Projections of orbital measures for classical Lie groups111This work is
supported by the Russian Science Foundation under grant 14-50-00005.

Dmitry Zubov

1\. Introduction. Let $G_{n}$ be a compact Lie group from the list:
$\mathrm{SO}(2n+1)$, $\mathrm{Sp}(2n),$ $\mathrm{O}(2n)$. Consider the adjoint
action of $G_{n}$ on its Lie algebra $\mathfrak{g}_{n}$. Since $G_{n}$ is a
compact group, for each orbit of the action there exists a unique
$G_{n}$-invariant probability measure supported on this orbit. We shall call
this measure the orbital measure.

Each orbit can be parametrized by a $n$-tuple $X=(x_{1}\leq...\leq x_{n})$,
$x_{1}\geq 0$ of weakly decreasing numbers, which corresponds to the canonical
form of the matrix from $\mathfrak{g}_{n}$. Let us denote by $\mathcal{X}_{n}$
the set of such $n$-tuples.

Now we consider a natural projection map
$p_{k}^{n}:\mathfrak{g}_{n}\to\mathfrak{g}_{k}$. Let $\mu_{X}$ be a
$G_{n}$-orbital measure; then the measure $p_{k}^{n}(\mu_{X})$ is invariant
under the action of the group $G_{k}$. Each invariant measure can be
represented as a continuous combination of orbital measures: in fact, for each
Borel subset $S\in\mathfrak{g}_{k}$ we have

$p_{k}^{n}(\mu_{X})(S)=\int\limits_{Y\in\mathcal{X}_{k}}\mu^{(k)}_{Y}(S)\cdot\nu_{X,k}(dY),$

where $\mu^{(k)}_{Y},Y\in\mathcal{X}_{k}$ are $G_{k}$-orbital measures, while
$\nu_{X,k}$ is a certain probability measure on the set $\nu_{X,k}$ of
$k$-tuples.

The measure $\nu_{X,k}$ is called the radial part of the measure
$p_{k}^{n}(\mu_{X})$. In the case of the unitary group $\mathrm{U}(n)$ the
measure $\nu_{X,k}$ was computed by Olshanski [5] and Faraut [3]. Using the
method of Faraut, we compute these measures for the groups
$\mathrm{SO}(2n+1)$, $\mathrm{Sp}(2n),$ $\mathrm{O}(2n)$, expressing the
radial parts of the orbital measures in terms of determinants of B-splines
with the knots that are symmetric with respect to 0.

2\. Main result Before we formulate the main result, let us give the necessary
definitions.

According to Curry and Schoenberg, [1], the B-spline with $n$ knots
$t_{1}<...<t_{n}$ is a ${\mathit{C}}^{n-3}$-smooth function
$M_{n}(t_{1},...,t_{n};t)$ on $\mathbb{R}$ with the following properties:

1) $\mathrm{supp}(M_{n}(t_{1},...,t_{n};t)=[t_{1},t_{n}]$;

2) The function $M_{n}(t_{1},...,t_{n};t)$ is the polynomial in $t$ of degree
$n-2$ on each subinterval $(t_{i},t_{i+1})$;

3) $\int_{\mathbb{R}}M_{n}(t_{1},...,t_{n};t)dt=1$.

Note that the conditions above define the B-spline $M_{n}(t_{1},...,t_{n};t)$
in a unique way.

Recall also that the divided differences of a function $f$ are defined by
induction as follows:

$f[t_{1},t_{2}]=\frac{f(t_{1})-f(t_{2})}{t_{1}-t_{2}};\,...;\,f[t_{1},...,t_{n}]=\frac{f[t_{1},t_{2},...,t_{n-1}]-f[t_{2},...,t_{n-1},t_{n}]}{t_{1}-t_{n}}.$

The Hermite-Genocchi formula connects B-splines with divided differences. This
formula tells that for a function $f$ which has a piecewise continuous
derivative of degree $n-1$ there is an equality

$f[t_{1},...,t_{n}]=\frac{1}{(n-1)!}\int_{\mathbb{R}}f^{(n-1)}(t)M_{n}(t_{1},...,t_{n};t)dt.$

Using the method of Faraut [3],for the groups
$\mathrm{SO}(2n+1),\mathrm{Sp}(2n),\mathrm{SO}(2n)$ we obtain the formulas for
$\nu_{X,k}$ in terms of determinants of B-splines (Theorem 1).

Let us denote by

$V_{n}(T):=V_{n}(t_{1},...,t_{n})=\prod\limits_{1\leq i<j\leq m}(t_{i}-t_{j})$

the Vandermonde determinant in the variables $T=(t_{1},...,t_{n})$, and let us
also set

$v_{n}(dT)=V_{n}(T^{2})\cdot\prod_{i=1}^{n}dt_{i}=V_{n}(t_{1}^{2},...,t_{n}^{2})\cdot\prod_{i=1}^{n}dt_{i}.$

Finally, for arbitrary $j,m\in\mathbb{N}$ we shall use a shortened notation

$M_{2m+2}(\pm
t|_{j}^{m+j};t):=M_{2m+2}(-t_{m+j},...,-t_{j},t_{j},...,t_{m+j};t).$

Theorem 1. The radial part $\nu_{X,k}$ of projection of the $G_{n}$-orbital
measure $\mu_{X}$ is given by the following formula:

$\nu_{X,k}(dY)=\frac{c(n,k)}{\prod\limits_{j-i\geq{n-k+1}}(x_{j}^{2}-x_{i}^{2})}\det\left[\Delta
M_{2n-2k+2}(\pm x|_{j}^{j+n-k};y_{i})\right]_{i,j=1}^{k}v_{k}(dY),$

where $\Delta$ is the following differential operator

$\Delta=-y\frac{d}{dy}+\varkappa(n,k),$

and $c(n,k),\varkappa(n,k)$ are the constants that depend on $n$ and $k$.

If $G_{n}=\mathrm{SO}(2n+1)$ or $G_{n}=\mathrm{Sp}(2n)$, then
$\varkappa(n,k)=0$ and

$c(n,k)=\frac{(2n-2k)!!}{(2n)!!}\prod_{i=0}^{k-1}{2n-2k+2i+2\choose 2i+1}.$

In the case when $G_{n}=\mathrm{O}(2n)$ we have $\varkappa(n,k)=2(n-k)$ and

$c(n,k)=\frac{(2n-2k-1)!!}{(2n-1)!!}\prod_{i=0}^{k-1}{2n-2k+2i+1\choose 2i}.$

Remark. Note that the derivative of the B-spline $M_{m},m\in\mathbb{N}$ can be
expressed as the difference of two splines of order $m-1$; namely, for any $m$
points $t_{1},...,t_{m}\in\mathbb{R}$ the following equality holds (see [6]):

$\frac{t_{m}-t_{1}}{m-1}\cdot\frac{d}{dt}M_{m}(t_{1},...,t_{m};t)=M_{m-1}(t_{1},...,t_{m-1};t)-M_{m-1}(t_{2},...,t_{m};t).$

3\. Proof of Theorem 1.

3.1. The Laplace transform of orbital measures. We define the Laplace
transform of the orbital measure $\mu$ on the Lie algebra $\mathfrak{g}_{n}$
of the group $G_{n}$ as the orbital integral

$\widehat{\mu_{X}}(T)=\int_{\mathrm{Orbit}(X)}e^{\mathrm{tr}(TX)}\mu(dX)=\int_{G_{n}}e^{\mathrm{tr}(T\,\mathrm{Ad}_{g}(X))}\,dg,$

where $dg$ is the Haar measure on $G_{n}$.

In general, the Laplace transform for the measures with a compact support is
defined on the complexification of the Lie algebra. But let us notice that the
funciton $\widehat{\mu_{X}}(T)$ is invariant under the adjoint action of
$G_{n}$. Thus we can think of $T$ as of the matrix of canonical form and
consider the Laplace transform of the orbital measure as a function on the
coordinate space $\mathbb{C}^{n}$.

Due to the Harish-Chandra theorem (see [4], Theorem 2), one can obtain the
following formulas for the Laplace transform of the orbital measure $\mu_{X}$
for $G_{n}$:

$\displaystyle\widehat{\mu_{X}}^{(B_{n},C_{n})}(t_{1},...,t_{n})$
$\displaystyle=\frac{(2n-1)!(2n-3)!...1!}{V_{n}(X^{2})V_{n}(T^{2})}\det\left[\frac{\sinh(t_{i}x_{j})}{t_{i}x_{j}}\right]_{i,j=1}^{n},$
(1) $\displaystyle\widehat{\mu_{X}}^{(D_{n})}(t_{1},...,t_{n})$
$\displaystyle=\frac{(2n-2)!...2!}{V_{n}(X^{2})V_{n}(T^{2})}\det\left[\cosh(t_{i}x_{j})\right]_{i,j=1}^{n}.$
(2)

In the formulas (1), (2) $B_{n},C_{n}$ and $D_{n}$ are the standard notations
for the classical series of Lie algebras of groups $\mathrm{SO}(2n+1)$,
$\mathrm{Sp}(2n)$ and $\mathrm{O}(2n)$ respectively.

These formulas imply that, up to multiplicative constants, the Laplace
transform of the orbital measure $\mu_{X}$ has the form

$D_{n}(f;T;X)=\frac{\det\left[f(t_{i}x_{j})\right]_{i,j=1}^{n}}{V_{n}(T^{2})V_{n}(X^{2})},$

where $f$ is an analytic function with the Taylor series

$f(z)=\sum_{m=0}^{\infty}\frac{z^{2m}}{2^{2m}m!(\alpha+1)_{m}}.$

Here $(\alpha+1)_{m}=(\alpha+1)(\alpha+2)...(\alpha+m)$ is the Pochhammer
symbol, and the parameter $\alpha$ is equal to $1/2$ in the case of the groups
$\mathrm{SO}(2n+1)$ and $\mathrm{Sp}(2n)$ and $\alpha=-1/2$ when
$G_{n}=\mathrm{O}(2n)$.

3.2. Projections and the Laplace transform. For any Borel measure $\mu$ on
$\mathfrak{g}_{n}$ the restriction of the Laplace transform to
$\mathfrak{g}_{k}$ is equal to the Laplace transform of the measure
$\mu^{(k0}$ which is the image of $\mu$ under projection onto
$\mathfrak{g}_{k}$. Thus the problem is reduced to the computation of the
quantities $D_{n}(f;t_{1},...,t_{k},0,...,0;X)$.

Lemma 1. If $f$ is an even analytic function in some neighbourhood of 0, then
the quantity $D_{n}(f;t_{1},...,t_{k},0,...,0;X)$ can be expressed in terms of
divided differences of the functions $\varphi_{i}(y)=f(t_{i}\sqrt{y})$ so that

$D_{n}(f;t_{1},...,t_{k},0,...,0;X)=\frac{a_{n-k}(f)}{V_{n}(X^{2})V_{k}(T^{2})(t_{1}...t_{k})^{2(n-k)}}\times$

$\times\prod\limits_{1\leq j-i\leq
n-k}(x_{j}^{2}-x_{i}^{2})\cdot\det\left[\varphi_{i}[x_{j}^{2},...,x^{2}_{j+n-k}]\right]_{i,j=1}^{k}.$
(3)

Here $a_{n}(f)=\prod_{j=0}^{n-1}c_{2j}$, where $c_{2j}$ are the even
coefficients of the Taylor series of $f$.

The proof of this lemma is similar to the ones of Theorem 4.1 and 5.3 in the
paper [3]. $\square$

3.3. Doubling the knots. Since the functions $\varphi_{i}$ from Lemma 1 depend
on $\sqrt{z}$, the formula (3) becomes inconvenient for applying the Hermite-
Gennochi formula. But we can overcome this difficulty, if we double the the
number of knots.

Lemma 2. Let $f(z)$ be an analytic function in a certain neighbourhood of $0$,
and let $\varphi(z)=f(\sqrt{z})$. Then, given $m$ points $0<z_{1}<...<z_{m}$,
the following equality for the divided differences holds:

$\varphi[z_{1}^{2},...,z_{m}^{2}]=g[-z_{m},...,-z_{1},z_{1},...,z_{m}],$

where $g(z)=zf(z)$.

Proof. By the definition of divided differences, we have

$\varphi[z_{1}^{2},...,z_{m}^{2}]=\sum\limits_{l=1}^{m}f(z_{l})\prod\limits_{r\neq
l}\frac{1}{z_{l}^{2}-z_{r}^{2}}=$
$=2\sum\limits_{l=1}^{m}\frac{z_{l}f(z_{l})}{2z_{l}}\prod\limits_{r\neq
l}\frac{1}{z_{l}^{2}-z_{r}^{2}}=g[-z_{m},...,-z_{1},z_{1},...,z_{m}],$

where $g(z)=zf(z)$. Lemma 2 is proved. $\square$

3.4. Applying the Hermite-Genocchi formula.

Due to Lemmas 1 and 2 the computation of $D_{n}(f;t_{1},...,t_{k},0,...,0;X)$
has been reduced to the computation of the determinant of divided differences
of the functions $g_{i}(z)=zf(t_{i}z)$.

Applying the Hermite-Genocchi formula, we get the following equality:

$g_{i}[-x_{j+n-k},...,-x_{j},x_{j},...,x_{j+n-k}]=\frac{1}{(2n-2k+1)!}\\\
\,\,\,\times\int\limits_{\mathbb{R}}(g_{i}(z))^{(2n-2k+1)}M_{2n-2k+2}(-x_{j+n-k},...,-x_{j},x_{j},...,x_{j+n-k};z)dz=\\\
=\frac{-1}{(2n-2k+1)!}\int\limits_{\mathbb{R}}(g_{i}(z))^{(2n-2k)}M^{\prime}_{2n-2k+2}(\pm
x|_{j}^{j+n-k};z)dz.$ (4)

Now we want to express $(g_{i}(z))^{(2n-2k)}=(zf_{i}(z))^{(2n-2k)}$ in terms
of $f_{i}(z)=f(t_{i}z)$.

By the definition of $f$, for any $m\in\mathbb{N}\cup\\{0\\}$ we have

$(g_{i}(z))^{(2n-2k)}=\sum_{m=0}^{\infty}\frac{t_{i}^{2m}z^{2m+1}}{2^{2m}m!(\alpha+1)_{m}}\cdot\gamma_{\alpha,m},$

where

$\,\gamma_{\alpha,m}=\gamma_{\alpha}(m):=\frac{(m+\frac{3}{2})_{n-k}}{(m+\alpha+1)_{n-k}}.$

The problem is now divided into two cases:

(B-C) $G_{n}=\mathrm{Sp}(2n)$ or $\mathrm{SO}(2n+1)$. In this case
$\alpha=1/2$, $\gamma_{\alpha,m}=1$ and

$(g_{i}(z))^{(2n-2k)}=t_{i}^{2n-2k}zf(t_{i}z).$

(D) $G_{n}=\mathrm{SO}(2n)$. In this case $\alpha=-1/2$,
$\gamma_{\alpha,m}=1+\frac{2(n-k)}{2m+1}$ and therefore,

$(g_{i}(z))^{(2n-2k)}=t_{i}^{2n-2k}\cdot\left(zf(t_{i}z)+(2n-2k)t_{i}^{-1}f^{(-1)}(t_{i}z)\right),$

where $f^{(-1)}(\cdot)$ is the primitive of $f$ such that $f^{(-1)}(0)=0$.

In both cases the divided differences of $g_{i}$’s can be expressed as
follows:

$g_{i}[-x_{j+n-k},...,-x_{j},x_{j},...,x_{j+n-k}]=\\\
=\frac{t_{i}^{2n-2k}}{(2n-2k+1)!}\cdot\int\limits_{\mathbb{R}}f(t_{i}y)\,\Delta
M_{2n-2k+2}(\pm x|_{j}^{j+n-k};y)dy,$ (5)

where the operator $\Delta$ has the form

$\Delta=\Delta_{1/2}=-y\frac{d}{dy}\,\text{\, (the (B-C) case)}$

or

$\Delta=\Delta_{-1/2}=-y\frac{d}{dy}+2(n-k)\,\,\text{\, (the (D) case)}.$

Now let us apply the Binet-Cauchy identity to the formula (5):

$\displaystyle\det\left[g_{i}[-x_{j},...,-x_{j+n-k},x_{j+n-k},...,x_{j}]\right]=\prod_{i=1}^{k}t_{i}^{2n-2k-1}\cdot\frac{1}{((2n-2k+1)!)^{k}}\times$
(6)
$\displaystyle\times\int\limits_{\mathbb{R}^{k}_{+}}\det\left[f(t_{i}y_{j}))\right]_{i,j=1}^{k}\det\left[\Delta
M_{2n-2k+2}(\pm x|_{j}^{j+n-k};y_{i})\right]_{i,j=1}^{k}dy_{1}...dy_{k}.$

To complete the proof of Theorem 1 it suffices to compare the LHS and RHS in
(3) and to apply the inverse Laplace transform to both sides of that equation.
Theorem 1 is proved. $\square$.

Remark. After this work was finished, the author became aware of an
interesting paper of Defosseux [2], where she considered, for the matrix
$X\in\mathfrak{g}_{n}$, the array $M(X)$ of the union of spectra of $X$ and of
all its images under projections $p_{k}^{n},k=1,...,n-1$. In turns out that,
the radial part of projection of the orbital measure $\mu_{X}$ can be derived
as a correlation function of the determinantal point process on arrays $M(X)$
with the kernel given explicitely (see Theorem 6.3 of the work of Defosseux).

Acknowledgement. I am deeply grateful to Grigori Olshanski, who suggested me
this problem, for many useful discussions and paying attention to this work.

Steklov Mathematical Institute of the Russian Academy of Sciences,
8, Gubkina str., Moscow, 119991, Russian Federation

e-mail<EMAIL_ADDRESS>
## References

  * [1] H.B. Curry, I.J.Schoenberg, On Polya frequency functions, J. Analyse Math, 17 (1966), 71-107.
  * [2] M. Defosseux, Orbit measures, random matrix theory and interlaced determinantal processes, Ann. Inst. H. Poincaré Probab. Statist. 46:1, (2010), 209-249.
  * [3] J. Faraut, Rayleigh theorem, projection of orbital measures and spline functions, Adv. Pure Appl. Math. 6:4 (2015), 261-283.
  * [4] Harish-Chandra, Differential operators on a semisiple Lie algebra, American Journal of Mathematics, Vol. 79:1, (1957), 87-120.
  * [5] G. Olshanski, Projections of orbital measures, Gelfand-Tsetlin polytopes, and splines Journal of Lie Theory 23:4, (2013),1011-1022.
  * [6] G.M. Phillips, Interpolation and approximation by polynomials, CMS Books in Math. vol.14. Springer-Verlag, New York, 2003


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# Signature of Scramblon Effective Field Theory in Random Spin Models

Zeyu Liu  Department of Physics, Fudan University, Shanghai, 200438, China
Pengfei Zhang  Department of Physics, Fudan University, Shanghai, 200438,
China  Center for Field Theory and Particle Physics, Fudan University,
Shanghai, 200438, China

###### Abstract

Information scrambling refers to the propagation of information throughout a
quantum system. Its study not only contributes to our understanding of
thermalization but also has wide implications in quantum information and black
hole physics. Recent studies suggest that information scrambling is mediated
by collective modes called scramblons. However, a criterion for the validity
of scramblon theory in a specific model is still missing. In this work, we
address this issue by investigating the signature of the scramblon effective
theory in random spin models with all-to-all interactions. We demonstrate
that, in scenarios where the scramblon description holds, the late-time
operator size distribution can be predicted from its early-time value,
requiring no free parameters. As an illustration, we examine whether Brownian
circuits exhibit a scramblon description and obtain a positive confirmation
both analytically and numerically. We also discuss the prediction of multiple-
quantum coherence when the scramblon description is valid. Our findings
provide a concrete experimental framework for unraveling the scramblon field
theory in random spin models using quantum simulators.

_Introduction.–_ Quantum thermalization Srednicki (1994); Deutsch (1991) in
many-body systems requires the initial local information to become scrambled
throughout the entire system under unitary evolutions Hayden and Preskill
(2007); Sekino and Susskind (2008); Shenker and Stanford (2015). This process,
known as information scrambling, serves as a crucial link connecting condensed
matter, quantum information, and gravity physics. In the Heisenberg picture,
information scrambling is described by the growth of operator size Roberts
_et al._ (2015); Nahum  _et al._ (2018); von Keyserlingk  _et al._ (2018);
Khemani  _et al._ (2018); Hunter-Jones (2018); Chen and Lucas (2021); Lucas
(2019); Chen  _et al._ (2020); Lucas and Osborne (2020); Yin and Lucas (2021);
Zhou and Swingle (2021); Dias  _et al._ (2021); Wu  _et al._ (2021); Zhang and
Yu (2022); Zhang and Gu (2022); Jian  _et al._ (2021), with its expectation
value being associated with out-of-time-order correlators (OTOCs). In
$N$-qubit chaotic systems with all-to-all interactions, the average operator
size exhibits universal exponential growth in the early-time regime $t\ll
t_{\text{sc}}$, serving as a signature of quantum chaos. Here
$t_{\text{sc}}\propto\ln N$ denotes the scrambling time. Conversely,
understanding information scrambling in the late-time regime with $t\sim
t_{\text{sc}}$ poses greater challenges due to its non-perturbative nature.

Recently, inspired by the diagrammatic analysis in the Sachdev-Ye-Kitaev (SYK)
model Kitaev (2015); Maldacena and Stanford (2016); Kitaev and Suh (2018);
Chowdhury  _et al._ (2022), the scramblon effective field theory (SEFT) was
proposed as a unified description of information scrambling in both regimes Gu
_et al._ (2022). The key assumption is that for long time separations, only
out-of-time-order correlations are essential, which are mediated by collective
modes called scramblons Gu and Kitaev (2019); Gu  _et al._ (2022). The SEFT is
specified by a set of Feynman rules. For systems with time-reversal symmetry,
this includes the scramblon propagator $\lambda=e^{\varkappa t}/C$ (with
quantum Lyapunov exponent $\varkappa$ and numerical factor $C\propto N$) and
the scattering vertex between a pair of Hermitian operators $\hat{O}$ and $m$
scramblons

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(2)

Here the wavy lines represent the scramblon modes. In the remaining part of
the manuscript, we omit the argument $t_{12}$ when $t_{12}=0$. In particular,
we have $\Upsilon^{m}_{O}=\langle\hat{O}^{2}\rangle$, where we define
$\langle\hat{M}\rangle=\mathcal{D}^{-1}\text{tr}[\hat{M}]$ with Hilbert space
dimension $\mathcal{D}$.

By employing the SEFT, analytical results for OTOCs Gu  _et al._ (2022); Zhang
(2023), operator size distribution Zhang and Gu (2022); Zhang and Yu (2022);
Zhou  _et al._ , and quantum teleportation Zhang  _et al._ can be obtained in
SYK-like models. However, the exact class of models where the scramblon theory
is valid remains largely unexplored. The main challenge lies in the absence of
a valid diagrammatic expansion in general $0+1$-D models. On the other hand,
significant advancements in the quantum simulation of many-body systems have
been witnessed in recent years. In particular, considerable attention has been
paid to the experimental study of information scrambling by measuring OTOCs
Islam  _et al._ (2015); Li  _et al._ (2017); Gärttner  _et al._ (2017);
Brydges  _et al._ (2019); Sánchez  _et al._ (2019); Landsman  _et al._ (2019);
Joshi  _et al._ (2020); Blok  _et al._ (2021); Domínguez  _et al._ (2021);
Domínguez and Álvarez (2021); Mi  _et al._ (2021); Cotler  _et al._ (2022);
Sánchez  _et al._ (2022). Furthermore, concrete protocols have been proposed
for measuring the operator size distribution Qi  _et al._ (2019).

Figure 1: The schematics illustrate the consistency relation between the
early-time operator size distribution and the late-time operator size
distribution, valid under the assumption of SEFT.

In this work, motivated by these developments, we propose a scheme to test the
validity of SEFT on quantum simulators. We focus on random spin models with
all-to-all interactions, which has been realized in various experimental
platforms. By generalizing the size-charge relation for Majorana systems, we
derive the scramblon theory prediction of the operator size distribution in
random spin models. The results show that the late-time operator size
distribution can be predicted from its early-time value without any prior
knowledge of $C$ or $\Upsilon^{m}_{O}$. This consistency relation serves as a
smoking gun for the SEFT. As an illustration, we apply our scheme to Brownian
circuits Zhou and Chen (2019), where the operator size distribution can be
efficiently simulated by solving classical differential equations. Both
analytical and numerical results confirm the validity of its SEFT description.
We also discuss the prediction of multiple-quantum coherence (MQC) when the
SEFT is valid. Our result is of fundamental interest for understanding the
scrambling dynamics in generic quantum systems.

_Operator size & total spin.–_ We focus on systems of $N$ spins
$\hat{\bm{s}}^{j}=\hat{\bm{\sigma}}^{j}/2$ with time-reversal-invariant all-
to-all interactions. Here $j=$1, 2, …, $N$ labels different spins. Protocols
without the time-reversal symmetry are discussed in later. An Hermitian
operator $\hat{O}(t)$ can be expanded using the complete basis of Pauli
operators

$\hat{O}(t)=\sum_{n}\sum_{j_{1}<\cdots<j_{n}}\sum_{\alpha_{1}\cdots\alpha_{n}}C_{j_{1}\alpha_{1}\cdots
j_{n}\alpha_{n}}\hat{\sigma}_{\alpha_{1}}^{j_{1}}\hat{\sigma}_{\alpha_{2}}^{j_{2}}\cdots\hat{\sigma}_{\alpha_{n}}^{j_{n}},$
(3)

where $\alpha_{k}\in\\{x,y,z\\}$ labels different Pauli matrices. The length
$n$ of $\sigma_{\alpha_{1}}^{j_{1}}\cdots\sigma_{\alpha_{n}}^{j_{n}}$ is
defined as the size of this basis. The operator size distribution $P(n,t)$
counts the operator weight on bases with size $n$ as

$P(n,t)=\sum_{j_{1}<\cdots<j_{n}}\sum_{\alpha_{1}\cdots\alpha_{n}}(C_{j_{1}\alpha_{1}\cdots
j_{n}\alpha_{n}})^{2},$ (4)

with $n\in\\{1,2,...,N\\}$. The total probability is then conserved
$\sum_{n}P(n,t)=1$ due to the unitarity if we normalize
$\langle\hat{O}^{2}\rangle=1$. In the thermodynamic limit $N\gg 1$, we further
introduce a continuous variable $s=n/N\in[0,1]$ with continuous distribution
$\mathcal{P}(s,t)\equiv NP(sN,t)$, which is normalized as
$\int_{0}^{1}ds~{}\mathcal{P}(s,t)=1$.

In Ref. Qi and Streicher (2019), the authors establish a connection between
the operator size in Majorana systems and the charge operator in the doubled
system. This formulation enables a systematic investigation of the
distribution of operator sizes within the framework of the SEFT. We present a
generalization of this relation to spin models. We first introduce an
auxiliary system consisting of $N$ spins $\hat{\bm{s}}^{\prime
j}=\hat{\bm{\tau}}^{j}/2$. The doubled system is prepared in the tensor
product of singlet states $|\text{EPR}\rangle=\prod_{\otimes
j}|\text{S}\rangle_{j}=\prod_{\otimes
j}(\ket{\uparrow}_{\sigma^{j}}\ket{\downarrow}_{\tau^{j}}-\ket{\downarrow}_{\sigma^{j}}\ket{\uparrow}_{\tau^{j}})$.
Consequently, we have
$\hat{\bm{s}}^{j}_{\text{tot}}|\text{EPR}\rangle=(\hat{\bm{s}}^{\prime
j}+\hat{\bm{s}}^{j})|\text{EPR}\rangle=0$. To study the size of an operator
$\hat{O}(t)$, we map the operator to a state by
$|O(t)\rangle=O(t)|\text{EPR}\rangle$ Qi and Streicher (2019). The Eq. (3)
then becomes an expansion in orthonormal states. The crucial observation is

$|\text{T},\alpha\rangle_{j}\equiv\hat{\sigma}^{j}_{\alpha}|\text{S}\rangle_{j},\
\ \ \ \
(\hat{\bm{s}}^{j}_{\text{tot}})^{2}|\text{T},\alpha\rangle_{j}=2|\text{T},\alpha\rangle_{j}.$
(5)

Here, T indicates that the state is a spin-triplet. From a group theoretical
perspective, this is due to $\hat{\sigma}^{j}_{\alpha}$ being a spin-$1$
spherical tensor operator. Consequently, the size of a string of Pauli
operators is equal to the number of triplet states, counting over different
site $j$. This gives

$\mathcal{P}(s,t)=\langle\hat{O}(t)|\delta(s-\sum_{j}(\hat{\bm{s}}^{j}_{\text{tot}})^{2}/2N)|\hat{O}(t)\rangle.$
(6)

When calculating the operator size distribution, it is more convenient to
introduce the generating function $\mathcal{S}(\nu,t)=\int
ds~{}\mathcal{P}(s,t)e^{-\nu s}$. Using Eq. (6), we find

$\mathcal{S}(\nu,t)=e^{-\frac{3\nu}{4}}\langle\hat{O}(t)|e^{-\frac{\nu}{4N}\sum_{j}\bm{\sigma}^{j}\cdot\bm{\tau}^{j}}|\hat{O}(t)\rangle.$
(7)

Note that Eq. (6) and Eq. (7) not only serve as a starting point for
theoretical calculations but also provide a concrete protocol for measuring
the operator size distribution on quantum simulators. This protocol requires
the ability to prepare the initial EPR state and reverse the Hamiltonian,
which are feasible with state-of-the-art cold atom systems and superconducting
qubits.

$\hat{O}^{\dagger}$$\hat{O}^{\ }$$\hat{\sigma}^{j_{1}}_{\alpha_{1}}$$\hat{\sigma}^{j_{1}}_{\alpha_{1}}$$\hat{\sigma}^{j_{3}}_{\alpha_{3}}$$\hat{\sigma}^{j_{3}}_{\alpha_{3}}$$\hat{\sigma}^{j_{2}}_{\alpha_{2}}$$\hat{\sigma}^{j_{2}}_{\alpha_{2}}$ | $\hat{O}(t)^{\dagger}$$\hat{O}(t)^{\ }$$\hat{\sigma}^{j_{1}}_{\alpha_{1}}$$\hat{\sigma}^{j_{1}}_{\alpha_{1}}$$\hat{\sigma}^{j_{2}}_{\alpha_{2}}$$\hat{\sigma}^{j_{2}}_{\alpha_{2}}$
---|---
(a) General diagram for $\mathcal{S}(\nu,t)$ | (b) The early-time limit $(\lambda\ll 1)$
Figure 2: Typical diagrams of $\mathcal{S}(\nu,t)$ are shown in (a) for the
full time range and (b) in the early-time limit $\lambda\ll 1$. Wavy lines
represent the propagator of scrambling modes, while solid lines represent the
microscopic operators $\hat{O}(t)$ or $\hat{\sigma}^{j}_{\alpha}$. The
vertices within the dashed red circle become identical for
$\hat{O}=\hat{\sigma}^{j_{1}}_{\alpha_{1}}$, indicating a consistency relation
between different time regimes.

_SEFT calculation.–_ Assuming the validity of the scramblon description, the
generating function $\mathcal{S}(\nu,t)$ can be computed in closed-form.
Performing the Taylor expansion of (7), the result contains general OTOCs
between $\hat{O}$ and Pauli strings

$\mathcal{S}=e^{-\frac{3\nu}{4}}\sum_{m}\left(\frac{\nu}{4N}\right)^{m}\sum_{\\{j_{k},\alpha_{k}\\}}\langle\hat{O}(t)\hat{\sigma}^{j_{1}}_{\alpha_{1}}...\hat{\sigma}^{j_{m}}_{\alpha_{m}}\hat{O}(t)\hat{\sigma}^{j_{1}}_{\alpha_{1}}...\hat{\sigma}^{j_{m}}_{\alpha_{m}}\rangle.$
(8)

We have neglected contributions from collision terms where
$j_{k}=j_{k^{\prime}}$, which become negligible in the thermodynamic limit
$N\rightarrow\infty$. In the SEFT picture, each pair of
$\sigma^{j_{k}}_{\alpha_{k}}$ operators can emit an arbitrary number of
scramblons, all of which are ultimately absorbed by $\hat{O}$ operators in the
future. A typical diagram with $m=3$ is shown in FIG. 2 (a). By summing up all
possible configurations, result can be expressed as

$\mathcal{S}=e^{-\frac{3\nu}{4}}\sum_{m}\frac{1}{m!}\left(\frac{\nu}{4}\right)^{m}\sum_{\\{n_{k},\alpha_{k}\\}}\frac{(-\lambda)^{\sum_{m}{n_{m}}}}{m!}\Upsilon^{\sum_{m}{n_{m}}}_{O}\Upsilon_{\alpha_{1}}^{n_{1}}...\Upsilon_{\alpha_{m}}^{n_{m}}.$
(9)

Here, we define averaged vertex functions as
$\Upsilon_{\alpha}^{m}\equiv\sum_{j}\Upsilon_{\sigma^{j}_{\alpha}}^{m}/N$ for
conciseness. In comparison to the results for Majorana fermions Zhang and Gu
(2022); Zhang and Yu (2022), we need to include an additional summation over
$\alpha$ to account for the contributions from different Pauli matrices. By
utilizing Eq. (2), we can perform the summation and obtain

$\mathcal{S}(\nu,t)=\int_{0}^{\infty}dy~{}h_{O}(y)e^{-\frac{\nu}{4}({3}-\sum_{\alpha}f_{\alpha}(\lambda
y))},$ (10)

where we have $f_{O}(x)\equiv\sum_{m}(-x)^{m}\Upsilon_{O}^{m}/m!=\int
dy~{}h_{O}(y)e^{-xy}$ describing the perturbed two-point functions with
scramblon fields Gu  _et al._ (2022). The operator size distribution can then
be obtained by taking the inverse Laplace transform
$\mathcal{P}(s,t)=\int_{0}^{\infty}dy~{}h_{O}(y)~{}\delta\Big{(}s-\frac{{3}-\sum_{\alpha}f_{\alpha}(\lambda
y)}{4}\Big{)}.$

_Consistency relation.–_ We are ready to show that Eq. (10) indicates non-
trivial consistency relation of the operator size distribution when the SEFT
description is valid. We take an average of $\mathcal{\mathcal{P}}(s,t)$ over
single Pauli operators $\hat{O}=\hat{\sigma}^{j}_{\alpha}$

$\overline{\mathcal{P}}(s,t)=\int_{0}^{\infty}dy~{}\overline{h}(y)\delta\Big{(}s-s_{\text{sc}}({{1}-\overline{f}(\lambda
y))}\Big{)},$ (11)

with
$\overline{g}(x)\equiv\sum_{\alpha}g_{\alpha}(x)/3\equiv\sum_{\alpha,j}g_{\sigma_{\alpha}^{j}}(x)/3N$.
For systems in which different spins are equivalent, the averaging over $j$
can be omitted. $s_{\text{sc}}=3/4$ is the typical size for a maximally
scrambled operator in spin models, where all possible operators appear with
equal probability. This can be seen by taking $\lambda\rightarrow\infty$ in
Eq. (11), which gives
$\overline{\mathcal{P}}(s,\infty)=\delta(s-s_{\text{sc}})$ for chaotic models
with $\overline{f}(\infty)=0$ Gu  _et al._ (2022). It is worth noting that a
similar expression holds for systems with Majorana fermions Zhang and Gu
(2022), with $s_{\text{sc}}=1/2$. Therefore, the consistency relation derived
in this section can also be applied to Majorana systems.

The key observation is that the averaged generating function at different
times $\nu$ and $t$ relies solely on a single function $\overline{h}(y)$, as
$\overline{f}(x)$ is its Laplace transform. Our strategy is to first express
$\overline{h}(y)$ in terms of the early-time size distribution with
$N^{-1}\ll\lambda\ll 1$, and then establish its relation to the distribution
at late times. A diagrammatic illustration is presented in FIG. 2. In the
early-time regime with $N^{-1}\ll\lambda_{0}=e^{\varkappa t_{0}}/C\ll 1$, we
expand $\overline{f}(\lambda_{0}y)=1-\lambda_{0}y\overline{\Upsilon}^{1}$.
Using Eq. (11), we find

$\displaystyle\overline{\mathcal{P}}(s,t_{0})$
$\displaystyle=({s_{\text{sc}}\lambda_{0}\overline{\Upsilon}^{1}})^{-1}\overline{h}\Big{(}s({s_{\text{sc}}\lambda_{0}\overline{\Upsilon}^{1}})^{-1}\Big{)},$
(12) $\displaystyle\overline{s_{0}}$
$\displaystyle=\int_{0}^{1}ds~{}s~{}\overline{\mathcal{P}}(s,t_{0})=s_{\text{sc}}\lambda_{0}(\overline{\Upsilon}^{1})^{2}.$

The main obstacle in extracting $\overline{h}(y)$ from
$\overline{\mathcal{P}}(s,t_{0})$ lies in the unknown coefficients $C$ and
$\overline{\Upsilon}^{1}$. Fortunately, these coefficients ultimately cancel
out, resulting in no free parameters in the consistency relation. To
demonstrate this, let us consider the distribution (11) for a general time
$\lambda$. By introducing
$y=s_{1}/({s_{\text{sc}}\lambda_{0}\overline{\Upsilon}^{1}})$, it becomes
straightforward to show that
¯P(s,t)=∫0∞ds1¯P(s1,t0)δ(s-ssc+ssc∫0∞ds2¯P(s2,t0)exp(-s1s2ssc¯s0eϰ(t-t0))).
This is the main result for this work. Note that we can safely extend the
integral to $[0,\infty]$ since the early-time distribution
$\overline{\mathcal{P}}(s,t_{0})$ is only support near $s\ll 1$. Since
(Signature of Scramblon Effective Field Theory in Random Spin Models) should
be valid for arbitrary $s$ and $t$, it is highly restrictive and serves as a
signature of SEFT in systems with all-to-all interactions. Eq. (Signature of
Scramblon Effective Field Theory in Random Spin Models) can also be applied to
predict the averaged operator size

$\frac{\overline{s}}{s_{\text{sc}}}=1-\int_{0}^{\infty}\mathrm{d}s_{1}ds_{2}\overline{\mathcal{P}}(s_{1},t_{0})\overline{\mathcal{P}}(s_{2},t_{0})e^{-\frac{s_{1}s_{2}}{s_{\text{sc}}\overline{s_{0}}}e^{\varkappa(t-t_{0})}}.$
(13)

Figure 3: The operator size distribution of the Brownian circuits is shown for
$N=10^{4}$. In this plot, different colors correspond to different times $t$.
The dashed lines represent the results obtained from numerical simulations,
while the solid lines depict the analytical solution given by Eq. (16). The
region shaded with light color is obtained by numerically evaluating the
integral in (Signature of Scramblon Effective Field Theory in Random Spin
Models) for various $t_{0}$ values, while ensuring that
$\overline{s_{0}}/s_{\text{sc}}\in[0.03,0.05]$.

Protocol: We summarize the concrete experimental protocol to test the validity
of the SEFT on quantum simulators:

Step 1: Perform measurements of the operator size distribution
$\mathcal{P}(s,t)_{\text{exp}}$ for different times $t$ with
$\hat{O}=\hat{\sigma}^{j}_{\alpha}$, following the size-total spin relation or
other existing protocols Qi  _et al._ (2019); Blocher  _et al._ (2023).
Compute the averaged distribution $\overline{\mathcal{P}}(s,t)_{\text{exp}}$
over sites $j$ and $\alpha\in\\{x,y,z\\}$.

Step 2: Choose a set of ${t_{0}^{k}}$ with $t_{0}^{k}\ll\log N$, compute the
averaged operator size $\overline{s_{0}}$, and extract $\varkappa$ by
performing a linear fit of the averaged size $\log\overline{s_{0}}$ for
different $t_{0}$ values.

Step 3: Select a specific $t_{0}\ll\log N$, compute the numerical integral in
Eq. (Signature of Scramblon Effective Field Theory in Random Spin Models)
using the experimental result $\overline{\mathcal{P}}(s,t_{0})_{\text{exp}}$
for a general time $t$ to obtain the theoretical prediction
$\overline{\mathcal{P}}(s,t)_{\text{th}}$. Compare this prediction with the
experimental data $\overline{\mathcal{P}}(s,t)_{\text{exp}}$.

We finally address the generalization to systems without time-reversal
symmetry or at finite temperatures. (i). In the absence of time-reversal
symmetry, the vertices for absorbing scramblons in the future and emitting
scramblons in the past become distinct, denoted as $\Upsilon^{\text{R},m}$ and
$\Upsilon^{\text{A},m}$, respectively. Similarly, the labels R$/$A should be
added to the functions $f_{O}^{\text{R/A}}$ and $h_{O}^{\text{R/A}}$.
Consequently, we can only determine the operator size at late-time $t\gg
t_{\text{sc}}$ by combining information of $\mathcal{P}(s,t_{0})$ and
$\mathcal{P}(s,-t_{0})$. (ii). At finite temperatures, an unambiguous
definition of the operator size distribution is lacking. To establish a
relationship between the operator sizes in different time regimes, we adopt a
specific definition. The operator size distribution of $\hat{O}(t)$ at an
inverse temperature $\beta$ is defined as the standard operator size
distribution of $\rho^{1/4}\hat{O}(t)\rho^{1/4}$, where
$\rho=e^{-\beta\hat{H}}$ represents the thermal density matrix. Under this
definition, an analog of (Signature of Scramblon Effective Field Theory in
Random Spin Models) holds. We leave a detailed derivation into the
supplementary material SM .

_Example.–_ As an illustration, let us consider the Brownian circuits for
spin-$1/2$ systems Zhou and Chen (2019). The Hamiltonian reads

${H_{s}}\Delta
t=J\sum_{i<j}\sum_{{\mu_{i}},{\mu_{j}}}\hat{\sigma}^{i}_{{\mu_{i}}}\hat{\sigma}^{j}_{{\mu_{j}}}\Delta
B^{i,j}_{{\mu_{i}},{\mu_{j}}},$ (14)

where $\mu_{i}\in\\{0,x,y,z\\}$. $\Delta B^{i,j}_{{\mu_{i}},{\mu_{j}}}$
represents independent Brownian variables with $(\Delta
B^{i,j}_{{\mu_{i}},{\mu_{j}}})^{2}=\Delta t$. We take the normalization
$J={({{{8}N}})^{-1/2}}$. The system exihibits the time-reversal symmetry. In
Ref. Zhou and Chen (2019), the authors solved the operator size distribution
of Brownian circuits by deriving a set of ordinary equations for $P(n,t)$
using Itô Calculus, which allows for efficient classical simulations with
large $N$. However, due to the lack of a diagrammatic analysis, it remains
unclear whether information scrambling in Brownian circuits can be described
by the SEFT.

Here, we apply our protocol to Brownian circuits to address this question. In
Brownian circuits, there is a permutation symmetry for different spins $j$ and
different directions $\alpha$. As a result, the average can be omitted
simplifying the analysis. The early-time operator size distribution $P(n,t)$
has been analytically computed in Zhou and Chen (2019). Taking the
thermodynamic limit with fixed $e^{\varkappa t}/N$ and $\varkappa=3$, we
discover

$\displaystyle\overline{\mathcal{P}}(s,t_{0})=Ne^{-\varkappa
t_{0}}\text{exp}\big{(}-sNe^{-\varkappa t_{0}}\big{)},\ \ \ \ \ \
\overline{s_{0}}=e^{\varkappa t_{0}}/N.$ (15)

We then try to predict the operator size distribution for general time $t$
using (Signature of Scramblon Effective Field Theory in Random Spin Models)
and (15). Performing the integration gives

$\overline{\mathcal{P}}(s,t)=\frac{N}{(1-s/s_{\text{sc}})^{2}}\exp(-\varkappa
t-\frac{sNe^{-\varkappa t}}{(1-s/s_{\text{sc}})}).$ (16)

In FIG. 3, we compare this result to numerical simulations with $N={10^{4}}$.
We find excellent agreement for arbitrary time $t$. This supports the validity
of the SEFT in Brownian circuits. Interestingly, this formula has not been
obtained explicitly within to our knowledge. Its moments match Eq. (20) in
Zhou and Chen (2019) . For example, we have
$\overline{s}/s_{\text{sc}}=1+ae^{a}\text{Ei}(-a)$ with
$a=s_{\text{sc}}Ne^{-\varkappa t}$. For smaller $N$, we should take finite
size corrections for Eq. (Signature of Scramblon Effective Field Theory in
Random Spin Models) into account, as established in the supplementary material
SM .

Up to now, we are using the analytical expression (15) for
$\mathcal{P}(s,t_{0})$. However, the extraction of $\mathcal{P}(s,t_{0})$
contains additional continuous limit, which may lead to large error. We
further test our protocol by direct application to numerical data of $P(n,t)$.
We choose $t_{0}$ by fixing $\overline{s_{0}}/s_{\text{sc}}\in[0.03,0.05]$.
The integral in (Signature of Scramblon Effective Field Theory in Random Spin
Models) is performed numerically by using a Gaussian approximation of the
Dirac delta function with standard deviation $\sigma={10^{-2}}$. The result
for different $t_{0}$ is plotted in FIG. 3 as the shaded region, which is
consistent with both numerics and analytical results (16).

_MQC.–_ After confirming the validity of the SEFT in a specific model, we can
proceed to make theoretical predictions for other experimental observables. As
an example, we present results for the MQC, which is extensively studied in
NMR experiments Abraham  _et al._ (1998). To obtain MQC of $\hat{O}$, one
first performs the measurement of

$F(\varphi,t)=\langle
e^{i\varphi\sum_{j}\hat{s}_{z}^{j}}\hat{O}(t)e^{-i\varphi\sum_{j}\hat{s}_{z}^{j}}\hat{O}(t)\rangle,\
\ \ \ \ \ \ \varphi\in[0,2\pi).$ (17)

The MQC, denoted as $I(m,t)$, is defined as the Fourier transform of
$F(\varphi,t)$ for integer values of $m$. In the thermodynamic limit $N\gg 1$,
$F(\varphi,t)$ rapidly concentrates around $\varphi=0$ with a width on the
order of $\sim{N}^{-\frac{1}{2}}$. To address this, we introduce the variable
$\xi\equiv\sqrt{N}\varphi$ and $\mathcal{F}(\xi,t)\equiv F(\sqrt{N}\phi,t)$.
The computation of $\mathcal{F}(\xi,t)$ can be done in a similar manner as for
$\mathcal{S}(\nu,t)$ utilizing the SEFT. The resulting expression is

$\mathcal{F}(\xi,t)=\int_{0}^{\infty}dy~{}h_{O}(y)e^{-\frac{\xi^{2}}{4}(1-f_{z}(\lambda
y))}.$ (18)

Unlike the operator size distribution, the MQC solely depends on $f_{z}$. By
performing the Fourier transform $\mathcal{I}(u,t)=\int d\xi
e^{iu\xi}\mathcal{F}(\xi,t)/2\pi$, the continuum limit of the MQC, denoted as
$\mathcal{I}(u,t)$ with $u\equiv m/\sqrt{N}$, can be expressed as

$\mathcal{I}(u,t)=\int_{0}^{\infty}dy~{}h_{O}(y)[\pi(1-f_{z}(\lambda
y))]^{-1/2}e^{-\frac{u^{2}}{1-f_{z}(\lambda y)}}.$ (19)

This can also be related to the operator size distribution at early times. For
instance, if we consider $O=\hat{\sigma}^{j}_{z}$ and take an average over
sites $j$, we can show that

$\mathcal{I}_{z}(u,t)=\int_{0}^{\infty}ds_{1}~{}\mathcal{P}_{z}(s_{1},t_{0})[\pi(1-\mathcal{S}_{z}(\eta
s_{1},t_{0}))]^{-1/2}e^{-\frac{u^{2}}{1-\mathcal{S}_{z}(\eta s_{1},t_{0})}},$

where we have introduced
$\eta(t,t_{0})={e^{\varkappa(t-t_{0})}}/{s_{\text{sc}}\overline{s_{0}}}$ for
conciseness. The subscripts $z$ are included as a reminder.

_Discussions.–_ In this work, we explore the signature of the SEFT in random
spin models with all-to-all interactions. Our result (Signature of Scramblon
Effective Field Theory in Random Spin Models) reveals a non-trivial
consistency relation between the distribution of operator sizes in the early-
time regime and their values at late-time. This finding serves as strong
evidence for the validity of SEFT in a specific model, which can be
experimentally tested on quantum simulator platforms. As an illustration, we
apply the scheme to Brownian circuits, where a diagrammatic calculation is not
available. Both analytical and numerical implements of our protocol confirms
the validity of SEFT in Brownian circuits. We also present discussions for the
MQC.

Recently, there are growing interest in quantum teleportation through emergent
traversable wormholes Gao  _et al._ (2017); Maldacena  _et al._ (2017);
Susskind and Zhao (2018); Gao and Liu (2019); Brown  _et al._ (2019); Gao and
Jafferis (2021); Nezami  _et al._ (2021); Schuster  _et al._ (2022), which has
been experimentally investigated in Ref. Jafferis  _et al._ (2022). The
teleportation fidelity has been proposed to be associated with the size
winding at finite temperatures. Our calculation can be generalized to
establish a relationship between size winding in different time regimes for
systems described by the SEFT. We would like to postpone a detailed study to
future works.

_Acknowledgement._ We thank Xiao Chen, Yingfei Gu, Xinhua Peng and Tian-Gang
Zhou for invaluable discussions.

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