On the localization regime of certain random operators within Hartree-Fock theory

In recent decades there has been intense activity regarding mathematical aspects of disordered systems. In the Anderson model in dimension two or higher, there is an extensive literature regarding localization in the regimes of large disorder or at spectral edges. In this context, proofs either follow the strategy of the multiscale analysis, see Bourgain-Kenig ; Carm-Klein-Mart ; Ding-Smart ; Fr-M-S-S85 ; Fr-Spenc ; Germ-Hislop-Klein ; GKBootstrap ; Kir-Stol-Stolz ; vonD-K and also the surveys Kirschsurv ; Klein-surv , or the method of fractional moments, dating back to A-Molc ; Aiz-weakdis and further developed in both discrete and continuous settings A-S-F-H ; Aiz-Elg-N-S-S and also in the context of non-monotone potentials Elg-Taut-Ves ; E-S-S , see also the survey Stolz-surv and the monograph A-W-B . Localization in the context of weak disorder and existence of the so-called Lisfshitz tails were also extensively studied, see Aiz-weakdis ; Elgart-LT3d ; RojasM-Geb1 ; RojasM-Geb2 ; Klopp-Lif ; Klopp-Nak ; Wang-univ and references therein. For results on complete localization in one dimension using large-deviation techniques we refer to Jit-Z ; Bucaj1dloc .

In the past years, there has been a number of developments in the context of many-body disordered systems such as systems with a finite number of particles A-W-Mult ; Chul-Suh ; Chu-Suh-ind ; the quantum $XY$ AR-N-S-S ; H-S-Stolz ; Klein-Perez and $XXZ$ E-K22 ; E-K-S-2 ; E-K-S-1 spin chains; systems of hardcore particles Beaud-W ; harmonic oscillators in the presence of disorder N-S-S-harmonic ; particle-oscillator interactions S-M-Holst . Unlike in the single-particle Anderson-type models, where the notions of localization aimed at are usually spectral or dynamical localization, the challenges in the context of true many-body quantum systems start at defining the correct objects and notions of localization for each model.

One alternative to explore interactive quantum systems while remaining closer to the single-particle Schrödinger operator setting is to approximate the true many-body Hamiltonian by an effective one, as in the case of mean field theories and the Hartree/Hartree-Fock approximations which are widely studied beyond the setting of disordered systems B-Lieb-S ; Bene-Porta-Schlein ; Canc-Lab-Lew ; CHENN2022109702 ; Hainzl-Lew-Spa ; Lah-1 ; Lieb-Simon .

In the disordered setting, Anderson localization in the Hartree-Fock approximation was first studied in Duc . There, through the multiscale analysis technique, spectral localization was obtained in the presence of a spectral gap at both large disorder and at spectral edges. Recently in M-S localization properties of the disordered Hubbard model at positive temperature within the Hartree-Fock approximation have been established via the Aizenman-Molchanov fractional moment technique. There, exponential dynamical localization ( in fact, decay of eigenfunction correlators) is shown to hold at large disorder in dimension $d\geq 2$ and at any disorder in dimension $d=1$ provided the interaction strength is sufficiently small. No assumption on the existence of a spectral gap is made but, in contrast, the interactions are modelled at positive temperature. The present manuscript is devoted to localization properties of random operators in the form

$$H_{\omega}=-\Delta+\lambda V_{\omega}+gV_{{\mathrm{eff}},\omega}$$

(I.1)

where $\{V_{\omega}(n)\}_{n\in\mathbb{Z}^{d}}$ are independent, identically distributed random variables and $V_{{\mathrm{eff}},\omega}$ is a multiplication operator implicitly defined by

$$V_{{\mathrm{eff}},\omega}(n)=\sum_{m\in\mathbb{Z}^{d}}a(n,m)\langle\delta_{m},F(H_{\omega})\delta_{m}\rangle\,\text{for all}\,n\in\mathbb{Z}^{d}.$$

(I.2)

Here $|a(m,n)|\leq C_{a}e^{-\gamma d(m,n)}$ will be assumed to decay sufficiently fast ( see 1-7 for the precise assumptions) with respect to a metric $d:\mathbb{Z}^{d}\times\mathbb{Z}^{d}\rightarrow\mathbb{R}$, $C_{a}>0$, $\gamma>0$ and $F$ is an analytic function on a strip $\{|\mathrm{Im}z|<\eta\}$ which is bounded. It is worth noting that the above setting allows for the decay of $|a(m,n)|$ to be of polynomial type. The above model is somewhat analogous (in the Hartree-Fock setting) to models in the single-particle setting with fast decaying potentials which still exhibit monotonicity properties (for instance, the ones studied in Kir-Stol-Stolz ). In the particular case where $F(z)=\frac{1}{1+e^{\beta(z-\bar{\mu})}}$ is the Fermi-Dirac function at temperature $\beta^{-1}>0$ and a chemical potential $\bar{\mu}\in\mathbb{R}$ and $a(m,n)=\delta_{mn}$ ( with $\delta_{mn}$ the Kronecker delta) (I.2) simplifies to operators already studied in M-S . There for a fixed $\beta>0$, dynamical localization is shown in any dimension provided $|g|<g_{0}$ and $\lambda>\lambda_{0}$ for certain constants $g_{0}$ and $\lambda_{0}$ which depend on $\beta$ and $d$ but a more concrete estimate for $\lambda_{0}$ was not pursued there. In this note, we generalized the large disorder result of M-S to the operator (I.2), obtain a novel result of localization at weak disorder/extreme energies and, moreover, study the question of stability of the large disorder threshold under ‘weak’ interactions, inspired by the analysis of Schenkl . In particular, it is proven here that there is a large disorder threshold $\lambda_{\mathrm{HF}}$ such that the operators given by (I.2) exhibit dynamical localization provided $\lambda>\lambda_{HF}$ as long as $|g|\norm{F}_{\infty}$ is sufficiently small. Moreover, we show that $\lambda_{HF}\to\lambda_{\mathrm{And}}$ as $|g|\norm{F}_{\infty}\to 0$ where $\lambda_{\mathrm{And}}$ is the solution of the transcendental equation

$$\lambda_{\mathrm{And}}=2\norm{\rho}_{\infty}\mu_{d}e\ln\left(\frac{\lambda_{\mathrm{And}}}{2\norm{\rho}_{\infty}}\right)$$

(I.3)

with $\mu_{d}$ the connectivity constant of $\mathbb{Z}^{d}$. For the uniform distribution in $[-1,1]$, in which case $2\norm{\rho}_{\infty}=1$, this value of $\lambda_{\mathrm{And}}$ was obtained for the Anderson model in Schenkl and coincides with Anderson’s original prediction in Anderson . To the best of our knowledge, in arbitrary dimension $\lambda_{\mathrm{And}}$ in (I.3) is the best rigorous large disorder threshold proved with current methods. It is worth noting that letting $|g|\norm{F}_{\infty}\to 0$ formally in (I.2) we obtain the Anderson model $H_{\mathrm{And}}=-\Delta+V_{\omega}$.

We now comment on other technical merits of the present work, for further technical aspects we refer to section II.1 below. Our first observation is that even for the non-interacting Anderson model $H_{\mathrm{And}}$, the fractional moment method requires the random variables $V_{\omega}$ to have a density $\rho$ which is “sufficiently regular". Thus, it is to be expected that a direct application of this technique to interacting models, which is the approach adopted here and also in M-S , will require further regularity of $\rho$. The previous paper M-S covers a large class of probability distributions with $\mathrm{supp}\rho=\mathbb{R}$ by making use of the symmetry $F(z)=1-F(-z)$ and decay properties of the Fermi-Dirac function when $\mathrm{Re}z\to\infty$ in order to obtain certain improved Combes-Thomas bounds. Such bounds reflect decay of the effective potential at a given site $V_{{\mathrm{eff}},\omega}(n)$ when the local potential $\omega(m)$ is changed at a site $m\neq n$. However, such bounds appear not to be available in the generality studied here. In fact, they seem not to be available even when one restricts (I.2) to the case of nearest neighbor lattice fermions, i.e., when $a(m,n)=1$ if and only if $|m-n|=1$ where $|m|=|m_{1}|+\cdots+|m_{d}|$ and thus $V_{{\mathrm{eff}},\omega}(n)=\sum_{n^{\prime}\sim n}\langle\delta_{n^{\prime}},F(H_{\omega})\delta_{n^{\prime}}\rangle$ with $n^{\prime}\sim n$ indicating that $n^{\prime}$ and $n$ are nearest neighbors on $\mathbb{Z}^{d}$. The key observation surrounding the present paper is that there is a trade off between the regularity/decay properties of $F$ on the real line, the decay properties of the interaction kernel $a(m,n)$ and the density $\rho$. Namely, by reducing the class of probability distributions covered by our main result, we are able to include interactions of a much longer range, including the case where $a(m,n)$ only decays in an algebraic fashion and where $F(z)$ is bounded of a strip but does not necessarily decay as $\mathrm{Re}z\to\infty$. In conclusion, even though the methods employed here to obtain the a-priori bound on fractional moments of the Green’s function follow the general scheme of M-S , in order to prove our stability result, we need to keep an explicit dependence on all parameters involved $\eta,\lambda,g,\norm{F}_{\infty}$ and now have the inclusion the decay rate $\gamma$ of $a(m,n)$ as well. Once an a-priori bound on fractional moments of the Green’s function is obtained, we follow the approach of Schenker in Schenkl in order to get the best large disorder threshold which seem to be available with current methods which turns out to converge to $\lambda_{\mathrm{And}}$ when $|g|\norm{F}_{\infty}\to 0$.

This paper is dedicated to Abel Klein in occasion of his 78th birthday. Klein’s contributions to the field go well beyond the aforementioned works and can hardly be overstated. Certain aspects of the present work were also inspired by Klein’s efforts. For instance, the idea of studying distributions near a suitable chosen density (for which explicit calculations are available) used below in assumption 6 is analogous to Acosta-Klein , where analyticity of the density of states on a strip is shown for distributions sufficiently close to the Cauchy distribution. Moreover, throughout the note, Combes-Thomas type bounds for kernels of analytic functions of $H_{\omega}$ are used extensively. In GK-CT such bounds are obtained in great generality which provides hope for future extensions of the results below.

This note concerns random operators of the form

$$H_{\omega}=A+\lambda V_{\omega}+gV_{{\mathrm{eff}},\omega}$$

(II.1)

acting on $\ell^{2}\left(\mathbb{Z}^{d}\right)$ as follows:

1. 1.

   $(A\psi)(n)=\sum_{|n^{\prime}-n|=1}\psi(n^{\prime})$ for each $\psi\in\ell^{2}\left(\mathbb{Z}^{d}\right)$, i.e. $A:\ell^{2}\left(\mathbb{Z}^{d}\right)\rightarrow\ell^{2}\left(\mathbb{Z}^{d}\right)$ is the adjacency operator of $\mathbb{Z}^{d}$.

2. 2.

   $(V_{\omega}\psi)(n)=\omega(n)\psi(n)$ for each $\psi\in\ell^{2}\left(\mathbb{Z}^{d}\right)$ where $\{\omega(n)\}_{n\in\mathbb{Z}^{d}}$ are independent, identically distributed random variables with a bounded density $\rho$.

3. 3.

   The effective potential $V_{{\mathrm{eff}},\omega}:\ell^{2}\left(\mathbb{Z}^{d}\right)\rightarrow\ell^{2}\left(\mathbb{Z}^{d}\right)$ is a multiplication operator implicitly defined by

   $$V_{{\mathrm{eff}},\omega}(n)=\sum_{m\in\mathbb{Z}^{d}}a(n,m)\langle\delta_{m},F(H_{\omega})\delta_{m}\rangle\,\,\,\text{for all}\,\,\,n\in\mathbb{Z}^{d}.$$

   (II.2)

   We impose the following conditions on $a(n,m)$ and $F$.

4. 4.

   There exists $\eta>\eta_{0}>0$ such that $F$ is an analytic function on the strip

   $$\mathcal{S}_{\eta}=\{|\mathrm{Im}z|<\eta\}$$

   and bounded on its closure $\overline{\mathcal{S}_{\eta}}$. Moreover, we assume that $F(\mathbb{R})\subset\mathbb{R}$.

5. 5.

   The values $a(m,n)$ are real numbers for all $m,n\in\mathbb{Z}^{d}$ and

   $$|a(m,n)|\leq C_{a}e^{-\gamma_{a}d(m,n)}$$

   (II.3)

   for constants $C_{a}>0$ and $\gamma_{a}>0$ and some metric $d:\mathbb{Z}^{d}\times\mathbb{Z}^{d}\rightarrow\mathbb{R}$ for which there exists $\delta\in(0,\gamma_{a}/2)$ such that

   $$\norm{S_{\delta-\gamma_{a}}}_{\infty,\infty}:=\sup_{n\in\mathbb{Z}^{d}}\sum_{m\in\mathbb{Z}^{d}}e^{(\delta-\gamma_{a})d(m,n)}<\infty.$$

   (II.4)

6. 6.

   We also assume that $\mathrm{supp}\rho=\mathbb{R}$ and that for some $c_{1}>0$ and $\varepsilon_{1}>0$

   $$\frac{\rho(v_{1})}{\rho(v_{2})}\geq e^{-c_{1}|v_{1}-v_{2}|},\,\,\,\text{for all}\,\,v_{1},v_{2}\in\mathbb{R}$$

   (II.5)

   and

   $$\sup_{v\in\mathbb{R}}\frac{\rho(v)}{\int^{\infty}_{-\infty}\rho(\alpha)e^{-\varepsilon_{1}\absolutevalue{v-\alpha}}\,d\alpha}<\infty$$

   (II.6)

For the results of localization at weak disorder/extreme energies we will make the following additional requirement.

1. 7.

   We further assume that $\rho(v)=h(v)e^{-c_{\rho}|v|}$ for some $c_{\rho}>0$ where

   $$\frac{h(v_{1})}{h(v_{2})}\geq e^{-\varepsilon_{2}|v_{1}-v_{2}|}\,\,\,\text{for all}\,\,v_{1},v_{2}\in\mathbb{R}$$

   (II.7)

   for some $\varepsilon_{2}\in(0,\frac{1}{2}c_{\rho})$.

Working with finite volume restrictions of both $H_{\omega}$ and also $V_{{\mathrm{eff}},\omega}$ will turn out convenient thus we let $\Lambda_{L}=[-L,L]^{d}\cap\mathbb{Z}^{d}$ and

$$V_{{\mathrm{eff}},\omega,L}(n)=\sum_{m\in\Lambda_{L}}a(n,m)\langle\delta_{m},F(H_{\omega,L})\delta_{m}\rangle\,\,\,\text{for all}\,n\in\Lambda_{L}.$$

(II.8)

where $H_{\omega,L}=\mathds{1}_{L}(A+V_{\omega}+V_{{\mathrm{eff}},\omega,L})\mathds{1}_{L}$ and $\mathds{1}_{L}:\ell^{2}\left(\mathbb{Z}^{d}\right)\rightarrow\ell^{2}\left(\Lambda_{L}\right)$ is the projection onto $\mathrm{span}\{\delta_{l}:\,\,\,l\in\Lambda_{L}\}$. We will often write

$$U(n)=\omega(n)+\frac{g}{\lambda}V_{{\mathrm{eff}},\omega,L},\,\,\,n\in\Lambda^{\prime}\subset\Lambda_{L}$$

(II.9)

to denote the “full potential" at site $n$. It will be shown below in Lemma 8 that under assumptions 1-6 the conditional distribution of $U(n_{0})=v$ at specified values of $\{U(n)\}_{n\in\Lambda^{\prime}\setminus\{n_{0}\}}$ has a density which is bounded with an upper bound independent of the parameters $\omega,\Lambda^{\prime}$ and $L$. This upper bound is denoted herein by $M_{\infty}$ and the conditional density by $\rho^{{\mathrm{eff}}}_{n_{0}}=\rho^{{\mathrm{eff}},\Lambda^{\prime}}_{n_{0},L}$. We also recall the definition of the eigenfunction correlators for an operator $H$:

$$Q_{I}(m,n):=\sup_{|\varphi|\leq 1}\absolutevalue{\langle\delta_{m},\varphi(H)\delta_{n}\rangle}$$

(II.10)

where the supremum is taken over Borel measurable functions $\varphi$ bounded by one and supported on the interval $I$. In case $I=\mathbb{R}$ we simply write $Q(m,n)$. In what follows we denote by $Q^{\Lambda^{\prime}}_{I,L}(m,n)$ the eigenfunction correlators of $H^{\Lambda^{\prime}}_{\omega,L}={\mathds{1}}_{\Lambda^{\prime}}H_{\omega,L}{\mathds{1}}_{\Lambda^{\prime}}$ for $\Lambda^{\prime}\subset\Lambda_{L}$ and by $\mathbb{E}(f)$ the expected value of $f$ with respect to the probability space in question.

Our first result is the following.

Theorem 1 above extends to the present context a result of Schenker Schenkl , who obtained the large disorder threshold $\lambda_{\mathrm{And}}$ which solves

$$\lambda_{\mathrm{And}}=2\norm{\rho}_{\infty}\mu_{d}e\ln\left(\frac{\lambda_{\mathrm{And}}}{2\norm{\rho}_{\infty}}\right).$$

(II.14)

for the Anderson model with a uniformly distributed potential on $[-1,1]$.

We also show that $\lambda_{\mathrm{HF}}$ is close to $\lambda_{\mathrm{And}}$ in a quantified fashion.

Before stating our second theorem we let, for each $n_{0}\in\Lambda^{\prime}$

$$\psi^{n_{0}}_{s}(z)=\int^{\infty}_{-\infty}\frac{|v|^{s}\rho^{{\mathrm{eff}},\Lambda^{\prime}}_{n_{0},L}(v)}{|v-z|^{s}}\,dv,$$

(II.15)

$$\phi^{n_{0}}_{s}(z)=\int^{\infty}_{-\infty}\frac{\rho^{\mathrm{eff},\Lambda^{\prime}}_{n_{0},L}(v)}{\absolutevalue{v-z}^{s}}\,dv$$

(II.16)

and

$$D_{s,1}=\sup_{L\in\mathbb{N}}\sup_{\Lambda^{\prime}\subset\Lambda_{L}}\sup_{z\in\mathbb{C},n_{0}\in\Lambda^{\prime}}\frac{\psi^{n_{0}}_{s}(z)}{\phi^{n_{0}}_{s}(z)}.$$

(II.17)

As we shall see below, under assumptions 1-7 the measure $\rho^{\mathrm{eff},\Lambda^{\prime}}_{n_{0},L}(v)\,dv$ is $1$-moment regular in the sense of (A-W-B, , Definition 8.5) meaning that $D_{s,1}<\infty$ for all $s\in(0,1)$. We also define the Green’s function of $H_{\omega}$ at $z\in\mathbb{C}\setminus\sigma(H_{\omega})$ by

$$G(m,n;z)=\langle\delta_{m},(H_{\omega}-z)^{-1}\delta_{n}\rangle$$

(II.18)

and let, for $\Lambda^{\prime}\subset\Lambda_{L}$, $G_{L}(m,n;z)$ and $G^{\Lambda^{\prime}}_{L}(m,n;z)$ be the Green’s function of $H_{\omega,L}$ and $H^{\Lambda^{\prime}}_{\omega,L}={\mathds{1}}_{\Lambda^{\prime}}H_{\omega,L}{\mathds{1}}_{\Lambda^{\prime}}$, respectively:

$$G_{L}(m,n;z)=\langle\delta_{m},(H_{\omega,L}-z)^{-1}\delta_{n}\rangle\,\,\text{and}\,\,G^{\Lambda^{\prime}}_{L}(m,n;z)=\langle\delta_{m},(H^{\Lambda^{\prime}}_{\omega,L}-z)^{-1}\delta_{n}\rangle.$$

(II.19)

We emphasize that the effective potential is $V_{{\mathrm{eff}},\omega,L}$ for both of the above operators. Finally, we denote by $G_{0}(m,n;z)$ the Green’s function of the “free” operator $A$, namely

$$G_{0}(m,n;z)=\langle\delta_{m},(A-z)^{-1}\delta_{n}\rangle.$$

(II.20)

We are now ready to state our second Theorem which yields localization at weak disorder/extreme energies provided the interaction strength is not too large relative to the remaining parameters.

Let us collect some basic facts which will be repeatedly used in this note. Firstly, if $H_{\omega,L}$ is as above we can write $F(H_{\omega,L})=\frac{1}{2\pi i}\int^{\infty}_{-\infty}\left(\frac{1}{H_{\omega,L}+t-i\eta}-\frac{1}{H_{\omega,L}+t+i\eta}\right)f(t)\,dt$ for $f=F_{+}+F_{-}+D\ast F$, where $F_{\pm}(u)=F(u\pm i\eta\mp i0)$ and $D(u)=\frac{\eta}{\pi\left(\eta^{2}+u^{2}\right)}$ the Poisson kernel, see (A-G, , Appendix D). In particular, the inequality $\|f\|_{\infty}\leq 3\|F\|_{\infty}$ holds. The formula

$$V_{{\mathrm{eff}},\omega,L}(n)=\frac{1}{2\pi i}\int^{\infty}_{-\infty}K_{L}(n,\omega;t)f(t)\,dt$$

(III.1)

with

$$K_{L}(n,\omega;t)=\sum_{m\in\mathbb{Z}^{d}}a(n,m)\left(G_{L}(m,m;t-i\eta)-G_{L}(m,m;t+i\eta)\right)$$

(III.2)

readily follows and is a useful representation for the effective potential. It is shown below that it yields, for each $n,l\in\Lambda_{L}$, self-consistent equations for the derivatives $\frac{\partial V_{{\mathrm{eff}},\omega,L}(n)}{\partial\omega(l)}$ which in turn imply the desired exponential decay in Step 1 of the proof strategies given earlier. We introduce $\nu>0$ such that

$$\sup_{n\in\mathbb{Z}^{d}}\sum_{|n^{\prime}-n|=1}e^{\nu d(n,n^{\prime})}<\eta/2.$$

(III.3)

The decay rate in the Lemma below will be dictated by $\nu$ and $\gamma_{a}$.

Proof.

Denote by $P_{l}:\ell^{2}\left(\mathbb{Z}^{d}\right)\rightarrow\mathrm{Span}\{\delta_{l}\}$ the projection onto $\mathrm{Span}\{\delta_{l}\}$. Using difference quotients, it is immediate to check that

$$\frac{\partial}{\partial\omega(l)}\frac{1}{H_{L}-z}=-\lambda\frac{1}{H_{L}-z}P_{l}\frac{1}{H_{L}-z}-g\frac{1}{H_{L}-z}\frac{\partial V_{{\mathrm{eff}},\omega,L}}{\partial\omega(l)}\frac{1}{H_{L}-z}.$$

(III.7)

Taking matrix elements we obtain from (III.2) that

$$\frac{\partial K_{L}(n,\omega;t)}{\partial\omega(l)}=\sum_{m\in\Lambda_{L}}a(n,m)\left(-\lambda r_{L}(m,l;t)-g\sum_{k\in\Lambda_{L}}r_{L}(m,k;t)\frac{\partial V_{{\mathrm{eff}},\omega,L}(k)}{\partial\omega(l)}\right)$$

(III.8)

with

$$r_{L}(u,v;t):=G_{L}(u,v;t-i\eta)G_{L}(v,u;t-i\eta)-G_{L}(u,v;t+i\eta)G_{L}(v,u;t+i\eta).$$

(III.9)

The above derivatives of the kernel $K_{L}(n,\omega;t)$ are shown to decay exponentially in $d(n,l)$ as follows. We first rewrite $r_{L}(u,v;t)$ as

	$\displaystyle r_{L}(u,v;t)=$	$\displaystyle\left(G_{L}(u,v;t-i\eta)-G_{L}(u,v;t+i\eta)\right)G_{L}(v,u;t-i\eta)$		(III.10)
		$\displaystyle+G_{L}(u,v;t+i\eta)\left(G_{L}(v,u;t-i\eta)-G_{L}(v,u;t+i\eta)\right).$

For the operators studied here the Combes-Thomas bound (A-W-B, , Theorem 10.5) yields

$$|G_{L}(u,v;z)|\leq\frac{2}{\eta}e^{-\nu d(u,v)},\,\,\,\,z\in\mathbb{C}\setminus\mathbb{R}$$

(III.11)

for all $\nu>0$ satisfying (III.3). Moreover, by (A-G, , Appendix D, Lemma 3) we have the following inequality:

$$|G_{L}(u,v;t+i\eta)-G_{L}(u,v;t-i\eta)|\leq 12\eta e^{-\nu d(u,v)}\langle\delta_{u},{\frac{1}{(H_{L}-t)^{2}+\eta^{2}/2}}\delta_{u}\rangle^{1/2}\langle\delta_{v},{\frac{1}{(H_{L}-t)^{2}+\eta^{2}/2}}\delta_{v}\rangle^{1/2}.$$

(III.12)

We remark that the above result, as the usual Combes-Thomas bound, may also be applied to the metric $d$ instead of the usual metric of $\mathbb{Z}^{d}$. One then obtains

$$\absolutevalue{r_{L}(u,v;t)}\leq 48e^{-2\nu d(u,v)}\langle\delta_{u},{\frac{1}{(H_{L}-t)^{2}+\eta^{2}/2}}\delta_{u}\rangle^{1/2}\langle\delta_{v},{\frac{1}{(H_{L}-t)^{2}+\eta^{2}/2}}\delta_{v}\rangle^{1/2}.$$

(III.13)

By the spectral measure representation and the Cauchy-Schwarz inequality, the right-hand side of (III.13) can be controlled via

$$\int^{\infty}_{-\infty}\langle\delta_{u},{\frac{1}{(H_{L}-t)^{2}+\eta^{2}/2}}\delta_{u}\rangle^{1/2}\langle\delta_{v},{\frac{1}{(H_{L}-t)^{2}+\eta^{2}/2}}\delta_{v}\rangle^{1/2}\,dt\leq\frac{\sqrt{2}\pi}{\eta}.$$

(III.14)

Therefore,

$$\frac{1}{2\pi}\int^{\infty}_{-\infty}\absolutevalue{r_{L}(u,v;t)f(t)}\,dt\leq\frac{72\sqrt{2}\norm{F}_{\infty}}{\eta}e^{-2\nu d(u,v)}.$$

(III.15)

Keeping in mind assumption 5 and combining (III.1), (III.8) and (III.15) we reach the inequality

	$\displaystyle\absolutevalue{\frac{\partial V_{{\mathrm{eff}},\omega,L}(n)}{\partial\omega(l)}}$	$\displaystyle\leq\frac{C_{a}72\sqrt{2}\norm{F}_{\infty}}{\eta}\sum_{m\in\Lambda_{L}}\lambda e^{-\gamma_{a}d(n,m)-2\nu d(m,l)}$		(III.16)
		$\displaystyle+\frac{C_{a}72\sqrt{2}\norm{F}_{\infty}}{\eta}|g|\sum_{k\in\Lambda_{L}}e^{-\gamma_{a}d(n,m)-2\nu d(m,k)}\absolutevalue{\frac{\partial V_{{\mathrm{eff}},\omega,L}(k)}{\partial\omega(l)}}.$

We now apply Lemma 4 with fixed $l\in\Lambda_{L}$ and the choices $\varphi(n)=\absolutevalue{\frac{\partial V_{{\mathrm{eff}},\omega,L}(n)}{\partial\omega(l)}}$,

$$\psi(n)=\frac{C_{a}72\sqrt{2}\norm{F}_{\infty}}{\eta}\lambda\sum_{m\in\Lambda_{L}}e^{-\gamma_{a}d(n,m)-2\nu d(m,l)}$$

(III.17)

$$K(n,u)=\frac{C_{a}72\sqrt{2}\norm{F}_{\infty}}{\eta}|g|\sum_{m\in\Lambda_{L}}e^{-\gamma_{a}d(n,m)-2\nu d(m,u)}$$

(III.18)

in the regime where $\norm{K}_{\infty,\infty}<1$, i.e. when

$$\frac{C_{a}72\sqrt{2}\norm{F}_{\infty}}{\eta}|g|S_{-\gamma}S_{-2\nu}<1.$$

(III.19)

In this context, introducing the weight function $W(n)=e^{\delta d(n,l)}$ with $\delta<\min\{\gamma_{a},2\nu\}$ we reach

$$b_{1}:=\sum_{n\in\mathbb{Z}^{d}}W(n)\psi(n)\leq\frac{C_{a}72\sqrt{2}\norm{F}_{\infty}}{\eta}\lambda S_{\delta-\gamma}S_{\delta-2\nu}$$

(III.20)

and

$$b_{2}=\sup_{n^{\prime}\in\mathbb{Z}^{d}}\sum_{n\in\mathbb{Z}^{d}}\frac{W(n)}{W(n^{\prime})}K(n,n^{\prime})\leq\frac{C_{a}72\sqrt{2}\norm{F}_{\infty}}{\eta}|g|S_{\delta-\gamma}S_{\delta-2\nu}.$$

(III.21)

In particular, under the more restrictive assumption

$$\frac{C_{a}72\sqrt{2}\norm{F}_{\infty}}{\eta}|g|S_{\delta-\gamma}S_{\delta-2\nu}<\frac{1}{2}$$

(III.22)

we find that $\frac{1}{1-b_{2}}\leq 2$ and thus $\varphi\leq 2b_{1}$, finishing the proof.