Supersymmetric Polar Coordinates with applications to the Lloyd model

TL;DR

This paper introduces supersymmetric polar coordinates to analyze spectral properties of random Schrödinger operators, providing a dual representation applicable to general distributions and applying it to the Lloyd model.

Contribution

It develops a new supersymmetric polar coordinate method for dual representation of Green's functions, extending analysis to general distributions and specific models like the Lloyd model.

Findings

Recovered the exact density of states for non-negative correlation in the Lloyd model.

Showed the density of states is well approximated for small negative interactions.

Results are uniform on the lattice  in volume.

Abstract

Spectral properties of random Schr\"odinger operators are encoded in the average of products of Greens functions. For probability distributions with enough finite moments, the supersymmetric approach offers a useful dual representation. Here we use supersymmetric polar coordinates to derive a dual representation that holds for general distributions. We apply this result to study the density of states of the linearly correlated Lloyd model. In the case of non-negative correlation, we recover the well-known exact formula. In the case of linear small negative interaction localized around one point, we show that the density of states is well approximated by the exact formula. Our results hold on the lattice $\mathbb{Z}^d$ uniformly in the volume.

Equations

$$H:=-{\Delta }_{{\mathbb{Z}}^d}+\lambda V,$$

$$H_{\Lambda }=-\Delta +\lambda V,$$

$$(\Delta \psi )(j)=\sum _{k\in \Lambda :|j-k|=1}(\psi (k)-\psi (j))+\text{mbox}eventualboundaryterms.$$

$$\mathbb{E}[G_{\Lambda }(z_1{)}_{j_1,k_1}\dots G_{\Lambda }(z_n{)}_{j_n,k_n}],$$

$${\int }_{\mathbb{R}}\frac{1}{z-E}{\bar{\rho }}_{\lambda }(E)\mathrm{d}E=\frac{1}{\pi |\Lambda |}\mathbb{E}[\mathrm{Tr}{G}_{\Lambda }(z)],$$

$${\bar{\rho }}_{\Lambda }(E):=-\frac{1}{\pi |\Lambda |}\underset{\epsilon \to 0^{+}}{\lim }\mathbb{E}[\mathrm{Im}\mathrm{Tr}{G}_{\Lambda }(E+i\epsilon )],$$

$${\mathcal{G}}_{\epsilon }(E,\tilde{E})=\mathbb{E}\left[\frac{\det ((E+i\epsilon )\text{mathds}{1}_{\Lambda }-H_{\Lambda })}{\det ((\tilde{E}+i\epsilon )\text{mathds}{1}_{\Lambda }-H_{\Lambda })}\right].$$

$$\mathrm{Tr}{G}_{\Lambda }(E+i\epsilon )=-{\partial }_{\tilde{E}}{\mathcal{G}}_{\epsilon }(E,\tilde{E}){|}_{\tilde{E}=E}={\partial }_E{\mathcal{G}}_{\epsilon }(E,\tilde{E}){|}_{\tilde{E}=E}.$$

$$\mathbb{E}[|G_{\Lambda }(E+i\epsilon {)}_{jk}{|}^2],\text{mbox}or{\rho }_2(E,E+\omega ):=\mathbb{E}[{\rho }_{\Lambda }(E){\rho }_{\Lambda }(E+\omega )]$$

$${\Phi }^{\ast }A\Phi :={\Phi }^{\ast }\mathrm{diag}(A,A)\Phi =\sum _{j,k\in \Lambda }{A}_{jk}{\Phi }_j^{\ast }{\Phi }_k,$$

$${\hat{\mu }}_j(a)=\mathbb{E}[e^{ia{V}_j}]:={\hat{\mu }}_j(b_a)+{\hat{\mu }}_j^{\prime }(b_a)n_a+\frac{1}{2}{\hat{\mu }}_j^{\prime \prime }(b_a)n_a^2.$$

$${\mathcal{G}}_{\epsilon }(E,\tilde{E})=\int [\mathrm{d}{\Phi }^{\ast }\mathrm{d}\Phi ]{\mathrm{e}}^{i{\Phi }^{\ast }(\mathbf{E}+i\epsilon +\Delta )\Phi }\prod _{j\in \Lambda }{\hat{\mu }}_j(\lambda {\Phi }_j^{\ast }{\Phi }_j),$$

$$\begin{array}{rl}\mathbb{E}[|G_{\Lambda }(E+iϵ{)}_{jk}{|}^2]=& \int [\mathrm{d}{\Phi }^{\ast }\mathrm{d}\Phi ][\mathrm{d}{\tilde{\Phi }}^{\ast }\mathrm{d}\tilde{\Phi }]{\mathrm{e}}^{i{\Phi }^{\ast }(\mathbf{E}+i\epsilon +\Delta )\Phi -i{\tilde{\Phi }}^{\ast }(\mathbf{E}-i\epsilon +\Delta )\tilde{\Phi }}\\ & \times z_j{\bar{z}}_k{\tilde{z}}_k{\overline{\tilde{z}}}_j\prod _{j\in \Lambda }{\hat{\mu }}_j(\lambda ({\Phi }_j^{\ast }{\Phi }_j-{{\tilde{\Phi }}_j}^{\ast }{\tilde{\Phi }}_j)).\end{array}$$

$$\mathcal{G}(E,\tilde{E})=\mathbb{E}\left[\int [\mathrm{d}{\Phi }^{\ast }\mathrm{d}\Phi ]{\mathrm{e}}^{i{\Phi }^{\ast }(\mathbf{E}+i\epsilon +\Delta -\lambda V)\Phi }\right]$$

$$\mathbb{E}[|G_{\Lambda }(E+i\epsilon {)}_{jk}{|}^2]=$$

$$\mathbb{E}\left[\int [\mathrm{d}{\Phi }^{\ast }\mathrm{d}\Phi ][\mathrm{d}{\tilde{\Phi }}^{\ast }\mathrm{d}\tilde{\Phi }]{\mathrm{e}}^{i{\Phi }^{\ast }(\mathbf{E}+i\epsilon +\Delta -\lambda V)\Phi -i{\tilde{\Phi }}^{\ast }(\mathbf{E}-i\epsilon +\Delta -\lambda V)\tilde{\Phi }}{z}_j{\bar{z}}_k{\tilde{z}}_k{\overline{\tilde{z}}}_j\right]$$

$$\Psi (r,\theta ,\bar{\rho },\rho )=\left(\begin{array}{c}z(r,\theta ,\bar{\rho },\rho )\\ \bar{z}(r,\theta ,\bar{\rho },\rho )\\ \chi (r,\theta ,\bar{\rho },\rho )\\ \bar{\chi }(r,\theta ,\bar{\rho },\rho )\end{array}\right):=\left(\begin{array}{c}{\mathrm{e}}^{i\theta }(r-\frac{1}{2}\bar{\rho }\rho )\\ {\mathrm{e}}^{-i\theta }(r-\frac{1}{2}\bar{\rho }\rho )\\ \sqrt{r}\rho \\ \sqrt{r}\bar{\rho }\end{array}\right)$$

$$I(f)={\int }_{{\mathbb{C}}^N}[\mathrm{d}{\Phi }^{\ast }\mathrm{d}\Phi ]f({\Phi }^{\ast },\Phi )=\sum _{\alpha \in \{0,1{\}}^N}{I}_{\alpha }(f)$$

$$I_{\alpha }(f)={\pi }^{-|1-\alpha |}{\int }_{({\mathbb{R}}^{+}\times (0,2\pi ){)}^{1-\alpha }}(\mathrm{d}r\mathrm{d}\theta \mathrm{d}\bar{\rho }\mathrm{d}\rho {)}^{1-\alpha }f\circ {\Psi }_{\alpha }(r,\theta ,\bar{\rho },\rho ),$$

$$\{\begin{array}{ll}{z}_j(r_j,{\theta }_j,{\bar{\rho }}_j,{\rho }_j)& ={\delta }_{{\alpha }_{j}0}{\mathrm{e}}^{i{\theta }_j}(r_j-\frac{1}{2}{\bar{\rho }}_j{\rho }_j),\\ {\bar{z}}_j(r_j,{\theta }_j,{\bar{\rho }}_j,{\rho }_j)& ={\delta }_{{\alpha }_{j}0}{\mathrm{e}}^{-i{\theta }_j}(r_j-\frac{1}{2}{\bar{\rho }}_j{\rho }_j),\\ {\chi }_j(r_j,{\theta }_j,{\bar{\rho }}_j,{\rho }_j)& ={\delta }_{{\alpha }_{j}0}\sqrt{r_j}{\rho }_j,\\ {\bar{\chi }}_j(r_j,{\theta }_j,{\bar{\rho }}_j,{\rho }_j)& ={\delta }_{{\alpha }_{j}0}\sqrt{r_j}{\bar{\rho }}_j.\end{array}$$

$$\phi (x)=\{\begin{array}{ll}{\mathrm{e}}^{-(1-2|x{|}^2{)}^{-1}}& \text{if }|x|<\frac{1}{\sqrt{2}}\\ 0& \text{otherwise}.\end{array}$$

$$I(f)$$

$$=\frac{1}{2\pi }{\int }_0^{\frac{1}{\sqrt{2}}}\mathrm{d}r{\int }_0^{2\pi }\mathrm{d}\theta 4r{\mathrm{e}}^{-(1-2r^2{)}^{-1}}(1-2r^2{)}^{-2}=e^{-1},$$

$$I(f)={\pi }^{-1}{\int }_{{\mathbb{R}}^{+}\times (0,2\pi )}\mathrm{d}r\mathrm{d}\theta \mathrm{d}\bar{\rho }\mathrm{d}\rho f\circ \Psi (r,\theta ,\bar{\rho },\rho )+f\circ \Psi (0)=e^{-1},$$

$${\mathcal{G}}_{\epsilon }(E,\tilde{E})=\sum _{\alpha \in \{0,1{\}}^{\Lambda }}{\int }_{({\mathbb{R}}^{+}\times (0,2\pi ){)}^{1-\alpha }}(\frac{\mathrm{d}r\mathrm{d}\theta \mathrm{d}\bar{\rho }\mathrm{d}\rho }{\pi }{)}^{1-\alpha }\hat{\mu }(\{\lambda r_j^2{\}}_{j\in \Lambda }){|}_{r^{\alpha }=0}g\circ {\Psi }_{\alpha }(r,\theta ,\bar{\rho },\rho )$$

$$\mathbb{E}[|G_{\Lambda }(E+i\epsilon {)}_{jk}{|}^2]$$

$$=\sum _{\begin{array}{c}\alpha \in \{0,1{\}}^{\Lambda }\\ \tilde{\alpha }\in \{0,1{\}}^{\Lambda }\end{array}}{\pi }^{-|1-\alpha |-|1-\tilde{\alpha }|}{\int }_{\begin{array}{c}({\mathbb{R}}^{+}\times (0,2\pi ){)}^{1-\alpha }\\ \times ({\mathbb{R}}^{+}\times (0,2\pi ){)}^{1-\tilde{\alpha }}\end{array}}(\mathrm{d}r\mathrm{d}\theta \mathrm{d}\bar{\rho }\mathrm{d}\rho {)}^{1-\alpha }(\mathrm{d}\tilde{r}\mathrm{d}\tilde{\theta }\mathrm{d}\overline{\stackrel{~}{\rho }}\mathrm{d}\tilde{\rho }{)}^{1-\tilde{\alpha }}$$

$$\hat{\mu }(\{\lambda (r_j^2-{\tilde{r}}_j^2){\}}_{j\in \Lambda }{)}_{r^{\alpha }=0={\tilde{r}}^{\tilde{\alpha }}}{g}^{+}\circ {\Psi }_{\alpha }(r,\theta ,\bar{\rho },\rho )g^{-}\circ {\Psi }_{\tilde{\alpha }}(r,\theta ,\bar{\rho },\rho ),$$

$${\mathcal{G}}_{\epsilon }(E,\tilde{E})=\int [\mathrm{d}{\Phi }^{\ast }\mathrm{d}\Phi ]g({\Phi }^{\ast },\Phi ){\mathrm{e}}^{-{\sum }_k\lambda {\sum }_j{T}_{jk}{\Phi }_j^{\ast }{\Phi }_j}.$$

$$\begin{array}{rl}{\mathcal{G}}_{\epsilon }(E,\tilde{E})=& \sum _{\beta \in \{++,+-,-+\}}{\int }_{{\mathcal{I}}_{\beta }}[\mathrm{d}{\Phi }^{\ast }\mathrm{d}\Phi ]h({\Phi }^{\ast },\Phi ){\mathrm{e}}^{-\lambda {\sum }_{j=1}^2{T}_j^{\beta }{\Phi }_j^{\ast }{\Phi }_j}+R(h)\end{array}$$

$$\begin{array}{rl}{\mathcal{I}}_{++}& =\{z\in {\mathbb{C}}^N:\delta |z_2|<|z_1|<|z_2|/\delta \},\\ {\mathcal{I}}_{+-}& =\{z\in {\mathbb{C}}^N:|z_1|>|z_2|/\delta \},\\ {\mathcal{I}}_{-+}& =\{z\in {\mathbb{C}}^N:|z_1|<\delta |z_2|\}.\end{array}$$

$$\begin{array}{rl}R(h)=& -\frac{1}{{\pi }^2}{\int }_{{\mathbb{R}}^{+}\times (0,2\pi {)}^2}\mathrm{d}{r}_2\mathrm{d}{\theta }_1\mathrm{d}{\theta }_2[\mathrm{d}{\hat{\Phi }}^{\ast }\mathrm{d}\hat{\Phi }]h\circ {\Psi }_{12}(r_2,{\theta }_1,{\theta }_2,{\hat{\Phi }}^{\ast },\hat{\Phi })\\ & \times \lambda r_2{\delta }^2\left[{\mathrm{e}}^{-\lambda (1-{\delta }^4)r_2^2}+{\delta }^2{\mathrm{e}}^{-\lambda (1-{\delta }^4){\delta }^2{r}_2^2}\right],\end{array}$$

$${\Psi }_{12}(r_2,{\theta }_1,{\theta }_2)=({\Phi }_1,{\Phi }_2)=\left(\begin{array}{cc}{\mathrm{e}}^{i{\theta }_1}\delta r_2& {\mathrm{e}}^{i{\theta }_2}{r}_2\\ 0& 0\end{array}\right)$$

Full text

1 Introduction

A major open problem in mathematical physics is the existence of an Anderson transition in dimension three and higher for random Schrödinger operators. These operators model transport in disordered media, a classical example being electrical conductivity in metals with impurities. In this paper, we consider the quantum mechanical problem of an electron moving on a lattice $\mathbb{Z}^{d}$ and interacting with a random potential. The corresponding mathematical model is the so-called discrete Random Schrödinger operator, or Anderson’s tight binding model [And58], acting on the Hilbert space $l^{2}(\mathbb{Z}^{d})$ and defined by

[TABLE]

where $\Delta_{\mathbb{Z}^{d}}$ is the lattice Laplacian $(\Delta\psi)(j)=\sum_{k:|j-k|=1}(\psi(j)-\psi(k))$, and $V$ is a multiplication operator $(V\psi)(j)=V_{j}\psi(j)$. Here, $\{V_{j}\}_{j\in\mathbb{Z}^{d}}$ is a collection of random variables (independent or correlated) and $\lambda>0$ is a parameter expressing the strength of disorder. Physical information are encoded in the spectral properties of $H$. For a large class of random potentials $V$ localization of the eigenfunctions has been proved in $d=1$ for arbitrary disorder and in $d\geq 2$ for large disorder or at the band edge. A localization - delocalization transition has been proved on tree graphs, and is conjectured to hold on $\mathbb{Z}^{d},$ for $d\geq 3.$ A detailed up-to-date review on the model, known results and tools can be found in the book by Aizenman and Warzel [AW15].

Finite volume criteria allow to reconstruct properties of $H$ from the Green’s function (or resolvent) of a finite volume approximation $H_{\Lambda},$ by taking the thermodynamic limit $\Lambda\uparrow\mathbb{Z}^{d}.$ More precisely, let $\Lambda\subset\mathbb{Z}^{d}$ be a finite cube centered around the origin with volume $|\Lambda|=N$. We define the Random Schrödinger operator $H_{\Lambda}\in l^{2}(\Lambda)$ on $\Lambda$ as

[TABLE]

where $\Delta=\Delta_{\Lambda}$ is the discrete Laplacian on $\Lambda$

[TABLE]

The relevant quantities are expressions of the form

[TABLE]

where $G_{\Lambda}(z)\coloneqq(z\mathds{1}_{\Lambda}-H_{\Lambda})^{-1},$ $z\in\mathbb{C}\setminus\sigma(H),$ and $\mathbb{E}$ denotes the average with respect to the random vector $V.$

In particular the (averaged) density of states $\bar{\rho}_{\lambda}(E)$ satisfies the relation,

[TABLE]

hence (see for example [AW15, Section 4 and Appendix B])

[TABLE]

where $E\in\mathbb{R}.$ Regularity properties of $\bar{\rho}_{\Lambda}(E)$ and its derivatives can be inferred from the generating function

[TABLE]

For example

[TABLE]

Information on the nature of the spectrum can be deduced from the thermodynamic limit of

[TABLE]

where the spectral parameter $\varepsilon$ and the energy difference $\omega$ must be taken of order $|\Lambda|^{-1}.$

A possible tool to analyse these objects is the so-called supersymmetric (SUSY) approach. It allows to rewrite averages of the form (1.2) as an integral involving only the Fourier transform of the probability distribution, at the cost of introducing Grassmann variables in the intermediate steps. A short introduction on Grassmann variables and their application in our context is given in Appendix A. For more details see for example the following monographs: [Var04, Ber87, Weg16, DeW92]. This formalism proved to be especially useful in the case of random operators arising from quantum diffusion problems [Efe99]. The supersymmetric approach was applied with success to study Anderson localization as well as phase transitions on tree-graphs [Wan01, Bov90, CK86, KMP86]. All these applications are based on variations of the following key fact.

Theorem 1.1**.**

Let $H_{\Lambda}$ be as in Eq. (1.1) and assume the $V_{j}$ are independent random variables with probability measure $\mu_{j}$ such that $\int v_{j}^{2}d\mu_{j}(v_{j})<\infty$ $\forall j$, i.e., its Fourier transform $\hat{\mu}_{j}(t)\coloneqq\int e^{-itv_{j}}d\mu_{j}(v_{j})$ is twice differentiable with bounded first and second derivatives.

Let $\mathcal{A}=\mathcal{A}[\{\chi_{j},\bar{\chi}_{j}\}_{j\in\Lambda}\}]$ be a Grassmann algebra, $z\in\mathbb{C}^{\Lambda}$ a family of complex variables and set $\Phi_{j}\coloneqq(z_{j},\chi_{j})^{t},$ $\Phi^{*}_{j}\coloneqq(\bar{z}_{j},\bar{\chi}_{j})$ such that $\Phi^{*}_{j}\Phi_{k}=\bar{z}_{j}z_{k}+\bar{\chi}_{j}\chi_{k}$ is an even element in $\mathcal{A}$ for all $j,k\in\Lambda$. For any matrix $A\in\mathbb{C}^{\Lambda\times\Lambda}$, we define

[TABLE]

where ${\rm diag\,}(A,A)$ is a $2|\Lambda|\times 2|\Lambda|$ block diagonal matrix. In particular $\Phi^{*}\Phi=\sum_{j\in\Lambda}\Phi^{*}_{j}\Phi_{j}.$ Finally, for any even element $a=b_{a}+n_{a}$ in $\mathcal{A}^{0}$ with $n_{a}^{3}=0$ we define (cf. Eq. (A.2))

[TABLE]

Then the generating function (1.3) can be written as

[TABLE]

where we defined $[\,\mathrm{d}\Phi^{*}\,\mathrm{d}\Phi]\coloneqq\prod_{j\in\Lambda}(2\pi)^{-1}\,\mathrm{d}\bar{z}_{j}\,\mathrm{d}z_{j}\,\mathrm{d}\bar{\chi}_{j}\,\mathrm{d}\chi_{j}$, $\Phi^{*}\varepsilon\Phi=\varepsilon\Phi^{*}\Phi$ and $\mathbf{E}={\rm diag\,}(\tilde{E}\mathds{1}_{|\Lambda|},E\mathds{1}_{|\Lambda|})$ is a diagonal matrix. Moreover

[TABLE]

A similar representation holds for the two-point function $\rho_{2}(E,\tilde{E})$.

Remark*.*

In the formulas above both $\hat{\mu}_{j}(\lambda(\Phi_{j}^{*}\Phi_{j}))$ and $\hat{\mu}_{j}(\lambda(\Phi_{j}^{*}\Phi_{j}-{\tilde{\Phi}_{j}}^{*}\tilde{\Phi}_{j}))$ are well defined. Indeed, the even elements $a_{1}\coloneqq\Phi_{j}^{*}\Phi_{j}$ and $a_{2}\coloneqq\Phi_{j}^{*}\Phi_{j}-{\tilde{\Phi}_{j}}^{*}\tilde{\Phi}_{j},$ have nilpotent part $n_{a_{1}}=\bar{\chi}_{j}\chi_{j}$ and $n_{a_{2}}=\bar{\chi}_{j}\chi_{j}-\bar{\tilde{\chi}}_{j}\tilde{\chi}_{j}$, respectively. The result then follows from $n_{a_{1}}^{2}=0=n_{a_{2}}^{3},$ together with Eq. (1.5).

Note that we have taken independent variables above only to simplify notations. In the general case, the product of one-dimensional Fourier transforms is replaced by a joint Fourier transform. The generalized formula will hold as long as the Fourier transform admits enough derivatives.

Proof.

We write $\mathcal{G}_{\varepsilon}(E,\tilde{E})$ and $\mathbb{E}[|G_{\Lambda}(E+i\varepsilon)_{jk}|^{2}]$ as a supersymmetric integral (cf. Theorem A.1)

[TABLE]

This step holds for any choice of $V\in\mathbb{R}^{\Lambda}.$ Note that we need two copies of SUSY variables to represent $\mathbb{E}[|G_{\Lambda}(E+i\varepsilon)_{jk}|^{2}]$. When $\,\mathrm{d}\mu_{j}$ admits two finite moments, we can move the average inside. The result follows. ∎

The aim of this paper is to extend this representation to probability distributions with less regularity. To this purpose we introduce a supersymmetric version of polar coordinates which allows to reexpress $e^{i\lambda V_{j}\Phi^{*}_{j}\Phi_{j}}$ as $e^{i\lambda V_{j}x_{i}},$ where $x_{j}\in\mathbb{R}$ is a real variable. As a result, the formula can be extended to any probability distribution on $N=|\Lambda|$ real variables. In contrast to the ordinary ones, supersymmetric polar coordinates introduce correction terms due to the boundary of the integration domain. The simple formula above will then be replaced by a sum of integrals.

As a concrete example, we consider the so-called Lloyd model, with $V$ defined as $V_{j}\coloneqq\sum_{k\in\Lambda}T_{jk}W_{k}$, where $\{W_{k}\}_{k\in\Lambda}$ is a family of i.i.d. random variables with Cauchy distribution $\,\mathrm{d}\mu(x)=\pi^{-1}(1+x^{2})^{-1}\,\mathrm{d}x.$ The standard (uncorrelated) Lloyd model corresponds to $T_{jk}=\delta_{jk}.$ In this case the variables $\{V_{j}\}_{j\in\Lambda}$ are independent and Cauchy distributed. Note that $\,\mathrm{d}\mu(x)$ has no finite moments. For this model, the averaged Green’s function (and hence the density of states) can be computed exactly whenever $T_{jk}\geq 0$ $\forall j,k$ (non-negative correlation) [Llo69, Sim83].

Using supersymmetric polar coordinates, we show here that for the non-negative linearly correlated Lloyd model Eq. (1.6) and (1.7) remain valid, with an appropriate redefinition of $\hat{\mu}(b_{a}+n_{a}).$ In this case, one can easily recover the exact formula for the averaged Green’s function. The formula remains valid also in the case of linear negative correlation, at the price of adding additional correction terms, due to boundary effects.

We expect the supersymmetric representation will help to study problems not yet accessible via other tools, such as negative correlations or the two point function at weak disorder. As a first test, we considered a simplified model with small negative correlations localized on one site. For this toymodel we used the supersymmetric representation to prove that the density of states remains in the vicinity of the exact formula. Our result holds in any dimension and arbitrary volume.

Overview of this article.

In Section 2 we state the main results of the paper, and give some ideas about the proofs. More precisely, Section 2.1 introduces supersymmetric polar coordinates (Theorem 2.1), with a general integrated function $f,$ not necessarily compactly supported. Applications to $\mathcal{G}_{\varepsilon}(E,\tilde{E})$ and $\mathbb{E}[|G_{\Lambda}(E+i\varepsilon)_{jk}|^{2}]$ are given in Theorem 2.2. The detailed proofs of both theorems can be found in Section 3. In Subsection 2.2 we consider the Lloyd model and give an application of the formula for a simple toymodel. The corresponding proofs are in Section 4.

2 Main results

2.1 Supersymmetric polar coordinates

For an introduction to the supersymmetric formalism see Appendix A.

Consider first $\mathcal{A}[\bar{\chi},\chi]$ a Grassmann algebra with two generators. The idea of supersymmetric polar coordinates is to transform between generators $(\bar{z},z,\bar{\chi},\chi)$ of $\mathcal{A}_{2,2}(\mathbb{C})$ and $(r,\theta,\bar{\rho},\rho)$ of $\mathcal{A}_{2,2}(\mathbb{R}^{+}\times(0,2\pi))$ 333cf. Definition A.2. Note that $\bar{\rho},\rho\in\mathcal{A}^{1}[\bar{\chi},\chi]$. such that $\bar{z}z+\bar{\chi}\chi=r^{2}$. A reasonable change is

[TABLE]

Indeed, we have $\bar{z}z+\bar{\chi}\chi=(r-\frac{1}{2}\bar{\rho}\rho)^{2}+r\bar{\rho}\rho=r^{2}$.

Note that [math] is a boundary point for polar coordinates since it maps $\mathbb{R}^{+}\times(0,2\pi)$ to $\mathbb{C}\backslash\{0\}$. For functions with compact support in $U=\mathbb{C}\backslash\{0\}$ a SUSY version of the standard coordinate change formula applies, where the Jacobian is replaced by a Berezinian, c.f. Theorem A.3. On the contrary, functions with $f(0)\neq 0$ have no compact support in the domain $U=\mathbb{C}\backslash\{0\}$ and we collect additional boundary terms as the following theorem shows.

Theorem 2.1** (Supersymmetric polar coordinates).**

Let $N\in\mathbb{N}$, $\mathcal{A}_{2N}$ the complex Grassmann algebra generated by $\{\bar{\chi}_{j},\chi_{j}\}_{j=1}^{N}$ and $\{\Phi_{j}^{*},\Phi_{j}\}_{j=1}^{N}$ a set of supervectors defined as in Theorem 1.1. Let $f\in\mathcal{A}_{2N,2N}(\mathbb{C}^{N})$ be integrable, i.e., all $f_{I}:\mathbb{C}^{N}\to\mathbb{C}$ are integrable. Then

[TABLE]

with multiindex $\alpha$ and

[TABLE]

where $(\,\mathrm{d}r)^{1-\alpha}=\prod_{j:\alpha_{j}=0}\,\mathrm{d}r_{j}$ and $\Psi_{\alpha}$ is given by $\Psi_{\alpha}:(r,\theta,\bar{\rho},\rho)\mapsto(z,\bar{z},\chi,\bar{\chi})$ with

[TABLE]

Proof.

See Section 3. ∎

Remark*.*

For $f$ compactly supported on $\mathbb{C}\backslash\{0\}$ (this means in particular $f(0)=0$), we recover the result of Theorem A.3. Namely for $\alpha=0$, we obtain the right-hand side of Theorem A.3 while all contributions from $\alpha\neq 0$ vanish.

Example*.*

To illustrate the idea behind the above result, consider the following simple example. Let $\varphi$ be the smooth compactly supported function $\varphi:\mathbb{R}\to\mathbb{R}$, given by

[TABLE]

Note that $\varphi(0)=\,\mathrm{e}^{-1}\neq 0,$ hence $f(\bar{z},z,\bar{\chi},\chi)=\varphi(\bar{z}z+\bar{\chi}\chi)$ is a smooth function without compact support in $\mathbb{C}\backslash\{0\}$. By a straightforward computation, we have

[TABLE]

where we expand the expression in the Grassmann variables and change to ordinary polar coordinates after integrating over the Grassmann variables. Applying formulas (2.2) and (2.3), we obtain directly

[TABLE]

where the first integral vanishes, since $f\circ\Psi$ is independent of $\bar{\rho}$ and $\rho$.

Now consider the generating function (1.3). In the case of an integrable density without other regularity conditions, we obtain the following result.

Theorem 2.2**.**

Let $\Lambda\subset\mathbb{Z}^{d}$ be a finite volume and $H_{\Lambda}=-\Delta+\lambda V$ be the Schrödinger operator introduced in Eq. (1.1), where $\{V_{j}\}_{j\in\Lambda}$ is a family of real random variables with integrable joint density $\mu$. Then the generating function (1.3) can be written as

[TABLE]

where $g(\Phi^{*},\Phi)=\,\mathrm{e}^{i\Phi^{*}(\mathbf{E}+i\varepsilon+\Delta)\Phi}$, $\mathbf{E}={\rm diag\,}(\tilde{E}\mathds{1}_{|\Lambda|},E\mathds{1}_{|\Lambda|})$ and $\hat{\mu}(\{\lambda r_{j}^{2}\}_{j\in\Lambda})$ is the $|\Lambda|$-dimensional, joint Fourier transform of $\mu$. Similarly

[TABLE]

where $g^{+}(\Phi^{*},\Phi)=\bar{z}_{k}z_{j}\,\mathrm{e}^{i\Phi^{*}(\mathbf{E}+i\varepsilon+\Delta)\Phi}$ and $g^{-}(\tilde{\Phi}^{*},\tilde{\Phi})=\overline{\tilde{z}_{j}}\tilde{z}_{k}\,\mathrm{e}^{-i{\tilde{\Phi}}^{*}(\mathbf{E}-i\varepsilon+\Delta)\tilde{\Phi}}$.

Idea of the proof.

Again we write $\mathcal{G}_{\varepsilon}(E,\tilde{E})$ and $|G_{\Lambda}(E+i\varepsilon)_{jk}|^{2}$ as a supersymmetric integral (Theorem A.1). Note that we need two copies of SUSY variables to represent $|G_{\Lambda}(E+i\varepsilon)_{jk}|^{2}$. Taking the average inside at this point would cause problems. Hence we apply first our polar-coordinate formula Theorem 2.1. Since $r$ is now real, the expression $\mathbb{E}[\,\mathrm{e}^{i\lambda\sum_{j}V_{j}r_{j}^{2}}]$ is the standard Fourier transform $\hat{\mu}(\{\lambda r_{j}^{2}\}_{j\in\Lambda})$. Details can be found in Section 3. ∎

2.2 Applications to the Lloyd model

As a concrete example, we consider the Lloyd model with linear correlated random potentials, i.e. $V_{j}=\sum_{k}T_{jk}W_{k}$, where $W_{k}\sim\text{Cauchy}(0,1)$ are i.i.d. random variables, $T_{jk}=T_{kj}\in\mathbb{R}$ and $\sum_{j}T_{jk}>0$.

We discuss three cases:

the classical Lloyd model, where $T_{jk}=\delta_{jk}$, hence $V_{j}\sim\text{Cauchy}(0,1)$ are i.i.d. 2. 2.

the (positive) correlated Lloyd model, where $T_{jk}\geq 0$ with $\sum_{j}T_{jk}>0$. 3. 3.

a toymodel with single negative correlation, i.e. $T_{jj}=1$ and $T_{21}=T_{12}=-\delta^{2}$ with $0<\delta<1$ and $T_{jk}=0$ otherwise. The indices $1$ and $2$ denote two fixed, nearest neighbour points $i_{1},i_{2}\in\Lambda$ with $|i_{1}-i_{2}|=1.$

Proposition 2.3**.**

When $T_{jk}\geq 0$ for all $j,k$ (Case 1. and 2. above) we have

[TABLE]

where $g(\Phi^{*},\Phi)\coloneqq\,\mathrm{e}^{i\Phi^{*}(\mathbf{E}-i\varepsilon+\Delta)\Phi}$. For the toymodel (Case 3. above) a similar formula holds with additional correction terms. Precisely

[TABLE]

where $h(\Phi^{*},\Phi)=g(\Phi^{*},\Phi)\,\mathrm{e}^{-\lambda\sum_{j\neq 1,2}\Phi^{*}_{j}\Phi_{j}}$, we defined $T^{++}=(1-\delta^{2})(1,1)$, $T^{+-}=(1+\delta^{2})(1,-1)$ and $T^{-+}=(1+\delta^{2})(-1,1)$ and

[TABLE]

Moreover, the additional boundary term is given by

[TABLE]

where $\hat{\Phi}=(\Phi_{j})_{j\in\Lambda\backslash\{1,2\}}$ and

[TABLE]

The same formulas hold for $\mathbb{E}[{\rm Tr\,}G_{\Lambda}(E+i\varepsilon)]$ with $g$ replaced by $g_{1}(\Phi^{*},\Phi)=\sum_{j}|z_{j}|^{2}\,\mathrm{e}^{i\Phi^{*}(E-i\varepsilon+\Delta)\Phi}.$

Idea of the proof.

We use the representation from Theorem 2.2 and insert the Fourier transform of the given density. For non-negative correlations we can then undo the coordinate change. When negative correlations are present this operation generates additional correction terms. For details see Section 4. ∎

In the case of non-negative correlations we recover exact formulas, as follows.

Theorem 2.4**.**

Let $T_{jk}=\delta_{jk}$ (classical Lloyd model). We have

[TABLE]

where $H_{0}=-\Delta$. In particular

[TABLE]

For $T_{jk}\geq 0$ (non-negative correlation) Eq. (2.7) and (2.8) still hold, with $\lambda\mathds{1}_{\Lambda}$ replaced by the diagonal matrix $\lambda\hat{T}$, where $\hat{T}_{ij}=\delta_{ij}\sum_{k}T_{jk}$.

In particular both, the classical and the (positive) correlated Lloyd model have the same (averaged) density of states as the free Laplacian $H_{0}=-\Delta$ with imaginary mass $\lambda$ and $\lambda\hat{T}$, respectively.

Idea of the proof.

Follows from Proposition 2.3. For details see Section 4. ∎

Note that the results on the density of states above can be derived also by other methods (cf. [Llo69] and [Sim83]).

In the case of localized negative correlation (the toymodel in Case 3. above) we obtain the following result.

Theorem 2.5** (Toymodel).**

Consider $T_{jk}$ be as in Case 3. above, $\lambda>0$ and $0<\delta\ll(1+\lambda^{-1})^{-1}.$ Then

[TABLE]

Idea of the proof.

Follows from Proposition 2.3 by integrating first over the uncorrelated variables in $\Lambda$ and estimating the remaining integral. For details see Section 4. ∎

3 Supersymmetric polar coordinates

3.1 Proof of Theorem 2.1

Proof of Theorem 2.1.

The idea is to apply the coordinate change $\Psi$ from Eq. (2.1) for each $j\in\{0,\dots,N\}$. To simplify the procedure, we divide it into $\Psi_{1}\circ\Psi_{2}\circ\Psi_{3}$, where $\Psi_{1}$ is a change from ordinary polar coordinates into complex variables, $\Psi_{2}$ rescales the odd variables and $\Psi_{3}$ translates the radii into super space. Note that only the last step mixes ordinary and Grassmann variables and produces boundary terms.

We first change the complex variables $z_{j},\bar{z}_{j}$ for all $j$ into polar coordinates

[TABLE]

The Jacobian is $\prod_{j=1}^{N}2r_{j}$ and by an ordinary change of variables

[TABLE]

where $\Psi_{1}=\psi_{1}\times\mathds{1}$. Note that no boundary terms arise. Now we rescale the odd variables by

[TABLE]

There are again no boundary terms since we have a purely odd transformation. The Berezinian is given by $\prod_{j=1}^{N}r_{j}^{-1}$. Since $\psi_{2}$ is a linear transformation, this can also be computed directly. This cancels with the Jacobian from $\Psi_{1}$ up to a constant. Hence

[TABLE]

where $\Psi_{2}=\mathds{1}\times\psi_{2}$. After these transformations, we have $\bar{z}_{j}z_{j}+\bar{\chi}_{j}\chi_{j}=r_{j}^{2}+r_{j}\bar{\rho}_{j}\rho_{j}=(r_{j}+\tfrac{1}{2}\bar{\rho}_{j}\rho_{j})^{2}$. We set $\Psi_{3}(r,\theta,\bar{\rho},\rho)=(r-\frac{1}{2}\bar{\rho}\rho,\theta,\bar{\rho},\rho)$. Hence $\Psi=\Psi_{1}\circ\Psi_{2}\circ\Psi_{3}$ is the $\Psi$ from Eq. (2.1):

[TABLE]

We expand $\tilde{f}=f\circ\Psi_{1}\circ\Psi_{2}\circ\Psi_{3}$ as follows

[TABLE]

Note that we can set $\rho_{j}=0$ and $\bar{\rho}_{j}=0$ for $\alpha_{j}=1$ in $\partial_{r}^{\alpha}\tilde{f}$. We use the short-hand notation $\partial_{r}^{\alpha}\tilde{f}(r,\theta,\bar{\rho},\rho)|_{\bar{\rho}^{\alpha}=\rho^{\alpha}=0}$. Inserting (3.2) into the integral (3.1), we can integrate out the variables $r^{\alpha},$ $\rho^{\alpha}$ and $\bar{\rho}^{\alpha},$ as follows:

[TABLE]

where in the second line we applied integration by parts in $r^{\alpha}$

[TABLE]

and

[TABLE]

Note that $\tilde{f}(r,\theta,\bar{\rho},\rho)|_{r^{\alpha}=\bar{\rho}^{\alpha}=\rho^{\alpha}=0}=f\circ\Psi_{\alpha}$ is independent of $\theta_{j}$ for $\alpha_{j}=1$ and we can integrate $\int(\,\mathrm{d}\theta)^{\alpha}=(2\pi)^{|\alpha|}$. This proves the theorem. ∎

3.2 Proof of Theorem 2.2

Proof of Theorem 2.2.

Applying Theorem A.1 to $\mathcal{G}_{\varepsilon}(E,\tilde{E})$ yields

[TABLE]

Note that we cannot interchange the average with the integral, since the average of the supersymmetric expression $\,\mathrm{e}^{i\lambda\Phi^{*}V\Phi}$ may be ill-defined if infinite moments are present. But after applying Theorem 2.1 we get

[TABLE]

where $g(\Phi^{*},\Phi)=\,\mathrm{e}^{i\Phi^{*}(\mathbf{E}+i\varepsilon+\Delta)\Phi}$. Now we can take the average inside the integral. The same arguments hold for $\mathbb{E}[|G_{\Lambda}(E+i\varepsilon)_{jk}|^{2}]$. ∎

4 Applications to the Lloyd model

4.1 Proof of Proposition 2.3

We will need the following well-known result for the proof of the proposition.

Lemma 4.1**.**

Let $A\sim\text{Cauchy}(0,1)$ and $t\in\mathbb{R}$. Then $\mathbb{E}[\,\mathrm{e}^{itA}]=\,\mathrm{e}^{-|t|}$.

Proof.

Let $t\geq 0$. We take the principal value and apply the residue theorem.

[TABLE]

where $\gamma(s)=R\,\mathrm{e}^{is}$ for $s\in[0,\pi]$. The case $t<0$ follows analogously by closing the contour from below. ∎

Proof of Proposition 2.3.

Starting from the representation (2.4) of Theorem 2.2, we use Lemma 4.1 to determine the Fourier transform

[TABLE]

As long as $r_{j}\in\mathbb{R}$, this is well-defined and the integral remains finite for arbitrary correlation $T$. When $T_{jk}\geq 0$ for all $j,k$, we can drop the absolute value and obtain

[TABLE]

where $\tilde{\mu}(\Phi^{*},\Phi)=\exp[-\sum_{k}\lambda\sum_{j}T_{jk}\Phi^{*}_{j}\Phi_{j}]$ is a smooth, integrable function in $\mathcal{A}_{2N,2N}(\mathbb{C}^{N})$, which can be transformed back to ordinary supersymmetric coordinates by Theorem 2.1.

In the case of the toymodel, our function is continuous but only piecewise smooth. We partition the integration domain into regions, where our function is smooth. In polar coordinates the regions (2.5) become

[TABLE]

Hence $(0,\infty)^{\Lambda}$ can be written as the disjoint union $\mathcal{I}_{++}\cup\mathcal{I}_{+-}\cup\mathcal{I}_{-+}\cup\mathcal{N}$, where $\mathcal{N}$ is a set of measure [math]. Using $T^{\beta}$ defined above, we can write

[TABLE]

where $\beta\in\{++,+-,-+\}$ and $h(\Phi^{*},\Phi)=g(\Phi^{*},\Phi)\,\mathrm{e}^{-\lambda\sum_{j\neq 1,2}\Phi_{j}^{*}\Phi_{j}}$ is independent of $\beta$. Finally, $\chi(\mathcal{I}_{\beta})$ is the characteristic function of $\mathcal{I}_{\beta}$ and $r^{\alpha}=0$ means $r_{j}=0$ for $\alpha_{j}=1$.

To transform back we need to repeat the proof of Theorem 2.1 on the different domains. Consider the integral

[TABLE]

where ${\mathcal{I}}_{\beta}$ are the corresponding subsets of $\mathbb{C}^{\Lambda}$ (cf. Eq. (2.5)). We will show that inserting polar coordinates in $I_{2}$, we recover $I_{1}$ plus some correction terms. In each region, the integrated function is smooth and we can apply the first two transformations $\Psi_{1}$ and $\Psi_{2}$ from the proof of Theorem 2.1 and obtain

[TABLE]

Replacing as in Eq. (3.2) the integrand by the Taylor-expansion of $\tilde{f}_{\beta}=\,\mathrm{e}^{-\lambda\sum_{j=1}^{2}(T_{\beta})_{j}r_{j}^{2}}\tilde{h}$, with $\tilde{h}=h\circ\Psi_{1}\circ\Psi_{2}\circ\Psi_{3}$, we obtain

[TABLE]

Applying now integration by parts as in Eq. (3.3) generates additional boundary terms. More precisely we distinguish three cases.

Case 1: $\boldsymbol{\alpha_{1}=\alpha_{2}=0}$.

Here no derivatives in $r_{1}$ and $r_{2}$ appear and $\mathcal{I}_{\beta}=\tilde{\mathcal{I}}_{\beta}\times(0,\infty)^{\Lambda\backslash\{1,2\}}$. Hence no additional terms arise and

[TABLE]

Case 2: $\boldsymbol{\alpha_{1}=1,\alpha_{2}=0}$ (or vice versa).

Here additional boundary terms do appear but cancel since the function is continuous:

[TABLE]

where $\,\mathrm{d}\hat{r}=\prod_{j\neq 1}\,\mathrm{d}r_{j}$ and $h^{(\alpha)}=\partial_{r}^{\hat{\alpha}}\tilde{h}$ and $\hat{\alpha}_{j}=\alpha_{j}$ for all $j\neq 1,2$, $\hat{\alpha}_{1}=\hat{\alpha}_{2}=0$. Note that in the second step all terms except the first one cancel because of continuity: $\sum_{j=1}^{2}T^{-+}_{j}r_{j}^{2}|_{r_{1}=\delta r_{2}}=\sum_{j=1}^{2}T^{++}_{j}r_{j}^{2}|_{r_{1}=\delta r_{2}}$ and $\sum_{j=1}^{2}T^{++}_{j}r_{j}^{2}|_{r_{1}=r_{2}/\delta}=\sum_{j=1}^{2}T^{+-}_{j}r_{j}^{2}|_{r_{1}=r_{2}/\delta}$. We can apply now integration by parts for $r^{\hat{\alpha}}$ as before. Note that for $r_{1}=0$ the sets $\mathcal{I}_{++}=\mathcal{I_{+-}=\emptyset}$ and we obtain only contributions from the set $\mathcal{I}_{-+}=\{r_{2}\in\mathbb{R}^{+}\}$ which is the same as writing $\sum_{\beta}\chi(\mathcal{I}_{\beta})|_{r_{1}=0}$.

Case 3: $\boldsymbol{\alpha_{1}=\alpha_{2}=1}$.

Here we obtain additional boundary terms which do not cancel. Applying integration by parts in $r_{1}$, we need to evaluate

[TABLE]

on the different boundaries. The contributions of $\partial_{r_{2}}h^{(\alpha)}\,\mathrm{e}^{-\lambda\sum_{j=1}^{2}(T_{\beta})_{j}r_{j}^{2}}$ cancel as above by continuity except for the term at $r_{1}=0$. The contributions from the second summand remain. Precisely we get

[TABLE]

where $R_{\alpha}(h)$ is defined below in Eq. (4.1). In the first integral, we can apply integration by parts in $r_{2}$ and $r^{\hat{\alpha}}$ as before and the result is independent of $\beta$. It remains to consider

[TABLE]

Here, we can integrate over $r^{\hat{\alpha}},$ $\theta^{\hat{\alpha}},$ $\rho^{\alpha}$ and $\bar{\rho}^{\alpha}$ first as follows:

[TABLE]

where we used $|\alpha|=|\hat{\alpha}|+2.$ The resulting integral is:

[TABLE]

After rescaling the second term $r_{2}\mapsto\delta^{2}r_{2}$, we obtain

[TABLE]

As a result we proved that $I_{2}-I_{1}=\sum_{\hat{\alpha}}R_{\alpha}(h).$ Note that in above representation for $R_{\alpha}(h)$ we can transform the variables of $\Lambda\backslash\{1,2\}$ back to flat coordinates by Theorem 2.1. This yelds (2.6) and thus completes the proof. ∎

4.2 Proof of Theorem 2.4

Proof of Theorem 2.4.

We start from the result of Propostion 2.3.

In both models, the classical and the positive correlated one, we have $T_{jk}\geq 0$ and $\sum_{k}T_{jk}>0$, hence the body of $\lambda\sum_{j}T_{jk}\Phi^{*}_{j}\Phi_{j}$ is strictly positive except on a set of measure [math]. We end up with

[TABLE]

where we can take the average $\varepsilon\to 0$ and go back to the original representation. ∎

4.3 Proof of Theorem 2.5

Proof of Theorem 2.5.

Using Eq. (1.4) and the result of Proposition 2.3, we obtain

[TABLE]

where for $\beta=(++),(+-)$ or $(-+)$ we have

[TABLE]

and for $h(\Phi^{*},\Phi)=\sum_{j}|z_{j}|^{2}\,\mathrm{e}^{-\lambda\sum_{j\neq 1,2}\Phi^{*}_{j}\Phi_{j}}\,\mathrm{e}^{i\Phi^{*}(E-i\varepsilon+\Delta)\Phi}$ the remainder $R(h)$ is defined in Eq. (2.6).

We will show that the main contribution comes from $\mathcal{I}_{++}$ and indeed

[TABLE]

In the following we show that $I_{+-}$ and $I_{-+},$ as well as $R(h)$ are small in terms of $\delta$.

Analysis of the $\boldsymbol{I_{\beta}}$ terms.

Integrating out the Grassmann variables, we obtain for all $\beta$

[TABLE]

where $C_{\beta}$ has the block structure

[TABLE]

and we defined the vectors $d_{1},d_{2}\in\mathbb{R}^{\Lambda\setminus\{1,2\}}$ as $d_{1}(j)=\delta_{|i_{1}-j|,1},$ $d_{2}(j)=\delta_{|i_{2}-j|,1},$ where $i_{1},i_{2}$ are the positions of $1,2.$ Note that the blocks $B$ and $D$ are independent of $\beta$ and ${\rm Re\,}B>0.$ On the contrary ${\rm Re\,}C_{\beta}>0$ holds only for $\beta=(+,+).$ We set then $\varepsilon=0$ in our formulas and reorganize $I_{++}+I_{+-}+I_{-+}$ as follows

[TABLE]

To estimate $\mathcal{E}_{+-}$ and $\mathcal{E}_{-+}$, we integrate over the variables $w=(z_{j})_{j\in\Lambda,j\neq 1,2}$ exactly. We define $z=(\hat{z},w),\hat{z}=(z_{1},z_{2}).$ Then

[TABLE]

Integrating over $w$ we get

[TABLE]

where we defined $M\coloneqq 1-DB^{-2}D^{T}.$ Then for $\beta=(+-),(-+)$ and $\beta^{\prime}=\beta$ or $\beta^{\prime}=(++)$ we have

[TABLE]

where $S_{\beta^{\prime}}=A_{\beta^{\prime}}+DB^{-1}D^{T}$ is the Schur complement of the $2\times 2$ block of $C_{\beta^{\prime}}$ corresponding to $1,2$. We also used $\det C_{\beta^{\prime}}=\det B\det S_{\beta^{\prime}}$. We consider now the error term $\mathcal{E}_{-+}.$ The error term $\mathcal{E}_{+-}$ works analogously. From the results above we get

[TABLE]

where we used $S_{++}^{-1}$ invertible and we defined

[TABLE]

Now we change the coordinate $z_{1}$ to $v=z_{1}z_{2}^{-1}\delta^{-1}.$ As a short-hand notation write $S=S_{++}$. We have

[TABLE]

where $\mathbf{v}=(\delta v,1)^{t}$ and $\mathbf{v}^{*}=(\delta\bar{v},1)$. Note that ${\rm Re\,}S>0$ and

[TABLE]

therefore we can perform the integral over $z_{2}$ exactly

[TABLE]

where we applied Lemma 4.2 below and

[TABLE]

Applying Lemma 4.2 again, together with Eq. (4.3) we get

[TABLE]

Analysis of $\boldsymbol{R(h)}$.

Note that we can set $\varepsilon=0$ in $R(h)$. By the notations in Eq. (4.2), we can write

[TABLE]

where $v_{\theta}^{t}=(\,\mathrm{e}^{i\theta_{1}}\delta,\,\mathrm{e}^{i\theta_{2}})$. Integrating over the Grassmann variables, we obtain

[TABLE]

Define $S_{0}=A_{0}+DB^{-1}D^{T}$. Integrating over $w$ and $r_{2}$, we obtain

[TABLE]

Similar to the estimates above, we insert absolute values and use Lemma 4.2 and Eq. (4.4) to bound the first term by $\delta^{2}(1+\mathcal{O}(|\Lambda|^{-1}))$ and the second one by $\delta^{2}\mathcal{O}(\lambda^{-1}|\Lambda|^{-1})$. ∎

Lemma 4.2**.**

Let $\eta>0$ and $\mu_{\lambda}=\frac{\lambda\eta}{\lambda+4d\,\mathrm{e}^{\eta}}$. Let $B,M,C_{++}$ and $S_{++}$ be the matrices in the proof above. Set $0<\delta\leq\frac{1}{2}.$ Then

1. (i)

$|B^{-1}_{ij}|\leq\frac{2}{\lambda}\,\mathrm{e}^{-\mu_{\lambda}|i-j|}$* and ${\rm Re\,}(f^{*}B^{-1}f)\geq\frac{\lambda}{\lambda^{2}+(4d)^{2}}\|f\|^{2}$ $\forall f\in\mathbb{C}^{\Lambda\setminus\{1,2\}}$ * 2. (ii)

$|{\rm Tr\,}C_{++}^{-1}|\geq\frac{|\Lambda|\lambda}{K(\lambda+1)^{2}}.$** 3. (iii)

${\rm Re\,}(f^{*}S_{++}f)\geq\frac{\lambda}{2}\|f\|^{2}$* $\forall f\in\mathbb{C}^{\Lambda\setminus\{1,2\}}.$ Moreover*

$|(S_{++})_{jk}|\leq K(\lambda+\frac{1}{\lambda})$* for all $j,k=1,2$ . *

Proof.

$(i)$ We have $B=i(-\Delta_{|\Lambda\setminus\{1,2\}}-(E+i\lambda))$. The upper bound follows by Combes-Thomas [AW15][Sect 10.3]. For the lower bound note that

[TABLE]

Moreover $\|Bg\|^{2}=\lambda^{2}\|g\|^{2}+g^{*}(E+\Delta_{|\Lambda\setminus\{1,2\}})g\leq(\lambda^{2}+(4d)^{2})\|g\|^{2}.$ The result follows setting $g=B^{-1}f$.

$(ii)$ As in $(i)$ above $f^{*}{\rm Re\,}C_{++}^{-1}f\geq\lambda(1-\delta^{2})\|C_{++}^{-1}f\|^{2}.$ We can write $C_{++}=\lambda-\lambda\delta^{2}\mathds{1}_{1,2}-i(E+\Delta),$ where $\mathds{1}_{1,2}$ is the diagonal matrix $(\mathds{1}_{1,2})_{ij}=\delta_{ij}[\delta_{ji_{1}}+\delta_{ji_{2}}].$ Hence

[TABLE]

The result follows by inserting this decomposition in $\|C_{++}g\|^{2}$ for $g=C_{++}^{-1}f.$

$(iii)$ Using $(i)$ we have

[TABLE]

The upper bound follows from $(i)$ too. ∎

Appendix A Super analysis

We collect here only a minimal set of definitions for our purpose. For details, see [Ber87, Var04, Weg16, DeW92].

A.1 Basic definitions

Let $q\in\mathbb{N}$. Let $\mathcal{A}=\mathcal{A}_{q}=\mathcal{A}[\chi_{1},\dots,\chi_{q}]$ be the Grassmann algebra over $\mathbb{C}$ generated by $\chi_{1},\dots,\chi_{q}$, i.e.

[TABLE]

where $V$ is the complex vector space with basis $(\chi_{1},\dots,\chi_{q})$, $V^{0}=\mathbb{C}$ and $V^{j}=V^{j-1}\wedge V$ for $j\geq 2$ with the anticommutative product $\wedge$

[TABLE]

As a short hand notation, we write in the following $\chi_{i}\chi_{j}=\chi_{i}\wedge\chi_{j}$ and for $I\subset\{1,\dots,q\}$ denote $\chi^{I}=\prod_{j\in I}\chi_{j}$ the ordered product of the $\chi_{j}$ with $j\in I$. Then each $a\in\mathcal{A}$ has the form

[TABLE]

where $\mathcal{P}(q)$ is the power set of $\{1,\dots,q\}$ and $a_{I}\in\mathbb{C}$ for all $I\in\mathcal{P}(q)$. We distinguish even and odd elements $\mathcal{A}=\mathcal{A}^{0}\oplus\mathcal{A}^{1}$, where

[TABLE]

The parity operator $p$ for homogeneous (i.e. purely even, resp. purely odd) elements is defined by

[TABLE]

Note that even elements commute with all elements in $\mathcal{A}$ and two odd elements anticommute. For an even element $a\in\mathcal{A}^{0}$, we write $a=b_{a}+n_{a}$, where $n_{a}$ is the nilpotent part and $b_{a}=a_{\emptyset}\in\mathbb{C}$ is called the body of $a$.

Let $U\subset\mathbb{R}$ open. For any function $f\in C^{\infty}(U)$, we define

[TABLE]

via its Taylor expansion. Note that the sum above is always finite.

A.2 Differentiation

Let $I^{\prime}\subset I$. We define the signs $\sigma_{l}(I,I^{\prime})$ and $\sigma_{r}(I,I^{\prime})$ via

[TABLE]

Then the left- resp. right-derivative of an element $a$ of the form (A.1) is defined as

[TABLE]

A.3 Integration

The integration over a subset of (odd) generators $\chi_{j},j\in I$ is defined by

[TABLE]

where $a$ has the form (A.1) and $\,\mathrm{d}\chi^{I}=\prod_{j\in I}\,\mathrm{d}\chi_{j}$ is again a ordered product. Note that the one forms $\,\mathrm{d}\chi_{i}$ are anticommutative objects and e.g. $\int\,\mathrm{d}\chi_{i}\,\mathrm{d}\chi_{j}\chi_{i}\chi_{j}=-\int\,\mathrm{d}\chi_{i}\,\mathrm{d}\chi_{j}\chi_{j}\chi_{i}=-1$.

Gaussian integral.

There is a useful Gaussian integral formula for Grassmann variables. We rename our basis as $(\chi_{1},\dots,\chi_{q},\bar{\chi}_{1},\dots\bar{\chi}_{q})$. Then for $M\in\mathbb{C}^{q\times q}$

[TABLE]

where $\,\mathrm{d}\bar{\chi}\,\mathrm{d}\chi=\prod_{j=1}^{q}\,\mathrm{d}\bar{\chi}_{j}\,\mathrm{d}\chi_{j}$. Combining this with complex Gaussian integral formulas, we obtain the following result.

Theorem A.1** (Supersymmetric integral representation).**

Let $A_{1},A_{2}\in\mathbb{C}^{n\times n}$ with ${\rm Re\,}A_{1}>0$. Let $\Phi=(z,\chi)^{t}\in\mathbb{C}^{n}\times V^{n}$ be a supervector and $\Phi^{*}=(\bar{z},\bar{\chi})\in\mathbb{C}^{n}\times V^{n}$ its transpose. With the notations $[\,\mathrm{d}\Phi^{*}\,\mathrm{d}\Phi]=(2\pi)^{-n}\,\mathrm{d}\bar{z}\,\mathrm{d}z\,\mathrm{d}\bar{\chi}\,\mathrm{d}\chi$ and $\Phi^{*}A\Phi=\sum_{j,k=1}^{n}\bar{z}_{j}(A_{1})_{jk}z_{k}+\bar{\chi}_{j}(A_{2})_{jk}\chi_{k}$ for a block matrix $A=\begin{pmatrix}A_{1}&0\\ 0&A_{2}\end{pmatrix}$ (a supermatrix with odd parts [math]) we can write

[TABLE]

and

[TABLE]

where $\hat{A}_{1}=\begin{pmatrix}A_{1}&0\\ 0&A_{1}\end{pmatrix}$.

Proof.

Combine Eq. (A.3) with the complex Gaussian integral formulas

[TABLE]

Note that while Eq. (A.3) holds for all matrices $A\in\mathbb{C}^{n\times n}$, we need the additional condition ${\rm Re\,}A>0$ for the complex ones to ensure that the complex integral is finite. ∎

A.4 Grassmann algebra functions and change of variables

In this section, we denote the body of an even element $a$ by $b(a)$ instead of $b_{a}$.

Definition A.2**.**

Let $U\subset\mathbb{R}^{p}$ open. The algebra of smooth $\mathcal{A}[\chi_{1},\dots,\chi_{q}]$-valued functions on a domain $U$ is defined by

[TABLE]

We call $y_{i}(x,\chi)$, $\eta_{j}(x,\chi)$, for ${i=1,\dots p,j=1,\dots,q}$ generators of $\mathcal{A}_{p,q}(U)$ if $p(y_{i})=0$, $p(\eta_{j})=1$ and

1. (i)

$\{(b(y_{1}(x,0)),\dots,b(y_{p}(x,0))),x\in U\}$ is a domain in $\mathbb{R}^{p}$, 2. (ii)

we can write all $f\in\mathcal{A}_{p,q}(U)$ as $f=\sum_{I}f_{I}(y)\eta^{I}$.

Note that $(x,\chi)$ are generators for $\mathcal{A}_{p,q}(U)$.

A change of variables is then a parity preserving transformation between systems of generators of $\mathcal{A}_{p,q}(U)$. A practical change of variable formula for super integrals is currently only known for functions with compact support, i.e. functions $f\in\mathcal{A}_{p,q}(U)$ such that $f_{I}\in C_{c}^{\infty}(U)$ for all $I\in\mathcal{P}(q)$.

Theorem A.3**.**

Let $U\subset\mathbb{R}^{p}$ open, $x,\chi$ and $y(x,\chi)$, $\eta(x,\chi)$ two sets of generators of $\mathcal{A}_{p,q}(U)$. Denote the isomorphism between the generators by

[TABLE]

and $V=b(\psi(U))=\{(b(y_{1}(x,0)),\dots,b(y_{p}(x,0))),x\in U\}$. Then for all $f\in\mathcal{A}_{p,q}(V)$ with compact support, we have

[TABLE]

where ${\rm Sdet\,}(J\psi)$ is called the Berezinian defined by

[TABLE]

Integration over even elements $x$ and $y$ means integration over the body $b(x)$ and $b(y)$ in the corresponding regions $U$ and $V$.

Proof.

See [Ber87, Theorem 2.1] or [Var04, Theorem 4.6.1]. ∎

Remark*.*

Applying an isomorphism $\psi$ that changes only the odd elements, Theorem A.3 holds also for smooth, integrable functions that are not necessarily compactly supported. Changing also the even elements for a non compactly supported function, boundary integrals can occur.