\lambda_{\bf R}$ presented graphically, which decays with a correlation length in all the three spacetime directions. }
\label{fig:2D_class_A_topo_inv_corr_fn_graphics}
\end{center}
\end{figure}

In the presence of weak interactions, we use Dyson's equation, Eq.~(\ref{Dyson_equation}), to obtain the full Green's function up to one-loop. The $\left(G_{0}\Sigma G\right)_{IJ}$ part of the Dyson's equation is detailed in Appendix \ref{appendix:self_energy}. The calculation of self-energy now requires a 2D integral over momentum. Together with the 3D integral in Eq.~(\ref{2D_C_interacting}), evidently the calculation of topological invariant ${\cal C}$ at a given tuning parameter ${\bf M}$ requires a total of 5D integration, obviously a tedius task. We will demonstrate how CRG circumvents the 3D integral in Eq.~(\ref{2D_C_interacting}) by using a concrete example in the next section.

\begin{figure}[ht]
\begin{center}
\includegraphics[clip=true,width=0.7\columnwidth]{Chern_ephonon_RGflow}
\includegraphics[clip=true,width=0.99\columnwidth]{Chern_ephonon_exponent_fixw0}
\caption{ (a) The RG flow $d{\bf M}/dl$ (black arrows) of 2D Chern insulator with electron-phonon interaction in the ${\bf M}=(M,\omega_{0})$ parameter space. The flow component $d\omega_{0}/dl$ is reduced by a factor of $0.1$ for the sake of presentation. The color scale indicates the high (green) and low (brown) flow rate on log scale $\ln|d{\bf M}/dl|$. The flow identifies a phase boundary (green line) between the topologically nontrivial ${\cal C}=-1$ and trivial ${\cal C}=0$ phase that is independent from the phonon frequency $\omega_{0}$. (b) and (c) show the $|F({\bf K}_{0},M)|^{1/2}$ and $\left\{\xi_{x}^{-1},\xi_{t}^{-1}\right\}$. At large $\omega_{0}$, $|F({\bf K}_{0},M)|^{1/2}$ shows linear behavior near the critical point $M_{0}=0$, indicating the critical exponent $\gamma\approx 2$, where as at small $\omega_{0}$ it deviates from linear and the exponent approaches $\gamma\approx 3$. The linear behavior of $\left\{\xi_{x}^{-1},\xi_{t}^{-1}\right\}$ near the critical point indicates the critical exponent $\nu_{x}\approx\nu_{t}\approx 1$. In addition, $\xi_{t}^{-1}$ features an offset at the critical point that is linear in $\omega_{0}$, as described by Eq.~(\ref{xi_ephonon_critical_behavior}). }
\label{fig:Chern_ephonon_RGflow}
\end{center}
\end{figure}

\subsection{2D Chern insulator with electron-phonon interaction}

As an example of frequency-dependent self-energy, we study 2D Chern insulator in the presence of electron phonon interaction ${\cal H}={\cal H}_{0}+{\cal H}_{e-ph}$. The unperturbed lattice Hamiltonian of 2D class A in momentum space is described by Eq.~(\ref{general_2by2_Dirac_Hamiltonian}) with\cite{Bernevig13}
\begin{eqnarray}
&&d_{0}=0,\;d_{1}=\sin k_{x},\;d_{2}=\sin k_{y},
\nonumber \\
&&d_{3}=M+2-\cos k_{x}-\cos k_{y}\;,
\label{2D_Chern_H0}
\end{eqnarray}
and the electron-phonon interaction is that in Eq.~(\ref{general_electron_phonon_interaction}). After calculating the self-energy using the formula in Appendix \ref{appendix:self_energy}, we calculate the renormalized ${\bf d}^{\prime}$-vector and its derivatives according to Eq.~(\ref{renormalized_d_vector_mu}), and subsequently put them into Eq.~(\ref{TrGGGGGG_to_dprime_to_FkM}) to obtain the curvature function. Numerically, we varify that the curvature function indeed takes the Ornstein-Zernike form of Eq.~(\ref{Ornstein_Zernike}) is all three directions in the momentum-frequency space. We will treat the mass term in the Dirac Hamiltonian and the optical phonon frequency as tuning parameters ${\bf M}=(M,\omega_{0})$.

To study the topological phase transition in the ${\bf M}=(M,\omega_{0})$ parameter space, we apply the CRG technique in Sec.~\ref{sec:RG} by putting the curvature function in Eq.~(\ref{TrGGGGGG_to_dprime_to_FkM}) into Eq.~(\ref{RG_procedure_general}), without explicitly performing the 3D integral of Eq.~(\ref{2D_C_interacting}). The resulting RG flow is shown in Fig.~\ref{fig:Chern_ephonon_RGflow}, which features a phase boundary between the topological nontrivial ${\cal C}=-1$ and trivial ${\cal C}=0$ phase that passes through the known critical point at the noninteracting limit ${\bf M}_{c}=(0,0)$. Interestingly, at a general phonon frequency $\omega_{0}$ the critical point is ${\bf M}_{c}=(0,\omega_{0})$ at any polar coupling strength $u$ in Eq.~(\ref{general_electron_phonon_interaction}), meaning that the topological phase transition is not influenced by the phonon frequency, at least in the weak coupling regime.

However, the phonon frequency influences the critical exponents. We found that the critical behavior of the correlation length $\left\{\xi_{x},\xi_{y}\right\}$ and correlation time $\xi_{t}$ is described by, having in mind that $M_{c}=0$,
\begin{eqnarray}
\xi_{x}\approx\frac{1}{c_{x}|M|}\;,
\;\;\xi_{y}\approx\frac{1}{c_{y}|M|}\;,
\;\;\xi_{t}\approx\frac{1}{c_{t\omega}\omega_{0}+c_{tM}|M|}\;,\;
\label{xi_ephonon_critical_behavior}
\end{eqnarray}
where the $\left\{c_{x},c_{y},c_{t\omega},c_{tM}\right\}$ are phenomenological fitting parameters. The correlation lengths along the two momentum directions diverge with critical exponents $\nu_{x}=\nu_{y}=1$. The correlation time, on the other hand, displays a more complicated dependence. Equation (\ref{xi_ephonon_critical_behavior}) indicates that in the noninteracting limit $\omega_{0}=0$, the correlation time $\xi_{t}$ has critical exponent $\nu_{t}=1$, consistent with that described after Eq.~(\ref{FkM_2D_class_A_noninteracting_limit}). However, at finite electron-phonon interaction, $\xi_{t}$ no longer diverges at $M\rightarrow M_{c}=0$ but saturates to a constant that is inversely proportional to the phonon frequency $\omega_{0}$, and consequently assigned a critical exponent $\nu_{t}=0$. Therefore according to the scaling law $\gamma=\nu_{x}+\nu_{y}+\nu_{t}$, the critical exponent for the extremum of curvature function should be $\gamma=2$ for the interacting case, and $\gamma=3$ for the noninteracting limit, which are indeed varified numerically and presented in Fig.~\ref{fig:Chern_ephonon_RGflow}. We are lead to conclude that in the presence of electron-phonon interaction, the critical exponents $\left\{\gamma,\nu_{t}\right\}$ are changed, and as $M\rightarrow M_{c}$ the scale invariance does not manifest along the time-direction but only along the two spatial directions. Note that because the curvature function does not diverge in the frequency direction, $\delta{\bf K}$ has to be chosen along either $\delta k_{x}$ or $\delta k_{y}$ in the CRG procedure.

\subsection{Frequency-independent self-energy}

We proceed to address the situation when self-energy is frequency-independent, since it is relevant to several types of density-density interaction. In this case, the ${\bf d}^{\prime}$-vector in Eq.~(\ref{renormalized_d_vector_mu}) is frequency-independent and hence $\partial_{\omega}d_{i}^{\prime}=0$, following which the form of the curvature function in Eq.~(\ref{TrGGGGGG_to_dprime_to_FkM}) is greatly simplified
\begin{eqnarray}
&&\frac{\pi}{3}\epsilon^{abc}{\rm Tr}\left[(G_{0}^{-1}\partial_{a}G_{0})(G_{0}^{-1}\partial_{b}G_{0})(G_{0}^{-1}\partial_{c}G_{0})\right]
\nonumber \\
&&=\frac{4\pi}{\left[-(i\omega+d_{0}^{\prime})^{2}+d^{\prime 2}\right]^{2}}\epsilon^{abc}d_{a}^{\prime}\partial_{x}d_{b}^{\prime}\partial_{y}d_{c}^{\prime}\;.
\label{TrGdGGdGGdG_to_Fxy}
\end{eqnarray}
As a result, the form of correlation function and the topological invariant expressed in terms of real space Green's function is simplified. The integration over frequency in Eq.~(\ref{2D_C_interacting}) can be performed analytically
\begin{eqnarray}
\int_{-\infty}^{\infty}\frac{d\omega}{\left[-(i\omega+d_{0}^{\prime})^{2}+d^{\prime 2}\right]^{2}}=\frac{\pi}{2d^{\prime 3}}\;\;({\rm assuming}\;|d_{0}^{\prime}|<d^{\prime}).
\nonumber \\
\end{eqnarray}
The assumption of $|d_{0}^{\prime}|<d^{\prime}$ is reasonable, since $d_{0}^{\prime}$ comes from the self-energy according to Eq.~(\ref{renormalized_d_vector_mu}), and in the weak coupling regime it should be smaller than the renormalized band width $d^{\prime}$. The topological invariant in Eq.~(\ref{2D_C_interacting}) then takes the form
\begin{eqnarray}
&&{\cal C}=\frac{1}{4\pi}\int_{BZ}d^{2}{\bf k}\frac{\epsilon^{abc}}{d^{\prime 3}}d_{a}^{\prime}\partial_{x}d_{b}^{\prime}\partial_{y}d_{c}^{\prime}|_{\left\{a,b,c\right\}=\left\{1,2,3\right\}}
\nonumber \\
&&=\frac{1}{4\pi}\int_{BZ} d^{2}{\bf k}\,{\hat{\bf d}}^{\prime}\cdot\left(\partial_{x}{\hat{\bf d}}^{\prime}\times\partial_{y}{\hat{\bf d}}^{\prime}\right)|_{\left\{a,b,c\right\}=\left\{1,2,3\right\}}\;.\;\;\;\;\;
\label{2D_C_Fxy_dprime}
\end{eqnarray}
The physical meaning of the topological invariant becomes clear: it simply counts the skyrmion number of the self-energy-renormalized $(d_{1}^{\prime},d_{2}^{\prime},d_{3}^{\prime})$-vector. Provided $|d_{0}^{\prime}|<d^{\prime}$, the $d_{0}^{\prime}$ component does not influece the topology. In the noninteracting limit $(d_{1}^{\prime},d_{2}^{\prime},d_{3}^{\prime})\rightarrow(d_{1},d_{2},d_{3})$, this skyrmion number is known to coincide with the Hall conductance\cite{Bernevig13}. Interaction modifies the profile of $(d_{1}^{\prime},d_{2}^{\prime},d_{3}^{\prime})$ in momentum space, yet the skyrmion number always takes an integer value.

\begin{figure}[ht]
\begin{center}
\includegraphics[clip=true,width=0.8\columnwidth]{2D_class_A_freq_ind_topo_inv_corr_fn_graphics}
\caption{ Graphic presentation of several quantities for 2D class A models with frequency-independent self-energy. (a) The frequency-integrated real space Green's function times a distance $r_{i}G({\bf r})$ as a thick line and the inverse of the frequency-integrated Green's function $G^{-1}({\bf r})$ as a thin line. (b) Topological invariant expressed in terms of the real space Green's function. The blue point denotes the origin, and the 2 black points denote the positions that are to be integrated. (c) The correlation function $\lambda_{\bf r}$ presented graphically, which decays with a correlation length. }
\label{fig:2D_class_A_freq_ind_topo_inv_corr_fn_graphics}
\end{center}
\end{figure}

For this case of frequency-independent self-energy, we discuss the criticality near topological phase transitions in the following manner according to the argument in Sec.~\ref{sec:criticality}. We observe that if we integrate out the frequency variable of the Green's function in Eq.~(\ref{2D_G_dprime}), the result only depends on the ${\bf d}^{\prime}$-vector
\begin{eqnarray}
G({\bf k})\equiv\int\frac{d\omega}{2\pi}G({\bf k},i\omega)
=-\frac{1}{2}\left(
\begin{array}{cc}
-1+\frac{d_{3}^{\prime}}{d^{\prime}} & \frac{d_{1}^{\prime}-id_{2}^{\prime}}{d^{\prime}} \\
\frac{d_{1}^{\prime}+id_{2}^{\prime}}{d^{\prime}} &
-1-\frac{d_{3}^{\prime}}{d^{\prime}}
\end{array}
\right),
\nonumber \\
\label{Gkw_integrate_out_w}
\end{eqnarray}
after substituting Eq.~(\ref{2D_G_dprime}). Consequently, the ${\bf d}^{\prime}$-vector can be written in terms of $G({\bf k})$
\begin{eqnarray}
-d_{1}^{\prime}/d^{\prime}&=&G_{AB}({\bf k})+G_{BA}({\bf k})\equiv \tilde{G}_{1}({\bf k})\;
\nonumber \\
id_{2}^{\prime}/d^{\prime}&=&G_{AB}({\bf k})-G_{BA}({\bf k})\equiv \tilde{G}_{2}({\bf k})\;
\nonumber \\
-d_{3}^{\prime}/d^{\prime}&=&G_{AA}({\bf k})-G_{BB}({\bf k})\equiv \tilde{G}_{3}({\bf k})\;.
\label{dprime_Gi}
\end{eqnarray}
In terms of this newly defined, frequency-independent Green's function elements $\left\{\tilde{G}_{1},\tilde{G}_{2},\tilde{G}_{3}\right\}$, the topological invariant in Eq.~(\ref{2D_C_Fxy_dprime}) is
\begin{eqnarray}
&&{\cal C}=\int\frac{d^{2}{\bf k}}{(2\pi)^{2}}F({\bf k},{\bf M})\;,
\nonumber \\
&&F({\bf k},{\bf M})=\frac{\pi}{i}\epsilon^{abc}\tilde{G}_{a}({\bf k})\partial_{x}\tilde{G}_{b}({\bf k})\partial_{y}\tilde{G}_{c}({\bf k})\;,
\label{2D_C_freq_ind_GaGbGc}
\end{eqnarray}
where the frequency-independent curvature function $F({\bf k},{\bf M})$ is introduced in terms of $\tilde{G}_{a}({\bf k})$, whose profile in momentum space obviously depends on the interacting or noninteracting parameters ${\bf M}=(M_{1},M_{2}...)$ in the Hamiltonian. We then consider the Fourier transform
\begin{eqnarray}
\tilde{G}({\bf k})=\sum_{\bf r}e^{-i{\bf k\cdot r}}\tilde{G}({\bf r})=\int d^{2}{\bf r}\,e^{-i{\bf k\cdot r}}\tilde{G}({\bf r})\;,
\nonumber \\
\tilde{G}({\bf r})=\frac{1}{N}\sum_{\bf k}e^{i{\bf k\cdot r}}\tilde{G}({\bf k})=\int \frac{d^{2}{\bf k}}{\left(2\pi\right)^{2}}e^{i{\bf k\cdot r}}\tilde{G}({\bf k})\;.
\label{Gk_Gr_Fourier}
\end{eqnarray}
to define the real space counterpart of $\tilde{G}({\bf k})$. In terms of these real space functions, the topological invariant in Eq.~(\ref{2D_C_freq_ind_GaGbGc}) reads, after a Fourier transform,
\begin{eqnarray}
{\cal C}&=&i\pi\int d^{2}{\bf r}_{1}\int d^{2}{\bf r}_{2}
\nonumber \\
&&\times\epsilon^{abc}\tilde{G}_{a}(-{\bf r}_{1}-{\bf r}_{2})r_{1x}\tilde{G}_{b}({\bf r}_{1})r_{2y}\tilde{G}_{c}({\bf r}_{2})\;.
\end{eqnarray}
We proceed to follow the principle mentioned in Sec.~\ref{sec:criticality} to introduce the correlation function as the Fourier transform of the curvature function. In this case the result is
\begin{eqnarray}
&&\lambda_{\bf r}=\int\frac{d^{2}{\bf k}}{(2\pi)^{2}}e^{i{\bf k\cdot r}}F({\bf k},{\bf M})
\nonumber \\
&&=i\pi\int d^{2}{\bf r}_{1}\int d^{2}{\bf r}_{2}
\nonumber \\
&&\times\epsilon^{abc}\tilde{G}_{a}({\bf r}-{\bf r}_{1}-{\bf r}_{2})r_{1x}\tilde{G}_{b}({\bf r}_{1})r_{2y}\tilde{G}_{c}({\bf r}_{2})\;,
\nonumber \\
\label{lambdaR_2D}
\end{eqnarray}
as present graphically in Fig.~\ref{fig:2D_class_A_freq_ind_topo_inv_corr_fn_graphics}. Note that in these expressions, $\tilde{G}_{i}({\bf r})$ is a function calculated from the matrix elements of $G({\bf k},i\omega)$ integrated out the frequency variable and then Fourier transformed to real space, according to Eqs.~(\ref{Gkw_integrate_out_w}), (\ref{dprime_Gi}), and (\ref{Gk_Gr_Fourier}), and ${\cal C}$ and $\lambda_{\bf r}$ are products of $\tilde{G}_{i}({\bf r})$ and its inverse propagated over several segments.

\begin{figure}[ht]
\begin{center}
\includegraphics[clip=true,width=0.7\columnwidth]{Chern_ee_RGflow}
\includegraphics[clip=true,width=0.99\columnwidth]{Chern_ee_exponent_fixM}
\includegraphics[clip=true,width=0.99\columnwidth]{Chern_ee_exponent_fixV}
\caption{ (a) The RG flow (black arrows) of 2D Chern insulator with density-density interaction in the ${\bf M}=(M,V)$ parameter space. The color scale indicates the high (green) and low (brown) flow rate on log scale $\ln|d{\bf M}/dl|$. The flow identifies a phase boundary (green line) between the topologically nontrivial ${\cal C}=-1$ and trivial ${\cal C}=0$ phase. The brown line indicates the fixed points in the topologically nontrivial phase. (b) and (c) show $|F({\bf k}_{0},{\bf M})|^{-1/2}$ and $\xi_{x}^{-1}$ along $V$ at several values of $M$, and (e) and (e) show them along $M$ at several values of $V$. The linear behavior near the critical point (where they vanish) indicate the critical exponents are $\gamma\approx 2$ and $\nu_{x}\approx 1$. }
\label{fig:Chern_ee_RGflow}
\end{center}
\end{figure}

\subsection{2D Chern insulator with density-density interaction}

As a concrete example for 2D class A with frequency-independent self-energy, we examine the 2D Chern insulator, with the unperturbed Hamiltonian described by Eqs.~(\ref{general_2by2_Dirac_Hamiltonian}) and (\ref{2D_Chern_H0}), in the presence of the weak density-density interaction of the form of Eq.~(\ref{general_density_density_interaction}).
We choose the $V_{\bf q}$ in Eq.~(\ref{general_density_density_interaction}) to have the form
\begin{eqnarray}
V_{\bf q}=V(2+\cos q_{x}+\cos q_{y})\;,
\end{eqnarray}
which may come from some nearest-neighbor density-density interaction between the two sublattices, by generalizing the argument in Sec.~\ref{sec:SSHnn} to 2D. Our aim is then to study the topological phase transition driven by the mass term in the unperturbed Hamiltonian and the nearest-neighbor interaction ${\bf M}=(M,V)$, which are treated as dimensionless parameters since the momentum and Fermi velocity are implicitly set to be dimensionless.

The CRG approach is applied to circumvent the 2D integral in Eq.~(\ref{2D_C_Fxy_dprime}). The curvature function $F({\bf k},{\bf M})$ is now only a function of momentum ${\bf k}=(k_{x},k_{y})$ since frequency has been integrated out. The RG flow of $M_{i}$ described by Eq.~(\ref{generic_RG_eq_discrete}) then requires to calculate the three mesh points $F({\bf k}_{0},{\bf M})$, $F({\bf k}_{0}+\Delta{\bf k},{\bf M})$, and $F({\bf k}_{0},{\bf M}+\Delta{\bf M}_{i})$, where $\Delta{\bf k}$ can be either $\Delta k_{x}{\hat{\bf k}}_{x}$ or $\Delta k_{y}{\hat{\bf k}}_{y}$. The result using ${\bf k}_{0}=(0,0)$ is shown in Fig.~\ref{fig:Chern_ee_RGflow}. Using the rule in Eq.~(\ref{identifying_Mc_Mf}), one identifies a phase boundary between the topologically nontrivial ${\cal C}=-1$ and trivial ${\cal C}=0$ phase in the ${\bf M}=(M,V)$ parameter space, which is a continuous line that passes through the well-known phase critical point in the noninteracting limit\cite{Bernevig13} ${\bf M}_{c}=(0,0)$ (slightly off in Fig.~\ref{fig:Chern_ee_RGflow} due to finite grid size). The fixed point in the nontrivial phase, at which the correlation length vanishes $\xi_{i}\rightarrow 0$, locates within the parameter space investigated. Figure \ref{fig:Chern_ee_RGflow} also shows the investigation of critical exponents defined in Eq.~(\ref{Fk0_xi_critical_exponent}). We extract $\gamma\approx 2$ and $\nu_{i}\approx 1$, same result as the noninteracting 2D Chern insulators and satisfy the scaling law $\gamma\approx \nu_{x}+\nu_{y}$\cite{Chen17}.

\section{Conclusions}

In summary, an RG approach is introduced into the Green's function formalism of weakly interacting TIs, as demonstrated particularly for 1D class BDI and 2D class A Dirac models under the influence of electron-electron and electron-phonon interactions. In the Green's function formalism, the topological invariant for these two symmetry classes is calculated from the integration of a curvature function calculated from the single-particle Green's functions. After the effect of interaction is included perturbatively into the Green's function, we show that the curvature function in these models diverges at the HSPs as the system approaches the topological phase transitions. We propose the CRG scheme based on this divergence, which is an iterative procedure to search for the trajectory in the parameter space along which the divergence reduces but the topological invariant remains unchanged, though which the topological phase transitions can be identified. The CRG scheme is proved to be a numerically efficient tool to investigate interacting models, since it circumvents the integration of topological invariant and only requires to calculate the curvature function on few points in the momentum or momentum-frequency space.

The divergence of curvature function also unveils a number of statistical aspects related to the quantum criticality in the Green's function formalism. The first is the correlation function introduced from the Fourier transform of the curvature function, which is generally a product of real space or spacetime Green's functions, their inverse, and the distance they travel over some segments. The correlation function decays exponentially due to the Ornstein-Zernike form of the curvature function. The divergence of curvature function further indicates that the correlation length or correlation time diverges at the topological phase transition, from which the scale invariance is interpreted. The critical exponents of the correlation length and correlation time, as well as that of the curvature function, are found to be constrained by a scaling law due to the conservation of topological invariant, and we find that interactions may or may not change the critical exponents. These results suggest that investigating the divergence of curvature function is a numerically efficient way to identify and characterize the topological phase transitions driven by either interacting or noninteracting parameters. We anticipate that the principles revealed in the article may also be applicable to the Green's function formalism of other symmetry classes, such as time-reversal invariant models with interactions\cite{Budich13,Lang13,Meng14}, which will be subject to further investigations.

\section{Acknowledgement}

The author acknowledges fruitful discussions with M. Sigrist, A. Schnyder, T. C. Lang, and R. Chitra.

redpajama_set_name: RedPajamaArXiv


text:
Humphry Marshall¹ (10 de octubre de 1722 Condado de Chester (Pensilvania) - 6 de septiembre de 1801, Marshallton) fue un botánico estadounidense. Marshall ha sido llamado el "Padre de la Dendrología de EE.UU.".

Era hijo de Abraham Marshall y de Mary Hunt; y primo de los botánicos John Bartram y de William Bartram. Como muchos de esa profesión en esa época, pertenecía a la Sociedad Religiosa de los Amigos o cuáquero.

Se casó con Sarah Pennock el 16 de septiembre de 1748; y el 10 de enero de 1788 con Margaret Minshall.

Por tradición su familia era de constructores y granjeros. Trabajó en una granja a partir de 1748, y en 1768 llevó a cabo la primera conservación de especies vegetales de la región.

Estableció un jardín botánico considerado el segundo del país (el primero era de sus primos botánicos).

Se vinculó científicamente con Peter Collinson (1694-1768) y con el médico John Fothergill (1712-1780).

Su sobrino Moses Marshall, también fue botánico. Pasó sus últimos años, parcialmente ciego por cataratas.

Algunas publicaciones
 Arbustrum American, the American Grove, 1785

Honores

Epónimos
El género botánico Marshallia Schreb. 1791, de la familia Asteraceae, fue nombrado en su honor.

Véase también
 Anexo:Naturalistas y epónimos

Referencias
Nota:: Ciertas fuentes indican Humphrey.

Enlaces externos

Botánicos de Estados Unidos
Dendrólogos de Estados Unidos
Escritores en inglés
Cuáqueros

redpajama_set_name: RedPajamaWikipedia


In Moratuwa, Sri Lanka, the tsunami disaster of 2004 forced new ways of working on both organizations of the urban poor and local authorities. Building on this experience, an emerging federation of the urban poor (built on community savings groups) has been collaborating with local authorities to secure land and adequate housing in two deprived settlements unaffected by the tsunami. This article examines these recent initiatives and their citywide relevance. The federation (the Women's Development Bank Federation) receives support from a local NGO (Janarukula) and is affi liated with an international confederation (Slum Dwellers International), whose principles are also evident in these improvement efforts. One of the two settlements is located on the periphery of Moratuwa, where land is plentiful and existing plots can be upgraded. The other is a central settlement, where land is scarcer and the federation has opted for multi-storey residences. Securing community control over the multi-storey development has proved diffi cult and has required far tighter collective action. Success in dense settlements is important, however, if the federation is to meet its goal of creating a citywide strategy.


Junk Jack Forum • View topic - [OUTDATED] Society's ultimate mob grinders!
[OUTDATED] Society's ultimate mob grinders!
Finally decided to share my mob grinding designs!
I've made countless prototypes to make the most efficient mob grinders and the finished products are finally here!
Since all my designs are the final products of countless prototypes, all of them are guaranteed to be satisfactory or better.
- MAKING MASSIVE FARMS ARE POINTLESS. Monsters spawn in a certain distance from the player regardless, so might aswell make a small farm where monsters take less time to travel from the spawning platforms to the desired trap location.
- The amount of monsters spawned is limited! Don't expect a kagillion zombies to spawn while leaving your phone on for an hour, it just doesn't work that way.
- Wood tools (eg. wood bridge, wood door, wood ladder) and weaker break in lava and acid.
- Iron bridges, ladders and doors are immune to lava and acid.
- Jail doors and Iron Doors are explosive-proof.
- Most of my designs require acid and lava. Black Demon Armor Legs work great with my designs.
Automatic - Requires no punching/hitting of the mobs.
Manual - Mobs are immune to drowning/acid and require manual handling.
Semi-Automatic - Requires some supervision.
Surface - Device is made on the surface to kill surface-spawned monsters.
Bedrock - Device is made next to bedrock to kill underground-spawned monsters.
Child's play - Even my unborn child can make this!
Easy - Takes no effort to build.
Average - Takes some time to build.
Difficult - Ghost zombie has been summoned. Building this will be bloody.
For the new players who wish to farm on PEACEFUL only. Does not work without peaceful unless you have acid!
Special feature: Filters out acid-proof slimeballs and incinerates them.
If you want to test out the grinders before-hand, I'd be more than happy to let you try it out yourself!
Enjoyed the farm? Made a little loitering area next to a farm to show off to your friends? How about showing us! Share your safe-areas to see who has the coolest pimped out farm!
Last edited by Society on Mon Jul 25, 2016 8:20 am, edited 16 times in total.
Re: Society's ultimate mob grinders!
ps: LOOK AT MY PIXLE AT ON THE CREATIVE SECION!!!!!
Was never a download to begin with. It's supposed to give you an idea of how to make efficient mob grinders.
Very, very nice! You must have put a lot iof time and effort in creating these, fantastic work!
I'd show you all the prototypes I trashed just to achieve these, it's a shame I can't.
Updated, added a grinder for Terra. Made for players who have just started out!
There's only four reasons it didn't work for you.
1. You didn't remove the torches near the spawning platforms.
2. You didn't recreate the design completely.
3. You didn't read the important notes before making the farm. ALL caves and possible spawning areas that are not wanted near the farm MUST be filled/lit up.
4. You're standing too far from the spawning platforms.
I placed pink wool to indicate the safe areas where you're able to afk. If you're too far from the spawning platforms, the monsters will not spawn.
Last edited by Society on Mon Sep 30, 2013 9:55 pm, edited 1 time in total.
Can you make a mob spawner that does not require a lot of thought. I want it very basic but also not require building a giant structure.
Wow.....filling every cave in the game.....anybody does that and can I have all my rares....EVERY SINGLE HOLE!!!!
Gunther4C wrote: Wow.....filling every cave in the game.....anybody does that and can I have all my rares....EVERY SINGLE HOLE!!!!
Not every hole. Just in a 13 block radius. If there are caves or holes near the farm, the monsters will spawn there instead.
Grettir wrote: Can you make a mob spawner that does not require a lot of thought. I want it very basic but also not require building a giant structure.
Depends on what type of farm you want.


text:
Rick and Morty co-creator Justin Roiland is facing domestic violence charges
Justin Roiland, star and co-creator of the animated comedy series "Rick and Morty," has been charged with felony domestic violence in Orange County, California, in connection with a 2020 incident, according to a criminal complaint filed in May 2020 by the Orange County District Attorney. He appeared in court Thursday for a preliminary hearing.
Roiland pleaded not guilty in 2020. Roiland and his attorneys did not respond to requests for comment.
NBC News obtained the criminal complaint in the case, which charges Roiland, 42, with one felony count of domestic battery with bodily injury and one felony count of false imprisonment by threat, violence, fraud and/or deception. NBC News is the first to report the case publicly.
The complaint states that the incident occurred on or about January 19, 2020, against an unidentified Jane Doe who was dating Roiland at the time.
According to Orange County Superior Court records, Roiland pleaded not guilty to both charges in October 2020. Since then, the case has been the subject of more than a dozen court hearings, including pretrial hearings. No trial date has been set at this time.
Many court records remain sealed, but available public documents seen by NBC News say Roiland was charged in May 2020. Roiland was arrested and released on $50,000 bail in August 2020 and arraigned in October 2020. .
A protective order filed in October 2020 and documented in court transcripts said Roiland does not harass, threaten or monitor the person named in the protective order, which is unknown to NBC News. According to the order, Roiland could not go within 100 feet of the person, who also ordered Roiland to turn over any firearms he or she possessed. The application runs until October 2023, and it is not known who applied.
Details of the case, including police camera footage, police reports, abuse investigation reports, medical reports and interview recordings, are currently being withheld from the public under a protective order. The affidavit supporting Roiland's arrest is also sealed.
During Thursday's hearing, Roiland's attorney confirmed that an offer to plead was available to Roiland, but did not discuss the details. The brief session concluded after a moment with an agreement to meet again on April 27. Roiland was ordered to attend the hearing.
Roiland is best known for Adult Swim's Rick and Morty, which has spawned a multi-billion dollar media and promotional franchise and is one of the most popular TV comedies. The show's sixth season aired in September 2022, and a seventh season has been ordered by Cartoon Network as part of a long-term deal for 70 new episodes. Roiland, in addition to co-creating the series with Dan Harmon, voices the two titular characters.
Adult Swim, whose parent company Warner Bros. owns the franchise, declined. Discovery also reported to Cartoon Network, for comment.
Over the past three years, as the domestic violence case has progressed, Roiland's career has advanced. He released an NFT art collection with more than a dozen works in 2021, which sold for more than $1.65 million. He released another NFT project in 2022. Roiland also sold his first painting via Sotheby's auction in 2021.
Roiland also co-created and stars as the main character in Hulu's "Solar Opposites," which debuted in 2020 and was renewed for a fifth season in October 2022. Most recently, he's behind Hulu's series "Koala Man," the first episode of which was released this month.
Roiland also released the video game "High on Life" in 2022, which he developed through his studio Squanch Games. It became the best-selling game on Steam and the fourth most-played Xbox game in its launch week. Roiland won an Emmy for "Rick and Morty" in 2020 and a Hollywood Critics Choice Association TV Award for the show in 2022.
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John Fogerty Reclaims His Song Rights – Rolling Stone

redpajama_set_name: RedPajamaCommonCrawl


Thyme is the October 2016 Plant of the Month for Boulder Colorado landscapes – Glacier View Landscape and Design, Inc.
Thyme (Thymus praecox) is a fabulous ground cover to use in a variety of landscape applications. There are 7 main types in use in Boulder Colorado landscapes and in much of the US, but there are hundreds of sub species. They are a versatile and adaptable species of ground cover with an abundance of flowers at certain times of the year. Thyme stays very low growing (1-2") and can even be used in cooking, as an insect repellent, and as an essential oil.
Generally a low moisture plant species, they are great to plant in moss rock dry stack walls, in between flagstone stepping stones, or even in an informal flagstone patio. The rhizomes will creep along underground and spread throughout the cracks between the stones, leaving a carpet-like appearance. Most are durable and can be stepped on without damaging the plant. They come in a variety of colors with white, red, and lavender as the most common. Once established with roots under rocks, they can survive on little to no additional moisture than the 16-18" or so per year here in Colorado.
People have been cooking with Thyme for hundreds of years. Use Thyme to marinate steak or hamburger, sprinkle on salads or use in salad dressing, and even as a potpourri. Add it to spaghetti sauce for that true Italian flavoring! It can be used fresh from the garden, or dried and stored as any leafy herb would be. Keep it in your kitchen cabinet all year long. There are even uses for the essential oils from Thyme including aromatherapy and as an insect repellent.
Thyme is best planted from flats of 32 in 2 ¼" pots, since they are easy to squeeze between rock cracks. Some soil amendments here in Colorado are helpful, but the plant will do OK in a rocky, nutrient poor soil also. Light but consistent watering when first planted until established is necessary, but be careful of overwatering and root rot. Cutting out winter kill in the spring will help the plant thrive the following winter. Plant a flat or two in your landscape and see where it takes you!
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Providing that all has gone well before, this becomes the time when the individual develops the spiritual strength and independence for his or her personal mission in life.
This individual becomes a human being who has attained a high level of moral conscience and responsibility, and can work for the good of humanity.
Thus the developmental life of a human being is a sequence of births.


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Social Data Analytics is an interdisciplinary major that teaches students to use the increasingly vast stores of information generated from social media, cell phones, "smart objects" and other technologies that capture moment-to-moment changes in where people are, what they are doing and thinking, and with whom they are associating. This data (often called "social data" or "big data") can help researchers and policy makers address a wide variety of political, economic and social problems. It is used to improve government services; to identify patterns of armed conflict, human rights abuses, and disease before they escalate; to enhance the efficiency of businesses; and to create more resilient communities in the face of climate change. Students in this major learn data analysis techniques and how to apply them to develop reliable answers to questions about the social and political world.
Why Study Social Data Analytics?
you want to improve how people live and work together.
Others may work for government agencies, such as the Department of Defense, National Institute of Health, Department of Energy, and Department of State, forecasting political change and coordinating resources to improve human health and security. The degree also prepares students to be strategists for political campaigns or to work in law enforcement.
This degree is excellent preparation for a variety of graduate programs, including social science, public policy, urban planning, and law.


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Homestead has been a mainstay in the North Shore, winning at least a share of the title in the past 16 seasons. Here's how they did it.
Nine White House press secretaries. Twenty-four Green Bay Packers playoff games. Three Mequon mayors. Approximately 5.6 million cream puffs sold at the Wisconsin State Fair. In the same time period where we've had these things, there has been one program that has won or shared every single North Shore Conference football championship.
For every year since 2001, the title has been Homestead's, including one run of 67 consecutive league victories from 2002-11. The North Shore got a facelift in advance of the 2017 season thanks to realignment, but one fact remains the same: Homestead is the team to beat.
2016 - After a Week 1 hiccup against Arrowhead, Homestead won 11 straight games before falling in Level 4 of the playoffs to Monona Grove. Sophomore Bradley Woldt led the North Shore in rushing with 1,329 rushing yards and 21 touchdowns.
2015 - After starting the season 2-2, Homestead outscored its opponents 343-99 en route to winning the Division 2 state title. Senior Matt Winters took the North Shore by force, running for almost 300 more yards than anyone else in the conference.
2014 - Homestead lost to Cedarburg in the regular season but still won the title, and the Highlanders avenged the loss to the Bulldogs in the playoffs but fell in Level 4 to Menasha in a gut-wrenching 8-7 buzzer-beater when the Bluejays went for the winning 2-point conversion (then went on to win the state title).
2013 - Six 20-plus point victories highlighted an 11-0 start to the year before the Highlanders fell to eventual state champion Kimberly in Level 3 of the playoffs. Senior Eric Wade ranked second in the conference with 1,103 rushing yards and 17 touchdowns.
2012 - The gold ball returned to Mequon Road after Homestead went 7-1 in regular-season play and 5-0 in postseason play. The rushing combination of Shaquille Cole and Jay Schneider helped Homestead rank second in the conference with 2,866 rushing yards. The only other North Shore foes to have more than 1,300 rushing yards were Whitefish Bay (2,891) and Cedarburg (2,086).
2011 - A loss to Germantown in the final week of the regular season spoiled a perfect conference record, but a six-game win streak was enough for the Highlanders' 11th straight conference championship. The team lost in Level 2 to Milwaukee Riverside.
2010 - Homestead had a perfect regular season record, but the Highlanders didn't necessarily make it look easy. Of the nine regular season wins, three of them were by three or fewer points. After falling behind in the playoffs to Germantown, 10-0, HHS never could never completely come back, losing 17-10.
2009 - The Highlanders had a perfect 9-0 regular season in which they never allowed more than 17 points. Senior Bryan Bronaugh finished second in the conference with 880 rushing yards. Homestead eventually lost to Menomonee Falls in the second round of the playoffs.
2008 - Casey Barnes' 1,855 passing yards and conference-high 19 touchdowns helped the Highlanders go undefeated in regular season play. Homestead went on to defeat Arrowhead, 13-11, for the Division 1 state title in a thriller that capped a wild three-year run in which the two programs met for the gold ball. Homestead won the series by taking two of three.
2007 - Homestead was perfect against in-state programs before running into a perfect Arrowhead team in the state finals, forcing the Highlanders to settle for silver. Brett Werner-Powell led the North Shore with 1,230 rushing yards and had almost 20 more yards per game than the second-best rusher.
2006 - The Highlanders' 35-0 win over Arrowhead in the state final epitomized the team's highly successful season. Nine of their last 10 wins were by at least 20 points. Close calls included a 20-15 win over Whitefish Bay and an 18-17 win over Germantown in the regular season.
2005 - Homestead coasted through the North Shore, going 7-0 in-conference and 9-0 overall in the regular season. The team never allowed more than 14 points until the state semifinals, where it lost to Racine Park, 20-13.
2004 - Homestead destroyed the North Shore, outscoring opponents, 280-53. The closest game in that stretch was a 21-7 win over Germantown. The Highlanders' luck ran out in the second round of the playoffs in a nail-biting 31-28 loss to Marquette.
2003 - Homestead surfed through the North Shore with another perfect 7-0 mark and a 9-0 regular season record. The only game to be within 10 points was the Highlanders' 23-20 win over rival Germantown.
2002 - A high-octane offense pushed the Highlanders to their second consecutive conference title. The team was 10-0 when the team scored at least 28 points. The road ended one weekend short of Camp Randall with a loss to eventual champ Marshfield for a second straight year in the state semifinal.
2001 - Homestead captured sole possession of the North Shore football title for the first time in five years, earning a three-seed in the playoffs. The Highlanders cruised past Oshkosh North, Menomonee Falls and Neenah before falling to Marshfield in the state semifinal.
2000 - Who was the last team to win the conference title that wasn't Homestead? There are three answers: Germantown, Port Washington and Whitefish Bay all went 5-1 to share the crown. Homestead finished 2-4 in league play. Ironically, the down year came after a 1999 season in which the Highlanders went 14-0 and won their first D1 title, defeating that pesky Marshfield in the championship game. It was only the first chapter of something special.


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In this seminar Prof. Petkovic will give an overview of research methods and planning that is critical for successfully and efficiently completing culminating experience projects. In addition, Prof. Petkovic will give guidance on how to write good culminating experience reports, based on his extensive experience of reading and evaluating these reports and his observations of repetitive patterns of common mistakes students make. This seminar is strongly recommended to all CS graduate students, domestic and international, who plan to graduate this or next semester.


The record-breaking and awe-inspiring waterfall of Angel Falls, known as Kerepacupai-meru to the local Pemon people.
One of the 17 most megadiverse countries in the world, Venezuela is home to Caribbean beaches, dense rainforests, high Andean valleys, mysterious table-top tepui mountains, and seasonally-flooded plains that are literally teeming with birds, beasts and creepy-crawlies.
Venezuela is also alive with the sound of folk music, from the Afro-Venezuelan tambores of the coast, to the harp-driven Joropo songs of Los Llanos and during his talk Russell will present several examples of the music to be found in the regions he discusses.
Rona Cant — Just do it – become the best you can be!
Once an unexceptional, single parent she took part in the BT Global Challenge Round the World Yacht Race (the toughest yacht race in the world); two days later she embarked on the West Coast Trail on Vancouver Island (one of the world's toughest), encountering bears and cougars. Rona helped organise and was on The Nordkapp Expedition, dogsledding in the Arctic Circle to the northernmost tip of Europe, a trail thought to be impossible.
Rona tells how she has achieved her goals using stories from her childhood, her marriage, how she entered into a life of adventure, her diverse adventures and what she has accomplished.


Tammy – Thank-you so much for this post that helps us know how to better interact with special needs kids and their families! We pray for you guys often! We still have lots of photos all over our house from the photo shoot you did for us many years ago in Kentucky. Blessings!
Thank you, Tammy! I love this post! 🙂 I really want to interact with all your super kids soon! You have been so helpful to me. In seeing through your eyes, I think I understand a little better how to love people!
You have got it "so together"! I am amazed at your blogs and your insight into your children's minds. As someone who has probably done all the "nots" you have listed, I do want to be a good person and not invade anyone's space, if I lean down it would be because my knees don't work to squat. I do admire people who have learned to adapt to their disabilities and reading your blogs I hope to learn more about interacting with them. God bless!
Sandra, Thank you for your kind words, but I must say that we are learning as we go by the grace of God. I appreciate your willing heart to learn as well.