$\mathsf{L}^{P}_{t,p}\lesssim\max(p,\,p^{\prime})$.

(iii) If $\\{T_{t}\\}_{t>0}$ is a symmetric diffusion semigroup, Stein’s
approach in [48, section IV.4] via Rota’s dilation yields
$\mathsf{L}^{P}_{c,p}\lesssim\max(p,\,p^{\prime\frac{3}{2}})$.

(iv) If $\\{T_{t}\\}_{t>0}$ is a symmetric submarkovian semigroup, Cowling
[11] proved that the negative generator $A$ of $\\{T_{t}\\}_{t>0}$ has a
bounded holomorphic functional calculus whose relevant constant is of
polynomial growth on $p$ as $p\to 1$ and $p\to\infty$. Using the equivalence
between bounded holomorphic functional calculus and square function
inequalities, one can then deduce a polynomial growth of
$\mathsf{L}^{T}_{c,p}$ and $\mathsf{L}^{T}_{t,p}$ too, the resulting orders
seem less good than those in Corollary 8.2.

######  Remark 8.3.

The orders on $\mathsf{L}^{P}_{c,p}$ in Theorem 8.1 are optimal both as
$p\to\infty$ and $p\to 1$ for they are already optimal for the classical
Poisson semigroup on ${\mathbb{R}}$ (see Proposition 8.5 below). Zhendong Xu
and Hao Zhang [61] proved that $\mathsf{L}^{T}_{t,p}\gtrsim p$ is as
$p\to\infty$ when $\\{T_{t}\\}_{t>0}$ is a symmetric diffusion semigroup, so
$\mathsf{L}^{P}_{t,p}\gtrsim p$ is as $p\to\infty$ for the subordinated
Poisson semigroup $\\{P_{t}\\}_{t>0}$ too. This shows that our method is
optimal.

However, at the time of this writing, we are unable to determine the optimal
orders of $\mathsf{L}^{P}_{t,p}$ as $p\to 1$ even when $\\{T_{t}\\}_{t>0}$ is
a symmetric diffusion semigroup.

######  Problem 8.4.

It would be interesting to determine the optimal orders of
$\mathsf{L}^{P}_{t,p}$ as $p\to 1$ when $\\{T_{t}\\}_{t>0}$ is a symmetric
submarkovian (or markovian) semigroup. In particular, does there exist a
constant $C$ (possibly depending on $\\{T_{t}\\}_{t>0}$) such that

$\|f-\mathsf{F}(f)\|_{L_{1}(\Omega)}\leq
C\big{\|}\mathcal{G}^{P}_{2}(f)\big{\|}_{L_{1}(\Omega)}\,,\;\forall\,f\in
L_{1}(\Omega)\,?$

The dual version of the above inequality is related to the BMO space
considered in [18]. It is true when $\\{P_{t}\\}_{t>0}$ is the Poisson or heat
semigroup on ${\mathbb{R}}^{d}$.

We conclude this section by the proof of the optimality of the growth orders
of the best constants in Corollary 1.6 in the scalar case, i.e.,
$X={\mathbb{C}}$ (see [60] for more related results). We will denote
$\mathsf{L}^{\mathbb{P}}_{c,q,p}({\mathbb{C}})$ and
$\mathsf{L}^{\mathbb{H}}_{c,q,p}({\mathbb{C}})$ simply by
$\mathsf{L}^{\mathbb{P}}_{c,q,p}$ and $\mathsf{L}^{\mathbb{H}}_{c,q,p}$,
respectively. It suffices to consider ${\mathbb{R}}$.

######  Proposition 8.5.

Let $1<p,q<\infty$. Then

$\mathsf{L}^{\mathbb{P}}_{c,q,p}\gtrsim\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}\;\text{
and
}\;\mathsf{L}^{\mathbb{H}}_{c,q,p}\gtrsim\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}.$

###### Proof.

By Remark 1.1, it suffices to show the assertion on the Poisson semigroup.

Let us first consider the case $p\leq q$. Fix $s>0$ and let
$f=\mathbb{P}_{s}$. Then

$t\frac{\partial}{\partial
t}\mathbb{P}_{t}(f)(x)=\frac{t}{\pi}\,\frac{x^{2}-(t+s)^{2}}{(x^{2}+(t+s)^{2})^{2}},\quad
x\in{\mathbb{R}}.$

For $x\geq 6s$, we have

$\displaystyle{\mathcal{G}}^{\mathbb{P}}_{q}(f)(x)\geq\Big{(}\int_{\frac{x}{3}-s}^{\frac{x}{2}-s}\big{|}t\frac{\partial}{\partial
t}\mathbb{P}_{t}(f)(x)\big{|}^{q}\,\frac{dt}{t}\Big{)}^{\frac{1}{q}}\gtrsim\frac{1}{x}.$

Thus

$\big{\|}{\mathcal{G}}^{\mathbb{P}}_{q}(f)\big{\|}_{L_{p}({\mathbb{R}})}\gtrsim\Big{(}\int_{6s}^{\infty}\frac{1}{x^{p}}\,dx\Big{)}^{\frac{1}{p}}\gtrsim\frac{s^{-\frac{1}{p^{\prime}}}}{p-1}.$

On the other hand,

$\big{\|}f\big{\|}_{L_{p}({\mathbb{R}})}\approx s^{-\frac{1}{p^{\prime}}}.$

Hence, $\mathsf{L}^{\mathbb{P}}_{c,q,p}\gtrsim p^{\prime}$.

Unfortunately, the above simple argument does not apply to the case $p>q$. Our
proof for the latter is much harder. By periodization, it is equivalent to
considering the torus ${\mathbb{T}}$ (equipped with normalized Haar measure).
The $g$-function relative to the Poisson semigroup on ${\mathbb{T}}$ is
defined by

$G_{q}^{\mathsf{P}}(f)=\Big{(}\int_{0}^{1}\big{|}(1-r)\frac{d}{dr}\mathsf{P}_{r}(f)\big{|}^{q}\,\frac{dr}{1-r}\Big{)}^{\frac{1}{q}}\,,$

where $\mathsf{P}_{r}$ denotes the corresponding Poisson kernel:

$\mathsf{P}_{r}(\theta)=\frac{1-r^{2}}{1-2r\cos\theta+r^{2}}\,.$

It is shown in [57] that the inequality

$\|G_{q}^{\mathsf{P}}(f)\|_{L_{p}({\mathbb{T}})}\leq\mathsf{L}^{\mathsf{P}}_{c,q,p}\|f\|_{L_{p}({\mathbb{T}})}$

is equivalent to the corresponding dyadic martingale inequality on
$\Omega=\\{-1,1\\}^{\mathbb{N}}$. It is well known that the relevant constant
in the latter martingale inequality for $q=2$ is of order $\sqrt{p}$ as
$p\to\infty$. To reduce the determination of optimal order of
$\mathsf{L}^{\mathsf{P}}_{c,q,p}$ to the martingale case, we need to refine an
argument in the proof of [57, Theorem 3.1].

Keeping the notation there, let $M=(M_{k})_{0\leq k\leq K}$ be a finite dyadic
martingale and

$M_{k}-M_{k-1}=d_{k}(\varepsilon_{1},\cdots,\varepsilon_{k-1})\,\varepsilon_{k},$

where $(\varepsilon_{k})$ are the coordinate functions of $\Omega$. The
transformation $\varepsilon_{k}={\rm sgn}(\cos\theta_{k})$ establishes a
measure preserving embedding of $\Omega$ into ${\mathbb{T}}^{\mathbb{N}}$.
Accordingly, define

$\displaystyle a_{k}(e^{{\rm i}\theta_{1}},\cdots,e^{{\rm i}\theta_{k-1}})$
$\displaystyle=d_{k}({\rm sgn}(\cos\theta_{1}),\cdots,{\rm
sgn}(\cos\theta_{k-1})),$ $\displaystyle b_{k}(e^{{\rm i}\theta_{k}})$
$\displaystyle={\rm sgn}(\cos\theta_{k}).$

Given $(n_{k})$ a rapidly increasing sequence of positive integers, put

$\displaystyle a_{k,(n)}(e^{{\rm i}\theta})$ $\displaystyle=a_{k,(n)}(e^{{\rm
i}\theta};e^{{\rm i}\theta_{1}},\cdots,e^{{\rm i}\theta_{k-1}})=a_{k}(e^{{\rm
i}(\theta_{1}+n_{1}\theta)},\cdots,e^{{\rm i}(\theta_{k-1}+n_{k-1}\theta)}),$
$\displaystyle b_{k,(n)}(e^{{\rm i}\theta})$ $\displaystyle=b_{k,(n)}(e^{{\rm
i}\theta};e^{{\rm i}\theta_{k}})=b_{k}(e^{{\rm i}(\theta_{k}+n_{k}\theta)}),$
$\displaystyle f_{(n)}(e^{{\rm i}\theta})$ $\displaystyle=f_{(n)}(e^{{\rm
i}\theta};e^{{\rm i}\theta_{1}},\cdots,e^{{\rm
i}\theta_{K}})=\sum_{k=1}^{K}a_{k,(n)}(e^{{\rm i}\theta})b_{k,(n)}(e^{{\rm
i}\theta}).$

The functions $f_{(n)}$, $a_{k,(n)}$ and $b_{k,(n)}$ are viewed as functions
on ${\mathbb{T}}$ for each $(\theta_{1},\cdots,\theta_{K})$ arbitrarily fixed.
Furthermore, by approximation, we can assume that all $a_{k}$ and $b_{k}$ are
polynomials. Then, if the sequence $(n_{k})$ rapidly increases, Lemmas 3.4 and
3.5 of [57] imply

$\frac{1}{2}\,G_{q}^{\mathsf{P}}(f_{(n)})\leq\Big{(}\sum_{k=1}^{K}|a_{k,(n)}|^{q}\,G_{q}^{\mathsf{P}}(b_{k,(n)})^{q}\Big{)}^{\frac{1}{q}}\leq
2G_{q}^{\mathsf{P}}(f_{(n)}).$

Therefore,

(8.1)
$\Big{\|}\Big{(}\sum_{k=1}^{K}|a_{k,(n)}|^{q}\,G_{q}^{\mathsf{P}}(b_{k,(n)})^{q}\Big{)}^{\frac{1}{q}}\Big{\|}_{L_{p}({\mathbb{T}})}\leq
2\mathsf{L}^{\mathsf{P}}_{c,q,p}\big{\|}f\big{\|}_{L_{p}({\mathbb{T}})}\,.$

The discussion so far comes from [57]. Now we require a finer analysis of the
$g$-function $G_{q}^{\mathsf{P}}(b_{k,(n)})$. To this end we write the Fourier
series of the function $b={\rm sgn}(\cos\theta)$:

$b(e^{{\rm
i}\theta})=\frac{2}{\pi}\,\sum_{j=0}^{\infty}\frac{(-1)^{j}}{2j+1}\,\big{[}e^{{\rm
i}(2j+1)\theta}+e^{-{\rm i}(2j+1)\theta}\big{]}.$

Then

$\frac{d}{dr}\mathsf{P}_{r}(b_{k,(n)})(e^{{\rm
i}\theta})=\frac{4}{\pi}\,n_{k}r^{n_{k}-1}{\rm
Re}\Big{(}\sum_{j=0}^{\infty}(-1)^{j}r^{2n_{k}j}e^{{\rm
i}(2j+1)(\theta_{k}+n_{k}\theta)}\Big{)}.$

Elementary computations show

$\left|\frac{d}{dr}\mathsf{P}_{r}(b_{k,(n)})(e^{{\rm i}\theta})\right|^{q}\geq
c^{q}n^{q}_{k}r^{q(n_{k}-1)}\cos^{q}(\theta_{k}+n_{k}\theta).$

Here and below, $c,\,C$ denote absolute positive constants. Thus

$\displaystyle G_{q}^{\mathsf{P}}(b_{k,(n)})^{q}$ $\displaystyle\geq
c^{q}\,\cos^{q}(\theta_{k}+n_{k}\theta)\,n^{q}_{k}\int_{0}^{1}(1-r)^{q-1}r^{q(n_{k}-1)}dr$
$\displaystyle\approx c^{q}\,\big{[}1+{\rm
O}(\frac{1}{n_{k}})\big{]}\cos^{q}(\theta_{k}+n_{k}\theta).$

Now lifting both sides of (8.1) to power $p$, then integrating the resulting
inequality over ${\mathbb{T}}^{K}$ with respect to
$(\theta_{1},\cdots,\theta_{K})$, we get

$\displaystyle\int_{\mathbb{T}}$
$\displaystyle\int_{{\mathbb{T}}^{K}}\Big{(}\sum_{k=1}^{K}|a_{k,(n)}(e^{{\rm
i}(\theta_{1}+n_{1}\theta)},\cdots,e^{{\rm
i}(\theta_{k-1}+n_{k-1}\theta)})|^{q}\big{[}1+{\rm
O}(\frac{1}{n_{k}})\big{]}\cos^{q}(\theta_{k}+n_{k}\theta)\Big{)}^{\frac{p}{q}}d\theta_{1}\cdots
d\theta_{K}d\theta$
$\displaystyle\leq\big{(}C\mathsf{L}^{\mathsf{P}}_{c,q,p}\big{)}^{p}\int_{\mathbb{T}}\int_{{\mathbb{T}}^{K}}\big{|}f_{(n)}(e^{{\rm
i}(\theta_{1}+n_{1}\theta)},\cdots,e^{{\rm
i}(\theta_{K}+n_{K}\theta)})\big{|}^{p}d\theta_{1}\cdots
d\theta_{K}d\theta\,.$

For each fixed $\theta$, the change of variables
$(\theta_{1},\cdots,\theta_{K})\mapsto(\theta_{1}-n_{1}\theta,\cdots,\theta_{K}-n_{K}\theta)$
being a measure preserving transformation of ${\mathbb{T}}^{K}$, we deduce

$\displaystyle\int_{{\mathbb{T}}^{K}}$
$\displaystyle\Big{(}\sum_{k=1}^{K}|a_{k,(n)}(e^{{\rm
i}\theta_{1}},\cdots,e^{{\rm i}\theta_{k-1}})|^{q}\,\big{[}1+{\rm
O}(\frac{1}{n_{k}})\big{]}\cos^{q}\theta_{k}\Big{)}^{\frac{p}{q}}\,d\theta_{1}\cdots
d\theta_{K}$
$\displaystyle\leq\big{(}C\mathsf{L}^{\mathsf{P}}_{c,q,p}\big{)}^{p}\int_{{\mathbb{T}}^{K}}\big{|}f_{(n)}(e^{{\rm
i}\theta_{1}},\cdots,e^{{\rm i}\theta_{K}})\big{|}^{p}\,d\theta_{1}\cdots
d\theta_{K}\,.$

Letting $n_{1}\to\infty$, we get

$\displaystyle\int_{{\mathbb{T}}^{K}}$
$\displaystyle\Big{(}\sum_{k=1}^{K}|d_{k}({\rm
sgn}(\cos\theta_{1}),\cdots,{\rm
sgn}(\cos\theta_{k-1}))|^{q}\,\cos^{q}\theta_{k}\Big{)}^{\frac{p}{q}}\,d\theta_{1}\cdots
d\theta_{K}$
$\displaystyle\leq\big{(}C\,\mathsf{L}^{\mathsf{P}}_{c,q,p}\big{)}^{p}\big{\|}M_{K}\big{\|}^{p}_{L_{p}(\Omega)}\,.$

Now we consider an elementary example where $M$ is simple random walk stopped
at $\pm 2$, namely

$d_{k}={\mathds{1}}_{\\{\tau\geq k\\}}\;\text{ with
}\;\tau=\inf\big{\\{}k:\big{|}\sum_{j=1}^{k}\varepsilon_{j}\big{|}=2\big{\\}}.$

Note that the probability of the event $\\{\tau=j\\}$ is zero for odd $j$ and
$2^{-\frac{j}{2}}$ for even $j$. On the other hand, recalling
$\varepsilon_{k}={\rm sgn}(\cos\theta_{k})$ and letting

$A_{j}=\big{\\{}\tau=j,\;|\cos\theta_{k}|\geq\frac{1}{\sqrt{2}}\,,\;1\leq
k\leq j\big{\\}},$

we easily check that the probability of $A_{j}$ is $8^{-\frac{j}{2}}$ for even
$j$. Thus

$\displaystyle\sum_{k=1}^{K}|d_{k}(\varepsilon_{1},\cdots,\varepsilon_{k-1}))|^{q}\,\cos^{q}\theta_{k}\geq{\mathds{1}}_{A_{j}}\sum_{k=1}^{j}{\mathds{1}}_{\\{\tau\geq
k\\}}\cos^{q}\theta_{k}\geq 2^{-\frac{q}{2}}j\,{\mathds{1}}_{A_{j}},$

consequently, for $K=2J$

$\displaystyle\int_{{\mathbb{T}}^{K}}$
$\displaystyle\Big{(}\sum_{k=1}^{K}|d_{k}({\rm
sgn}(\cos\theta_{1}),\cdots,{\rm
sgn}(\cos\theta_{k-1}))|^{q}\,\cos^{q}\theta_{k}\Big{)}^{\frac{p}{q}}\,d\theta_{1}\cdots
d\theta_{K}\geq c^{q}\sum_{m=1}^{J}j^{\frac{p}{q}}8^{-j}\geq
c^{p}p^{\frac{p}{q}}.$

Noting that $|M_{K}|\leq 2$ and combining all the previous inequalities
together, we finally obtain

$\mathsf{L}^{\mathsf{P}}_{c,q,p}\gtrsim p^{\frac{1}{q}}.$

This completes the proof. ∎

##  Appendix A Examples

There exist plenty of examples of semigroups to which the results of this
article apply. Many second order differential operators in analysis generate
such semigroups. In the following we will only discuss the cotype case since
the type one can be dealt with by duality. Note that it is obvious that if $X$
is of Lusin cotype $q$ relative to $\\{T_{t}\\}_{t>0}$, it is so relative to
the subordinated Poisson semigroup $\\{P_{t}\\}_{t>0}$ of $\\{T_{t}\\}_{t>0}$.

A main task in the study of the vector-valued Littlewood-Paley-Stein theory
would be the following

######  Problem A.1.

Determine the family of semigroups $\\{T_{t}\\}_{t>0}$ such that a Banach
space $X$ is of Lusin type (resp. cotype) $q$ relative to $\\{T_{t}\\}_{t>0}$
or its subordinated Poisson semigroup $\\{P_{t}\\}_{t>0}$ iff $X$ is of
martingale type (resp. cotype) $q$.

######  Example A.2.

Laplacian operators. The classical heat semigroup on ${\mathbb{R}}^{d}$ is
given by $\mathbb{H}_{t}=e^{t\Delta}$, where $\Delta$ is the Laplacian
operator. It is well known (and easy to check) that
$\\{\mathbb{H}_{t}\\}_{t>0}$ is analytic on $L_{p}({\mathbb{R}}^{d};X)$ for
any $1\leq p<\infty$ and any Banach space $X$ (with relevant constant
depending only on $d$). By [24, 59], if $X$ is of martingale cotype $q$, then
$X$ is of Lusin cotype $q$ relative to $\\{\mathbb{H}_{t}\\}_{t>0}$.
Conversely, suppose that $X$ is of Lusin cotype $q$ relative to
$\\{\mathbb{H}_{t}\\}_{t>0}$, then $X$ is also of Lusin cotype $q$ relative to
the classical Poisson semigroup $\\{\mathbb{P}_{t}\\}_{t>0}$, thereby $X$ is
of martingale cotype $q$ by virtue of [34]. Thus the Lusin cotype relative to
the classical heat semigroup is equivalent to the martingale cotype.

######  Example A.3.

Schrödinger operators. Let $\Omega$ be a region in ${\mathbb{R}}^{d}$ equipped
with Lebesgue measure. Let $a(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ be a positive
matrix whose entries are locally integrable real functions on $\Omega$ such
that

$\alpha(x)\leq a(x)\leq\beta(x)$

for two positive continuous functions $\alpha$ and $\beta$ on $\Omega$. We
consider the following elliptic operator

$L(f)=-\sum_{i,j=1}^{d}\frac{\partial}{\partial
x_{i}}\Big{(}a_{ij}\frac{\partial f}{\partial x_{j}}\Big{)}.$

Given $V$ a nonnegative locally integrable function on $\Omega$, define
$A(f)=L(f)+Vf$. It is well known that $-A$ generates a symmetric submarkovian
(markovian for $V=0$) semigroup $\\{T_{t}\\}_{t>0}$ on $\Omega$ (cf. [14,
Theorem 1.8.1]). In particular, $\\{T_{t}\\}_{t>0}$ is analytic on
$L_{p}(\Omega)$ for any $1<p<\infty$. Thus if $X$ is of martingale cotype $q$,
then it is of Lusin cotype $q$ relative to the Poisson semigroup
$\\{P_{t}\\}_{t>0}$ subordinated to $\\{T_{t}\\}_{t>0}$ on $L_{p}(\Omega;X)$
for any $1<p<\infty$.

Assume in addition that $L$ is uniformly elliptic, namely, the above two
functions $\alpha$ and $\beta$ are constant. Then the integral kernel
$K^{0}_{t}(x,y)$ of $e^{-tL}$ satisfies the following Gaussian upper bound
(cf. [14, Theorem 3.2.8])

$K^{0}_{t}(x,y)\leq
C_{\delta,\alpha}t^{-\frac{d}{2}}\exp\left(-\frac{|x-y|^{2}}{(1+\delta)\beta
t}\right),\quad t>0,\;x,y\in\Omega,\;0<\delta<1.$

By the Trotter formula

$e^{-tA}(f)=\lim_{n\to\infty}\big{(}e^{-\frac{tL}{n}}\,e^{-\frac{tV}{n}}\big{)}^{n}(f),$

we deduce that the integral kernel $K_{t}(x,y)$ of $e^{-tA}$ is majorized by
$K^{0}_{t}(x,y)$:

$K_{t}(x,y)\leq K^{0}_{t}(x,y).$

Thus $K_{t}(x,y)$ satisfies the same Gaussian upper bound as $K^{0}_{t}(x,y)$.
Let $z\in{\mathbb{C}}$ with ${\rm Re}z>0$. By [14, Theorem 3.4.8], the complex
time heat kernel of $e^{-zA}$ satisfies

$|K_{z}(x,y)|\lesssim_{\delta,\alpha}({\rm
Re}z)^{-\frac{d}{2}}\exp\left(-\frac{|x-y|^{2}\,{\rm
Re}(z^{-1})}{(1+\delta)\beta}\right).$

Then we easily show that $\\{T_{t}\\}_{t>0}$ extends to an analytic semigroup
of type $\frac{\pi}{2}$ on $L_{p}(\Omega;X)$ for any $X$ and any $1<p<\infty$
($p$ can be equal to $1$ too). It then follows that $X$ is of Lusin cotype $q$
relative to $\\{T_{t}\\}_{t>0}$ whenever $X$ is of martingale cotype $q$.

As in the case of Laplacian operators, it is likely that the Lusin cotype
relative to the semigroups generated by Schrödinger operators characterizes
the martingale cotype. This is indeed the case if $\Omega={\mathbb{R}}^{d}$
with $d\geq 3$, $L=-\Delta$ and the potential $V$ satisfies a reverse Hölder
inequality (see [1]). Let us formulate the general case explicitly as a
conjecture.

######  Conjecture A.4.

Let $\\{P_{t}\\}_{t>0}$ be the Poisson semigroup subordinated to the heat
semigroup $\\{T_{t}\\}_{t>0}$ generated by a Schrödinger operator as above. If
a Banach space $X$ is of Lusin cotype $q$ relative to $\\{P_{t}\\}_{t>0}$,
then $X$ is of martingale cotype $q$. The same conjecture is stated for
$\\{T_{t}\\}_{t>0}$ itself when the underlying differential operator $L$ is
uniformly elliptic.

######  Example A.5.

Laplace-Beltrami operators. The preceding examples can be extended to the
setting of Riemannian manifolds. Let $M$ be a complete $d$-dimensional
Riemannian manifold with metric $g$. Let $a(x)=(a_{ij}(x))_{1\leq i,j\leq d}$
be a positive matrix smoothly depending on $x\in M$. The associated second
order elliptic operator $L$ is represented as

$L(f)=-g^{-\frac{1}{2}}\sum_{i,j=1}^{d}\frac{\partial}{\partial
x_{i}}\Big{(}g^{\frac{1}{2}}a_{ij}\frac{\partial f}{\partial x_{j}}\Big{)}$

in local coordinates. Then $T_{t}=e^{-tL}$ extends to a symmetric diffusion
semigroup on $M$ for all $1\leq p\leq\infty$ (cf. [13]). Thus our previous
results apply to the associated subordinated Poisson semigroup. The most
important case is the one where $-L=\Delta$ is the Laplace-Beltrami operator.
Then the semigroup $\\{e^{\Delta t}\\}_{t>0}$ enjoys many properties of the
classical heat semigroup on ${\mathbb{R}}^{d}$.

######  Example A.6.

Hermite operators. The Hermite operator on ${\mathbb{R}}^{d}$ is a particular
Schrödinger operator: $A=-\Delta+|x|^{2}$. The associated semigroup
$\\{T_{t}\\}_{t>0}$ is a symmetric submarkovian semigroup on
${\mathbb{R}}^{d}$. The integral kernel of $T_{t}$ is given by

$K_{t}(x,y)=\Big{(}\frac{2}{\pi\sinh(2t)}\Big{)}^{\frac{d}{2}}\exp\Big{(}-\frac{1}{4}\big{[}|x-y|^{2}\coth
t+|x+y|^{2}\tanh t\big{]}\Big{)}.$

Using the Trotter formula, we see that $K_{t}(x,y)$ is less than or equal to
the heat kernel:

$K_{t}(x,y)\leq\Big{(}\frac{1}{4\pi
t}\Big{)}^{\frac{d}{2}}\,e^{-\frac{|x-y|^{2}}{t}}\,.$

This Gaussian upper bound can be deduced from the above explicit formula of
$K_{t}$. It then follows that $\\{T_{t}\\}_{t>0}$ is an analytic semigroup of
type $\frac{\pi}{2}$ on $L_{p}({\mathbb{R}}^{d};X)$ for any Banach space $X$
and $1<p<\infty$. Betancor et al showed in [6] that $X$ is of Lusin cotype $q$
relative to $\\{T_{t}\\}_{t>0}$ iff $X$ of martingale cotype $q$ (see also [8]
for related results).

######  Example A.7.

Laguerre operators. For simplicity, we only consider the Laguerre semigroup on
${\mathbb{R}}_{+}$, the multi-dimensional case can be treated by tensor
product. In this example, ${\mathbb{R}}_{+}$ is equipped with Lebesgue
measure, contrarily to our usual convention. Let $\alpha>-1$ and

$A=\frac{1}{2}\Big{(}-\frac{d^{2}}{dx^{2}}+x^{2}+\frac{1}{x^{2}}(\alpha^{2}-\frac{1}{4})\Big{)}.$

Then $-A$ generates a symmetric semigroup $\\{T_{t}\\}_{t>0}$ of positive
contractions on $L_{2}({\mathbb{R}}_{+})$. It is proved in [41] that
$\\{T_{t}\\}_{t>0}$ is contractive on $L_{p}({\mathbb{R}}_{+})$ for all $1\leq
p\leq\infty$ iff $\alpha=-\frac{1}{2}$ or $\alpha\geq\frac{1}{2}$, and that
$\\{T_{t}\\}_{t>0}$ is a bounded semigroup on $L_{p}({\mathbb{R}}_{+})$ for
all $1\leq p\leq\infty$ if $-\frac{1}{2}<\alpha<\frac{1}{2}$. However, for
$-1<\alpha<-\frac{1}{2}$, $T_{t}$ is unbounded on $L_{p}({\mathbb{R}}_{+})$
for $p\leq p_{\alpha}=\frac{2}{2\alpha+3}$ and $p\geq p_{\alpha}^{\prime}$.

On the other hand, [6] shows that for $\alpha>-\frac{1}{2}$, a Banach space
$X$ is of Lusin cotype $q$ relative to $\\{T_{t}\\}_{t>0}$ iff $X$ is of
martingale cotype $q$; as a byproduct, [6] also shows that for the same range
of $\alpha$, $\\{T_{t}\\}_{t>0}$ is analytic on $L_{p}({\mathbb{R}}_{+};X)$
for any $X$.

In the remaining case of $-1<\alpha<-\frac{1}{2}$, it is quite easy to show
that $\\{T_{t}\\}_{t>0}$ is a bounded semigroup on $L_{p}({\mathbb{R}}_{+})$
for $p_{\alpha}<p<p_{\alpha}^{\prime}$. Let us outline the argument for the
convenience of the reader. Recall that the integral kernel of $T_{t}$ is given
by

$K_{t}(x,y)=\frac{1}{\sinh t}\,\sqrt{xy}\,I_{\alpha}\big{(}\frac{xy}{\sinh
t}\big{)}\exp\big{(}-\frac{1}{2}(x^{2}+y^{2})\coth t\big{)},$

where $I_{\alpha}$ is the modified Bessel function of the first kind and order
$\alpha$:

$I_{\alpha}(z)=2^{-\alpha}z^{\alpha}\,\sum_{k=0}^{\infty}\frac{z^{2k}}{2^{2k}\Gamma(k+1)\Gamma(k+\alpha+1)}\,.$

By dilation invariance via the change of variables $u={x}/{\sqrt{\sinh t}}$
and $v={y}/{\sqrt{\sinh t}}$, the kernel $K_{t}$ is brought to

$\varphi(x,y)=\sqrt{xy}\,I_{\alpha}(xy)\exp\big{(}-\frac{1}{2}(x^{2}+y^{2})\cosh
t\big{)}.$

Let $\Phi$ be the associated integral operator:

$\Phi(f)=\int_{0}^{\infty}\varphi(x,y)f(y)dy.$

To estimate the norm of $\Phi$ in $B(L_{p}({\mathbb{R}}_{+};X))$ for
$p_{\alpha}<p<p_{\alpha}^{\prime}$, we appeal to the following estimates of
the Bessel function (cf. [31, Chapter 5])

$I_{\alpha}(z)\approx\frac{z^{\alpha}}{2^{\alpha}\Gamma(\alpha+1)}\;\text{ as
}\;z\to 0\;\text{ and }\;I_{\alpha}(z)\approx\frac{e^{z}}{\sqrt{\pi
z}}\;\text{ as }\;z\to\infty.$

Accordingly, $I_{\alpha}$ is decomposed as

$I_{\alpha}(xy)=I_{\alpha}(xy){\mathds{1}}_{xy\leq
1}+I_{\alpha}(xy){\mathds{1}}_{xy>1}\lesssim(xy)^{\alpha}{\mathds{1}}_{xy\leq
1}+(xy)^{-1/2}e^{xy}{\mathds{1}}_{xy>1}$

with the relevant constant depending only on $\alpha$. Thus

$\varphi(x,y)\lesssim\varphi_{1}(x,y)+\varphi_{2}(x,y),$

where

$\displaystyle\varphi_{1}(x,y)=(xy)^{\alpha+\frac{1}{2}}\exp\big{(}-\frac{1}{2}(x^{2}+y^{2})\cosh
t\big{)}{\mathds{1}}_{xy\leq 1}\,,$
$\displaystyle\varphi_{2}(x,y)=e^{xy}\exp\big{(}-\frac{1}{2}(x^{2}+y^{2})\cosh
t\big{)}{\mathds{1}}_{xy>1}\,.$

Let $\Phi_{i}$ be the integral operator corresponding to $\varphi_{i}$. Then
by the Hölder inequality, we have

$\displaystyle\big{\|}\Phi_{1}\big{\|}_{B(L_{p}({\mathbb{R}}_{+};X))}$
$\displaystyle\leq\Big{(}\int_{0}^{\infty}x^{p(\alpha+\frac{1}{2})}e^{-px^{2}\frac{\cosh
t}{2}}dx\Big{)}^{\frac{1}{p}}\Big{(}\int_{0}^{\infty}y^{p^{\prime}(\alpha+1/2)}e^{-{p^{\prime}}y^{2}\frac{\cosh
t}{2}}dy\Big{)}^{\frac{1}{p^{\prime}}}$ $\displaystyle=C_{\alpha,p}(\cosh
t)^{-(\alpha+1)}\leq C_{\alpha,p}\,,$

where we have used the assumption that $p_{\alpha}<p<p_{\alpha}^{\prime}$.
Noting that $\varphi_{2}(x,y)\leq e^{-(x-y)^{2}/2}$, we see that
$\big{\|}\Phi_{2}\big{\|}_{B(L_{p}({\mathbb{R}}_{+};X))}\leq 1$.

In particular, $\\{T_{t}\\}_{t>0}$ is a symmetric semigroup of positive
contractions on $L_{2}({\mathbb{R}}_{+})$, so analytic. Applying the previous
sections to the associated subordinated Poisson semigroup $\\{P_{t}\\}_{t>0}$,
we recover the result of [8] that $X$ is of Lusin cotype $q$ relative to
$\\{P_{t}\\}_{t>0}$ on $L_{2}({\mathbb{R}};X)$ whenever $X$ is of martingale
cotype $q$. Moreover, [8] shows that the converse is also true. Note that [8]
also extends this result to all $p\in(p_{\alpha},\,p_{\alpha}^{\prime})$.

We do not know, however, to determine the analyticity of $\\{T_{t}\\}_{t>0}$
on $L_{p}({\mathbb{R}};X)$ for the range $-1<\alpha<-\frac{1}{2}$.

######  Remark A.8.

We would like to point out an interesting phenomenon revealed by this example.
It is easy to get a semigroup of contractions on $L_{2}$ thanks to spectral
theory. If the contractions are further positive (or regular), the results of
the previous sections apply. In many concrete examples, one can then
extrapolate $L_{2}$ to $L_{p}$ using tools from harmonic analysis. This is
indeed the case for all previous examples.

######  Example A.9.

Ornstein-Uhlenbeck semigroup. Now ${\mathbb{R}}^{d}$ is equipped with the
canonical Gaussian measure $\gamma_{d}$. Let $\\{T_{t}\\}_{t>0}$ be the
Ornstein-Uhlenbeck semigroup on ${\mathbb{R}}^{d}$ whose negative generator is
given by $A=-\Delta+x\cdot\nabla$. This is again a symmetric diffusion
semigroup. By [45], $\\{T_{t}\\}_{t>0}$ is analytic on
$L_{p}({\mathbb{R}}^{d},\gamma_{d};X)$ iff $X$ is K-convex (a property weaker
than the finite martingale cotype). On the other hand, by [34], $X$ is of
Lusin cotype $q$ relative to the Poisson semigroup subordinated to
$\\{T_{t}\\}_{t>0}$ iff $X$ is of martingale cotype $q$. We then deduce that
the Lusin cotype $q$ of $X$ relative to $\\{T_{t}\\}_{t>0}$ itself on
$L_{p}({\mathbb{R}}^{d},\gamma_{d};X)$ characterizes the martingale cotype $q$
of $X$. It is worth noting that in contrast to [34], all estimates obtained by
the present method or by [59] are dimension free.

######  Example A.10.

Walsh semigroup. Let $\Omega=\\{-1,1\\}^{\mathbb{N}}$ be the dyadic group
equipped with normalized Haar measure. The coordinate functions
$\\{\varepsilon_{n}\\}_{n\geq 1}$ on $\Omega$ form an independent sequence of
symmetric random variables (Rademacher functions). We introduce the Walsh
system $(w_{A})$: for any finite subset $A\subset{\mathbb{N}}$ let

$w_{A}=\prod_{k\in A}\varepsilon_{k}\,.$

If $A=\emptyset$, $w_{\emptyset}=1$. All such $w_{A}$’s form an orthonormal
basis of $L_{2}(\Omega)$. Any $f\in L_{2}(\Omega)$ admits the following
Fourier expansion:

$f=\sum_{A}\alpha_{A}w_{A}.$

Define

$T_{t}(f)=\sum_{A}e^{-t|A|}\alpha_{A}w_{A}.$

Then $\\{T_{t}\\}_{t>0}$ is a symmetric diffusion semigroup on $\Omega$, it
can be viewed as a baby model of the Ornstein-Uhlenbeck semigroup. Again, by
[45], $\\{T_{t}\\}_{t>0}$ is analytic on $L_{p}(\Omega;X)$ iff $X$ is
K-convex.

######  Remark A.11.

The Lusin cotype relative to $\\{T_{t}\\}_{t>0}$ characterizes the martingale
cotype.

Indeed, assume that $X$ is of Lusin cotype $q$ relative to
$\\{T_{t}\\}_{t>0}$. Then by an argument via the central limit theorem as in
[3], we can show that $X$ is of Lusin cotype $q$ relative to the Ornstein-
Uhlenbeck semigroup too, so by the previous example, $X$ is of martingale
cotype $q$.

It would be interesting to show the analogue of Corollary 1.6 for Ornstein-
Uhlenbeck or Walsh semigroup.

######  Problem A.12.

Let $\\{T_{t}\\}_{t>0}$ be the Ornstein-Uhlenbeck or Walsh semigroup as above
and $X$ be of martingale cotype $q$. Does one have

$\mathsf{L}^{T}_{c,q,p}(X)\lesssim\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}\mathsf{M}_{c,q}(X)?$

It would, of course, suffice to consider the Walsh case. On the other hand, it
is likely that in the Ornstein-Uhlenbeck setting one could get a dimension
dependent estimate
$\mathsf{L}^{T}_{c,q,p}(X)\lesssim_{d}\max\big{(}p^{\frac{1}{q}},\,p^{\prime}\big{)}\mathsf{M}_{c,q}(X)$
by standard techniques on the Ornstein-Uhlenbeck semigroup. However, here a
dimension free estimate is more important than the corresponding one for the
heat semigroup in ${\mathbb{R}}^{d}$ in view of analysis in Wiener space.

######  Example A.13.

Translation semigroup. We have already used the translation semigroup
$\\{\tau_{t}\\}_{t>0}$ in in the proof of Theorem 1.2. It is not analytic on
$L_{p}({\mathbb{R}})$ for any $p$. By (7.6), if $X$ is of Lusin cotype $q$
relative to the subordinated Poisson semigroup $\\{P^{\tau}_{t}\\}_{t>0}$ on
$L_{p}({\mathbb{R}};X)$, so is $X$ relative to the Poisson semigroup
subordinated to any semigroup of regular contractions $\\{T_{t}\\}_{t>0}$ on
$L_{p}(\Omega;X)$. Consequently, the Lusin cotype relative to
$\\{P^{\tau}_{t}\\}_{t>0}$ is equivalent to the martingale cotype.

######  Example A.14.

$L_{2}$-theory. Let $A$ be a positive densely defined operator on
$L_{2}(\Omega)$. Then $-A$ generates an analytic semigroup $\\{T_{t}\\}_{t>0}$
of contractions on $L_{2}(\Omega)$. The positivity preservingness can be
characterized by means of the Dirichlet form associated to $A$ (cf. [14,
Theorem 1.3.2]). Many classical examples are built in this way. Note, however,
that it can happen that $\\{T_{t}\\}_{t>0}$ extends to a semigroup of bounded
operators on $L_{p}(\Omega)$ for $p$ only in a small symmetric interval around
$2$ as shown by the Laguerre semigroup. Even worse, $\\{T_{t}\\}_{t>0}$ cannot
extrapolate to $L_{p}(\Omega)$ for any $p\neq 2$.

######  Example A.15.

A final example. Given any $1<p<\infty$ we can artificially construct a
semigroup of regular contractions on $L_{p}$ but not bounded on $L_{q}$ for
any $q\neq p$. Let $u$ be a unimodular measurable function on ${\mathbb{R}}$.
Define

$T_{t}(f)(x)=u(x)e^{\frac{t+x}{p}}f(e^{t+x}),\quad x\in{\mathbb{R}},t>0.$

Then $T_{t}$ is a regular contraction on $L_{p}({\mathbb{R}})$ but unbounded
on $L_{q}({\mathbb{R}})$ for any $q\neq p$.

Acknowledgements. I am extremely grateful to Assaf Naor for many inspiring
communications as well as for pointing the problems in his paper [40] joint
with Robert Young; the resolution of their problems has forced me to invent
new techniques, thereby has lead to Theorem 1.5, Corollary 1.6 and a
significant improvement on the orders of the constants in a preliminary
version of Theorem 1.2 and Theorem 1.4; our communications have also been a
special impulse to the work [60] on the optimal orders of the best constants
in the classical Littlewood-Paley inequalities (Assaf also asked himself these
questions on the classical case in his own research). I also wish to thank
Alexandros Eskenazis for useful discussions and a careful reading of a
preliminary version of the article. This work is partially supported by the
French ANR project (No. ANR-19-CE40-0002).

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# Universality class of Ising critical states with long-range losses

Jamir Marino  Institut für Physik, Johannes Gutenberg Universität Mainz,
D-55099 Mainz, Germany  Kavli Institute for Theoretical Physics, University of
California, Santa Barbara, CA 93106-4030, USA

(September 3, 2024)

###### Abstract

We show that spatial resolved dissipation can act on Ising lattices molding
the universality class of their critical points. We consider non-local spin
losses with a Liouvillian gap closing at small momenta as $\propto
q^{\alpha}$, with $\alpha$ a positive tunable exponent, directly related to
the power-law decay of the spatial profile of losses at long distances. The
associated quantum noise spectrum is gapless in the infrared and it yields a
class of soft modes asymptotically decoupled from dissipation at small
momenta. These modes are responsible for the emergence of a critical scaling
regime which can be regarded as the non-unitary analogue of the universality
class of long-range interacting Ising models. In particular, for $0<\alpha<1$
we find a non-equilibrium critical point ruled by a dynamical field theory
ascribable to a Langevin model with coexisting inertial ($\propto\omega^{2}$)
and frictional ($\propto\omega$) kinetic coefficients, and driven by a gapless
Markovian noise with variance $\propto q^{\alpha}$ at small momenta. This
effective field theory is beyond the Halperin-Hohenberg description of
dynamical criticality, and its critical exponents differ from their unitary
long-range counterparts. Furthermore, by employing a one-loop improved RG
calculation, we estimate the conditions for observability of this scaling
regime before incoherent local emission intrudes in the spin sample, dragging
the system into a thermal fixed point. We also explore other instances of
criticality which emerge for $\alpha>1$ or adding long-range spin
interactions. Our work lays out perspectives for a revision of universality in
driven-open systems by employing dark states supported by non-local
dissipation.

## Introduction –

The search for non-equilibrium criticality in driven open quantum systems has
become an exciting research frontier, both for its fundamental relevance in
statistical mechanics, and for the variety of AMO platforms where it can be
concretely explored Hartmann  _et al._ (2008); Syassen  _et al._ (2008); Deng
_et al._ (2010); Baumann  _et al._ (2010); Klaers  _et al._ (2010); Blatt and
Roos (2012); Britton  _et al._ (2012); Houck  _et al._ (2012); Carusotto and
Ciuti (2013); Carr  _et al._ (2013); Zhu  _et al._ (2014); Klinder  _et al._
(2015); Fitzpatrick  _et al._ (2017); Dogra  _et al._ (2019); Keßler  _et al._
(2020); Ferri  _et al._ (2021); Nagy  _et al._ (2011); Nagy and Domokos
(2015); Lundgren  _et al._ (2020). The aim is the discovery of universality
classes which cannot be encompassed by established classifications of
dynamical criticality Hohenberg and Halperin (1977); Sondhi  _et al._ (1997);
Vojta (2003); Sachdev (2011) nor can be related to out-of-equilibrium scaling
in isolated systems Orioli  _et al._ (2015); Berges  _et al._ (2015); Prüfer
_et al._ (2018); Erne  _et al._ (2018); Eigen  _et al._ (2018); Schmied  _et
al._ (2020); Glidden  _et al._ (2021); Chiocchetta  _et al._ (2017); Sciolla
and Biroli (2010); Schiró and Fabrizio (2010); Eckstein  _et al._ (2009);
Žunkovič  _et al._ (2018), where, in sharp contrast, both energy and total
number of particles are conserved. A common obstruction against the
realization of this program is the occurrence of an effective thermal
behaviour for the soft modes relevant at the critical points of driven-
dissipative systems Mitra  _et al._ (2006); Dalla Torre  _et al._ (2012);
Torre  _et al._ (2013); Maghrebi and Gorshkov (2016); Foss-Feig  _et al._
(2017); Kirton  _et al._ (2019): although the full momentum distribution of
the non-equilibrium steady state manifestly breaks detailed balance, low
momenta can thermalize at an effective temperature set by the interplay of
drive, noise and losses. This forces several instances of driven-open
criticality to fall into known equilibrium universality classes Sieberer  _et
al._ (2016); Hohenberg and Halperin (1977), with few exceptions represented by
the appearance of novel independent anomalous exponents associated to
decoherence Sieberer  _et al._ (2013, 2014); Täuber and Diehl (2014), or by
exotic features as non-equilibrium multi-critical points Young  _et al._
(2020).

In a nutshell, the culprit for effective thermalization in driven-open
systems, is a spectrum of noise variance with a non-vanishing gap at small
momenta and frequencies, which sets the temperature of infrared modes in
several platforms of interest Mitra  _et al._ (2006); Dalla Torre  _et al._
(2012); Torre  _et al._ (2013); Maghrebi and Gorshkov (2016); Foss-Feig  _et
al._ (2017); Sieberer  _et al._ (2013, 2016). Softening such gap and allowing
the noise spectrum to scale down to zero at small momenta, is the route for
instances of driven-open criticality without thermal counterpart. In quadratic
fermionic models Eisert and Prosen ; Höning  _et al._ (2012a) or interacting
quantum wires Dalla Torre  _et al._ (2010); Marino and Diehl (2016a); Baldwin
_et al._ (2021), dissipation with non-local support in real space acting on
neighboring sites in a correlated fashion Diehl  _et al._ (2008), has been
employed to achieve non-equilibrium quantum criticality. Such dissipation
induces a structured noise spectrum in momentum space, that is vanishing in
the infrared, and which exposes a set of modes which are asymptotically
decoupled, for $q\to 0$, from the decohering and thermalizing effect of the
environment. These forms of non-local dissipation can steer a system into a
many-particle dark state with non-trivial quantum correlations – a state
preparation protocol with interesting perspectives for applications in quantum
information and technology Eisert and Prosen (2010); Verstraete  _et al._
(2009); Marcos  _et al._ (2012); Diehl  _et al._ (2011); Höning  _et al._
(2012b); Buča  _et al._ (2019); Barreiro  _et al._ (2011); Lin  _et al._
(2013); Ma  _et al._ (2019); Olmos  _et al._ (2014); Reiter  _et al._ (2016);
Parmee and Cooper (2018).

In this work, we consider spin losses with a controllable spatial profile
decaying algebraically at long distances González-Tudela  _et al._ (2015);
Hung  _et al._ (2016); Buyskikh  _et al._ (2016); Seetharam  _et al._ (2021).
We show how to employ their noise spectrum, which scales softly as $\propto
q^{\alpha}$ in the infrared, in order to realize (i) non-equilibrium Ising
critical points beyond currently available categorizations of equilibrium or
dynamical universality classes. The instances of criticality we find can be
regarded as (ii) non-unitary analogues of the universality class of long-range
interacting Ising models. (iii) For $0<\alpha<1$, the effective field theory
determining the static and dynamical exponents, is a Langevin theory where
inertial and frictional kinetic coefficients coexist and the driving noise is
gapless and scales proportionally to $q^{\alpha}$ in the infrared. (iv) Such
non-equilibrium scaling regime is bounded at long distances and times by the
effect of local spontaneous emission, which are unavoidably present in any
experimental realization and compete with ’long-range’ losses, eventually
driving the system into a thermal fixed point. Our analysis serves as a step-
stone for revising established instances of driven-open criticality in the
presence of tunable non-local dissipation, and it offers an alternative angle
on dissipative engineering of critical states in modern AMO or solid state-
quantum optics platforms.

## Ising criticality with non-local losses –

We consider a quantum Ising chain in $d$ dimensions

$H=-\sum_{\langle
i,j\rangle}\sigma^{x}_{i}\sigma^{x}_{j}+h\sum_{i}\sigma^{z}_{i},$ (1)

subject to a spin loss Lindblad channel which is non-local and shaped by a
spatial structure factor $\gamma_{i,j}$:

$\leavevmode\resizebox{420.89862pt}{}{
$\dot{\rho}=i[\rho,{H}]+\sum_{i,j}\gamma_{i,j}\left(\sigma^{-}_{i}\rho\sigma_{j}^{+}-\frac{1}{2}\left\\{\sigma_{j}^{+}\sigma^{-}_{i},\rho\right\\}\right)$}.$
(2)

A schematic portrait is provided in Fig. 1. Dissipators as in Eq.(2) can be
realized in cavity QED Seetharam  _et al._ (2021); Buyskikh  _et al._ (2016)
or photonic crystal waveguides González-Tudela  _et al._ (2015); Hung  _et
al._ (2016), where losses between pairs of spins at arbitrary distances can be
controlled through a combination of spatial-dependent ground state energy
shifts and external pump fields Seetharam  _et al._ (2021). The former
typically results from a background magnetic field gradient and gives spatial
resolution to dissipation, while a Raman drive with several sidebands enables
control on the functional form of $\gamma_{i,j}$, which can be made
translational invariant, $\gamma(r)$ with $r=|i-j|$.

Figure 1: Schematic portrait of a spin lattice subject to non-local losses,
$\gamma_{i,j}$, acting on pairs of spins at positions $i$ and $j$. In the
right inset we show three possible distance dependent loss channels which have
in common a Liouvillian gap vanishing as $q^{\alpha}$ at small momenta (in the
figure $0<\alpha<1$). This is the class of spatial dependent spin losses
considered in this work.

In the following we analyze the impact of a class of $\gamma(r)$ on Ising
critical systems. Using the frequencies and amplitudes of the Raman sidebands
Hung  _et al._ (2016), we can engineer a loss channel (2) with a decay rate
$\Gamma_{q}\sim\Gamma_{1}q^{\alpha}$ with $\alpha>0$, at small momenta (see
the note 111 The relation between $\Gamma_{q}$ and the amplitudes,
$\Omega_{b}$, of the Raman sidebands is
$\Omega_{b}\sim\sum_{q}e^{iqb}\sqrt{\Gamma_{q}}$ (see Ref. Seetharam  _et al._
(2021)). In order to have $\Gamma_{q}\sim q^{\alpha}$ for $q\to 0$, the
amplitudes have to scale as $\Omega_{b}\sim b^{-(d+\alpha/2)}$ for $b\gg 1$.).
In such situation, the infrared modes are ’dark’, asymptotically decoupled
from dissipative effects, since $\Gamma_{q}$ vanishes as $q\to 0$. Following
the analogy with criticality in long-range interacting spin systems where
instead it is the Hamiltonian spectral gap which scales as $m\sim q^{\alpha}$
Fisher  _et al._ (1972); Dutta and Bhattacharjee (2001); Maghrebi  _et al._
(2016), we can interpret the instance of criticality inspected in this work as
a non-unitary analogue of critical long-range interacting Ising systems.
Indeed, at long distances we have $\gamma(r)\propto 1/r^{\alpha+d}$ for its
Fourier transform, $\Gamma_{q}$, to behave as $\Gamma_{q}\sim q^{\alpha}$ in
the infrared. The anti-commutator in (2) can be therefore regarded as a non-
hermitian long-range interacting hamiltonian Seetharam  _et al._ (2021). The
shape of $\gamma(r)$ at short or intermediate distances is however
inconsequential for our results which will be pertinent to soft, infrared
modes at criticality. Therefore, our analysis applies to a broad class of
spatial profiles $\gamma(r)$, provided they entail the infrared scaling of the
Liouvillian gap mentioned above, $\Gamma_{q}\sim\Gamma_{1}q^{\alpha}$ (cf.
with the inset in Fig. 1). In this respect, our setup also extends previous
instances on preparation and criticality of open quantum systems with dark
states in cold atoms or quantum optics platforms, where, in contrast, the
shape of non-local losses is not tunable Eisert and Prosen ; Höning  _et al._
(2012a); Marcos  _et al._ (2012); Dalla Torre  _et al._ (2010); Marino and
Diehl (2016a). In the case discussed here, the exponent $\alpha$ can be
flexibly varied by a proper choice of the amplitudes of the Raman sidebands
(see note [69]). In the following we analyze the two distinct cases of
$0<\alpha<1$, and $\alpha>1$ (see also Table 1 for a summary of the results).

## Canonical scaling with long-range losses –

We now map the lattice model in Eq. (2) into a long wavelength non-equilibrium
field theory Kamenev (2011); Sieberer  _et al._ (2016). In particular, we will
discuss how the effective field theory governing non-equilibrium critical
behaviour for $0<\alpha<1$ is ascribable to a Langevin model with coexisting
inertial and frictional terms, driven by a gapless noise $\propto q^{\alpha}$
at small momenta.

Following the usual prescription Cardy (1996); Amit and Martin-Mayor (2005);
Kardar (2007) we map the spin operators in terms of bosons $\sigma^{-}_{i}\to
a_{i}$ and $\sigma^{z}_{i}\to 2a^{\dagger}_{i}a_{i}$, and we implement the
hard-core constraint with a large on site non-linearity
$a^{\dagger}_{i}a^{\dagger}_{i}a_{i}a_{i}$. By taking the continuum limit and
coarse graining over short wavelengths Cardy (1996); Amit and Martin-Mayor
(2005); Kardar (2007); Maghrebi and Gorshkov (2016), the Ising interaction in
Eq. (1) yields a second derivative in space within a leading order derivative
expansion ($Kq^{2}$ in momentum space), while the non-linearity yields the
usual $\varphi^{4}$ potential. As detailed in Refs. Lee  _et al._ (2013);
Maghrebi and Gorshkov (2016); Sieberer  _et al._ (2016), the quantum master
equation for an Ising model with losses (2) can be mapped into a Keldysh path
integral in terms of the classical and quantum components of the real field,
$\varphi_{c/q}(x,t)$, resulting in the following Gaussian action in momentum
and frequency $Q=(\mathbf{q},\omega)$ space,

$\mathcal{S}_{G}=\int_{Q}\left(\varphi_{c}(-Q),\varphi_{q}(-Q)\right)\begin{pmatrix}0&P^{A}\\\
P^{R}&P^{K}\end{pmatrix}\begin{pmatrix}\varphi_{c}(Q)\\\
\varphi_{q}(Q)\end{pmatrix}$ (3)

where

$\begin{split}P^{R/A}&=-\omega^{2}-2Kq^{2}\mp
2i\Gamma_{q}\omega+m-\Gamma^{2}_{q}/2,\\\ P^{K}&=i\Gamma_{q}\equiv
i(\Gamma_{0}+\Gamma_{1}q^{\alpha}),\end{split}$ (4)

are the retarded/advanced and Keldysh sectors of the action Kamenev (2011);
Sieberer  _et al._ (2016). The kinetic coefficients therein contained, as well
as the spectral gap $m$, can be directly related to microscopic couplings Lee
_et al._ (2013); Maghrebi and Gorshkov (2016). In the Liouvillian gap
$\Gamma_{q}$ we have included a constant term $\Gamma_{0}$ which takes into
account the spontaneous emission of the spins into free space and which is
often the main adversary in schemes implementing dissipative engineering with
dark states Sørensen and Mølmer (2002); Dalla Torre  _et al._ (2013). Its
detrimental role is to locally measure the atoms and therefore suppress the
entangling effect of non-local dissipation at long times (or small wave-
vectors). We will inspect the critical properties of our model in the regime
where $\Gamma_{1}q^{\alpha}\gg\Gamma_{0}$ and therefore $P^{K}$ scales
effectively as $q^{\alpha}$, and, at the same time, also the retarded/advanced
sectors become gapless, $P^{R/A}\sim q^{2\alpha}$, since we tune the spectral
mass to zero as well. The former can be implemented in the RG scheme via the
scaling ansatz $\Gamma_{0}\sim\Gamma_{1}q^{\alpha}$ (see also Ref. Marino and
Diehl (2016b)). As pointed out in Paz and Maghrebi (2019, 2021a, 2021b), weak
dissipation as the one considered here can expose novel critical behaviour for
a long temporal window before thermalizing effects set it. In our setup, this
is mirrored by the fact that for $q\ll(\Gamma_{0}/\Gamma_{1})^{1/\alpha}$
incoherent emission takes over, and the Gaussian action in (3) reduces to a
Langevin action with $\Gamma_{q}\simeq\Gamma_{0}$ and no $\omega^{2}$ term
which notoriously thermalizes Hohenberg and Halperin (1977); Täuber (2014).
This crossover is analogous to the suppression of equilibrium quantum
criticality at distances larger than the de Broglie thermal length Sondhi  _et
al._ (1997); Vojta (2003); Sachdev (2011).

| Critical exponent | $d_{l}<d<d_{u}$ | Effective field theory
---|---|---|---
LR losses with $\alpha<1$ | $\nu=1/(2\alpha)+\epsilon/(12\alpha^{2})$ | $2\alpha<d<4\alpha$ | coexisting inertial/frictional derivatives + soft noise
LR losses with $\alpha>1$ | $\nu=1/2+\epsilon/12$ | $3-\alpha<d<5-\alpha$ | short-range Ising model + soft noise
LR losses and interactions | $\nu=1/\alpha+\epsilon/(3\alpha^{2})$ | $\alpha/2<d<3\alpha/2$ | long-range Ising model + soft noise
Table 1: Overview of non-equilibrium critical regimes in the presence of
’long-range’ (LR) losses. The third column displays the lower ($d_{l}$) and
upper ($d_{u}$) critical dimensions, while the fourth one summarizes the
effective dynamical field theory valid at criticality.

Approaching criticality, $m\to 0$, and for $\Gamma_{0}\to 0$, we can adopt the
canonical scaling ansatz for the dynamical critical exponent $\omega\sim
q^{\alpha}$, with the result that both the contribution
$\propto\omega\Gamma_{q}$ and $\propto\omega^{2}$ scale as $\sim q^{2\alpha}$
in the infrared. This is contrast to relaxational Langevin models, where the
inertial term proportional to the second derivative in time ($\sim\omega^{2}$)
is subleading compared to the frictional first order time derivative
($\sim\omega$). Therefore, we have a scalar dynamical field theory with
coexisting inertial and frictional kinetic coefficients, driven by a gapless
noise, which is a model beyond the Halperin-Hohenberg classification Hohenberg
and Halperin (1977).

We now focus on kinetic coefficients proportional to spatial derivatives. At
the level of canonical power counting, there is a threshold value $\alpha<1$
at which the second derivative in space ($\propto Kq^{2}$ term) is subleading
in the infrared compared to the $\Gamma^{2}_{1}q^{2\alpha}$ fractional
derivative resulting from ’long-range’ losses in the spectral sector
($P^{R/A}$). This should be contrasted with critical long-range interacting
Ising models where such threshold is set at $\alpha=2$ Fisher  _et al._
(1972); Dutta and Bhattacharjee (2001); Vodola  _et al._ (2014); Maghrebi  _et
al._ (2016) besides small corrections resulting from anomalous dimensions Sak
(1973); Luijten and Blöte (2002); Angelini  _et al._ (2014); Defenu  _et al._
(2015); Fey and Schmidt (2016); Defenu  _et al._ (2017); Behan  _et al._
(2017). These different thresholds occur because hermitian long-range
interactions compete with $Kq^{2}$ through a $q^{\alpha}$ term in the R/A
sector (see Fisher  _et al._ (1972); Dutta and Bhattacharjee (2001); Vodola
_et al._ (2014); Maghrebi  _et al._ (2016)), while non-hermitian ones through
$q^{2\alpha}$ terms resulting from $\omega\Gamma_{q}$ and $\Gamma^{2}_{q}$
(cf. with Eq. (4)).

We notice that the RG procedure generates only analytical terms and thus
cannot renormalize the terms scaling with the exponent $\alpha$ (see also
Fisher  _et al._ (1972); Dutta and Bhattacharjee (2001); Maghrebi  _et al._
(2016)). The only term which can acquire an anomalous dimension is the kinetic
coefficient of the inertial term ($\sim\omega^{2}$), as we will further
discuss below. This makes unviable a fine compensation of the anomalous
dimensions of the retarded and Keldysh sectors, which would signal, whenever
occurring, effective infrared thermalization Sieberer  _et al._ (2013); Marino
and Diehl (2016b). Therefore, the RG fixed point discussed in the following
explicitly breaks fluctuation-dissipation relations and cannot have an
equilibrium counterpart, distinctly from other instances of non-equilibrium
open criticality Mitra  _et al._ (2006); Dalla Torre  _et al._ (2012); Torre
_et al._ (2013); Maghrebi and Gorshkov (2016); Foss-Feig  _et al._ (2017);
Kirton  _et al._ (2019). We now study the critical regime $m\to 0$ for
$\alpha<1$. We will focus on other values of $\alpha$ in the second part of
the article.

## RG fixed point and criticality for $\alpha<1$ –

We will now complement the Gaussian action in Eq. (3) with non-linear terms,
following the canonical power counting just outlined. For $\alpha<1$, we have
$P^{R/A}\sim q^{2\alpha}$ and $P^{K}\sim q^{\alpha}$, with canonical scaling
dimensions for the classical and quantum fields $\varphi_{c}\sim
q^{d/2-\alpha}$, $\varphi_{q}\sim q^{d/2}$, and accordingly a lower critical
dimension of $d_{l}=2\alpha$. Below the upper critical dimension
$d_{u}=4\alpha$ the classical non-linear term
$(u_{c}/4!)\varphi^{3}_{c}\varphi_{q}$ is relevant. The next RG leading non-
linearity appears at $d<3\alpha$ where the additive noise term
$i(\kappa/2)\varphi^{2}_{c}\varphi^{2}_{q}$ and the sextic term
$(\lambda/5!)\varphi^{5}_{c}\varphi_{q}$ are both RG relevant. Quantum
vertices with higher powers of quantum fields are always irrelevant hinting at
the semi-classical nature of the fixed point, and marking a difference with
previous studies on quantum criticality induced by dark states Eisert and
Prosen ; Höning  _et al._ (2012a); Dalla Torre  _et al._ (2010); Marino and
Diehl (2016a). Notice that similarly to the long-range interacting model A of
the Halperin-Hohenberg classification Halimeh and Maghrebi (2021) we have a
dynamical critical exponent $z=\alpha$, but different lower and upper critical
dimensions due to the gapless nature of the noise, suggesting that the scaling
regime studied here belongs to a different universality class. Similarly there
are neat differences with the canonical power counting of the zero-temperature
critical long-range Ising model, where $z=\alpha/2$. Such dissimilarities
imply a different canonical counting of non-linearities, and therefore
qualitatively different critical behaviour.

In order to find the interacting fixed point, we run a one-loop improved RG
Amit and Martin-Mayor (2005); Peskin (2018) on the effective potential
$\mathcal{U}$ including relevant non-linearities. We employ a sharp cutoff in
momentum space $k<q<\Lambda$; here $k$ is the running RG scale and $\Lambda$
an UV regulator. We assume the couplings in the action to run with the RG time
$t=\ln k$, and we obtain the RG flow of the effective potential following
Peskin (2018),

$\partial_{t}\mathcal{U}=\frac{S_{d}}{(2\pi)^{d}}k^{d}\left[\frac{i}{2}\int_{\omega}\ln\det\mathcal{S}^{(2)}\right]$
(5)

with $S_{d}=2\pi^{d/2}/\Gamma(d/2)$, where $\mathcal{S}^{(2)}$ is the second
order variation of the action with respect to the classical and quantum fields

$\mathcal{S}^{(2)}\equiv\frac{\delta^{2}\mathcal{S}}{\delta\varphi_{n}(-Q)\delta\varphi_{m}(Q)}$
(6)

with $n,m=c,q$. From Eq. (5) we get the RG flow equations for the couplings by
taking the appropriate derivatives (e.g. second derivatives for the retarded
and Keldysh masses, fourth order derivatives for the non-linear terms, etc)
and evaluating them at homogeneous vanishing background fields Amit and
Martin-Mayor (2005); Peskin (2018); Kamenev (2011); Sieberer  _et al._ (2016).
We first consider a leading order $\epsilon$-expansion, right below the upper
critical dimension $\epsilon\equiv d_{u}-d\ll 1$ (where $d_{u}=4\alpha$). We
follow the canonical rescaling discussed above, $\tilde{m}\sim m/k^{2\alpha}$,
$\tilde{\Gamma}_{0}\sim\Gamma_{0}/\Gamma_{1}k^{\alpha}$ and $\tilde{u}_{c}\sim
u_{c}/k^{4\alpha-d}$, and we find from the following rescaled beta functions

$\begin{split}&\partial_{t}\tilde{m}=-2\alpha\tilde{m}+\frac{\tilde{u}_{c}(-2\tilde{m}+(1+\tilde{\Gamma}_{0})^{2})}{4(1+\tilde{\Gamma}_{0})^{6}},\\\
&\partial\tilde{u}_{c}=-\epsilon\tilde{u}_{c}-\frac{3\tilde{u}^{2}_{c}}{2(1+\tilde{\Gamma}_{0})^{6}},\quad\partial_{t}\tilde{\Gamma}_{0}=-\alpha\tilde{\Gamma}_{0},\end{split}$
(7)

a Wilson-Fisher (WF) fixed point at
$(\tilde{m}^{*},\tilde{\Gamma}^{*}_{0},\tilde{u}^{*}_{c})=(-\epsilon/(12\alpha),0,-2\epsilon/3)$,
with a correlation length critical exponent $\xi\sim m^{-\nu}$,
$\nu=1/(2\alpha)+\epsilon/(12\alpha^{2})$. This fixed point has an additional
unstable RG direction corresponding to perturbations around the fixed point
value $\tilde{\Gamma}^{*}_{0}=0$, in agreement with the requirement to fine
tune the Liouvillian gap ($\Gamma_{0}\to 0$) in addition to the closing of the
spectral one ($m\to 0$). This is in full analogy with the RG relevance of
temperature at equilibrium quantum critical points, which is as well
responsible for the onset of an additional RG unstable direction Sondhi  _et
al._ (1997); Vojta (2003); Sachdev (2011). At $\mathcal{O}(\epsilon^{2})$, we
find $z\simeq\alpha+\epsilon^{2}/(24(1+4\alpha^{2}))$. A positive correction
to the canonical value of the dynamical critical exponent, $\omega\sim
q^{\alpha}$, indicates that the inertial term becomes slightly subleading in
the infrared compared to the frictional one, suggesting that the latter
dominates at criticality.

In Eqs. (7) the flow of $\tilde{\Gamma}_{0}$ is solely governed by its
canonical dimension. To find a non-trivial WF fixed point for $\Gamma_{0}$, we
need the multiplicative noise $i(\kappa/2)\varphi^{2}_{c}\varphi^{2}_{q}$ to
be RG relevant. As discussed above, this occurs for $\alpha>d/3$, giving to
the Gaussian noise sector, $P^{K}$, a one-loop dressing proportional to
$\sim\kappa\int_{Q}G^{K}(Q)$. For consistency with the canonical power
counting we have also included the sextic vertex $\propto\lambda$. By
evaluating, for instance, the RG flow of the one-loop effective potential in
(5) at $d=2$ and $\alpha=0.7$, we find a WF fixed point
$(\tilde{m}^{*},\tilde{\Gamma}^{*}_{0},\tilde{u}^{*}_{c},\tilde{g}^{*},\tilde{\lambda}^{*})=(0.04,0.23,2.53,-1.98,-0.93)$
with still two unstable directions; the one associated to the spectral mass
yields $\nu\simeq 0.71$. From the beta equation of $\Gamma_{0}$, we can
estimate the growth of the spontaneous emission, $\Gamma_{0}$, due to loop
corrections in the vicinity of the fixed point. We can compactly write that as

$\partial_{t}\Gamma_{0}\equiv
k\partial_{k}\Gamma_{0}=\mathcal{F}(\tilde{m}^{*},\tilde{\Gamma}^{*}_{0},\tilde{g}^{*})(\Gamma_{0}+\Gamma_{1}k^{\alpha}),$
(8)

with $\mathcal{F}(\tilde{m}^{*},\tilde{\Gamma}^{*}_{0},\tilde{g}^{*})$ a
function of the fixed point values of the couplings. By solving Eq. (8) and
requiring that $\Gamma_{0}(k^{*})\simeq\Gamma_{1}k^{*\alpha}$, we find the
momentum scale, $k^{*}$, where incoherent emission in free space is comparable
in magnitude to the noise induced by non-local losses. To assign an initial
condition to Eq. (8), we assume that at the Ginzburg scale Amit and Martin-
Mayor (2005), $k\simeq\Lambda_{G}$, the flow of the RG couplings has set to
their fixed point values; for the incoherent emission rate we have
$\Gamma_{0}(k\simeq\Lambda_{G})\simeq\tilde{\Gamma}^{*}\Gamma_{1}\Lambda^{\alpha}_{G}$.
After straightforward algebra, we find $k^{*}\simeq 10^{-6}\Lambda_{G}$, which
indicates that, despite loop corrections, it is still possible to find a broad
window of momenta where the effect of non-local losses dominates before the
disentangling effect of spontaneous emission takes over. This is an important
a posteriori check on the observability of the scaling regime discussed in
this work.

## Fixed point for $\alpha>1$ –

By inspection of Eqs. (4) we notice that for $\alpha>1$ the kinetic
coefficient $\sim Kq^{2}$ in the advanced/retarded sector dominates over the
$\propto\Gamma^{2}_{1}q^{2\alpha}$ term resulting from non-local dissipation.
This leads to a dynamical critical exponent $z=1$, with the term
$\propto\omega\Gamma_{q}$ now negligible in the infrared; in other words, we
have an Ising model with short range interactions and a
$\propto\Gamma_{1}q^{\alpha}$ Markovian noise drive. This changes the upper
and lower critical dimensions into $d_{l}=3-\alpha$ and $d_{u}=5-\alpha$, and
in leading order epsilon expansion we find a correlation length critical
exponent of $\nu=1/2+\epsilon/12$, with $\epsilon\equiv 5-\alpha-d\ll 1$. At
this WF fixed point quantum terms such as the quartic
$u_{q}\varphi^{3}_{q}\varphi_{c}$ are irrelevant, unless $\alpha>2$. However,
as $\alpha$ increases the spatial support of losses quickly shrinks Seetharam
_et al._ (2021), retrieving uninteresting local dissipation effects similar to
$\Gamma_{0}$.

## Competing long-range interactions and losses –

Finally, we consider the scenario where long-range Ising interacitons,
$\sum_{\langle
i,j\rangle}\frac{J}{|i-j|^{1+\alpha}}\sigma^{x}_{i}\sigma^{x}_{j}$, compete
with ’long-range’ losses. Such term adds a $Jq^{\alpha}$ contribution to
$P^{R/A}$ Fisher  _et al._ (1972); Dutta and Bhattacharjee (2001); Vodola  _et
al._ (2014); Maghrebi  _et al._ (2016). This situation can be explicitly
engineered following the Raman techniques mentioned in the introduction, in an
optical cavity where the atoms couple to a photon which is not far detuned
from resonance Hung  _et al._ (2016); Seetharam  _et al._ (2021), or, more
directly, considering non-local losses in trapped ions where long-range
interactions naturally occur . By inspection of Eqs. (4), we realize that for
$\alpha>2$ we recover the same scaling discussed above for ’long-range’ losses
with $\alpha>1$. For $\alpha<2$, instead, we find a leading scaling
$P^{R/A}\sim Jq^{\alpha}$, since long-range interactions suppress at small
momenta the contribution of non-local losses in the spectral sector
($z=\alpha/2$). This is equivalent to the critical scaling of a long-range
interacting Ising model driven by a $\propto\Gamma_{1}q^{\alpha}$ Markovian
noise, and it is a limit where classical and quantum vertices scale alike,
$u_{c,q}\sim q^{3\alpha/2-d}$. Such quantum scaling regime is formally
equivalent to a critical zero-temperature long-range interacting Ising model
Fisher  _et al._ (1972); Dutta and Bhattacharjee (2001), and it shares with
that the same upper and lower critical dimensions. Consistently, we find the
same correlation length critical exponent in leading order $\epsilon$
expansion, $\nu=1/\alpha+\epsilon/(3\alpha^{2})$.

## Conclusions –

An overview of non-equilibrium criticality in the presence of ’long-range’
losses is displayed in Tab. 1. The central result is the existence of a non-
equilibrium scaling regime governed by a Langevin equation with a noise
$\propto q^{\alpha}$, and with competing inertial and frictional kinetic
terms. We have argued that this can be regarded as a dissipative analogue of
the universality class of critical long-range Ising models.

On the whole, our work shows that spatial tunable dissipation could offer a
route to guide the exploration of criticality in open systems beyond current
paradigms. For instance, a promising direction of research consists in
focusing on richer driven-open platforms, where incoherent losses/pumps and
dephasing channels compete and can acquire non-local spatial character. For
example, an extension of the universality class of driven-open condensates
with $O(2)$ order parameter Sieberer  _et al._ (2013, 2014); Altman  _et al._
(2015); Marino and Diehl (2016a); Zamora  _et al._ (2017), naturally appears
as an intriguing perspective.

## Acknowledgements

We acknowledge N. Defenu, M. Maghrebi, A. Piñeiro-Orioli, K. Seetharam for
useful discussions and S. Diehl, S. Kelly for proof reading the manuscript.
This project has been supported by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation), Project-ID 429529648, TRR 306 QuCoLiMa (’Quantum
Cooperativity of Light and Matter’) and in part by the National Science
Foundation under Grant No. NSF PHY-1748958 (KITP program ’Non-Equilibrium
Universality: From Classical to Quantum and Back’).

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##  Appendix A Gaia Archive Queries

In order to crossmatch 2MASS to GDR3 sources with XP spectra, we run the
following ADQL query on the Gaia Archive:

⬇

1SELECT

2 g.random_index, g.source_id,

3 g.ra, g.dec,

4

5 tmass.designation,

6 tmass.ra as tm_ra, tmass.dec as tm_dec,

7 tmass.j_m, tmass.j_msigcom,

8 tmass.h_m, tmass.h_msigcom,

9 tmass.ks_m, tmass.ks_msigcom,

10 tmass.ph_qual,

11

12 xmatch.angular_distance

13 as gaia_tmass_angular_distance

14

15FROM gaiadr3.gaia_source AS g

16JOIN gaiadr3.tmass_psc_xsc_best_neighbour

17AS xmatch

18 USING (source_id)

19JOIN gaiadr3.tmass_psc_xsc_join AS xjoin

20 USING (clean_tmass_psc_xsc_oid)

21JOIN gaiadr1.tmass_original_valid AS tmass

22 ON xjoin.original_psc_source_id

23 = tmass.designation

24

25WHERE

26 (g.has_xp_continuous = ’true’)

27 AND (

28 (tmass.ph_qual LIKE ’A__’)

29 OR (tmass.ph_qual LIKE ’_A_’)

30 OR (tmass.ph_qual LIKE ’__A’)

31 )

32 AND (tmass.ext_key is NULL)

33 AND (g.source_id BETWEEN sid0 AND sid1)

34

35ORDER BY g.source_id

Above, sid0 and sid1 are integers used to divide our query into manageable
chunks.

We use the following query to fetch information from the GDR3 gaia_source
catalog and from the astrometric fidelity catalog (Rybizki et al., 2022) on
each source with an XP spectrum:

⬇

1SELECT

2 g.random_index, g.source_id, g.ref_epoch,

3

4 g.ra, g.ra_error, g.dec, g.dec_error,

5 g.parallax, g.parallax_error,

6 g.pmra, g.pmra_error, g.pmdec, g.pmdec_error

7

8 g.ruwe, g.astrometric_excess_noise,

9

10 spur.fidelity_v2, spur.norm_dg,

11

12 g.phot_g_mean_mag,

13 g.phot_g_mean_flux, g.phot_g_mean_flux_error,

14 g.phot_bp_mean_mag,

15 g.phot_bp_mean_flux, g.phot_rp_mean_flux_error,

16 g.phot_rp_mean_mag,

17 g.phot_rp_mean_flux, g.phot_bp_mean_flux_error,

18

19 g.has_xp_continuous,

20 g.phot_bp_n_obs, g.phot_rp_n_obs,

21 g.phot_bp_rp_excess_factor,

22 g.visibility_periods_used

23

24FROM gaiadr3.gaia_source as g

25 JOIN gaiadr3.astrophysical_parameters as p

26 ON g.source_id = p.source_id

27 LEFT OUTER JOIN external.gaiaedr3_spurious as spur

28 ON g.source_id = spur.source_id

29

30WHERE

31 (g.has_xp_continuous = ’True’)

32 AND (g.source_id BETWEEN sid0 AND sid1)

33

34ORDER BY g.source_id

##  Appendix B Stellar atmospheric parameter confidence estimates

As described in Section 4.4, we train a classifier for each stellar
atmospheric parameter ($T_{\rm eff}$, $\mathrm{[Fe/H]}$, $\log{g}$), which
assigns a “confidence” between 0 and 1 to each parameter estimate. Here, we
list the input features used by the classifiers, and describe the neural-
network structure of the classifiers, both of which are uniform across all
three classifiers.

We use the following features:

  * •

ln_rchi2_opt

  * •

ln_dplx2

  * •

ln_prior

  * •

teff_est

  * •

logg_est

  * •

feh_est

  * •

teff_est_err

  * •

logg_est_err

  * •

feh_est_err

  * •

asinh_plx_snr

  * •

asinh_g_snr

  * •

asinh_bp_snr

  * •

asinh_rp_snr

  * •

ln_phot_bp_rp_excess_factor

  * •

phot_g_mean_mag

  * •

ln_ruwe

  * •

fidelity_v2

  * •

norm_dg

  * •

ln_bp_chi_squared

  * •

ln_rp_chi_squared

Features of the form asinh_X_snr are computed using
$\mathrm{asinh}\left(X/\sigma_{X}\right)$, using the GDR3 parallax and flux
measurements. Features of the form ln_X are calculated as
$\ln\big{[}\max\big{(}X,10^{-7}\big{)}\big{]}$. The ln_dplx2 feature is
computed from $\left|\varpi_{\mathrm{GDR3}}-\varpi_{\rm
est}\right|/\sigma_{\varpi,\mathrm{GDR3}}$. The fields fidelity_v2 and norm_dg
are indicators of the quality of the GDR3 astrometric measurements and of
crowding, respectively (Rybizki et al., 2022). Features of the form X_est and
X_est_err are based on our stellar parameter estimates and their associated
uncertainties. In addition to the above features, we additionally provide the
network with information about the flux residuals between our best-fit models
and the observed XP spectra. In detail, we provide the network with
$\mathrm{asinh}\big{[}\big{(}f_{\rm pred}-f_{\rm
obs}\big{)}/\sigma_{f}\big{]}$ at each sampled wavelength.

We concatenate all of our input features into a single vector for each source,
and feed it into a feed-forward neural network with three densely-connected
hidden layers, each with 64 neurons and ReLU activation, and a single output
neuron with a sigmoid activation. In order to regularize the network during
training, we use a dropout fraction of 0.1 before each hidden layer
(Srivastava et al., 2014), and impose an L2 penalty of $10^{-3}$ on the
weights of each hidden layer. The output neuron represents the probability of
the given stellar atmospheric parameter estimate being reliable. We use the
binary cross-entropy between our output neuron and the training label as our
loss function. We train each classifier using the Adam optimizer (Kingma & Ba,
2014), with a batch size of 4096, and an initial learning rate of $10^{-3}$,
which we decrease by a factor of 10 whenever the validation loss fails to
decrease over a span of 256 training epochs. We terminate the training
procedure when the validation loss fails to decrease over a span of 1024
epochs.