\section{Introduction}
Amorphous solids (Glasses), which are known to have non-crystalline structure like fluids, possess solid-like behaviors strikingly different from crystalline solids such as saturation\cite{Graebner1983}, universality of internal friction\cite{Pohl2002}, linear heat capacity\cite{Zeller1971} etc.. One of these properties is called avalanche phenomena, in which the glass stress-strain curve presents a steep drop to a lower value at certain critical external strain. To explain these universal properties, Anderson, Halperin and Varma\cite{Anderson1972} group and Phillips\cite{Phillips1987} independently proposed a phenomenological model known as tunneling-two-level-system (TTLS). It not only explained several of existing experimental observations, but also predict new phenomena such as phonon echo\cite{Golding1976}. However, at least to the author's knowledge, there was no direct explaination to the mechanical avalanche problem in glass according to the original TTLS model papers\cite{Anderson1972, Phillips1987}. 



The glass mechanical avalanche phenomena is referred to a broad distribution of irreversible (inelastic) stress drops in non-quantum, amorphous materials undergoing quasi-static external strain\cite{Langer1998, Dahmen2009}. One of the experiments on the glass fracture behavior was in silica glasses and polymers\cite{Fineberg1991} at room temperatures. Based on two-dimensional molecular dynamics simulations, M. L. Falk and J. S. Langer\cite{Langer1998} developed Shear-Transformation-Zone (STZ) model to explain low-temperature shear deformations in metallic glasses.






The purpose of this paper is to develop a microscopic field theory model, namely the generic coupled block model, to investigate the mechanical avalanche phenomena in three-dimensional insulating glass. More specifically, we want to prove the ``existence of inelastic stress drops" in quantum amorphous materials under the deformation of external static, uniform strain. Consider an external static strain $\bm{e}$. The amorphous material will provide a corresponding stress response $\bm{T}(\bm{e})$. Since we want to prove the ``existence of steep stress drops in stress-strain relation", it is more convenient to prove the ``existence of positive-negative transitions in glass mechanical susceptibility-strain relation", where the glass mechanical susceptibility is defined as the derivative of stress with respect to strain: $\bm{\chi}(\bm{e})=\delta\bm{T}(\bm{e})/\delta \bm{e}$. Therefore, the main purpose of this paper is actually to investigate the ``existence of positive-negative transitions in glass mechanical susceptibility $\bm{\chi}(\bm{e})$" when external strain exceeds certain critical values, $\bm{e}_{\rm crit}$.








The reader should be aware that it is the first time to apply our ``generic coupled block model" in glass mechanical avalanche problems. Therefore our purpose is not to solve the entire avalanche problems from microscopic point of view; instead we want to provide some first-step results for future people to continue studying this problem from our model. 



The set up of our model begins from the generalization of tunneling-two-level-system (TTLS) model to generic multiple-level-system (MLS) model. Similar as TTLS model, phonon strain field could couple to multiple-level-systems (MLSs). As the coupling with phonon strain field, multiple-level-systems (MLSs) must generate a mutual RKKY-type interaction\cite{Joffrin1976} due to virtual phonon exchange process. Finally, our glass Hamiltonian is the summation of long-wavelength phonon contribution, a set of MLSs, the coupling between MLS and phonon strain field, and the mutual RKKY-type interaction between MLSs. Since we do not take conducting electrons into consideration, the model only applys for insulating glasses. Further considerations regarding conducting electron Hamiltonian, electron-phonon coupling and electron-MLS coupling are required to explore the ductility of metallic glass. 


We want to use this model to carry out macroscopic length scale glass mechanical susceptibility from microscopic length scale. The effective starting microscopic length scale is of order $\sim 50\AA$, corresponding to the characteristic thermal phonon wavelength with the temperature of order $60$K. Since the thermal phonon wavelength can be no smaller than the starting microscopic length scale, our theory is only valid below 60K. We will discuss the reasoning in details in section 3. However, at least to the author's knowledge, all of glass avalanche experiments are taken under room temperatures or glass transition temperatures\cite{Argon2005, Gauthier2004, Dahmen2014, Fineberg1991, Fineberg1992} ($T\sim 300$K). We hope more glass avalanche experiments could be taken below 60K to test the validity of our model.  







The paper is organized as follows: in section 2 we first generalize glass two-level-system model to multiple-level-system model. Then we give a detailed derivation of our generic coupled block model by introducing the mutual RKKY-type interaction between multiple-level-systems. We introduce the most important concept of this paper, namely non-elastic stress-stress susceptibility at the end of section 2(A). In section 3 we treat RKKY-type interaction as a perturbation, to set up the relation between small and large length scale non-elastic stress-stress susceptibilities. By repeating such recursion relation, we eventually carry out non-elastic stress-stress susceptibility at macroscopic length scale. In section 4 we prove the existence of positive-negative transitions in non-elastic stress-stress susceptibility with the increase of external static strain. We further prove the existence of positive-negative transitions in glass total mechanical susceptibility.





\section{The Model}




\subsection{The Set up of Problem}
We consider a block of amorphous material under the deformation of static, uniform strain. With the slowly increasing strain the bulk glass behaves elastically until it reaches critical strain value. The stress ($\bm{T}$) v.s. strain ($\bm{e}$) curve shows a steep drop. Let us consider the glass mechanical susceptibility, defined as follows: ${\chi}_{ijkl}(\bm{e})=\delta {T}_{ij}(\bm{e})/\delta e_{kl}$. At critical strain field when irreversable process happens, glass mechanical susceptibility presents an abrupt positive-negative transition. In this paper our main goal is to prove the existence of such positive-negative transitions in mechanical susceptibility, which is shown in Fig.1 as follows:
\begin{figure}[hp
\centering
\includegraphics[scale=0.27]{Illustration.png}
\quad
\includegraphics[scale=0.27]{Illustration2.png}
\caption{As an illustration of stress-strain curve, the left picture shows a steep drop of stress. As an illustration of susceptibility-strain curve, the right picture shows a positive-negative susceptibility transition, where the mechanical susceptibility is the first order derivative of stress with respect to external strain field.}
\end{figure}







Our purpose is to prove the existence of such positive-negative transition of mechanical susceptibility in glass materials. To explore this problem, we begin our discussion from the famous tunneling-two-level-system model (TTLS model)\cite{Phillips1987}. In this model we assume that there are a group of TTLSs randomly embedded in the glass material, with the location $\vec x_i$ for the $i$-th TTLS. The effective glass Hamiltonian $\hat{H}^{\rm tot}$ in TTLS theory is the summation of two-level-system Hamiltonian, long wavelength phonon Hamiltonian, and the coupling between TTLS and strain field (phonon field):
\begin{eqnarray}\label{0.1}
\hat{H}^{\rm tot} & = & \hat{H}^{\rm el}+\sum_i\frac{1}{2}\left(
\begin{array}{cc}
E_i & 0\\
0 & -E_i\\
\end{array}
\right)\nonumber \\
 & {} & +\frac{1}{2}\sum_i \left(
\begin{array}{cc}
D_i & M_i\\
M_i & -D_i\\
\end{array}
\right)\cdot\bm{e}(\vec x_i)
\end{eqnarray}
where the first, second and third terms stand for long wavelength phonon Hamiltonian (we will also call it ``purely elastic Hamiltonian $\hat{H}^{\rm el}$"), the Hamiltonian of a group of two-level-systems, and the coupling between every two-level-system and phonon strain field at corresponding position $\vec x_i$, respectively. The two-level-system Hamiltonian is written in the representation of energy eigenvalue basis, with $E_i=\sqrt{\Delta_i^2+\Delta_{0i}^2}$ the energy splitting; $D_i=\Delta_i/E_i$ and $M_i=\Delta_{0i}/E_i$ are diagonal and off-diagonal matrix elements of the coupling between two-level-system and strain field, and by definition they are no greater than 1; $\bm{e}(\vec x_i)$ is the local intrinsic strain field at the position of the $i$-th two-level-system. 








The purpose of this subsection is to develop the multiple-level-system from the generalization of 2-level-system model. At this moment, we have not applied any external strain field yet. We will consider external strain in subsection 2(C). We begin our model by considering a single block of glass with the length scale $L$ much greater than the atomic distance $a\sim 10\AA$. In the next subsection 2(B), we will combine a group of such single blocks to form a ``super block". We will consider the RKKY-type interaction between these single blocks, which is generated by virtual phonon exchange process. For now, we do not consider RKKY-type interaction and focus only on the Hamiltonian of single block glass. 




We further define intrinsic strain field $e_{ij}(\vec x)$ at position $\vec x$: if $\vec u(\vec x)$ denotes the displacement relative
to some arbitrary reference frame of the matter at point $\vec x$, then strain field is defined as follows
\begin{eqnarray}\label{1}
e_{ij}(\vec x)=\frac{1}{2}\left(\frac{\partial u_i(\vec x)}{\partial x_j}+\frac{\partial u_j(\vec x)}{\partial x_i}\right)
\end{eqnarray}
We write down the general Hamiltonian of glass as $\hat{H}^{\rm tot}$. Let us separate out from the glass general Hamiltonian $\hat{H}^{\rm tot}$, the purely elastic contribution $\hat{H}^{\rm el}$. It can be represented either by phonon creation-annihilation operators or strain fields:
\begin{eqnarray}\label{6}
 & {} & \hat{H}^{\rm el}
 = \sum_{k\alpha}\hbar\omega_{k\alpha}\left(\hat{a}_{k\alpha}^{\dag}\hat{a}_{k\alpha}+\frac{1}{2}\right)
 = {\rm Const}+\nonumber \\
 & {} & \frac{1}{2}\int d^3x\left(\sum_{ijkl}\chi_{ijkl}^{\rm el}e_{ij}(\vec x)e_{kl}(\vec x)+\sum_i\rho \dot{u}_i^2(\vec x)\right)
\end{eqnarray}
where $\alpha=l,t$ represents phonon polarization, i.e., longitudinal and transverse phonons. We define the elastic stress tensor $\hat{T}_{ij}^{\rm el}(\vec x)$ by 
\begin{eqnarray}\label{6.1}
\hat{T}_{ij}^{\rm el}(\vec x)=\frac{\delta \hat{H}^{\rm el}}{\delta e_{ij}(\vec x)}
\end{eqnarray}
From the elastic Hamiltonian Eq.(\ref{6}), the elastic stress tensor is further given by $
\hat{T}_{ij}^{\rm el}(\vec x)=\sum_{kl}\chi_{ijkl}^{\rm el}e_{kl}(\vec x)
$, where $\chi_{ijkl}^{\rm el}$ is the purely elastic part of susceptibility. Since by definition the properties of an isotropic amorphous material must be invariant under real space SO(3) rotational group, symmetry considerations require $\chi_{ijkl}^{\rm el}$ to have the generic form\cite{Zeller1971}
\begin{eqnarray}\label{7}
{\chi}^{\rm el}_{ijkl}=\left(\rho c_l^2-2\rho c_t^2\right)\delta_{ij}\delta_{kl}+\rho c_t^2\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right)\quad
\end{eqnarray}
where $\rho$ is the mass density and $c_{l,t}$ is the longitudinal/transverse sound velocity. 




Subtracting the purely elastic Hamiltonian $\hat{H}^{\rm el}$, we name the left-over glass Hamiltonian $(\hat{H}^{\rm tot}-\hat{H}^{\rm el})$ as ``the non-elastic part of glass Hamiltonian, $\hat{H}^{\rm non}$". We expand the left-over Hamiltonian $\hat{H}^{\rm non}$ up to the first order expansion of long wavelength intrinsic phonon strain field. We name the coefficient of the first order expansion to be ``non-elastic stress tensor $\hat{T}_{ij}^{\rm non}(\vec x)$", defined as follows:
\begin{eqnarray}\label{9}
 & {} & \hat{H}^{\rm non} = \hat{H}^{\rm tot}-\hat{H}^{\rm el}\nonumber \\
 & {} & \hat{H}^{\rm non} = \hat{H}^{\rm non}_0+\int d^3x\sum_{ij}e_{ij}(\vec x)\hat{T}^{\rm non}_{ij}(\vec x)+\mathcal{O}(e_{ij}^2)\nonumber \\
 & {} & \hat{T}_{ij}^{\rm non}(\vec x) = \frac{\delta \hat{H}^{\rm non}}{\delta e_{ij}(\vec x)}
\end{eqnarray}
Now let us stop for a moment and compare Eq.(\ref{9}) with Eq.(\ref{0.1}): $\hat{H}^{\rm tot}$ and $\hat{H}^{\rm el}$ in Eq.(\ref{9}) corresponds to the glass total Hamiltonian and purely elastic Hamiltonian in Eq.(\ref{0.1}) of TTLS model, respectively; the zeroth order expansion of non-elastic Hamiltonian $\hat{H}^{\rm non}$ with respect to strain field $e_{ij}$, $\hat{H}^{\rm non}_0$, is the generalization from two-level-system Hamiltonian to multiple-level-system Hamiltonian; non-elastic stress tensor $\hat{T}_{ij}^{\rm non}$ is the multiple-level generalization of the term which couples to strain field in TTLS model. In the rest of this paper, we denote $\hat{H}^{\rm non}_0$ to be the non-elastic Hamiltonian excluding the coupling between intrinsic phonon strain $e_{ij}(\vec x)$ and non-elastic stress tensor $\hat{T}_{ij}^{\rm non}(\vec x)$. We denote $\hat{H}^{\rm non}$ to be the non-elastic Hamiltonian including the stress tensor--intrinsic phonon strain coupling (see the second equation of Eq.(\ref{9})).







Let us denote $|m\rangle$ and $E_m$ to be the $m$-th eigenstate and eigenvalue of the non-elastic Hamiltonian $\hat{H}_0^{\rm non}$. Such set of eigenbasis $|m\rangle$ is a generic multiple-level-system. Note that in general, in the representation in which $H_0^{\rm non}$ is diagonal, $\hat{T}_{ij}^{\rm non}$ will have both diagonal and off-diagonal matrix elements. 





Now we can define the most important quantity of this paper, namely the non-elastic stress-stress susceptibility (i.e., linear response function). Consider an external infinitesimal sinusoidal strain field, $e_{ij}(\vec x, t)=e_{ij}(e^{i(\vec k\cdot \vec x-\omega t)}+e^{-i(\vec k\cdot \vec x-\omega t)})$, where $e_{ij}$ is real. The non-elastic Hamiltonian $\hat{H}^{\rm non}$ will provide a corresponding response $\langle \hat{T}_{ij}^{\rm non}\rangle(\vec x, t)=(\langle \hat{T}_{ij}^{\rm non}\rangle e^{i(\vec k\cdot \vec x-\omega t)}+c.c)$, where $\langle \hat{T}_{ij}^{\rm non}\rangle$ is in general complex. Then we can define the complex response function\cite{Anderson1986} (non-elastic stress-stress susceptibility) $\chi_{ijkl}^{\rm non}(\vec k, \omega)$
\begin{eqnarray}\label{4}
\chi_{ijkl}^{\rm non}(\vec k, \omega) & = & \frac{\delta  \langle\hat{T}^{\rm non}_{ij}\rangle}{\delta e_{kl}}(\vec k,\omega)
\end{eqnarray}
In the rest of this paper we will always use $\hat{H}$, $\hat{H}_0$, $\chi_{ijkl}$ and $\hat{T}_{ij}$ to represent non-elastic Hamiltonians $\hat{H}^{\rm non}$, $\hat{H}_0^{\rm non}$, susceptibility $\chi_{ijkl}^{\rm non}$ and stress tensor $\hat{T}^{\rm non}_{ij}$ respectively, while we use $\hat{H}^{\rm el}$, $\chi_{ijkl}^{\rm el}$ and $\hat{T}_{ij}^{\rm el}$ to represent the elastic Hamiltonian, susceptibility and stress tensor, respectively.







In Eq.(\ref{4}) the ``average" of non-elastic stress tensor operator, $\langle \hat{T}_{ij}\rangle(\vec x, t)$, is defined as follows: (please note from now on we use $\hat{T}_{ij}$ to stand for $\hat{T}_{ij}^{\rm non}$)
\begin{eqnarray}\label{100}
\langle \hat{T}_{ij}\rangle(\vec x, t)=\sum_m\frac{e^{-\beta E_m}}{\mathcal{Z}}\langle m,t|\hat{T}_{ij}(\vec x)|m,t\rangle\quad
\end{eqnarray}
where $|m, t\rangle$ is the $m$-th eigenstate wave function of $\hat{H}_0$ perturbed by externally induced infinitesimal perturbation $\int d^3x\,e_{ij}(\vec x, t)\hat{T}_{ij}(\vec x)$, and $\mathcal{Z}$ is the partition function $\mathcal{Z}=\sum_me^{-\beta E_m}$. In the rest of this paper, we use ``non-elastic susceptibility" to short for ``non-elastic stress-stress susceptibility". Combining the definitions Eq.(\ref{4}) and Eq.(\ref{100}), we see non-elastic susceptibility is the function of temperature. However, for notational simplicity we write $\chi_{ijkl}(\vec k,\omega; T)$ as $\chi_{ijkl}(\vec k,\omega)$ in the rest of this paper. By using linear response theory, we expand $\langle \hat{T}_{ij}\rangle(\vec x, t)$ up to the first order of perturbation $\int d^3x\,e_{ij}(\vec x, t)\hat{T}_{ij}(\vec x)$, to calculate non-elastic susceptibility Eq.(\ref{4}) with the following process:










We use the same language as TTLS model, that the non-elastic susceptibility can be expressed in the relxation and resonance parts. The relaxation susceptibility comes from the energy eigenvalue shift due to the diagonal matrix elements of perturbation $e_{ij}(\vec x, t)\hat{T}_{ij}$, while the resonance susceptibility comes from the off-diagonal matrix elements of it. We use $\chi_{ijkl}^{\rm rel}(\vec x, \vec x'; \omega)$ and $\chi_{ijkl}^{\rm res}(\vec x, \vec x'; \omega)$ to stand for the relaxation and resonance susceptibilities respectively. Let us denote $\tau$ to be the effective thermal relaxation time for the glass single block Hamiltonian at temperature $T$. The non-elastic susceptibility is therefore expressed as follows,
\widetext
\begin{eqnarray}\label{11.1}
\chi_{ijkl}(\vec x, \vec x'; \omega) & = &  {\chi_{ijkl}^{\rm rel}(\vec x, \vec x'; \omega)}+ \chi_{ijkl}^{\rm res}(\vec x, \vec x'; \omega)\nonumber \\
\chi_{ijkl}^{\rm rel}(\vec x, \vec x'; \omega) & = & \frac{\beta}{{1-i\omega\tau}}\bigg(\sum_{nm}P_nP_m\langle n|\hat{T}_{ij}(\vec x)|n\rangle\langle m|\hat{T}_{kl}(\vec x')|m\rangle-\sum_n P_n\langle n|\hat{T}_{ij}(\vec x)|n\rangle \langle n|\hat{T}_{kl}(\vec x')|n\rangle\bigg)\nonumber \\
\chi_{ijkl}^{\rm res}(\vec x, \vec x'; \omega) & = & -\frac{1}{\hbar}\sum_{n}\sum_{m\neq n}P_m\frac{\langle n|\hat{T}_{ij}(\vec x)|m\rangle\langle m|\hat{T}_{kl}(\vec x')|n\rangle}{\omega+(E_n-E_m)/\hbar+i\eta}
+
\frac{1}{\hbar}\sum_{m}\sum_{n\neq m}P_n\frac{\langle n|\hat{T}_{ij}(\vec x)|m\rangle\langle m|\hat{T}_{kl}(\vec x')|n\rangle}{\omega+(E_n-E_m)/\hbar+i\eta}
\end{eqnarray}
\endwidetext
where $P_n=e^{-\beta E_n}/\mathcal{Z}$ is the $n$-th eigenstate probability function, $\mathcal{Z}$ is the partition function, $\beta$ is the inverse of temperature $\beta=(k_BT)^{-1}$, and $\eta$ is a phenomenological parameter to represent the higher order corrections of non-elastic susceptibility due to the coupling between intrinsic strain field and non-elastic stress tensor: $\sum_{ij}e_{ij}(\vec x)\hat{T}_{ij}(\vec x)$. 


Please note it is an approximation that we use the parameter $\tau$ to represent the effective thermal relaxation time of glass non-elastic part of Hamiltonian $\hat{H}_0$: in principle, the relaxation process of the $n$-th state is the summation of all relaxation processes between $m$-th state and $n$-th state, $\forall m\neq n$. The effective thermal relaxation time $\tau_n$ is different for different quantum number $n$. Generally speaking, one cannot use a simple parameter $\tau$ to stand for the thermal relaxation process for an arbitrary multiple-level-system. However, in this paper we want to discuss the mechanical properties of amorphous materials under the deformation of external static strain with $\omega=0$. Therefore, we always have the important relation, $\omega\tau_n=0$, for arbitrary quantum number $\forall n=0,1,2,\ldots$. In the special case of static external strain, using a simple parameter $\tau$ to represent the thermal relaxation process does not harm our theory. In the rest of this paper, we will always set $\omega$ to be zero  in non-elastic susceptibility for simplicity (see Eq.(\ref{11.1})) .



We further define the space-averaged non-elastic stress tensor and susceptibility for a single block of glass with the volume $V=L^3$, as follows, 
\begin{eqnarray}
& {} & \chi_{ijkl}(\omega) = \frac{1}{V}\int_{V} d^3xd^3x'\,\chi_{ijkl}(\vec x, \vec x'; \omega)\nonumber \\
& {} & \hat{T}_{ij} = \int_{V} d^3x\, \hat{T}_{ij}(\vec x)
\end{eqnarray}
In the rest of this paper let us use $\chi_{ijkl}$, $\chi_{ijkl}^{\rm rel}$ and $\chi_{ijkl}^{\rm res}$ to stand for $\lim_{\omega \to 0}\chi_{ijkl}(\omega)$, $\lim_{\omega\to 0}\chi_{ijkl}^{\rm rel}(\omega)$ and $\lim_{\omega\to 0}\chi_{ijkl}^{\rm res}(\omega)$, respectively. The space-averaged non-elastic susceptibility for a certain block of glass at zero-frequency limit is given as follows:

\begin{eqnarray}\label{11}
\chi_{ijkl} & = &  \chi_{ijkl}^{\rm rel}+ \chi_{ijkl}^{\rm res}\nonumber \\
\chi_{ijkl}^{\rm rel} & = & \frac{\beta}{V}\bigg(\sum_{nm}P_nP_m\langle n|\hat{T}_{ij}|n\rangle\langle m|\hat{T}_{kl}|m\rangle\nonumber \\
 & {} & \quad-\sum_n P_n\langle n|\hat{T}_{ij}|n\rangle \langle n|\hat{T}_{kl}|n\rangle\bigg)\nonumber \\
\chi_{ijkl}^{\rm res} & = & -\frac{1}{V\hbar}\sum_{n}\sum_{m\neq n}P_m\frac{\langle n|\hat{T}_{ij}|m\rangle\langle m|\hat{T}_{kl}|n\rangle}{(E_n-E_m)/\hbar+i\eta}\nonumber \\
 & {} & +
\frac{1}{V\hbar}\sum_{m}\sum_{n\neq m}P_n\frac{\langle n|\hat{T}_{ij}|m\rangle\langle m|\hat{T}_{kl}|n\rangle}{(E_n-E_m)/\hbar+i\eta}\nonumber \\
\end{eqnarray}



Finally, recall the original purpose of our paper: we want to prove the existence of positive-negative transitions in glass mechanical susceptibility when external strain passes through certain critical value (see the r.h.s. of Fig.1). The mechanical susceptibility for a single block of glass is expressed by the summation of two parts:
\begin{eqnarray}\label{5}
\chi_{ijkl}^{\rm tot} & = & \chi_{ijkl}^{\rm el}+\chi_{ijkl}
\end{eqnarray}
where for elastic susceptibility $\chi_{ijkl}^{\rm el}$ please see Eq.(\ref{7}); for non-elastic susceptibility $\chi_{ijkl}$ please see Eq.(\ref{11}). We want to investigate if there is any positive-negative transition in glass mechanical susceptibility $\chi^{\rm tot}_{ijkl}$, Eq.(\ref{5}).















\subsection{Virtual Phonon Exchange Process: Non-Elastic Stress-Stress Interaction $\hat{V}$}
The previous discussions are within the considerations of single-block glass Hamiltonian. There was not much difference between 2-level-system and generalized multiple-level-system. However, if we combine a set of $N_0^3$ single blocks together to form a ``super block", the interactions between single blocks must be taken into account. Since the non-elastic stress tensors are coupled to intrinsic strain field, if we allow virtual phonons to exchange with each other, it will generate a RKKY-type many-body interaction between single blocks. This RKKY-type interaction is the product of stress tensors of single blocks at different locations:
\begin{eqnarray}\label{12}
\hat{V}=\int d^3xd^3x'\sum_{ijkl}\Lambda_{ijkl}(\vec x-\vec x')\hat{T}_{ij}(\vec x)\hat{T}_{kl}(\vec x')
\end{eqnarray}
where the coefficient $\Lambda_{ijkl}(\vec x-\vec x')$ was first derived by J. Joffrin and A. Levelut\cite{Joffrin1976}. For detailed discussions regarding this coefficient, please see Appendix (D):

\begin{eqnarray}\label{13}
 & {} & {\Lambda}_{ijkl}(\vec x-\vec x') = \sum_{\vec k}\Lambda_{ijkl}(\vec k)e^{i\vec k\cdot (\vec x-\vec x')}\nonumber \\
 & {} & \Lambda_{ijkl}(\vec k) = \frac{\alpha}{2\rho c_t^2}\left(\frac{k_ik_jk_kk_l}{k^4}\right)\nonumber \\
 & {} & -\frac{1}{8\rho c_t^2}\left(\frac{k_jk_l\delta_{ik}+k_jk_k\delta_{il}
+k_ik_l\delta_{jk}+k_ik_k\delta_{jl}
}{k^2}\right)\nonumber \\
\end{eqnarray}
where $\alpha=1-{c_t^2}/{c_l^2}$.
$i,j,k,l$ runs over $x,y,z$ cartesian coordinates. We name Eq.(\ref{12}) the non-elastic stress-stress interaction. In the rest of this paper for simplicity we will always use the approximation to replace $\vec x-\vec x'$ by $\vec x_{s}-\vec x_{s'}$ for the $s$-th and $s'$-th blocks, in which $\vec x_{s}$ denotes the center of the $s$-th block, and that $\int_{V^{(s)}}\hat{T}_{ij}(\vec x)d^3x=\hat{T}_{ij}^{(s)}$ is the uniform stress tensor of the $s$-th block. From this definition, the uniform stress tensor operator $\hat{T}_{ij}^{(s)}$ is a volume proportional extensive quantity. Also, from now on we use $e_{ij}^{(s)}$ to denote the strain field $e_{ij}(\vec x)$ located at the $s$-th block. By combining $N_0\times N_0\times N_0$ identical $L\times L\times L$ single blocks, we get a $N_0L\times N_0L\times N_0L$ super block. The non-elastic part of super block Hamiltonian (without the presence of external strain field) is given by the summation of single block non-elastic Hamiltonian and non-elastic stress-stress interaction:
\begin{eqnarray}\label{15}
\hat{H}_0^{\rm super}=\sum_{s=1}^{N_0^3}\hat{H}_0^{(s)}+\sum_{s\neq s'}^{N_0^3}\sum_{ijkl}\Lambda_{ijkl}^{(ss')}\hat{T}_{ij}^{(s)}\hat{T}_{kl}^{(s')}
\end{eqnarray}
From now on we apply the most important assumption of this paper: to assume that the correlation function of block uniform stress tensors $\hat{T}_{ij}^{(s)}$ are diagonal in spacial coordinates: ${\chi}^{(ss')}_{ijkl}=\frac{1}{L^3}\langle \hat{T}_{ij}^{(s)}\hat{T}_{kl}^{(s')}\rangle =\frac{1}{L^3}\langle \hat{T}_{ij}^{(s)}\hat{T}_{kl}^{(s)}\rangle\delta_{ss'}$. We assume that the uniform stress tensors lose spacial correlation for blocks at different locations $\vec x_s\neq \vec x_s'$. To further explain this assumption, we use the argument by D. C. Vural and A. J. Leggett\cite{Leggett2011}: in the absence of non-elastic stress-stress interaction $\hat{V}$, terms like $\langle n|\hat{T}_{ij}^{(s)}|m\rangle \langle m|\hat{T}_{kl}^{(s')}|n\rangle$ were originally uncorrelated. However, it is not obvious that they remain uncorrelated after the interaction $\hat{V}=\sum_{s\neq s'}^{N_0^3}\sum_{ijkl}\Lambda_{ijkl}^{(ss')}\hat{T}_{ij}^{(s)}\hat{T}_{kl}^{(s')}$ is taken into account (see super block Hamiltonian, Eq.(\ref{15})). However, we argue that on average, terms like $\langle n|\hat{T}_{ij}^{(s)}|m\rangle \langle m|\hat{T}_{kl}^{(s')}|n\rangle$ are likely to be small compared to terms like $\langle n|\hat{T}_{ij}^{(s)}|m\rangle \langle m|\hat{T}_{kl}^{(s)}|n\rangle$, because the interaction $\hat{V}$ is a highly frustrated interaction between stress tensor operators. 















\subsection{Glass Non-elastic Hamiltonian with the Presence of External Strain}
In previous subsections, we used the notation $\bm{e}(\vec x)$ to stand for intrinsic strain field. In this subsection we turn on external static, uniform strain, and use the notation $\bm{e}$ to represent it. As the simplest case, we consider the external strain as $e_{xx}=e$, $e_{yy}=e_{zz}=e_{xy}=e_{yz}=e_{zx}=0$. For other kinds of external strain ${\bm e}=e_{ij}$, similar positive-negative transition behaviors of glass mechanical susceptibility will be carried out as well. We consider glass single block and super block non-elastic Hamiltonians $\hat{H}_0^{(s)}(\bm{e})$ and $\hat{H}_0^{\rm super}(\bm{e})$ with the presence of external static strain $\bm{e}$ in this subsection. 





Let us combine a set of single blocks to form a super spherical glass with radius $r$. It is deformed by external strain to become an ellipsoid. The $xy$ and $xz$ plane cross sections are ellipses with eccentricity $\epsilon=\frac{\sqrt{e^2+2e}}{(1+e)}$  while the $yz$ cross section remains circular (see Fig.2 below).
\begin{figure}[hp
\centering
\includegraphics[scale=0.35]{Deform.png}
\caption{An isotropic (spherical) glass deformed by strain $e_{xx}=e$ to become an ellipsoid.}
\end{figure}





With the presence of external static strain $\bm{e}$, the $s$-th glass single block non-elastic Hamiltonian is given by 
\begin{eqnarray}\label{16}
\hat{H}^{(s)}_0(\bm{e})=\hat{H}_0^{(s)}+e_{xx}\hat{T}_{xx}^{(s)}(\bm{e})
\end{eqnarray}
Let us define $|n(\bm{e})\rangle$ and $E_n(\bm{e})$ to be the $n$-th eigenstate and eigenvalue of single block non-elastic Hamiltonian $\hat{H}_0^{(s)}(\bm{e})$. The single block volume $V(\bm{e})=(1+e_{xx})L^3$. We define $P_n(\bm{e})=e^{-\beta E_n(\bm{e})}/\mathcal{Z}(\bm{e})$ to be the probability function of $n$-th eigenstate of $\hat{H}_0^{(s)}(\bm{e})$. The glass single block zero-frequency non-elastic susceptibility $\chi_{ijkl}(\bm{e})$ is given by 
\widetext
\begin{eqnarray}\label{17}
\chi_{ijkl}(\bm{e}) & = &  \chi_{ijkl}^{\rm rel}(\bm{e})+ \chi_{ijkl}^{\rm res}(\bm{e})\nonumber \\
\chi_{ijkl}^{\rm rel}(\bm{e}) & = & \frac{\beta}{V(\bm{e})}\bigg(\sum_{nm}P_n(\bm{e})P_m(\bm{e})\langle n(\bm{e})|\hat{T}_{ij}(\bm{e})|n(\bm{e})\rangle\langle m(\bm{e})|\hat{T}_{kl}(\bm{e})|m(\bm{e})\rangle\nonumber \\
 & {} & \qquad\,\,-\sum_n P_n(\bm{e})\langle n(\bm{e})|\hat{T}_{ij}(\bm{e})|n(\bm{e})\rangle \langle n(\bm{e})|\hat{T}_{kl}(\bm{e})|n(\bm{e})\rangle\bigg)\nonumber \\
\chi_{ijkl}^{\rm res}(\bm{e}) & = & \frac{1}{V(\bm{e})}\sum_{n}\sum_{m\neq n}P_n(\bm{e})\frac{\langle n(\bm{e})|\hat{T}_{ij}(\bm{e})|m(\bm{e})\rangle\langle m(\bm{e})|\hat{T}_{kl}(\bm{e})|n(\bm{e})\rangle}{E_n(\bm{e})-E_m(\bm{e})+i\eta}\nonumber \\
 & - & \frac{1}{V(\bm{e})}\sum_{m}\sum_{n\neq m}P_m(\bm{e})\frac{\langle n(\bm{e})|\hat{T}_{ij}(\bm{e})|m(\bm{e})\rangle\langle m(\bm{e})|\hat{T}_{kl}(\bm{e})|n(\bm{e})\rangle}{E_n(\bm{e})-E_m(\bm{e})+i\eta}
\end{eqnarray}
For details of obtaining the results, Eq.(\ref{16}, \ref{17}), please see Appendix (A). 
\endwidetext





On the other hand, the super block glass non-elastic Hamiltonian with the presence of external strain $\bm{e}$ is given by 
\begin{eqnarray}\label{18}
\hat{H}_0^{\rm super}(\bm{e}) & = & \sum_{s}\bigg(\hat{H}_0^{(s)}+\sum_{ij}e_{xx}\hat{T}_{xx}^{(s)}(\bm{e})\bigg)\nonumber \\
 & + & \sum_{s\neq s'}\sum_{ijkl}\Lambda_{ijkl}^{(ss')}(\bm{e})\hat{T}_{ij}^{(s)}(\bm{e})\hat{T}_{kl}^{(s')}(\bm{e})
\end{eqnarray}
We denote $|n^*(\bm{e})\rangle$ and $E_n^*(\bm{e})$ to be the $n^*$-th eigenstate and eigenvalue of $\hat{H}_0^{\rm super}(\bm{e})$, which will be useful in later discussions.



According to the definition of non-elastic stress tensor, Eq.(\ref{9}), the super block non-elastic stress tensor is defined as the derivative of $\hat{H}^{\rm super}(\bm{e})$ with respect to infinitesimal intrinsic strain: $\hat{T}_{ij}^{\rm super}(\bm{e})={\delta  \hat{H}^{\rm super}(\bm{e})}/{\delta e_{ij}}$. Because of the strain field dependence of $\Lambda_{ijkl}^{(ss')}(\bm{e})$, an extra term proportional to ${\delta \Lambda_{abcd}^{(ss')}(\bm{e})}/{\delta e_{ij}}$ appears in super block stress tensor:
\begin{eqnarray}\label{19}
\hat{T}_{ij}^{\rm super}(\bm{e}) & = & \sum_s\hat{T}_{ij}^{(s)}(\bm{e}) +\sum_{ss'}\sum_{abcd}\frac{\delta \Lambda_{abcd}^{(ss')}(\bm{e})}{\delta e_{ij}}\hat{T}_{ab}^{(s)}(\bm{e})\hat{T}_{cd}^{(s')}(\bm{e})\nonumber \\
\end{eqnarray}








Smilar with super block non-elastic stress tensor, the super block non-elastic susceptibility also receives an extra term. The super block non-elastic susceptibility is given as follows 
\widetext
\begin{eqnarray}\label{20}
\chi_{ijkl}^{\rm super}(\bm{e})
 & = & \frac{\beta}{N_0^3V(\bm{e})}\bigg(\sum_{n^*m^*}P_n^*(\bm{e})P_m^*(\bm{e})\langle n^*(\bm{e})|\sum_s\hat{T}^{(s)}_{ij}(\bm{e})|n^*(\bm{e})\rangle\langle m^*(\bm{e})|\sum_{s'}\hat{T}^{(s')}_{kl}(\bm{e})|m^*(\bm{e})\rangle\nonumber \\
 & {} & \qquad\qquad -\sum_{n} P_n^*(\bm{e})\langle n^*(\bm{e})|\sum_{s}\hat{T}^{(s)}_{ij}(\bm{e})|n^*(\bm{e})\rangle \langle n^*(\bm{e})|\sum_{s'}\hat{T}^{(s')}_{kl}(\bm{e})|n^*(\bm{e})\rangle\bigg)\nonumber \\
 & + & \frac{1}{N_0^3V(\bm{e})}\bigg(\sum_{n^*}\sum_{m^*\neq n^*}P_n^*(\bm{e})\frac{\langle n^*(\bm{e})|\sum_s\hat{T}^{(s)}_{ij}(\bm{e})|m^*(\bm{e})\rangle\langle m^*(\bm{e})|\sum_{s'}\hat{T}^{(s')}_{kl}(\bm{e})|n^*(\bm{e})\rangle}{E_n^*(\bm{e})-E^*_m(\bm{e})+i\eta}\nonumber \\
 & {} & \qquad\qquad -\sum_{m^*}\sum_{n^*\neq m^*}P_m^*(\bm{e})\frac{\langle n^*(\bm{e})|\sum_s\hat{T}^{(s)}_{ij}(\bm{e})|m^*(\bm{e})\rangle\langle m^*(\bm{e})|\sum_{s'}\hat{T}^{(s')}_{kl}(\bm{e})|n^*(\bm{e})\rangle}{E_n^*(\bm{e})-E^*_m(\bm{e})+i\eta}\bigg)\nonumber \\
 & + & \frac{1}{N_0^3V(\bm{e})}\sum_{abcd}\sum_{ss'}\left<\frac{\delta^2\Lambda_{abcd}^{(ss')}(\bm{e})}{\delta e_{ij}\delta e_{kl}}\hat{T}_{ab}^{(s)}(\bm{e})\hat{T}_{cd}^{(s')}(\bm{e})\right>
\end{eqnarray}
\endwidetext
In the above result, $P_n^*(\bm{e})=e^{-\beta E_n^*(\bm{e})}/\mathcal{Z}^*(\bm{e})$ is the probability function of the $n^*$-th eigenstate of $\hat{H}_0^{\rm super}(\bm{e})$. The first and second lines stand for super block zero-frequency relaxation and resonance susceptibilities, while the third line is the extra term which comes from the second term of Eq.(\ref{19}). For details of discussions on super block non-elastic Hamiltonian and susceptibility with the presence of external strain, please see Appendix (B). In the next section we will prove that the last term in Eq.(\ref{20}) in real space renormalization irrelevant. 




























\section{Real Space Renormalization Procedure of Glass Non-Elastic Susceptibility}
Let us combine $N_0^3$ glass single blocks with the dimension $L\times L\times L$ to form a super block with the dimension $N_0L\times N_0L\times N_0L$. Because the super block is $N_0^3$ times the volume of single block, repeating such process from microscopic length scale will eventually carry out the glass Hamiltonian, non-elastic stress tensor and susceptibility at experimental length scale. Our main purpose of this section is to set up the recursion relation (i.e., real space renormalization equation) between single block and super block non-elastic susceptibilities.


According to the argument by D. C. Vural and A. J. Leggett\cite{Leggett2011}, the suggested starting microscopic length scale of renormalization procedure is, for example, $L_1\sim 50\AA$. Since the final result only logarithmically depends on this choice, it will not be sensitive. The chacteristic thermal phonon wavelength must be no less than the starting length scale $L_1$, which means the effective temperature of our model is no greater than $60$K. However, at least to the author's knowledge, all of the experiments on glass avalanche are taken under room temperatures or glass transition temperatures\cite{Argon2005, Gauthier2004, Dahmen2014, Fineberg1991, Fineberg1992} ($T\sim 300$K). We hope more experiments on mechanical properties of glass could be taken at low-temperatures below 60K to further test the validity of our theory.





In the $n$-th step renormalization, we combine $N_0^3$ single blocks with the side $L_n$ to form the $n$-th step super block glass with the side $N_0L_n$.  The single block in the $(n+1)$-th step is actually the super block in the $n$-th step, i.e., $L_{n+1}=N_0L_n$. Before we turn on non-elastic stress-stress interaction $\hat{V}(\bm{e})$, such a group of non-interacting single blocks have the Hamiltonian $\sum_{s=1}^{N_0^3}\left(\hat{H}^{(s)}_0+e_{xx}\hat{T}_{xx}^{(s)}(\bm{e})\right)$, eigenstates $|n(\bm{e})\rangle = \prod_{s=1}^{N_0^3}|n^{(s)}(\bm{e})\rangle$ and eigenvalues $E_n(\bm{e})=\sum_{s=1}^{N_0^3}E_n^{(s)}(\bm{e})$, where $|n^{(s)}(\bm{e})\rangle$ and $E_n^{(s)}(\bm{e})$ are the $n^{(s)}$-th eigenstate and eigenvalue of the $s$-th single block non-elastic Hamiltonian $\hat{H}_0^{(s)}(\bm{e})=\hat{H}_0^{(s)}+e_{xx}\hat{T}_{xx}^{(s)}(\bm{e})$. We combine single blocks into a super block and turn on interaction $\hat{V}(\bm{e})=\sum_{s\neq s'}^{N_0^3}\Lambda_{ijkl}^{(ss')}(\bm{e})\hat{T}^{(s)}_{ij}(\bm{e})\hat{T}^{(s')}_{kl}(\bm{e})$. We assume interaction $\hat{V}(\bm{e})$ is relatively weak compared to $\hat{H}_0(\bm{e})=\sum_{s=1}^{N_0^3}\left(\hat{H}^{(s)}_0+e_{xx}\hat{T}_{xx}^{(s)}(\bm{e})\right)$, so that the interaction can be treated as a perturbation. In fact, non-elastic stress-stress interaction is $\hat{V}(\bm{e})\propto 1/r^3$. At small length scales we know it always dominates glass Hamiltonian. However, if the non-elastic susceptibility decreases logarithmically as the increase of length scale (which will be proved in Eq.(\ref{3.23.4})), then that means the non-elastic stress-stress interaction $\hat{V}(\bm{e})$ can be treated as a perturbation at the late stages. The assumption that $\hat{V}(\bm{e})$ can be treated as a perturbation is qualitatively correct. In the last section we define the eigenstates and eigenvalues of super block non-elastic Hamiltonian $\hat{H}_0^{\rm super}(\bm{e})$ to be $|n^*(\bm{e})\rangle$ and $E_n^*(\bm{e})$. Their relations with $|n(\bm{e})\rangle$ and $E_n(\bm{e})$ are given as 

\begin{eqnarray}\label{21}
|n^*(\bm{e})\rangle & = & |n(\bm{e})\rangle +\sum_{p\neq n}\frac{\langle p(\bm{e})|\hat{V}(\bm{e})|n(\bm{e})\rangle}{E_n(\bm{e})-E_p(\bm{e})}|p(\bm{e})\rangle+\mathcal{O}(V^2)\nonumber \\
E_n^*(\bm{e}) & = & E_n(\bm{e})+\langle n(\bm{e})|\hat{V}(\bm{e})|n(\bm{e})\rangle +\mathcal{O}(V^2)
\end{eqnarray}
With the relations in Eq.(\ref{21}) one can expand super block non-elastic susceptibility Eq.(\ref{20}) in orders of $\hat{V}(\bm{e})$. Up to the first order in $\hat{V}(\bm{e})$ we can write super block non-elastic susceptibility ${\chi}^{\rm super}_{ijkl}(\bm{e})$ expansions in terms of single block non-elastic susceptibilities ${\chi}_{ijkl}(\bm{e})$. The recursion relation for single block and super block non-elastic susceptibilities are given as follows:

\widetext
\begin{eqnarray}\label{23}
\chi_{ijkl}^{\rm super}(\bm{e}) & = & \chi_{ijkl}^{\rm super\,rel}(\bm{e})+\chi_{ijkl}^{\rm super\, res}(\bm{e})\nonumber \\
 & = & 
\chi_{ijkl}^{\rm rel}(\bm{e})
 -\frac{V_n(\bm{e})}{N_0^3}\left[-\sum_{mnpq}\sum_{ss'}\Lambda_{mnpq}^{(ss')}(\bm{e})\right]\left(\chi_{ijmn}^{\rm rel}(\bm{e})\chi_{pqkl}^{\rm rel}(\bm{e})+\chi_{ijmn}^{\rm rel}(\bm{e})\chi_{pqkl}^{\rm res}(\bm{e})+\chi_{ijmn}^{\rm res}(\bm{e})\chi_{pqkl}^{\rm rel}(\bm{e})\right)\nonumber \\
 & + & \chi_{ijkl}^{\rm res}(\bm{e}) -\frac{V_n(\bm{e})}{N_0^3}\left[-\sum_{mnpq}\sum_{ss'}\Lambda_{mnpq}^{(ss')}(\bm{e})\right] \chi_{ijmn}^{\rm res}(\bm{e})\chi_{pqkl}^{\rm res}(\bm{e})\nonumber \\
 & + & \frac{1}{N_0^3V_n(\bm{e})}\sum_{mnpq}\sum_{ss'}\left<\frac{\delta^2\Lambda_{mnpq}^{(ss')}(\bm{e})}{\delta e_{ij}\delta e_{kl}}\hat{T}_{mn}^{(s)}(\bm{e})\hat{T}_{pq}^{(s')}(\bm{e})\right>
\end{eqnarray}
\endwidetext
For details of calculations of obtaining Eq.(\ref{23}), please see Appendix (C). The last term of Eq.(\ref{23}) is renormalization irrelevant. Compared to other terms in Eq.(\ref{23}) the last term decreases  ($\propto L^{-3}$) as the increase of length scale $L$. To prove this result let us provide a qualitative analysis: denote $\Lambda_{ijkl}^{(ss')}(\bm{e})=-\tilde{\Lambda}_{ijkl}(\bm{e})/8\pi\rho c_t^2x_{ss'}^3$ where $x_{ss'}=|\vec x_s-\vec x_s'|$ and $\tilde{\Lambda}_{ijkl}(\bm{e})$ is a dimensionless number of order 1 (see Eq.(\ref{13})). By applying linear response theory on the last term of Eq.(\ref{23}) with respect to infinitesimal external perturbation $\sum_{ij}\sum_se_{ij}^{(s)}(t)\hat{T}_{ij}^{(s)}$, the last term of Eq.(\ref{23}) is given by








\begin{eqnarray}\label{23.1}
\sum_{mnpq}\int d\Omega\,{\rm Im}\,{\chi}_{ijmn}^{\rm res}(\Omega)\left(\sum_{ss'}\frac{\hbar L_n^3{\lambda}_{mnpq}}{ 8\pi\rho^2 c_t^4x_{ss'}^6}\right)\,{\rm Im}\,{\chi}^{\rm res}_{pqkl}(-\Omega)\nonumber \\
\end{eqnarray}
where $\lambda_{mnpq}(\bm{e})$ is the second order derivative of $\tilde{\Lambda}_{mnpq}(\bm{e})$ with respect to strain. It is also a dimensionless number of order 1. According to the argument by R. O. Pohl and his group and D. C. Vural and A. J. Leggett\cite{Pohl2002, Leggett2011}, we assume that the ``reduced imaginary part of non-elastic resonance susceptibility", ${\rm Im}\,\tilde{\chi}_{l,t}^{\rm res}(\omega, T)=\left(1-e^{-\beta\hbar\omega}\right)^{-1}\,{\rm Im}\,\chi_{l,t}^{\rm res}(\omega, T)$ is approximately a constant of frequency and temperature up to $\omega_c\sim 10^{15}$Hz and around the temperature of order 1K. Since the imaginary part of resonance susceptibility is always smaller than it's reduced version: ${\rm Im\,}{\chi}^{\rm res}_{ijkl}(\omega)<{\rm Im\,}\tilde{\chi}^{\rm res}_{ijkl}(\omega)$, integrating over $\Omega$ gives the upper limit of Eq.(\ref{23.1}): $-{\rm C}{\hbar\omega_c}\,\left({\rm Im}\,\tilde{\chi}_t^{\rm res}\right)^2/{ \rho^2 c_t^4L_n^3}$, where ${\rm C}$ is also a dimensionless constant of order 1. If we require that there is a critical length scale $L_c$, below which the last term of Eq.(\ref{23}) is comparable to the other terms, the order of magnitude for $L_c$ is 
\begin{eqnarray}\label{23.2}
L_c< \left(\frac{\hbar\omega_c}{\rho c_{l,t}^2}\right)^{\frac{1}{3}}\approx 4.6\AA< L_1=50\AA
\end{eqnarray}
which means the upper limit of $L_c$ is even smaller than the starting effective length scale of renormalization technique. Throughout the entire renormalization procedure the last term in Eq.(\ref{23}) is always negligible. In the following discussions we drop the last term in Eq.(\ref{23}).


With the above simplifications one can rewrite the non-elastic susceptibility renormalization equation as follows, 
\begin{eqnarray}\label{3.23.1}
 & {} & \chi_{ijkl}^{\rm super}(\bm{e}) 
= \nonumber \\
 & {} & \chi_{ijkl}(\bm{e}) 
 -\frac{V_n(\bm{e})}{N_0^3}\left[-\sum_{mnpq}\sum_{ss'}\Lambda_{mnpq}^{(ss')}(\bm{e})\right]\chi_{ijmn}(\bm{e}) \chi_{pqkl}(\bm{e}) \nonumber \\
\end{eqnarray}
where the zero-frequency non-elastic susceptibilities are given by $\chi_{ijkl}(\bm{e}) =\chi_{ijkl}^{\rm rel}(\bm{e}) +\chi_{ijkl}^{\rm res}(\bm{e}) $, and $\chi_{ijkl}^{\rm super}(\bm{e}) =\chi_{ijkl}^{\rm super\,rel}(\bm{e}) +\chi_{ijkl}^{\rm super\,res}(\bm{e}) $.




The renormalization equation for non-elastic susceptibility can be further simplified with the following three steps. First of all, we define a 4-indice quantity $M_{mnpq}$, given by 
\begin{eqnarray}\label{3.23.3}
M_{mnpq}(\bm{e}) =\frac{V_n(\bm{e})}{N_0^3}\left[-\sum_{ss'}\Lambda_{mnpq}^{(ss')}(\bm{e})\right]
\end{eqnarray}
So the non-elastic susceptibility renormalization relation is simplified as 
\begin{eqnarray}\label{3.23.2}
\chi_{ijkl}^{\rm super} (\bm{e}) 
 & = & 
\chi_{ijkl}(\bm{e}) 
 -\sum_{mnpq}\chi_{ijmn}(\bm{e}) M_{mnpq}(\bm{e}) \chi_{pqkl}(\bm{e}) \nonumber \\
\end{eqnarray}

Second, let us denote the 2-fold indices in Eq.(\ref{3.23.2}), $(ij), (kl), (mn), (pq)$  to be $(ij)\to A$, $(kl)\to B$, $(mn)\to C$, $(pq)\to D$. With this simplification, we rewrite 4-indice quantities $\chi_{ijkl}$ and $M_{mnpq}$ into 2-indice matrix forms: $\bm{\chi}(\bm{e})={\chi}_{AB}(\bm{e})$ and $\bm{M}(\bm{e})={M}_{CD}(\bm{e})$. They are $6\times 6$ matrices, for example, $M_{CD}$ has the indices $C \,({\rm or}\, D)=(xx), (xy), (xz), (yy), (yz), (zz)$. Third, let us define the ``change of non-elastic susceptibility", $\delta\bm{\chi}(\bm{e})=\bm{\chi}^{\rm super}(\bm{e})-\bm{\chi}(\bm{e})$. We integrate over the length scale to calculate Eq.(\ref{3.23.2}). The macroscopic length scale non-elastic susceptibility is therefore given as follows:
\begin{eqnarray}\label{3.23.4}
\bm{\chi}^{-1}(R)=\bm{M}(\bm{e})\log_{N_0}\left(\frac{R}{L_1}\right)+\bm{\chi}^{-1}(L_1)
\end{eqnarray}
where the experimental length scale $R$ is the actual size of glass sample. We have no idea about the value of the starting microscopic non-elastic susceptibility $\bm{\chi}(L_1)$. However, we would like to argue, that the term $\bm{M}(\bm{e})\log_{N_0}\left({R}/{L_1}\right)$ is much greater than $\bm{\chi}^{-1}(L_1)$ becasue of the factor $\log_{N_0}\left({R}/{L_1}\right)$. In the rest of this paper, we neglect the starting microscopic length scale non-elastoc susceptibility $\bm{\chi}(L_1)$ in Eq.(\ref{3.23.4}).












\section{The Positive-Negative Transitions of Non-elastic Susceptibility}
We want to prove the existence of positive-negative transitions in glass mechanical susceptibility with the increase of external static strain $\bm{e}$. Since the positive elastic susceptibility $\chi_{ijkl}^{\rm el}=(\rho c_l^2-2\rho c_t^2)\delta_{ij}\delta_{kl}+\rho c_t^2(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$ is a constant, our hope is to find positive-negative transitions in non-elastic susceptibility, Eq.(\ref{11}). This is the main purpose of this section. 



The spherical glass is deformed by external static strain $\bm{e}=e_{xx}$ to become an ellipsoid with the eccentricity $\epsilon=\sqrt{({e^2+2e})/({1+e})}$ in $xy, xz$ plane cross sections. In Eq.(\ref{3.23.3}) let us take continuum limit and change the variables of spacial integrals $\vec r_s+\vec r_s'=\vec R$ and $\vec r_s-\vec r_s'=\vec r$ to calculate the 4-indice quantity $M_{mnpq}(\bm{e})$:
\begin{eqnarray}\label{26}
M_{mnpq}(\bm{e})=-\int_{V(\bm{e})} d^3r\,{\Lambda}_{mnpq}(\vec x_s-\vec x_s')
\end{eqnarray}
where the integral region $V(\bm{e})$ is the space of ellipsoid glass. After integrating over the ellipsoid region, the quantity $\bm{M}(\bm{e})\log_{N_0}(R/L_1)$ which appears in Eq.(\ref{3.23.4}) is then given by the following form
\widetext
\begin{eqnarray}\label{27}
 & {} & \bm{M}(\bm{e})\log_{N_0}\left(\frac{R}{L_1}\right)=\frac{1}{2\rho c_t^2}\ln\left(\frac{R}{L_1}\right)\nonumber \\
 & {} & \left(\begin{array}{cccccc}
A+\frac{4\pi}{3}\left(1-\frac{1}{5}\alpha\right) & 0 & 0 & B-\frac{8\pi}{90}\alpha & 0 & B-\frac{8\pi}{90}\alpha\\
 0 & C+\frac{2\pi}{3}\left(1-\frac{4}{30}\alpha\right) & 0 & 0 & 0 & 0\\
 0 & 0 & C+\frac{2\pi}{3}\left(1-\frac{4}{30}\alpha\right) & 0 & 0 & 0\\
B-\frac{8\pi}{90}\alpha & 0 & 0 & D+\frac{4\pi}{3}\left(1-\frac{1}{5}\alpha\right) & 0 & E-\frac{8\pi}{90}\alpha\\
 0 & 0 & 0 & 0 & F+\frac{2\pi}{3}\left(1-\frac{4}{30}\alpha\right) & 0\\
B-\frac{8\pi}{90}\alpha & 0 & 0 & E-\frac{8\pi}{90}\alpha & 0 & D+\frac{4\pi}{3}\left(1-\frac{1}{5}\alpha\right)\\
\end{array}\right)\nonumber \\
\end{eqnarray}
\endwidetext
To obtain the above matrix form of $\bm{M}(\bm{e})$, first we write the 4-indice quantity $M_{mnpq}(\bm{e})$ as $M_{(mn)(pq)}(\bm{e})$. Then we calculate the entire 36 matrix elements of it to obtain the above result, Eq.(\ref{27}). $A$, $B$, $C$, $D$, $E$ and $F$ are given as follows, 
\begin{eqnarray}\label{28}
A & = & 1-3\overline{n_x^2}+\frac{1}{2}\alpha\left(-3+18\overline{ n_x^2}-15\overline{ n_x^4}\right)\nonumber \\
B & = & \frac{1}{2}\alpha\left[-1-15\overline{ n_x^2n_y^2}+3\left(\overline{ n_x^2}+\overline{ n_y^2}\right)\right]\nonumber \\
C & = & \frac{1}{4}\left(2-3\overline{ n_x^2}-3\overline{ n_y^2}\right)\nonumber \\
 & {} & +\frac{1}{2}\alpha\left[-1-15\overline{ n_x^2n_y^2}+3\left(\overline{ n_x^2}+\overline{ n_y^2}\right)\right]\nonumber \\
D & = & 1-3\overline{ n_y^2}+\frac{1}{2}\alpha\left(-3+18\overline{ n_y^2}-15\overline{ n_y^4}\right)\nonumber \\
E & = & \frac{1}{2}\alpha\left(-1-15\overline{ n_y^2n_z^2}+6\overline{ n_y^2}\right)\nonumber \\
F & = & \frac{1}{2}\left(1-3\overline{ n_y^2}\right)+\frac{1}{2}\alpha\left(-1-15\overline{ n_y^2n_z^2}+6\overline{ n_y^2}\right)
\end{eqnarray}
In the above result we have applied rotational invariance of the integral region $V(\bm{e})$ around $x$-axis, and the experimentally measurable quantity $\alpha$ is $\alpha=1-c_t^2/c_l^2$. The definition of average values $\overline{ n_{x,y}^2}$, $\overline{ n_{x,y}^4}$, $\overline{ n_x^2n_y^2}$, $\overline{ n_y^2n_z^2}$ are given as follows: for an arbitrary function $f(\vec r)$, the average value $\overline{ f(\vec r)}$ is defined as  
\begin{eqnarray}\label{26}
\overline{ f(\vec r)}=\frac{\int_{V(\bm{e})} d^3r \, f(\vec r)/r^3}{\int_{V(\bm{e})} d^3r \, 1/r^3}
\end{eqnarray}
Integrate over the ellipsoid region $V(\bm{e})$, the unit vector average values are displayed as follows, 


\begin{eqnarray}\label{29}
 & {} & \overline{ n_x^2}  = \frac{\epsilon\sqrt{1-\epsilon^2}(-1+2\epsilon^2)+\arcsin\epsilon}
{4\epsilon^2\left(\epsilon\sqrt{1-\epsilon^2}+\arcsin\epsilon\right)}\nonumber \\
 & {} & \overline{ n_x^4}  = \frac{\epsilon\sqrt{1-\epsilon^2}(-3-2\epsilon^2+8\epsilon^4)+3\arcsin\epsilon}
{24\epsilon^4\left(\epsilon\sqrt{1-\epsilon^2}+\arcsin\epsilon\right)}\nonumber \\
 & {} & \overline{ n_y^2n_z^2} =
\frac{\epsilon\sqrt{1-\epsilon^2}(-3+10\epsilon^2+8\epsilon^4)}{192\epsilon^4\left(\epsilon\sqrt{1-\epsilon^2}+\arcsin\epsilon\right)}\nonumber \\
 & {} & \quad\quad\quad\,\,\,+\frac{3(1-4\epsilon^2+8\epsilon^4)\arcsin\epsilon}{192\epsilon^4\left(\epsilon\sqrt{1-\epsilon^2}+\arcsin\epsilon\right)}
\end{eqnarray}
where $0\le\epsilon\le 1$ is the eccentricity of the $xy$ and $xz$ cross sections of ellipsoid glass (see Fig.2). 
























Eq.(\ref{27}) is a $6\times 6$ matrix, so it has 6 eigenvalues and eigenvectors. We want to figure out which of these eigenvalues show positive-negative transitions. First of all we list a series of variable changes:
\begin{eqnarray}\label{30}
 & {} & A' = A+\frac{4\pi}{3}\left(1-\frac{1}{5}\alpha\right)\quad\,\,\,\, B' = B-\frac{8\pi}{90}\alpha\nonumber \\
 & {} & C' = C+\frac{2\pi}{3}\left(1-\frac{4}{30}\alpha\right)\quad D' = D+\frac{4\pi}{3}\left(1-\frac{1}{5}\alpha\right)\nonumber \\
 & {} & E' = E-\frac{8\pi}{90}\qquad \qquad\qquad\,\,\, F' = F+\frac{2\pi}{3}\left(1-\frac{4}{30}\alpha\right)\nonumber \\
 & {} & \Delta' = 8B'^2+(A'-D'-E')^2
\end{eqnarray}





Among the 6 eigenvalues of $\bm{M}(\bm{e})\log_{N_0}(\frac{R}{L_1})=\bm{\chi}^{-1}(R)$, 3 of them keep positive, while other 3 show positive-negative transitions we demand. The eigenvalues and corresponding eigenvectors of $\bm{\chi}^{-1}(R)$ are listed as follows:
\begin{eqnarray}\label{31}
\begin{array}{|c|c|}
\hline
 {\rm eigenvalue} & {\rm eigenvector}\\
\hline
C' & (0,0,1,0,0,0)\\
C' & (0,1,0,0,0,0)\\
({A'+D'+E'+\sqrt{\Delta'}})/2 & \left(\frac{A'-D'-E'+\sqrt{\Delta'}}{2B'},0,0,1,0,1\right)\\
({A'+D'+E'-\sqrt{\Delta'}})/2 & \left(\frac{A'-D'-E'-\sqrt{\Delta'}}{2B'},0,0,1,0,1\right)\\
D'-E' & (0,0,0,-1,0,1)\\
F' & (0,0,0,0,1,0)\\
\hline
\end{array}
\nonumber \\
\end{eqnarray}
The first three eigenvalues $C', C'$ and $(A'+D'+E'+\sqrt{\Delta'})/2$ stay positive for arbitrary positive $\chi_{l,t}(L_1)$ and $\epsilon$; other three eigenvalues, $(A'+D'+E'-\sqrt{\Delta'})/2$, $D'-E'$ and $F'$ decrease from positive to negative with the increase of eccentricity.





We choose the typical value of $\alpha=1-c_t^2/c_l^2=0.7$ and $R=1$m, so $\ln(R/L_1)\approx 20$.
The 6 eigenvalues of non-elastic susceptibility $\bm{\chi}(R)$ are $C'^{-1}$, $C'^{-1}$, ${2}/({A'+D'+E'+\sqrt{\Delta'}})$, ${2}/({A'+D'+E'-\sqrt{\Delta'}}) $, $\left(D'-E'\right)^{-1}$ and $F'^{-1}$. The first, second and third eigenvalues $C'^{-1}$, $C'^{-1}$ and ${2}/({A'+D'+E'+\sqrt{\Delta'}})$ stay positive for eccentricities vary from 0 to 1. The plots of eigenvalues versus eccentricities are displayed in Figs.3 and 4. As the external static deformation $e_{xx}=e$ increases, glass is hardening against the external strains in the directions of $e_{xy}$, $e_{xz}$ and $\frac{A'-D'-E'+\sqrt{\Delta'}}{2B'}e_{xx}+e_{yy}+e_{zz}$. The coefficient $\frac{A'-D'-E'+\sqrt{\Delta'}}{2B'}$ stays negative for arbitrary eccentricity varies from 0 to 1. We plot the coefficient $\frac{A'-D'-E'+\sqrt{\Delta'}}{2B'}$ in Fig.5 below.



\begin{figure}[hp
\centering
\includegraphics[scale=0.4]{eigenvalue1.png}
\caption{The first and second eigenvalues of non-elastic susceptibility, $C'^{-1}$ as the function of eccentricity. It stays positive.}
\end{figure}



\begin{figure}[hp
\centering
\includegraphics[scale=0.4]{eigenvalue2.png}
\caption{The third eigenvalue $2/(A'+D'+E'+\sqrt{\Delta'})$ of non-elastic susceptibility as the function of eccentricity. It stays positive.}
\end{figure}




\begin{figure}[hp
\centering
\includegraphics[scale=0.4]{coefficient1.png}
\caption{The \textbf{coefficient} in the third eigenvector, $\frac{A'-D'-E'+\sqrt{\Delta'}}{2B'}$ as the function of eccentricity. It stays negative.}
\end{figure}





On the other hand, the fourth, fifth and sixth eigenvalues of non-elastic susceptibility $\bm{\chi}(R)$, $2/({A'+D'+E'-\sqrt{\Delta}})$, $\left(D'-E'\right)^{-1}$ and $F'^{-1}$ present positive-negative transitions at certain critical eccentricities. We plot them as the function of eccentricity in Figs. 6, 7 and 8 as follows. The coefficient of the fourth eigenvector, $\frac{A'-D'-E'-\sqrt{\Delta}}{2B'}$ is always positive. We show this plot in Fig.9. Fig.6-8 indicate when external static deformations $e_{xx}=e$ exceed certain critical values, glass is fragile against the external strain fields in the directions of $\frac{A'-D'-E'-\sqrt{\Delta}}{2B'}e_{xx}+e_{yy}+e_{zz}$, $-e_{yy}+e_{zz}$ and $e_{yz}$.



















\begin{figure}[hp
\centering
\includegraphics[scale=0.45]{eigenvalue4.png}
\caption{The fourth eigenvalue of non-elastic susceptibility, $2/({A'+D'+E'-\sqrt{\Delta}})$ as the function of eccentricity. It presents a sudden positive-infinite to negative-infinite transition at the critical eccentricity $\epsilon_{\rm crit}^{(1)}=1.70$ (with the input of parameters we mentioned before). }
\end{figure}



\begin{figure}[hp
\centering
\includegraphics[scale=0.45]{eigenvalue5.png}
\caption{The fifth eigenvalue of non-elastic suscptibility, $\left(D'-E'\right)^{-1}$ as the function of eccentricity. It presents a sudden positive-infinite to negative-infinite transition at the critical eccentricity $\epsilon_{\rm crit}^{(2)}=0.239$. }
\end{figure}



\begin{figure}[hp
\centering
\includegraphics[scale=0.45]{eigenvalue6.png}
\caption{The sixth eigenvalue $F'^{-1}$ as the function of eccentricity. It presents a positive-negative transition at the critical eccentricity $\epsilon_{\rm crit}^{(3)}=\epsilon_{\rm crit}^{(2)}=0.239$. }
\end{figure}























\begin{figure}[hp
\centering
\includegraphics[scale=0.45]{coefficient2.png}
\caption{The \textbf{coefficient} in the sixth eigenvector, $\frac{A'-D'-E'-\sqrt{\Delta}}{2B'}$ as the function of eccentricity. It stays positive with the value varying from $0.4$ to $1$.}
\end{figure}









Let us discuss the fourth, fifth and sixth eigenvalues of non-elastic susceptibility in details. First of all, the eigensvector which corresponds to the fourth eigenvalue is $\left(\frac{A'-D'-E'-\sqrt{\Delta}}{2B'},0,0,1,0,1\right)$. From Fig.9, the coefficients of $e_{xx}$, $e_{yy}$ and $e_{zz}$ have the same signs. When the external static strain $e_{xx}$ exceeds critical value $e_{\rm crit}^{(1)}=1.70$, glass is fragile against an additional expansion or contraction. Second, the eigenvector which corresponds to the fifth eigenvalue is $(0,0,0,-1,0,1)$. When the external static strain exceeds critical value $e_{\rm crit}^{(2)}=0.239$, glass is fragile against an additional strain $\pm(e_{yy}-e_{zz})$. Third, the eigenvector which corresponds to the sixth eigenvalue is $e_{yz}$. When the external static strain exceeds critical value $e_{\rm crit}^{(3)}=e_{\rm crit}^{(2)}=0.239$, glass is fragile against an additional shear deformation in $yz$ plane. Since the glass is symmstric around the $x$-axis, it is obvious that the critical value should be the same for external strain fields $\pm(e_{yy}-e_{zz})$ and $e_{yz}$. However, compared to $e_{\rm crit}^{(2), (3)}=0.239$, the first critical strain $e_{\rm crit}^{(1)}=1.70$ is too big to be observed. Before reaching such great value, glass has already reached the other two critical strains to break. From the above discussions, we would like to predict, that the amorphous materials are more likely to crack for shear deformations rather than dilation deformations. 








\widetext

\begin{figure}[hp
\centering
\includegraphics[scale=0.35]{Deform1.png}
\includegraphics[scale=0.35]{Deform2.png}
\includegraphics[scale=0.35]{Deform3.png}
\caption{Three external strain field directions to crack the glass. (1) An expansion or contraction; (2) pull in $e_{yy}$ strain direction while squeeze in $e_{zz}$ direction, or vice versa; (3) shear in $yz$ plane, please note $\partial u_y/\partial z$ and $\partial u_z/\partial y$ not necessarily the same for this case. }
\end{figure}
\endwidetext








Finally, to verify the existence of positive-negative transitions in glass total susceptibility, we need to sum up elastic susceptibility (Eq.(\ref{7})) and non-elastic susceptibility: $\bm{\chi}^{\rm tot}=\bm{\chi}^{\rm el}+\bm{\chi}$. Please note that the elastic susceptibility does not provide positive-negative transitions. When external strain $e_{xx}$ approaches the critical values, non-elastic susceptibility presents sharp positive-negative transitions. It becomes overwhelmingly larger than the elastic susceptibility. The positive-negative transitions of total mechanical susceptibility are determined by non-elastic susceptibility. Our theory cannot explain multiple small slips in stress-strain curve, because once a slip occurs, groups of glass molecules shift positions macroscopically. We need to rewrite the entire glass Hamiltonian after the slip happens. 












\section{Discussions}
In this paper we develop a generic coupled block model in glass to prove the existence of positive-negative transitions of glass mechanical susceptibility. Our Hamiltonian contains long-wavelength phonon Hamiltonian $\hat{H}^{\rm el}$, non-elastic part of Hamiltonian $\hat{H}$, stress tensor-strain field coupling $e_{ij}\hat{T}_{ij}$, and non-elastic stress-stress interaction $\hat{V}$. We use real space renormalization procedure to set up the relation between microscopic and macroscopic non-elastic susceptibilities. The most important assumption in this paper is that the correlation function of non-elastic stress tensors are diagonal in spacial coordinates: $V^{-1}\langle \hat{T}_{ij}^{(s)}\hat{T}_{kl}^{(s')}\rangle=\chi_{ijkl}\delta_{ss'}$. With the presence of external static deformation, the spherical glass loses isotropicity, leading to the anisotropicity of non-elastic susceptibility at experimental length scale. Non-elastic susceptibility presents steep positive-negative transitions when external strain passes through critical values. However, this is still a tentative result, since it is based on the values of microscopic length scale non-elastic susceptibilities $\chi_t(L_1)$ and $\chi_l(L_1)$ which we do not know.



The basic requirement of our real space renormalization technique is that the characteristic thermal phonon wavelength $\lambda_T$ cannot exceed the starting microscopic length scale: $\lambda_T>50\AA$. Therefore, our renormalization technique is only valid below the temperature of $60$K. However,  up to now we cannot find any glass avalanche experiment below the temperature of 60K\cite{Argon2005, Gauthier2004, Dahmen2014, Fineberg1991, Fineberg1992}. We hope more experiments on the mechanical properties of amorphous materials at low temperatures could be carried out to test the validity of our theory.





















\section{Acknowledgement}
D. Zhou wishes to express his deepest thanks to his advisor Anthony J. Leggett. D. Zhou also thanks Karin A. Dahmen and Xueda Wen for insightful discussions. D. Zhou would like to especially thank the anonymous referee for precious suggestions. This work is supported by the National Science Foundation under Grant No. NSF-DMR 09-06921 at the University of Illinois.
























\widetext
