Abstract
We study discrete random Schrödinger operators via the supersymmetric formalism. We develop a cluster expansion that converges at both strong and weak disorder. We prove the exponential decay of the disorder-averaged Green’s function and the smoothness of the local density of states either at weak disorder and at energies in proximity of the unperturbed spectrum or at strong disorder and at any energy. As an application, we establish Lifshitz-tail-type estimates for the local density of states and thus localization at weak disorder.
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Acknowledgements
I express my sincere gratitude to Marcello Porta for inspiring discussions and valuable guidance. This work has been supported by the Swiss National Science Foundation via the grant “Mathematical Aspects of Many-Body Quantum Systems” and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, ERC Starting Grant MaMBoQ, Grant Agreement No. 802901. I thank the anonymous referee for the numerous useful comments on a previous version of the paper.
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Appendix A: Disorder Distribution
Appendix A: Disorder Distribution
We discuss two examples that connect the properties of the density ν with the IMB contained in hypothesis (H2-I)τ, see Definition 3.1 and below:
Example I.1
We can state the following lemma.
Lemma A.1
Let \(\nu \in {\mathscr{S}}(\mathbb {R})\) be analytic in the strip |Imt| < W. Then, if τ < W, Fτγ satisfies IMB with M = (W/2 − τ/2)− 1/2 and p = 1/2.
Proof
The assumptions on ν imply that \(\|\hat {\nu }({\Phi }^{+}{\Phi }^{-}) \| \leq C \mathrm {e}^{-W \phi ^{+}\phi ^{-}}\), for some universal constant C. Using that \(|(\phi ^{+}_{\sigma })^{n_{\sigma }^{+}}(\phi ^{-}_{\sigma })^{n_{\sigma }^{-}}| \leq (n_{\sigma }/2)! M^{-n_{\sigma }/2} \mathrm {e}^{M\phi ^{+}_{\sigma }\phi ^{-}_{\sigma }}\), where \(n_{\sigma } =n_{\sigma }^{+} + n_{\sigma }^{-} \) and M = W/2 − τ/2 the claim follows. Here K grows with M− 1. □
Notice that Example I.1 include Gaussian distributions.
Example I.2
The decay of the Fourier transform of test functions can be quantitatively characterised, see, e.g., [37]. For instance, if \(\nu (t) = \mathrm {e}^{-\frac {1}{1-t^{2}}}\mathbf {1}_{|t|\leq 1}\), so that F0 satisfies IMB with M = 2 and p = 1.
We then discuss two examples that connect the properties of the density ν with the IDB contained in hypothesis (H2-II)τ, see Definition 4.1 and below. Both examples include Gaussian distributions.
Example II.1
Introduce the following seminorm on functions of a supervector:
where \( \mathbb {R}_{W}^{2\mathsf {S}} = \left \{ \phi \in \mathbb {C}^{2\mathsf {S}} | |\text {Im} \phi _{{\scriptscriptstyle {i}},\sigma }| \leq W , i = 1,2 , \sigma \in \mathsf {S} \right \} \). We state the following lemma.
Lemma A.2
Assume that for any \(\alpha \in \mathbb {R}\), eα|t|ν(t) is bounded and that ∥Fτγ∥1,W is finite for some 0 < W ≤ 1. Then Fτγ satisfies IDB with K = ∥Fτγ∥1,W, M = W− 1 and p = 1.
Proof
We notice that \(\hat {\nu }\) is entire and so is Fτγ in ϕ. We thus apply the multi-variable Cauchy integral formula in \(\mathbb {R}_{W}^{2\mathsf {S}}\). Accordingly,
where \(D(0,W) = \left \{ w \in \mathbb {C} \big | |w| \leq W \right \}\) and dw = ×i,σdwi,σ; therefore:
where \(n = {\sum }_{i,\sigma } n_{i,\sigma }\). The claim follows because \(\partial /\partial \phi ^{\varepsilon }_{\sigma } = 1/2 (\partial /\partial \phi _{{\scriptscriptstyle {1}},\sigma }\) −iε∂/∂ϕ2,σ). □
Example II.2
Introduce the following seminorm on functions of a supervector:
where \(D(0,W) \subset \mathbb {C}\) was defined above. We state the following lemma.
Lemma A.3
Assume eW|t|ν(t) is bounded and ∥Fτγ∥1,W is finite for some 0 < W ≤ 1. Then Fτγ satisfies IDB with K = |S|!2|s|∥|Fτγ∥|1,W, M = 4(W− 1 + τ), p = 2.
Proof
We notice that \(\hat {\nu }(t)\) is holomorphic on the strip |Imt|≤ W which we shall use to estimate its derivative. We think of Fτγ(Φ) as the composite function f(Φ+Φ−) and compute its derivative accordingly. For simplicity, we carry out the computation in the case of |S| = 1 and we take, e.g., \(n_{\sigma }^{+}\geq n_{\sigma }^{-}\) and \(n_{\sigma } = n_{\sigma }^{+}+n_{\sigma }^{-}\). We accordingly bound the norm of the derivative as follows:
We apply the Cauchy integral representation to estimate the derivatives of f(⋅) and we use the bound \(|\phi |^{n_{\sigma }-2i} \leq (n_{\sigma }-2i)! \mathrm {e}^{(\phi ^{+}_{\sigma }\phi ^{-}_{\sigma })^{1/2}}\). Integrating in dϕσ and taking the superior over the contour variable gives the claim. □
To conclude this appendix, we will show that requiring either IMB or IDB over [0,τ], respectively in (H2-I)τ and (H2-II)τ, with K, M and p depending on τ, is very natural. In fact, if \(F_{\tau ^{\prime }\gamma }({\Phi }) = \mathrm {e}^{\tau {\Phi }^{+}{\Phi }^{-}} \hat {\nu }({\Phi }^{+}{\Phi }^{-})\) satisfies either IMB or IDB for \(\tau ^{\prime } = \tau \), then so does it for any \(\tau ^{\prime } \in [0,\tau ]\). In the case of IMB, this is a simple consequence of the following inequality
In the case of IDB more manipulations are needed. We write
where we have abridged the notation in an intuitive way: \(\underline {n}\) and \(\underline {\ell }\) stand for \(\{n^{\varepsilon }_{\sigma } \}\) and \(\{\ell ^{\varepsilon }_{\sigma } \}\) respectively, \(\begin {pmatrix} \underline {n} \\ \underline {\ell } \end {pmatrix} = {\prod }_{\varepsilon , \sigma } \begin {pmatrix} n^{\varepsilon }_{\sigma } \\ \ell ^{\varepsilon }_{\sigma } \end {pmatrix}\), \(\partial _{\phi }^{\underline {\ell }} = {\prod }_{\varepsilon ,\sigma }(\partial /\partial \phi _{\sigma }^{\varepsilon })^{\ell ^{\varepsilon }_{\sigma }}\) and \(\partial _{\phi }^{\underline {n} - \underline {\ell }} = {\prod }_{\varepsilon ,\sigma }(\partial /\partial \phi _{\sigma }^{\varepsilon })^{n^{\varepsilon }_{\sigma }-\ell ^{\varepsilon }_{\sigma }}\). The derivative of the exponential can be explicitly computed and by using the properties of the Grassmann norms and the bound \(\left [(\tau -\tau ^{\prime })|\phi |^{2}\right ]^{m} \leq m! \mathrm {e}^{(\tau - \tau ^{\prime })|\phi |^{2}}\) one can check that \(\left \| \partial _{\phi }^{\underline {\ell }} \mathrm {e}^{-(\tau - \tau ^{\prime }) {\Phi }^{+}{\Phi }^{-}} \right \| \leq 2^{|\mathsf {S}|} \mathrm {e}^{(\tau - \tau ^{\prime })|\mathsf {S}|} \underline {\ell }! (\tau - \tau ^{\prime })^{\ell /2}\), where \(\underline {\ell }! = {\prod }_{\varepsilon ,\sigma } \ell ^{\varepsilon }_{\sigma }!\) and \(\ell = {\sum }_{\varepsilon ,\sigma }\ell ^{\varepsilon }_{\sigma }\). If Fτγ satisfies IDB, then integration of (A.7) in dϕ proves that for any \(\tau ^{\prime } \in [0,\tau ]\) \(F_{\tau ^{\prime }\gamma }\) indeed satisfies IDB for some K, M and p depending on τ.
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Fresta, L. Supersymmetric Cluster Expansions and Applications to Random Schrödinger Operators. Math Phys Anal Geom 24, 4 (2021). https://doi.org/10.1007/s11040-021-09375-5
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DOI: https://doi.org/10.1007/s11040-021-09375-5