Abstract
We are interested in the phase transition of the correlation function of local densities of states at mesoscopic scales of random band matrices of width W in dimension 2. As a result, we show that the local densities of states are alternately positively and negatively correlated in the diffusive regime \(O(\log (L/W))\) times, L being the size of the system.
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1 Introduction
Impurities in a disordered quantum system can be modelised by a random band Hermitian matrix H corresponding to the Hamiltonian of the system. Pure materials have an underlying lattice structure which can be represented by a finite lattice in \({\mathbb {Z}}^d\), say the torus \({\mathbb {T}}=\left[ -\frac{L}{2}\right. ,\left. \frac{L}{2}\right) ^d\cap {\mathbb {Z}}^d\). Given \(x,y\in {\mathbb {T}}\), corresponding to the lattice sites, the matrix \(H_{xy}\) represents the quantum system in a d-dimensional discrete box of length L. Following the model introduced by Erdős and Knowles in [2, 3], we assume that the matrix H is an Hermitian matrix whose upper triangular entries are independent random variables with zero mean. Let \(S_{xy}:={\mathbb {E}}|H_{xy}|^2\) be a deterministic matrix given by an arbitrary profile function f on the scale W, that is,
where \(\left[ x\right] _L\) denotes the canonical representative of \(x\in {\mathbb {Z}}^d\) in the torus \({\mathbb {T}}\) and \(f:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) is an even, bounded, nonnegative, piecewise \(C^1\) function such that f and \(|\nabla f|\) are integrable and \(\int _{{\mathbb {R}}^d}\textrm{d}xf(x)|x|^{4+c_1}<\infty \) for some constant \(c_1>0\). Thus, for all \(x\in {\mathbb {T}}\),
We assume that the law of \(H_{xy}\) is symmetric, i.e. \(H_{xy}\) and \(-H_{xy}\) have the same law and that \(A_{xy}:=(S_{xy})^{-1/2}H_{xy}\) have uniform subexponential decay, that is,
for some constants \(c_2,c_3>0\) and for any \(\xi >0\). Let \(\phi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) be a test function defined as a smooth function verifying the following conditions (C): \(\int _{{\mathbb {R}}}\phi (E)\textrm{d}E=2\pi \) and such that for every \(q>0\), there exists a constant \(C_q\) satisfying
We are interested in the correlation of the number of eigenvalues around two energies \(E_1<E_2\) such that \(\omega =E_2-E_1\) is much larger than the energy window \(\eta \). More precisely, we are concerned with the following correlation
and \(Y_{\phi _i}^\eta (E)\) is the smoothed local density of states around energy E on the scale \(\eta \), defined by
\(\phi _i^\eta \) being the rescaled test function \(\phi _i^\eta (E):=\eta ^{-1}\phi _i(\eta ^{-1}E)\) and \(\phi _i\) a test function satisfying conditions (C). On the mesoscopic energy scale which corresponds to energy scales much larger than the eigenvalue spacing and much smaller than the total macroscopic energy scale of the system, we observe a phase transition at the critical energy, the Thouless energy, given by \(\eta _c=W^2/L^2\).
Throughout the paper, we make the following assumptions
for some constant C, \(\kappa \) a fixed positive constant, \(c_*\) a small enough positive constant depending on \(\kappa \). We introduce the covariance matrix of \(S_{x0}\), its fourth moment and the covariance matrix of f
and assume that \(c_4<D_0<c_5\) in the sense of quadratic forms for some positive constants \(c_4,c_5\). Here, we choose the matrix D as a positive multiple of the \(d\times d\) identity matrix \(I_d\), i.e. \(D={\mathcal {D}}I_d\), \({\mathcal {D}}>0\). We also introduce the parameters
Let \(\nu \equiv \nu (E)=2\sqrt{1-E^2}/\pi \) and \(E:=(E_1+E_2)/2\). Then, u and \(\zeta \) expand as
The diffusive regime is defined by \(\eta \gg \eta _c\) which corresponds to large samples, i.e. L is large with respect to the diffusion length \(W/\sqrt{\eta }\). In terms of our parameter b behaving as \(\omega L^2/W^2\) using (2), it means that \(b\gg 1\) from assumption (1) \(\omega \gg \eta \). On the contrary, the mean-field regime is defined by \(\eta \ll \eta _c\) corresponding to small samples.
The main contribution of the present paper is an exact asymptotic expression for the local density–density correlation function in dimension 1 valid in both regimes, while in dimension 2 we provide a more precise expression of it with respect to the asymptotics derived in [3] in the diffusive regime improving [3, Proposition 3.5]. In their paper, the authors estimate a series appearing in the main term using a Riemann sum estimate, while the key point here consists in no longer making this approximation. A curious fact is that we observe that for large b but less than \((\log (L/W))^2\) the correlation function has an interesting oscillatory behaviour. The following theorem states our results.
Theorem 1.1
There exists a constant \(c_0>0\) such that the local density–density correlation satisfies:
-
(i)
In dimension 1 for all \(b>0\),
$$\begin{aligned}&\frac{\langle Y_{\phi _1}^\eta (E_1)\ \!;Y_{\phi _2}^\eta (E_2)\rangle }{\langle Y_{\phi _1}^\eta (E_1)\rangle \langle Y_{\phi _2}^\eta (E_2)\rangle }=\\&-\frac{1}{16(\pi \nu )^{5/2}\sqrt{{\mathcal {D}}}\omega ^{3/2}LW}\\&\times \left( \frac{\sinh (\pi \sqrt{2b}) +\sin (\pi \sqrt{2b})}{\sinh ^2(\pi \sqrt{b/2})+\sin ^2(\pi \sqrt{b/2})}+ \pi \sqrt{2b}\frac{\sinh (\pi \sqrt{2b})\sin (\pi \sqrt{2b})}{(\sinh ^2(\pi \sqrt{b/2})+\sin ^2(\pi \sqrt{b/2}))^2}\right) \\&+O\left( \frac{\omega ^{-1/2}}{LW}\left( 1+\frac{\eta }{\omega ^2}\right) +\frac{1}{\omega L^2}\left( 1+\frac{\eta }{\omega ^2}\right) \right. \\&\left. +\frac{1}{\omega W^3}e^{-\pi \sqrt{2\omega }L/W}+\frac{W^{-c_0-1}}{L(\omega +\eta )^{1/2}}+\frac{W^{-c_0}}{L^2(\omega +\eta )}\right) . \end{aligned}$$ -
(ii)
In dimension 2 for \(b\gg 1\),
$$\begin{aligned}&\frac{\langle Y_{\phi _1}^\eta (E_1)\ \!;Y_{\phi _2}^\eta (E_2)\rangle }{\langle Y_{\phi _1}^\eta (E_1)\rangle \langle Y_{\phi _2}^\eta (E_2)\rangle }=\\&-\frac{1}{2\pi ^5{\mathcal {D}}\nu ^4L^2W^2}\left[ \frac{L^2}{\pi {\mathcal {D}}W^2}e^{-\pi \sqrt{2b}}b^{-3/4}\sin \left( \pi \sqrt{2b}-\frac{\pi }{8}\right) -(Q-1)|\log \omega |\right. \\&\left. +O\left( 1+\frac{\eta ^2}{\omega ^3}+\omega |\log \omega |+e^{-\pi \sqrt{2b}}b^{1/4}\left( 1+\frac{1}{\omega W^2}+\frac{L^2}{W^2}b^{-3}\right) \right. \right. \\&\left. \left. +\frac{1}{W^{c_0}}|\log (\omega +\eta )|+\frac{W^{2-c_0}}{L^2(\omega +\eta )}\right) \right] . \end{aligned}$$
We emphasise that for the two-dimensional case, the second term has been calculated in [3], while the first term is new and is dominant for \(b\ll (\log (L/W))^2\), showing that the correlation function oscillates around zero \(O(\log (L/W))\) times. Then, for \(b\gtrsim (\log (L/W))^2\) it is dominated by the logarithmic term. This will be shown in Proposition 2.5 and Corollary 2.6.
An interesting question would be to see whether this oscillatory behaviour is as well observed in other systems and its physical interpretation.
2 Calculation of the correlation function
In [3, Theorem 6.1], it has been shown that, under the assumptions (1), there exists a constant \(c_0>0\) such that for any \(E_1,E_2\in \left[ -1+\kappa ,1-\kappa \right] \) for small enough \(c_*>0\), the local density–density correlation satisfies
where \(R(s):=1+{\textbf{1}}(d=1)s^{-1/2}+{\textbf{1}}(d=2)|\log s|\) and the leading term \(\Theta _{\phi _1,\phi _2}^\eta \) is given by
The proof of the above relation is given in the appendix. For \(q\in \left[ -\pi W,\pi W\right) ^d\), let \({\widehat{S}}_W(q)\) denote \({\widehat{S}}(q/W)\) where \({\widehat{S}}(p):=\sum _{x\in {\mathbb {T}}}e^{-ip\cdot x}S_{x0}\) is defined for all \(p\in \left[ -\pi ,\pi \right) ^d\). Also define the following quantity
Recall from [3, Lemma B.1] that for any \(\epsilon >0\), there is a \(\delta _\epsilon >0\) such that \({\widehat{S}}_W(q)\) is bounded, \(|{\widehat{S}}_W(q)|\leqslant 1-\delta _\epsilon \) if \(|q|\geqslant \epsilon \) for large enough W and has the following expansion
which comes from a fourth-order Taylor’s expansion of
Let \(\epsilon >0\) be such that \({\widehat{S}}_W(q)={\mathcal {I}}-a(q)\) where a is a function satisfying \(c_4|q|^2\leqslant a(q)\leqslant c_5|q|^2\leqslant 1\) for \(|q|\leqslant \epsilon \). Equation [3, B.15] states that
where \({\mathbb {T}}^*=\frac{2\pi }{L}{\mathbb {T}}\).
In Subsection 2.1, we study the above trace in the one-dimensional case, while the two-dimensional case is treated in Subsection 2.2.
2.1 Dimension 1
Proposition 2.1
In dimension 1, the following relation holds
Proof
With the notations introduced above, equation [3, B.16] reads
and equation [3, B.17]
which together with equation (5) lead to the following expression for the trace appearing in the \(\Theta _{\phi _1,\phi _2}^\eta (E_1,E_2)\) term in (4)
Let
then
In dimension 1,
Using Poisson summation formula,
Thus,
By differentiating the above relation with respect to z, we obtain
Hence,
The result then follows from (7) and (6) and the fact that \({\mathcal {I}}=1+O(W^{-1})\). \(\square \)
Proof of Theorem 1.1(i)
We have
Hence,
The result follows using Proposition 2.1. \(\square \)
A plot of the following function at the transition \(b\simeq 1\) is given in Fig. 1.
Corollary 2.2
In the mean-field regime \(b\ll 1\),
which recovers [2, Theorem 2.9 (ii)] up to a multiplicative constant in the leading term and gives the next term in the asymptotic expansion. In the diffusive regime \(1\ll b\ll \min \left( (\log (L/W))^2,(\eta /\omega +\omega )^{-2}\right) \),
which recovers [3, Theorem 2.12 (i)] and gives the next term in the asymptotic expansion.
Proof
In the mean-field regime \(b\ll 1\), \(f(b)=-\frac{4\sqrt{2}}{\pi \sqrt{b}}+\frac{4\sqrt{2}}{45}\pi ^3b\sqrt{b}+O(b^{7/2})\) so that the following asymptotic formula holds
In the diffusive regime \(1\ll b\ll \min \left( (\log (L/W))^2,(\eta /\omega +\omega )^{-2}\right) \), \(f(b)=-2-8\sqrt{2}\pi e^{-\pi \sqrt{2b}}\sqrt{b}\sin (\pi \sqrt{2b})+O(e^{-\pi \sqrt{2b}})\), so that the following asymptotic formula holds
The results follow from (4) and (3). \(\square \)
2.2 Dimension 2
From equation (5), it follows
where in the second equality we used the fact that
Expanding the denominator yields to
In [3], the authors estimated this trace by using a Riemann sum approximation for each summation. The real part of the first term then vanishes, while the next terms give a logarithmic contribution. Here, we no longer make the Riemann sum approximation and give a precise estimation of this term. More precisely, rewriting the first summation over the torus \((2\pi W/L){\mathbb {Z}}^2\) as a summation over \({\mathbb {Z}}^2\), it can be expressed in terms of an inhomogeneous Epstein zeta function
where
To express the leading term in terms of \({\mathcal {S}}_2(b)\) defined by
we expand \(\zeta \) using (2) implying that
We have
Thus,
and
Proposition 2.3
The real part of \({\mathcal {S}}_2(b)\) is given by
Proof
Let \(\theta \) be the third Jacobi theta function defined by \(\theta (t)=\sum _{n\in {\mathbb {Z}}}e^{-n^2t}\). Then, \({\mathcal {S}}_2(b)\) is the Laplace–Mellin transform of the squared Jacobi theta function
which can be rewritten in a more convenient way using a Jacobi identity (see, for example, [4])
Hence,
To calculate the second sum, we rewrite equation (8) as
which gives by differentiation with respect to z
or similarly, for real z,
The result follows by replacing z by \(b/(2n+1)\) in (10). \(\square \)
Proposition 2.4
For \(b\gg 1\), the real part of \({\mathcal {S}}_2(b)\) has the following asymptotic behaviour
Proof
Given an analytic function g in \(\{z\in {\mathbb {C}}\vert \ \textrm{Re}(z)\geqslant 0\}\) such that
uniformly in x on every finite interval in \(\left[ 0,\infty \right. )\) and such that
exists for all \(x\geqslant 0\) and tends to 0 as \(x\rightarrow \infty \), then Abel–Plana summation formula gives an integral representation of an alternating series through the following relation (see, for example, [1])
Define \(g_b(n):=1-\left( \frac{\pi b/(2n+1)}{\sinh (\pi b/(2n+1))}\right) ^{\!2}\) which is analytic in \(\{z\in {\mathbb {C}}\vert \ \textrm{Re}(z)\geqslant 0\}\). We have
verifying condition (i). Let \(\varphi =\frac{\pi b(2x+1)}{(2x+1)^2+4y^2}\) and \(\psi =\frac{2\pi by}{(2x+1)^2+4y^2}\). Also,
Using that \(\sinh \varphi \geqslant \varphi \), \(\sin \psi \leqslant \psi \) and \(\cosh \varphi /\sinh ^4\varphi \leqslant 16\sinh ^4(3\varphi /4)\), for all \(x,y\geqslant 0\), we have
for some positive constants \(\gamma _i\), \(i=1,\ldots ,5\), which shows that condition (ii) is satisfied. Hence, Abel–Plana summation formula implies that
We have
Let \(h(y):=-\pi (2/(2iy+1)+y/b)\) and
The function \(\textrm{Re}\,h(y)\) attains his maximum at \(y_0=\sqrt{b}e^{-i\pi /4}+i/2\) which satisfies \(h(y_0)= -\pi \sqrt{2/b}+i(\pi \sqrt{2/b}-\pi /(2b))\)and \(h''(y_0)=2\pi b^{-3/2}e^{-3i\pi /4}\). From the saddle point method, it follows that
Putting Proposition 2.3, (11) and (12) together, it comes
showing that the function \(\textrm{Re}\,{\mathcal {S}}_2(b)\) changes sign infinitely many times. \(\square \)
Proof of Theorem 1.1(ii)
Putting (3), (4), (9), Proposition 2.4 and [3, Proposition 3.3(ii)] together proves Theorem 1.1 (ii). \(\square \)
Figure 2 is a plot of \(\textrm{Re} \, \mathcal {S}_2(b)\) and Fig. 3 is a plot of \((4\pi ^2)^{-1}b^{3/4}e^{\pi \sqrt{2b}}\,\textrm{Re}\,\mathcal {S}_2(b)\) compared to its asymptotic expression.
Proposition 2.5
The two leading terms of the local density–density correlation function are of the same order for \(b\sim (\log (L/W))^2\). More precisely,
Proof
Let \(L=W^{1+a}\) and \(\omega =W^{-c}\) for some positive constants a and c. Using that \(b\sim \omega L^2/W^2>1\), it implies that \(2a>c\). Write \(b=\gamma _0(\log W)^2\) for some \(\gamma _0\), then the solution to the equation below
corresponds to the root of the following function
which is decreasing in \(\gamma _0\) and is such that \(f(a^2/(2\pi ^2))>0\). Indeed
where
Thus, \(g^\textrm{v}(a)=\beta _1^5e^{a\beta _1}>0\) implying that \(g^\textrm{iv}\) is increasing, and \(g^\textrm{iv}(0)=\beta _1^4-24\beta _2=(\log W)^4(1-24/(\sqrt{2}\pi ^3))>0\), so that \(g^\textrm{iv}(a)>0\) for all \(a>0\). We deduce that \(g(a)>0\) for all \(a>0\). Also, we have
Thus, there is a constant \(\gamma _0\in \left[ \frac{1}{2\pi ^2},\frac{2}{\pi ^2}\right] \) such that \(f(\gamma _0a^2)=0\) showing the proposition. \(\square \)
As a consequence of the above proposition, we deduce the following corollary.
Corollary 2.6
In dimension 2, the densities of states are alternately positively and negatively correlated \(O(\log (L/W))\) times.
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References
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Acknowledgements
The author thanks Antti Knowles for suggesting this problem to her and useful discussions. The author gratefully acknowledges the anonymous referees for a careful reading of the manuscript and valuable comments which greatly improved the readability of the paper.
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Appendix
Appendix
For the sake of comprehensiveness, we present the main steps of the proof of relation (4) following [3]. For more details, we refer to Section 3 of Erdős and Knowles’ paper. Let introduce the following parameters \(\rho \in (0,1/3)\), \(\mu \) such that \(\rho<\mu <1/3\) and \(\delta >0\) satisfying \(2\delta<\mu -\rho <3\delta \) and write \(\eta :=M^{-\rho }\). We also introduce the following notations taken from [3]
\(J_\nu \) denoting the \(\nu \)-th Bessel function of the first kind. For \(n\in {\mathbb {N}}\), \(E\in {\mathbb {R}}\) and \(\phi \) a test function, let \(\gamma _n(E)\) and \({\widetilde{\gamma }}_n(E,\phi )\) be defined by
\({\widehat{\phi }}\) being the Fourier transform of \(\phi \)
Then, from [2, Lemma 3.2]
The leading term \(\Theta _{\phi _1,\phi _2}^\eta \) is defined by [3, (4.60)]
where the main term \({\mathcal {V}}_{\text {main}}\) is given in [3, (3.23)]
where \({\mathcal {A}}\) is the set \({\mathcal {A}}:=(\{1,2,\ldots \}\times \{0,1,\ldots \})\backslash \{(2,0),(1,1)\}\). Let \(\psi _i(E):=\phi _i(-E)\), \(i=1,2\), where \(\phi _i\), \(i=1,2\), are test functions satisfying conditions (C). It is shown that the above expands as [3, (3.64)]
where \(\psi _i^{\leqslant ,\eta }(E):=\eta ^{-1}\psi _i(\eta ^{-1}E)\chi (M^{-\tau /2}E)\), \(i=1,2\), \(\chi \) being a smooth nonnegative symmetric function bounded by 1 satisfying \(\chi (E)=1\) for \(|E|\leqslant 1\) and \(\chi (E)=0\) for \(|E|\geqslant 2\) and \(\tau \) is a positive constant such that \(\eta \leqslant M^{-\tau }\omega \). The set \({\mathcal {A}}_\mu \) is the subset \({\mathcal {A}}_\mu :=(\{1,2,\ldots ,\lfloor M^\mu \rfloor \}\times \{0,1,\ldots ,\lfloor M^\mu \rfloor -1\})\backslash \{(2,0),(1,1)\}\), and the convolution of two functions \(\phi \) and \(\psi \) is defined by
Using relation \((2\,\text {Re}\,x_1)(2\,\text {Re}\,x_2)=2\,\text {Re}\,(x_1{\overline{x}}_2+x_1x_2)\) in (14), the authors in [3] then split \({\mathcal {V}}_{\text {main}}\) into two parts as \({\mathcal {V}}_{\text {main}}=2\,\text {Re}\,({\mathcal {V}}'_{\text {main}}+{\mathcal {V}}''_{\text {main}})+O_q(L^dM^{-q})\) where \({\mathcal {V}}'_{\text {main}}\) identifies with the \(x_1{\overline{x}}_2\) part and \({\mathcal {V}}''_{\text {main}}\) with the \(x_1x_2\) part, with \(x_1=\gamma _{2b_1+b_3+b_4}*\psi _1^{\leqslant ,\eta }\) and \(x_2=\gamma _{2b_2+b_3+b_4}*\psi _2^{\leqslant ,\eta }\). On the one hand, they show in [3, (3.76)] that
and on the other hand they split again \({\mathcal {V}}'_{\text {main}}\) into two terms \({\mathcal {V}}'_{\text {main}}={\mathcal {V}}'_{\text {main},0}-{\mathcal {V}}'_{\text {main},1}\) where [3, (3.69)]
Then, \({\mathcal {V}}'_{\text {main},0}\) is shown to be equal to [3, (3.73)]
where \(A_i:=\arcsin {E_i}\) and
Using the estimate
and by definition of the convolution product, we have
On the domain of integration, we have
which implies
Moreover by definition of \(\psi _i^{\leqslant ,\eta }\) and the test functions, we have for \(i=1,2\)
and
implying that
Also,
Thus, \({\mathcal {V}}'_{\text {main},0}\) becomes
and
since \(\alpha =1+O(\omega )\). Recall from [3, Lemma 4.17] that the expectation value of the local density of states around energy \(E\in \left[ -1+\kappa ,1-\kappa \right] \) is given by
Putting equations (13), (15) and (16) together ends the proof of (4).
We emphasise that in the above proof, which is derived in [3], is valid in all regimes, i.e. while the authors are considering the diffusive regime in their paper, the hypothesis \(\eta \gg \eta _c\) is not used at any stage of it and is thus valid in our case.
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Louis, J. Correlation of local densities of states on mesoscopic energy scales in random band matrices. Lett Math Phys 113, 79 (2023). https://doi.org/10.1007/s11005-023-01690-y
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DOI: https://doi.org/10.1007/s11005-023-01690-y