# On the GREM approximation of TAP free energies

Giulia Sebastiani  Giulia Sebastiani
J.W. Goethe-Universität Frankfurt, Germany<EMAIL_ADDRESS>and  Marius Alexander Schmidt  M. A. Schmidt
J.W. Goethe-Universität Frankfurt, Germany<EMAIL_ADDRESS>frankfurt.de

###### Abstract.

We establish both a Boltzmann-Gibbs principle and a Parisi formula for the
limiting free energy of an abstract GREM (Generalized Random Energy Model)
which provides an approximation of the TAP (Thouless-Anderson-Palmer) free
energies associated to the Sherrington-Kirkpatrick (SK) model.

###### Key words and phrases:

Mean Field Spin Glasses, Large Deviations, Gibbs-Boltzmann and Parisi
Variational Principles, Random Energy Models

###### 2000 Mathematics Subject Classification:

60J80, 60G70, 82B44

We are grateful to Nicola Kistler for leading the way in this line of
research.

##  1\. Introduction

Introduced by Derrida [12, 13] in the 80s, the Generalized Random Energy
Models (GREMs) serve as simplified, abstract models that are completely
solvable and have significantly contributed to our comprehension of specific
facets within the Parisi theory for mean field spin glasses [23].

The simplest of Derrida’s models is the Random Energy Model (REM) [12], whose
Hamiltonian at volume $N\in\mathbb{N}$ is given by a vector
$\\{H_{N}(\sigma)\\}_{\sigma\in\\{-1,+1\\}^{N}}$ of independent Gaussian
random variables with variance $N$. The GREMs [13] generalize the REM,
featuring hierarchically correlated Gaussian fields that inherently exhibit
the ultrametricity predicted by the Parisi theory.

In [21] we related the REM and the Thouless-Anderson-Palmer (TAP) free
energies [33] for the Sherrington-Kirkpatrick (SK) model [30], the
prototypical mean field spin glass. The linkage is based on the observation
that the free energy of TAP-solutions for the SK can be expressed as a highly
nonlinear functional of the local fields (see (4) and (5) below). Through a
replacement of these fields we obtain abstract models involving only the
alleged geometrical properties of the relevant TAP-solutions, that are
solvable using Boltzmann formalism through a classical, Sanov-type large
deviation analysis.

To see how this unfolds and for the reader’s convenience, we briefly review
the setting given in Section 2 of [21]. The starting point is a non-rigorous
result [33] of Thouless, Anderson, and Palmer, which expresses the free energy
associated to the SK Hamiltonian (by means of a diagrammatic expansion of the
partition function) as a constrained maximum of a certain disorder-dependent
function, the so-called TAP free energy (1). This function is defined on the
space of possible magnetizations of the spins, given as $[-1,1]^{N}$ where $N$
is the system size.

With $\beta\in\mathbb{R}^{+}$ the inverse temperature parameter,
$h\in\mathbb{R}$ an external magnetic field and $J_{ij}$ i.i.d. standard
Gaussians issued on some probability space $(\Omega,\mathcal{F},\mathbb{P})$,
namely the quenched disorder, the SK Hamiltonian at volume $N$ is given by the
Gaussian field
$\sigma\in\\{-1,+1\\}^{N}\mapsto(2N)^{-\frac{1}{2}}\beta\sum_{i\neq
j}J_{ij}\sigma_{i}\sigma_{j}+h\sum_{i=1}^{N}\sigma_{i}$. The TAP free energy
in ${\bf m}=(m_{i})_{i=1}^{N}$ can be written, shortening
$\bar{J}_{ij}:=(J_{ij}+J_{ji})/\sqrt{2}$, as

$\frac{\beta}{\sqrt{N}}\sum\limits_{i<j}\bar{J}_{ij}m_{i}m_{j}+h\sum\limits_{i=1}^{N}m_{i}-\sum\limits_{i=1}^{N}I(m_{i})+\frac{\beta^{2}N}{4}(1-\frac{1}{N}\sum\limits_{i=1}^{N}m_{i}^{2})^{2}$
(1)

where $I$ is the classic coin tossing rate function
$I(m):=\frac{1+m}{2}\log(1+m)+\frac{1-m}{2}\log(1-m)$, which can also be
expressed as

$I(m)=m\tanh^{-1}(m)-\log\cosh\tanh^{-1}(m).$ (2)

The TAP formulation is notably compelling, providing a physically meaningful
representation consisting of three discernible components: internal energy,
entropic contribution and Onsager correction. Hence, the construction and
study of simplified models incorporating the structural elements of the TAP
formulation gives a natural avenue to enhance the framework that ideally
bridges the classical Boltzmann-Gibbs theory with the powerful and intricate,
yet as of now still not fully understood, Parisi theory for mean field spin
glasses.

A quick computation shows that the critical points of the TAP free energy (1)
must be solutions of the TAP equations

$m_{i}=\tanh[\,h+\beta h_{i}({\bf m})\,],\qquad i=1,\dots,N,$ (3)

where

$h_{i}({\bf m}):=\frac{1}{\sqrt{N}}\sum_{j\neq
i}\bar{J}_{ij}m_{j}-\beta(1-\frac{1}{N}\sum_{j=1}^{N}m^{2}_{j})m_{i}.$ (4)

######  Remark 1.1.

For an analysis of the TAP equations, see e.g. [16, 20, 32]. A proof that a
constrained maximum of the TAP free energy yields the exact SK free energy can
be found in [9, Theorem 1]; the formulation of the constraint however strongly
relies on the celebrated Parisi formula for the SK free energy. For more on
the TAP-Plefka approximation within Guerra’s interpolation scheme [17], see
[10] and references therein.

If we assume to have a solution to (3), say ${\bf m}^{\alpha}\in[-1,1]^{N}$,
and evaluate the TAP free energy (1) in ${\bf m}^{\alpha}$; we find it to be
equal to $\text{TAP}_{N}({\bf m}^{\alpha})$, where

$\displaystyle\text{TAP}_{N}({\bf m})$
$\displaystyle:=\frac{\beta^{2}}{4}N\left[\,1-\left(\frac{1}{N}\sum_{j=1}^{N}\tanh^{2}(h+\beta
h_{j}({\bf m}))\right)^{\,2}\,\right]+$ (5)
$\displaystyle\quad+\sum_{i=1}^{N}\left(\,-\frac{\beta}{2}h_{i}({\bf
m})\tanh(h+\beta h_{i}({\bf m}))+\log\cosh(h+\beta h_{i}({\bf m}))\,\right).$

This is easily seen, as follows. If ${\bf m}^{\alpha}$ is a solution to (3),
then the equality

$\beta h_{i}({\bf m}^{\alpha})=\tanh^{-1}(m_{i}^{\alpha})-h$

must hold for every $i=1,\dots,N$. Using the definition (4), the latter
implies

$\frac{\beta}{2}\frac{1}{\sqrt{N}}\sum_{j\neq
i}\bar{J}_{ij}m_{j}^{\alpha}=\frac{1}{2}\tanh^{-1}(m^{\alpha}_{i})-\frac{h}{2}+\frac{\beta^{2}}{2}(1-q_{N}^{\,\alpha})m^{\alpha}_{i},$
(6)

where we shortened
$q_{N}^{\,\alpha}:=\frac{1}{N}\sum_{j=1}^{N}(m^{\alpha}_{j})^{2}$.
Consider now the TAP free energy (1) evaluated in ${\bf m}={\bf m}^{\alpha}$,
which can be rewritten as

$\frac{\beta^{2}N}{4}(1-q_{N}^{\,\alpha})^{2}+\sum\limits_{i=1}^{N}\left[\left(\frac{\beta}{2}\frac{1}{\sqrt{N}}\sum\limits_{j\neq
i}\bar{J}_{ij}m^{\alpha}_{j}+h\right)m^{\alpha}_{i}-I(m^{\alpha}_{i})\right].$

Plugging (6) into this expression, we get

$\displaystyle\frac{\beta^{2}N}{4}(1-q_{N}^{\,\alpha})^{2}+\frac{\beta^{2}N}{2}(1-q_{N}^{\,\alpha})q_{N}^{\,\alpha}+\sum\limits_{i=1}^{N}\left[\frac{1}{2}m^{\alpha}_{i}\tanh^{-1}(m^{\alpha}_{i})+\frac{h}{2}m^{\alpha}_{i}-I(m_{i}^{\alpha})\right]$
(7)
$\displaystyle=\frac{\beta^{2}N}{4}[1-(q_{N}^{\,\alpha})^{2}]+\sum\limits_{i=1}^{N}\left[-\frac{1}{2}m_{i}^{\alpha}\tanh^{-1}(m^{\alpha}_{i})+\frac{h}{2}m_{i}^{\alpha}+\log\cosh\tanh^{-1}(m^{\alpha}_{i})\right],$

where we used the representation (2) for the function $I$.

Expressing each $m^{\alpha}_{i}$ as $\tanh(h+\beta h_{i}({\bf m}^{\alpha}))$,
specifically $q_{N}^{\alpha}=\frac{1}{N}\sum_{i=1}^{N}\tanh(h+\beta h_{i}({\bf
m}^{\alpha}))$, the latter becomes

$\displaystyle\frac{\beta^{2}}{4}N\left[\,1-\left(\frac{1}{N}\sum_{i=1}^{N}\tanh^{2}(h+\beta
h_{i}({\bf m}^{\alpha}))\right)^{\,2}\,\right]+$ (8)
$\displaystyle\qquad\qquad+\sum_{i=1}^{N}\left(\,-\frac{\beta}{2}h_{i}({\bf
m}^{\alpha})\tanh(h+\beta h_{i}({\bf m}^{\alpha}))+\log\cosh(h+\beta
h_{i}({\bf m}^{\alpha}))\,\right),$

which is exactly $\text{TAP}_{N}({\bf m}^{\alpha})$ as defined in (5).

The REM approximation of TAP free energies [21] is achieved by replacing the
local fields ${h_{i}({\bf m}^{\alpha})}$ with a set of REM-like variables,
specifically independent standard Gaussians ${X_{\alpha,i}}$, where
$i=1,\dots,N$ and $\alpha=1\dots 2^{N}$.

Positioned as the second natural step beyond the REM-approximation, this paper
explores the GREM approximation of TAP free energies, wherein the local fields
are substituted with GREM-like variables (see (10) below). For the GREM
approximation of TAP free energies, we can establish an orthodox Boltzmann-
Gibbs principle and introduce a dual formula (given in Theorem 2.2 below) for
the free energy, akin to the Parisi formula, which closely resembles the one
given by the Parisi theory for models within the 1-step replica symmetry
breaking approximation. The highlight is that the inherent non-linearities of
(5) bring the Parisi formulation for our GREM-like models into closer
correspondence with that for the free energy of the SK, presenting a departure
from conventional GREM formulations (for more on this, see Remark (2.3)
below). Our original result, Theorem 2.2 below, suggests that the correction
to the Parisi formula, caused by the inherent nonlinearities, remains
unchanged if we perform a GREM-replacement instead of a REM-replacement; that
is, if we admit more than one level of hierarchy. Specifically, this
correction does not vary for models within the $n$-steps replica symmetry
breaking approximation, where $n\in\mathbb{N}$ is the number of observable
levels of hierarchy.

In the next section, we provide the necessary definitions and present our
results, which we then prove in the subsequent two sections.

##  2\. Definitions and results

Given $\Theta\in\mathbb{R}^{+}$ and $n\in\mathbb{N}$; we investigate, in an
abstract setting, the following GREM-approximation of the TAP free energy (5)

$\lim_{N\to\infty}\frac{1}{N}\log\sum_{\alpha=1}^{2^{\Theta N}}\exp
N{\bf\Phi}_{\text{TAP}}[L_{N,\alpha}]$ (9)

where, given positive numbers
$\\{\gamma_{1},\dots,\gamma_{n}\\}\,:\,\sum_{k=1}^{n}\gamma_{k}=1$, each
configuration
$\alpha\in\\{1,\dots,2^{\Theta N}\\}$ is identified with an $n$-tuple

$\alpha\equiv(\alpha_{1},\dots,\alpha_{n}):\quad\alpha_{k}\in\\{1,\dots,2^{\gamma_{k}\Theta
N}\\},\qquad k\leq n$

and

$L_{N,\alpha}:=\frac{1}{N}\sum\limits_{i=1}^{N}\delta_{X_{\alpha,i}}$

are the empirical measures associated to sequences $\\{X_{\alpha,i}\\}$ of
random vectors

$X_{\alpha,i}=(X^{(1)}_{\alpha_{1},i},X^{(2)}_{\alpha_{1},\alpha_{2},i},\dots,X^{(n-1)}_{\alpha_{1},\alpha_{2}\dots,\alpha_{n-1},i},X^{(n)}_{\alpha,i})\in\mathbb{R}^{n},$
(10)

with independent entries, being each
$\\{X^{(k)}_{\alpha_{1},\dots,\alpha_{k},i}\\}_{\alpha_{1},\dots,\alpha_{k},i}$
i.i.d. on $\mathbb{R}$.111We stress that $\mathbb{R}$ can be replaced with any
Polish space, without this changing the validity of our proofs. We choose in
this paper to just refer to the case in which this space is the real line for
the sake of simplicity and with the aim of improving readability. For the
couple of topological considerations necessary to present the proofs in the
case of variables defined on a generic Polish space, we refer the interested
reader to previous works [6, 7, 21].

Indicating with $\mathcal{M}^{+}_{1}(\mathbb{R}^{n})$ the space of all regular
Borel probability measures on $\mathbb{R}^{n}$ endowed with the topology of
weak convergence of measures; the functional

${\bf\Phi}_{\text{TAP}}:\mathcal{M}^{+}_{1}(\mathbb{R}^{n})\to\mathbb{R}$

is continous and given by

$\displaystyle{\bf\Phi}_{\text{TAP}}[\nu]=$
$\displaystyle\Phi_{\text{SK}}\left(\,\int\varphi_{\text{SK}}\left(\sum_{k=1}^{n}x_{k}\right)\nu(d{\bf
x}\,)\right)+\int f_{\text{SK}}\left(\sum_{k=1}^{n}x_{k}\right)\nu(d{\bf x})$
(11)

where, with ${\bf x}=(x_{1},\dots,x_{n})$ a generic point in $\mathbb{R}^{n}$

$\displaystyle\Phi_{\text{SK}}:\mathbb{R}\to\mathbb{R}\,,$
$\displaystyle\Phi_{\text{SK}}(x):=\frac{\beta^{2}}{4}(1-x^{2})\,;$ (12)
$\displaystyle\varphi_{\text{SK}}:\mathbb{R}\to\mathbb{R}\,,$
$\displaystyle\varphi_{\text{SK}}(x):=\tanh^{2}(h+\beta x)\,;$ $\displaystyle
f_{\text{SK}}:\mathbb{R}\to\mathbb{R}\,,$ $\displaystyle
f_{\text{SK}}(x):=-\frac{\beta}{2}x\tanh(h+\beta x)+\log\cosh(h+\beta x)\,.$

It is readily checked that $\boldsymbol{\Phi}_{\text{TAP}}[L_{N,\alpha}]$ is
obtained taking (5) for ${\bf m}={\bf m}^{\alpha}$ and substituting the fields
$h_{i}({\bf m}^{\alpha})$ with the GREM-like variables

$\sum_{k=1}^{n}X^{(k)}_{\alpha_{1},\dots,\alpha_{k},i}.$ (13)

As established in [7], if to given $\nu\in\mathcal{M}_{1}^{+}(\mathbb{R}^{n})$
we indicate with $\nu^{(k)}\in\mathcal{M}_{1}^{+}(\mathbb{R}^{k})$ its
marginal on the first $k\in\\{1,\dots,n\\}$ coordinates, and consider the
relative entropy

$H(\nu^{(k)}\mid\mu^{(k)}):=\begin{cases}\int\log\frac{d\nu^{(k)}}{d\mu^{(k)}}\,d\nu^{(k)}&\quad\text{if}\,\,\nu^{(k)}<<\mu^{(k)}\\\
+\infty&\quad\text{otherwise}\,\,,\end{cases}$ (14)

where

$\mu=\bigotimes_{k=1}^{n}\mu_{k}\in\mathcal{M}^{+}_{1}(\mathbb{R}^{n})$

is the common law of the vectors (10); the following can be proved within a
classical large deviation treatment via second moment method and Sanov’s
theorem (see Section 3 below).

######  Theorem 2.1 (Boltzmann-Gibbs principle).

Let ${\bf\Phi}:\mathcal{M}^{+}_{1}(\mathbb{R}^{n})\to\mathbb{R}$ be a
continous functional and consider

$f_{N}:=\frac{1}{N}\log\sum_{\alpha=1}^{2^{\Theta N}}\exp
N{\bf\Phi}[L_{N,\alpha}].$

Then the limit $\lim_{N\to\infty}f_{N}$ exists, is non-random and given by

$f:=\sup_{\nu\in\mathcal{K}}\quad{\bf\Phi}[\nu]-H(\nu\mid\mu)+\Theta\log 2\\\
$ (15)

with

$\mathcal{K}:=\left\\{\nu\in\mathcal{M}^{+}_{1}(\mathbb{R}^{n}):\quad
H(\nu^{(k)}\mid\mu^{(k)})\leq\,\Theta\log
2\sum_{j=1}^{k}\gamma_{j}\quad\forall k\in\\{1,\cdots,n\\}\right\\}.$ (16)

Given bounded and continous functions $\varphi,f:\mathbb{R}^{n}\to\mathbb{R}$,
as well as a differentiable function $\Phi:\mathbb{R}\to\mathbb{R}$; our
result Theorem 2.2 below, concerns a specific case in which the functional
$\boldsymbol{\Phi}$ is of the form222Notice that choosing $\varphi({\bf
x})\equiv\varphi_{\text{SK}}(\sum_{k=1}^{n}x_{k})$, $f({\bf x})\equiv
f_{\text{SK}}(\sum_{k=1}^{n}x_{k})$ and $\Phi(x)\equiv\Phi_{\text{SK}}(x)$ the
generic functional (17) becomes the specific TAP-functional (11).

${\bf\Phi}[\nu]\equiv\Phi(\int\varphi d\nu)+\int fd\nu$ (17)

and extends to $n\geq 2$ the duality between the Boltzmann-Gibbs infinite
dimensional principle of Theorem 2.1 and a finite dimensional, Parisi
principle. Precisely, we prove (see Section 4 below) that, considering the
following recursively constructed functions

$\displaystyle P^{\,q,\bf m}_{n}(x_{1},\dots,x_{n})$
$\displaystyle=\Phi^{\prime}(q)\varphi(x_{1},\dots,x_{n})+f(x_{1},\dots,x_{n}),$
(18) $\displaystyle P^{\,q,\bf m}_{j-1}(x_{1},\dots,x_{j-1})$
$\displaystyle=\frac{1}{m_{j}}\log\int\exp{m_{j}P^{\,q,\bf
m}_{j}(x_{1},\dots,x_{j})}\mu_{j}(dx_{j})\qquad\forall j\leq n$

parametrized by $(q,{\bf m})$ such that $q\in[\inf\varphi,\,\sup\varphi]$ and

${\bf m}=(m_{1},\dots,m_{n})\in\Delta:=\\{{\bf m}\in[0,1]^{n}:0<m_{1}\leq
m_{2}\leq\dots\leq m_{n}\leq 1\\};$

the following holds:

######  Theorem 2.2 (Parisi principle).

If $\varphi,f:\mathbb{R}^{n}\to\mathbb{R}$ are bounded;
$\Phi:\mathbb{R}\to\mathbb{R}$ is a twice differentiable function with
$\Phi^{\prime\prime}(x)<0$ for all $x\in[\inf\varphi,\,\sup\varphi]$ and

${\bf\Phi}[\nu]:=\Phi(\int\varphi d\nu)+\int fd\nu;$ (19)

then

$\sup_{\nu\in\mathcal{K}}\,{\bf\Phi}[\nu]-H(\nu\mid\mu)=\inf_{(q,{\bf
m})\in[\inf\varphi,\,\sup\varphi]\times\Delta}P(q,{\bf m}),$ $P(q,{\bf
m}):=\Phi(q)-q\Phi^{\prime}(q)+P^{\,q,\bf
m}_{0}+\sum_{k=1}^{n}\frac{\gamma_{k}\Theta\log 2}{m_{k}}-\Theta\log 2.$ (20)

######  Remark 2.3.

The case $n=1$ corresponds to the REM-approximation of the TAP free energy
which, as pointed out in the introduction, is the subject of study in [21].
The novelty of the present work and [21] w.r.t to previous works ([6, 7]) is
the nonlinearity of the functional (19). In fact, (19) is continous, but not
linear on $\mathcal{M}^{+}_{1}(\mathbb{R}^{n})$; namely not of the form
$\int\varphi d\nu$. Specifically in (20), the nonlinearity causes the presence
of the same correction term $\Phi(q)-q\Phi^{\prime}(q)$, for any
$n\in\mathbb{N}$. This term is zero in the classic / linear cases (e.g. for
$\Phi$ the identity function) while reproducing, for the choices (12), the
unsettling quadratic (in the inverse temperature) appearing in the 1RSB free
energy for the SK model. For more details on the comparison between the REM-
approximation of TAP free energies and the 1RSB Parisi solution for the free
energy of the SK, we refer the reader to Subsection 3.1 of [21].

In [7] the authors focus on continous and linear functionals
$\boldsymbol{\Phi}$ and establish both the Boltzmann-Gibbs principle and the
Parisi formula for the limiting free energy. Their proof of the Boltzmann-
Gibbs principle, whose formulation is entirely analogous to that given in
Theorem 2.1, is rather powerful and does not require the linearity assumption.
We refer to [7, Theorem 2.3] for a detailed proof, but give in Section 3 a
comprehensive outline for the reader’s convenience. In the last Section 4 we
prove Theorem 2.2, namely we see how the finite dimensional Parisi principle
encodes the optimal measure for the infinite dimensional Boltzmann-Gibbs
principle.

##  3\. The Boltzmann-Gibbs principle: proof of Theorem 2.1

Some infrastructure is necessary: if a sequence of events
$\left\\{E_{N}\right\\}_{N\in\mathbb{N}}\subset\mathcal{A}$ is such that there
exists $\delta>0$ s.t. for $N$ large enough $\mathbb{P}(E_{N})\geq
1-e^{-\delta N}$ we say that $E_{N}$ holds with overwhelming probability
(o.p.). With $B_{r}(\nu)$ we denote the open ball centered in
$\nu\in\mathcal{M}^{+}_{1}(\mathbb{R}^{n})$ with radius $r>0$ in one of the
standard metrics (e.g. Prokhorov’s metric); a neighborhood or an open subset
are conceived in the corresponding topology. This turns
$\mathcal{M}^{+}_{1}(\mathbb{R}^{n})$ into a complete, separable metric space.
The argument follows a second moment method performed on the counting variable

$M_{N}(U):=\\#\\{\alpha\in\\{1,\dots,2^{\Theta N}\\}:L_{N,\alpha}\in U\\}$
(21)

defined for every $U\subset\mathcal{M}^{+}_{1}(\mathbb{R}^{n})$. Estimating
its first moments on exponential scale through Sanov’s Theorem, we can easily
prove that for any $\epsilon^{\prime}>0$

  * $i)$

if $H(\nu\mid\mu)<+\infty$, then we can find an arbitrary small neighborhood
$U$ of $\nu$:

$M_{N}(U)\leq\exp{N[\Theta\log
2-H(\nu\mid\mu)+\epsilon^{\prime}]}\,\,\text{with o.p.}\,;$ (22)

  * $ii)$

if $H(\nu^{(k)}\mid\mu^{(k)})>\Theta\log 2\sum_{j=1}^{k}\gamma_{j}$ for some
$k\in\\{1,\dots,n\\}$, i.e. if $\nu\not\in\mathcal{K}$, then we can find an
arbitrary small neighborhood $U$ of $\nu$:

$M_{N}(U)=0\,\,\text{with o.p.}\,.$ (23)

The statement $i)$ follows from a straightforward application of Markov’s
inequality; while in order to show $ii)$ we need to consider, given
$\epsilon^{\prime}>0$ and a neighborhoods $U^{\prime}$ of $\nu$, the subset

$U:=\\{\nu\in\mathcal{M}^{+}_{1}(\mathbb{R}^{n}):\nu\in
U^{\prime},\nu^{(k)}\in U^{\prime\prime}\\}$ (24)

where $U^{\prime\prime}\subset\mathcal{M}^{+}_{1}(\mathbb{R}^{k})$ is such
that

$\inf_{\nu^{\prime}\in\text{cl}U^{\prime\prime}}H(\nu^{\prime}\mid\mu^{(k)})=\inf_{\nu^{\prime}\in
U^{\prime\prime}}H(\nu^{\prime}\mid\mu^{(k)}),\qquad\inf_{\nu^{\prime}\in
U^{\prime\prime}}H(\nu^{\prime}\mid\mu^{(k)})\geq\Theta\log
2\sum_{j=1}^{k}\gamma_{j}+\eta.$ (25)

for some $\eta>0$. Then, noticing that the marginal of $L_{N,\alpha}$ on the
first $k$ coordinates only depend on the first $k$ entries of $\alpha$, to wit
$L_{N,\alpha}^{(k)}=N^{-1}\sum_{i\in[N]}\delta_{(X^{1}_{\alpha_{1},i},\dots,X^{k}_{\alpha_{1},\dots,\alpha_{k},i})}$;
we get

$\displaystyle\mathbb{P}(\exists\alpha:L_{N,\alpha}\in U)$
$\displaystyle\leq\mathbb{P}(\exists(\alpha_{1},\dots,\alpha_{k}):L_{N,\alpha}^{(k)}\in
U^{\prime\prime})\leq
2^{\Theta\sum_{j=1}^{k}\gamma_{j}N}\mathbb{P}(L_{N,1}^{(k)}\in
U^{\prime\prime}).$ (26)

The latter, together with Sanov’s Theorem and (25), imply $ii)$. Now, as
$\boldsymbol{\Phi}$ is continous on $\mathcal{M}^{+}_{1}(\mathbb{R}^{n})$ and
$\mathcal{K}$ is compact, it is easy to see that $i)$, $ii)$, together with
the Borel-Cantelli Lemma, imply the upper bound. Specifically that there
exists a finite covering of $\mathcal{K}$, say
$\\{B_{r_{l}}(\nu_{l})\\}_{l=1}^{M}$, such that $\mathbb{P}$-a.s.

$\displaystyle\limsup\limits_{N\to\infty}f_{N}$
$\displaystyle\leq\limsup\limits_{N\to\infty}\frac{1}{N}\log\sum\limits_{l=1}^{M}\sum\limits_{\alpha:L_{N,\alpha}\in
B_{r_{l}}(\nu_{l})}\exp\,N\boldsymbol{\Phi}[L_{N,\alpha}]$ (27)
$\displaystyle\leq\limsup\limits_{N\to\infty}\frac{1}{N}\log\sum\limits_{l=1}^{M}M_{N}(B_{r_{l}}(\nu_{l}))\exp\\{\,N\boldsymbol{\Phi}[\nu_{l}]+\epsilon^{\prime}N\\}$
$\displaystyle\leq\limsup\limits_{N\to\infty}\frac{1}{N}\log\sum\limits_{l=1}^{M}\exp\\{\,N\boldsymbol{\Phi}[\nu_{l}]+N\Theta\log
2-NH(\nu_{l}\mid\mu)+2\epsilon^{\prime}N\\}$

which, as each $\nu$ is in $\mathcal{K}$, implies the upper bound:

$\limsup_{N\to\infty}f_{N}\leq\sup_{\nu\in\mathcal{K}}\,\boldsymbol{\Phi}[\nu]-H(\nu\mid\mu)+\Theta\log
2.$ (28)

The key to the claim (2.1), i.e. to the lower bound, is now a concentration
property of the counting variable (21). Specifically, by independence, for any
$U\subset\mathcal{M}^{+}_{1}(\mathbb{R}^{n})$

$\displaystyle{\mathbb{E}}\,M^{2}_{N}(U)$
$\displaystyle\leq[{\mathbb{E}}M_{N}(U)]^{2}+\sum_{k=1}^{n}\sum_{\stackrel{{\scriptstyle\alpha,\alpha^{\prime}:\alpha_{i}=\alpha^{\prime}_{i}\,\forall
i\leq k}}{{\alpha_{k+1}\neq\alpha^{\prime}_{k+1}}}}\mathbb{P}(L_{N,\alpha}\in
U,L_{N,\alpha^{\prime}}\in U)$ (29)

so that a relative version of Sanov Theorem (see [7, Lemma 3.2]) implies from
the latter that if $H(\nu^{(k)}\mid\mu^{(k)})<\Theta\log
2\sum_{j=1}^{k}\gamma_{j}$ for all $k\in\\{1,\dots,n\\}$, then for any small
enough neighborhood $U$ of $\nu$:

$\mathbb{V}\text{ar}M_{N}(U)\leq e^{-\delta
N}[{\mathbb{E}}M_{N}(U)]^{2}\quad\text{for some}\quad\delta>0.$ (30)

A classic application of Chebyshev inequality and the Borel-Cantelli Lemma now
imply333For more details see the proof of [7, Lemma 3.5] and that of [29,
Proposition 3.2].

$\lim\limits_{N\to\infty}\frac{1}{N}\log
M_{N}(U)=\lim\limits_{N\to\infty}\frac{1}{N}\log{\mathbb{E}}M_{N}(U)=\Theta\log
2-H(\nu\mid\mu)\,.$ (31)

To see how (31) leads to the lower bound, first notice that
$\mathcal{K}\subset\mathcal{M}^{+}_{1}(\mathbb{R}^{n})$ is compact and convex
and $\nu\to{\bf\Phi}[\nu]-H(\nu\mid\mu)$ is upper-semicontinous and concave,
therefore the Boltzmann-Gibbs principle (15) admits a unique solution.
Indicating with $\nu^{*}\in\mathcal{K}$ the solution to (15), consider the
convex combination

$\nu_{a}:=a\mu+(1-a)\nu^{*}\in\mathcal{K}\quad\text{for}\quad a\in(0,1).$ (32)

One has that $\nu_{a}\to\nu^{*}$ weakly as $a\to 0$, and
$\boldsymbol{\Phi}[\nu_{a}]\to\boldsymbol{\Phi}[\nu^{*}]$,
$H(\nu_{a}\mid\mu)\to H(\nu^{*}\mid\mu)$. Given $\epsilon^{\prime}>0$ we
choose $a=a(\epsilon^{\prime})>0$, and a neighborhood $U$ of $\nu_{a}$:

$\boldsymbol{\Phi}[\nu_{a}]-H(\nu_{a}\mid\mu)\geq\boldsymbol{\Phi}[\nu^{*}]-H(\nu^{*}\mid\mu)-\epsilon^{\prime},$
(33)
$\boldsymbol{\Phi}[\nu]\geq\boldsymbol{\Phi}[\nu_{a}]-\epsilon^{\prime}\quad\forall\nu\in
U.$ (34)

Then

$\displaystyle f_{N}$
$\displaystyle\geq\frac{1}{N}\log\sum_{\alpha:L_{N,\alpha}\in
U}\exp{N\boldsymbol{\Phi}[L_{N,\alpha}]}\geq\boldsymbol{\Phi}[\nu_{a}]-\epsilon^{\prime}+\frac{1}{N}\log
M_{N}(U)$ (35)

so that

$\displaystyle\liminf\limits_{N\to\infty}f_{N}$
$\displaystyle\geq\boldsymbol{\Phi}[\nu_{a}]-H(\nu_{a}\mid\mu)+\Theta\log
2-\epsilon^{\prime}\geq\boldsymbol{\Phi}[\nu^{*}]-H(\nu^{*}\mid\mu)+\Theta\log
2-2\epsilon^{\prime},$ (36)

which gives the lower bound. This, together with (28), proves Theorem 2.1.

##  4\. The Parisi principle: proof of Theorem 2.2

In this Section we prove Theorem 2.2. First, we show that the Parisi function
(20) has a minum point on $[\inf\varphi,\sup\varphi]\times\Delta$. To this
aim, notice that applying Jensen’s inequality and Fubini’s theorem $n$ times
we get

$\displaystyle P_{0}^{\,q,{\bf m}}$
$\displaystyle=\frac{1}{m_{1}}\log\int\exp{m_{1}P^{\,q,{\bf
m}}_{1}(x_{1})}\mu_{1}(dx_{1})\geq\int P_{1}^{(q,{\bf
m})}(x_{1})\mu_{1}(dx_{1})$
$\displaystyle=\int\frac{1}{m_{2}}\log\left(\int\exp{m_{2}P_{2}(x_{1},x_{2})}\mu_{2}(dx_{2})\right)\,\mu_{1}(dx_{1})\geq\int
P_{2}^{\,q,{\bf m}}(x_{1},x_{2})\mu_{1}\otimes\mu_{2}(dx_{1},dx_{2})$
$\displaystyle\geq\int P^{\,q,{\bf m}}_{n}({\bf x})\mu(d{\bf
x})=\int\,\Phi^{\prime}(q)\varphi({\bf x})+f({\bf x})\,\mu(d{\bf x});$

so that

$\displaystyle P(q,{\bf m})$
$\displaystyle=\Phi(q)-q\Phi^{\prime}(q)+P^{\,q,\bf
m}_{0}+\sum_{k=1}^{n}\frac{\gamma_{k}\Theta\log 2}{m_{k}}-\Theta\log 2$
$\displaystyle\geq\Phi(q)-q\Phi^{\prime}(q)+\sup_{{\bf
x}\in\mathbb{R}^{n}}\mid\Phi^{\prime}(q)\varphi({\bf x})+f({\bf
x})\mid+\sum_{k=1}^{n}\frac{\gamma_{k}\Theta\log 2}{m_{k}}-\Theta\log 2,$

which implies that $P(q,{\bf m})\to+\infty$ as $(q,{\bf m})$ tends in
$[\inf\varphi,\sup\varphi]\times\Delta\subset\mathbb{R}^{n+1}$ to any point
$(q^{\prime},0,m^{\prime}_{2},\cdots,m^{\prime}_{n})$. The infimum of $P$ on
$[\inf\varphi,\sup\varphi]\times\Delta$, where we recall $\Delta=\\{{\bf
m}=(m_{1},\dots,m_{n})\in[0,1]^{n}:0<m_{1}\leq\dots\leq m_{n}\leq 1\\}$, must
be therefore equal to the infimum of $P$ on
$[\inf\varphi,\sup\varphi]\times\\{{\bf m}\in\Delta:m_{1}\geq\epsilon\\}$, for
some $\epsilon>0$.
Specifically, as $P$ achieves minimum on the compact set
$[\inf\varphi,\sup\varphi]\times\\{{\bf m}\in\Delta:m_{1}\geq\epsilon\\}$,
being there continous; it must achieve minimum on
$[\inf\varphi,\sup\varphi]\times\Delta$. To wit

$\text{argmin}_{[\inf\varphi,\sup\varphi]\times\Delta}P\neq\emptyset,$

and we can state the following Lemma, whose simple proof shows how the Parisi
functional enables finding the solution to the Boltzmann-Gibbs principle.

######  Lemma 4.1.

Let $\bar{q}\in\mathbb{R},\bar{{\bf
m}}=(\bar{m}_{j})_{j=1}^{n}\in\mathbb{R}^{n}$ be s.t.
$(\bar{q},{\bf\bar{m}})\in\text{argmin}_{[\inf\varphi,\sup\varphi]\times\Delta}P$,
and consider the probability measure $G=G^{\bar{q},{\bar{\bf
m}}}\in\mathcal{M}^{+}_{1}(\mathbb{R}^{n})$ whose Radon-Nikodym derivative
w.r.t. $\mu=\bigotimes_{j=1}^{n}\mu_{j}$ is

$\frac{dG}{d\mu}(x_{1},\dots,x_{n}):=\prod_{j=1}^{n}\frac{e^{\bar{m}_{j}P^{\,\bar{q},{\bf\bar{m}}}_{j}(x_{1},\dots,x_{j})}}{\int
e^{\bar{m}_{j}P^{\,\bar{q},{\bf\bar{m}}}_{j}(x_{1},\dots,x_{j-1},x^{\prime}_{j})}\mu_{j}(dx^{\prime}_{j})}.$
(37)

Then $G\in\mathcal{K}$ and

$\boldsymbol{\Phi}[\nu]-H(\nu\mid\mu)\leq\boldsymbol{\Phi}[G]-H(G\mid\mu)$
(38)

holds $\forall\nu\in\mathcal{K}$.

###### Proof.

For ${\bf x}=(x_{1},\dots,x_{n})$ a generic point in $\mathbb{R}^{n}$ and
every $j\in\\{1,\cdots,n\\}$, we write ${\bf x}_{j}:=(x_{1},\dots,x_{j})$ for
its projection on the first $j$ coordinates and shorten

$P_{j}({\bf x}_{j}):=P_{j}^{\,\bar{q},{\bar{\bf m}}}({\bf x}_{j}).$

We also set for every couple $(q,{\bf
m})\in[\inf\varphi,\sup\varphi]\times\Delta$ and given a
$\mu^{(j)}-$measurable function $v:S^{j}\to\mathbb{R}$

$\displaystyle{\sf{E}}^{\,q,\bf m}_{j}(v)$ $\displaystyle:=\int v({\bf
x}_{j})\frac{e^{m_{1}P^{\,q,\bf m}_{1}(x_{1})}}{\int e^{m_{1}P^{\,q,\bf
m}_{1}(x^{\prime}_{1})}\mu_{1}(dx_{1}^{\prime})}\dots\frac{e^{m_{j}P^{\,q,\bf
m}_{j}({\bf x}_{j})}}{\int e^{m_{j}P^{\,q,\bf m}_{j}({\bf
x}_{j-1},x_{j}^{\prime})}\mu_{j}(dx^{\prime}_{j})}\mu^{(j)}(d{\bf x}_{j})$
(39)

and

$\displaystyle H^{\,q,\bf m}_{j}({\bf x}_{j-1})$
$\displaystyle:=\int\log\left(\frac{e^{m_{j}P^{\,q,\bf m}_{j}({\bf
x}_{j})}}{\int e^{m_{j}P^{\,q,\bf m}_{j}({\bf
x}_{j-1},x^{\prime}_{j})}\mu_{j}(dx^{\prime}_{j})}\right)\,\frac{e^{m_{j}P^{\,q,\bf
m}_{j}({\bf x}_{j})}}{\int e^{m_{j}P^{\,q,\bf m}_{j}({\bf
x}_{j-1},x^{\prime}_{j})}\mu_{j}(dx^{\prime}_{j})}\mu_{j}(dx_{j}),$ (40)

with $H^{\,q,\bf m}_{1}$ dependent only on $(q,{\bf m})$. It is readily
checked that if $(q,{\bf m})=(\bar{q},{\bf\bar{m}})$ then
${\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{j}$ is the expectiation w.r.t the marginal
$G^{(j)}$ of the measure $G$, and that therefore the relation

${\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{j}(v)=\int v({\bf x}_{j})\,G^{(j)}(d{\bf
x}_{j})$ (41)

holds for every $j\in\\{1,\cdots,n\\}$ and every $\mu^{(j)}-$measurable
function $v$. Upon closer inspection, it also becomes evident that

$H^{\,\bar{q},{\bf\bar{m}}}_{1}=H(G^{(1)}\mid\mu_{1})$ (42)

and that if $j\geq 2$ and $(X_{1},\dots,X_{j})$ is a random vector distributed
according to $G^{(j)}$, then $H^{\,q,\bf m}_{j}({\bf x}_{j-1})$ is the
Kullback-Leibler divergence between $\mu^{(j)}$ and the conditional
probability distribution of $X_{j}$ given $X_{1}=x_{1},\dots,X_{j-1}=x_{j-1}$.

This notation enables us to express partial derivatives of $P$ in a compact
and convenient manner. Specifically, a straightforward computation shows that,
if $\partial_{q}P$ is the partial derivative of $P$ w.r.t the variable $q$
and, for every $j\in\\{1,\cdots,n\\}$, $\partial_{m_{j}}P$ is the one w.r.t to
the variable $m_{j}$; then

$\begin{cases}\partial_{q}P(q,{\bf
m})=\Phi^{\prime\prime}(q)\left[{\sf{E}}^{\,q,\bf m}_{n}(\varphi)-q\right]\\\
\\\ \partial_{m_{1}}P(q,{\bf m})=\frac{1}{m_{1}^{2}}\left[H^{\,q,\bf
m}_{1}-\gamma_{1}\Theta\log 2\right]\\\ \\\ \partial_{m_{j}}P(q,{\bf
m})=\frac{1}{m^{2}_{j}}\left[{\sf{E}}^{\,q,\bf m}_{j-1}(H^{\,q,\bf
m}_{j})-\gamma_{j}\Theta\log 2\right]\quad\forall
j\in\\{2,\cdots,n\\}.\end{cases}$ (43)

Through the first of these relations, the minimality of $\bar{q}$ for the
univariate function $q\to P(q,{\bf\bar{m}})$ on $[\inf\varphi,\sup\varphi]$,
together with the fact that $\Phi^{\prime\prime}$ is strictly negative on
$[\inf\varphi,\sup\varphi]$; imply

$\bar{q}={\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{n}(\varphi)\stackrel{{\scriptstyle\eqref{relation_E_Gj}\,\text{with}\,j\equiv
n}}{{=}}\,\,\,\int\varphi({\bf x})G(d{\bf x}).$ (44)

Similarly, the minimality of $\bar{m}_{1}\neq 0$ for the univariate function
$m_{1}\to P(\bar{q},m_{1},\bar{m}_{2},\dots,\bar{m}_{n})$ on $(0,\bar{m}_{2}]$
implies that $\partial_{m_{1}}P(\bar{q},{\bf\bar{m}})\leq 0$, i.e.

$\gamma_{1}\Theta\log 2\geq
H^{\,\bar{q},{\bf\bar{m}}}_{1}\,\,\,\stackrel{{\scriptstyle\eqref{relation_H_HG}}}{{=}}H(G^{(1)}\mid\mu_{1});$
(45)

which amounts to say that the probability measure $G$ satisfies the  first
entropic condition appearing in the definition (16) of the set $\mathcal{K}$.
Given (45) as a base case, the fact that the couple $G$ actually satisfy all
the entropic conditions, i.e. that

$H(G^{(j)}\mid\mu^{(j)})\leq\Theta\log 2\sum_{k\in[j]}\gamma_{k}$ (46)

for every $j\in\\{1,\cdots,n\\}$, and therefore $G\in\mathcal{K}$; can be
easily proved by induction. To see how this comes about, consider
$j\in\\{2,\cdots,n\\}$ and assume that for any $k<j$

$H(G^{(k)}\mid\mu^{(k)})\leq\Theta\log 2\sum_{l\in[k]}\gamma_{l}$ (47)

i.e. the inductive hypothesis. Then consider

$\text{min}_{j}:=\min\\{k:\quad 1\leq k\leq
j:\quad\bar{m}_{k}=\bar{m}_{j}\\},$

and, setting $\bar{m}_{0}:=0,\bar{m}_{n+1}:=1$, the interval
$(\bar{m}_{\text{min}_{j}-1},\bar{m}_{j+1}]$. Notice that one has
$0\\!\\!<\\!\\!\bar{m}_{1}\\!\leq\\!\bar{m}_{k}\,\,\forall k\leq n$, so that
the interval $(\bar{m}_{\text{min}_{j}-1},\bar{m}_{j+1}]$ is non-empty and we
can therefore define the real function
$m\in(\bar{m}_{\text{min}_{j}-1},\bar{m}_{j+1}]\to g_{j}(m)$ given by

$g_{j}(m):=\begin{cases}P(\bar{q},m,\dots,m)&\text{if}\,\,\min_{j}=1,\,j=n\,;\\\
P(\bar{q},\bar{m}_{1},\dots,\bar{m}_{\text{min}_{j}-1},m,\dots,m)&\text{if}\,\,\min_{j}>1,\,j=n\,;\\\
P(\bar{q},m,\dots,m,\bar{m}_{j+1},\dots,\bar{m}_{n})&\text{if}\,\,\min_{j}=1,\,j<n\,;\\\
P(\bar{q},\bar{m}_{1},\dots,\bar{m}_{\text{min}_{j}-1},m,\dots,m,\bar{m}_{j+1},\dots,\bar{m}_{n})&\text{if}\,\,\min_{j}>1,\,j<n\,.\\\
\end{cases}$ (48)

By construction, $g_{j}$ must have a minimum point in
$m=\bar{m}_{j}\in(\bar{m}_{\text{min}_{j}-1},\bar{m}_{j+1}]$ which is a
critical point for $g_{j}$ if $\bar{m}_{j}\neq\bar{m}_{j+1}$ while, possibly,
such that $g^{\prime}_{j}(\bar{m}_{j})<0$ if $\bar{m}_{j}=\bar{m}_{j+1}$.
As
$g^{\prime}_{j}(\bar{m}_{j})=\sum\limits_{k=\text{min}_{j}}^{j}\partial_{m_{k}}P(\bar{q},{\bf{\bar{m}}})$,
this implies that it must hold
$\sum\limits_{k=\text{min}_{j}}^{j}\partial_{m_{k}}P(\bar{q},{\bf{\bar{m}}})\leq
0$ in any case.
In order to include the case $\text{min}_{j}\equiv 1$ we extend our notation
setting
${\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{0}(H^{\,\bar{q},{\bf\bar{m}}}_{1})\equiv
H^{\,\bar{q},{\bf\bar{m}}}_{1}$ and $H(G^{(0)}\mid\mu^{(0)})\equiv 0$.
Specifically, through the relations (43), we obtain

$\displaystyle\sum\limits_{k=\text{min}_{j}}^{j}\partial_{m_{k}}P(\bar{q},{\bf{\bar{m}}})$
$\displaystyle=\sum\limits_{k=\text{min}_{j}}^{j}\frac{1}{\bar{m}^{2}_{k}}\left[{\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{k-1}(H^{\,\bar{q},{\bf\bar{m}}}_{k})-\Theta\gamma_{k}\log
2\right]$ (49)
$\displaystyle=\frac{1}{\bar{m}^{2}_{j}}\sum\limits_{k=\text{min}_{j}}^{j}\left[{\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{k-1}(H^{\,\bar{q},{\bf\bar{m}}}_{k})-\Theta\gamma_{k}\log
2\right]\leq 0.$

The well known chain-rule for Kullback-Leibler divergences is now basically
everything we need. With our notation, it is equivalent to the fact that for
each $k$

$H(G^{(k)}\mid\mu^{(k)})=H(G^{(k-1)}\mid\mu^{(k-1)})+{\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{k-1}(H^{\,\bar{q},{\bf\bar{m}}}_{k}),$
(50)

which in turn implies that

$H(G^{(j)}\mid\mu^{(j)})=H(G^{(\text{min}_{j}-1)}\mid\mu^{(\text{min}_{j}-1)})+\sum_{k=\text{min}_{j}}^{j}{\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{k-1}(H^{\,\bar{q},{\bf\bar{m}}}_{k}).$

Using the latter together with the last inequality in (49), we get

$\displaystyle H(G^{(j)}\mid\mu^{(j)})\leq
H(G^{(\text{min}_{j}-1)}\mid\mu^{(\text{min}_{j}-1)})+\Theta\log
2\sum\limits_{k=\text{min}_{j}}^{j}\gamma_{k}$ (51)

so that the induction step follows using the inductive hypotesis (47) for
$k\equiv\text{min}_{j}-1<j$. All in all, we proved that (46) holds for every
$j\in\\{1,\cdots n\\}$, therefore that $G\in\mathcal{K}$.

We now prove (38), i.e. our claim is that for any fixed $\nu\in\mathcal{K}$

$\boldsymbol{\Phi}[G]-H(G\mid\mu)-\boldsymbol{\Phi}[\nu]+H(\nu\mid\mu)\geq 0.$
(52)

In order to lighten notation, we set

$\displaystyle
D_{n}:=\boldsymbol{\Phi}[G]-H(G\mid\mu)-\boldsymbol{\Phi}[\nu]+H(\nu\mid\mu);$
$\displaystyle D_{j}:=\int
P_{j}\,dG^{(j)}-\frac{1}{\bar{m}_{j+1}}H(G^{(j)}\mid\mu^{(j)})-\int
P_{j}\,d\nu^{(j)}+\frac{1}{\bar{m}_{j+1}}H(\nu^{(j)}\mid\mu^{(j)}),\quad
j\in\\{1,\cdots,n-1\\}.$

We will prove that $D_{1}\geq 0$ and that $D_{k}\geq D_{k-1}$ for any
$k\in\\{2,\cdots,n\\}$, which implies $D_{n}\geq 0$, namely the claim (52).
Both proofs follow a slight extension of the argument we used to show that
$G\in\mathcal{K}$. We start by proving that $D_{1}\geq 0$.
First notice that, considering the two possible cases
$\bar{m}_{1}=\bar{m}_{2}$ and $\bar{m}_{1}<\bar{m}_{2}$, we have:

  1. (1)

if $\bar{m}_{1}=\bar{m}_{2}$ then

$\displaystyle D_{1}=$ $\displaystyle\int
P_{1}\,dG^{(1)}-\frac{1}{\bar{m}_{1}}H(G^{(1)}\mid\mu_{1})-\int
P_{1}\,d\nu^{(1)}+\frac{1}{\bar{m}_{1}}H\left(\nu^{(1)}\mid\mu_{1}\right)\,;$
(53)

  2. (2)

if $\bar{m}_{1}<\bar{m}_{2}$ (as $\bar{m}_{1}\neq 0$) the partial derivative
$\partial_{m_{1}}P$ must vanish when evaluated in $(\bar{q},{\bf{\bar{m}}})$,
and this implies $H(G^{(1)}\mid\mu_{1})=\gamma_{1}\Theta\log 2$.
Therefore, since $\nu\in\mathcal{K}\Rightarrow
H(\nu^{(1)}\mid\mu_{1})\leq\gamma_{1}\Theta\log 2$, it holds

$H(G^{(1)}\mid\mu_{1})-H(\nu^{(1)}\mid\mu_{1})\geq 0,$

and hence

$\frac{1}{\bar{m}_{1}}\left[H(G^{(1)}\mid\mu_{1})-H(\nu^{(1)}\mid\mu_{1})\right]\geq\frac{1}{\bar{m}_{2}}\left[H(G^{(1)}\mid\mu_{1})-H(\nu^{(1)}\mid\mu_{1})\right].$
(54)

It follows from (53), (54) that in both cases it holds

$\displaystyle D_{1}$ $\displaystyle\geq\int
P_{1}\,dG^{(1)}-\frac{1}{\bar{m}_{1}}H(G^{(1)}\mid\mu_{1})-\int
P_{1}\,d\nu^{(1)}+\frac{1}{\bar{m}_{1}}H(\nu^{(1)}\mid\mu_{1}).$ (55)

Therefore, using the standard definition of relative entropy (14) to express
$H(G^{(1)}\mid\mu_{1})$, we get

$\displaystyle D_{1}$
$\displaystyle\geq\frac{1}{\bar{m}_{1}}\log\int\exp{\left(m_{1}P_{1}\right)}d\mu_{1}-\int
P_{1}\,d\nu^{(1)}+\frac{1}{\bar{m}_{1}}H(\nu^{(1)}\mid\mu_{1})$ (56)
$\displaystyle=\frac{1}{\bar{m}_{1}}\left\\{H\left(\nu^{(1)}\mid\mu_{1}\right)-\left[\int\bar{m}_{1}P_{1}d\nu^{(1)}-\log\int\exp{\left(m_{1}P_{1}\right)}d\mu_{1}\right]\right\\}\geq
0$

where in the last step we used the Donsker and Varadhan variational formula
for relative entropies

$H(\nu\mid\mu)=\sup_{f\in C_{b}}\\{\int fd\nu-\log\int e^{f}d\mu\\}$

(see [14, 22] for details) and the fact that the hypotesis $\varphi,f\in
C_{b}(\mathbb{R}^{n})$ imply that $P_{j}\in C_{b}(S^{j})$ for every
$j\in\\{1,\cdots,n\\}$.
Now, we only need to prove that $D_{k}\geq D_{k-1}$ for any
$k\in\\{1,\cdots,n\\}$.
Notice that if $K\in\\{1,\cdots,n\\}$ is the total number of different values
assumed by the (positive and non-decreasing w.r.t the indices) entries of
${\bar{\bf m}}=(\bar{m}_{1},\dots,\bar{m}_{n})$; setting $\text{max}_{0}:=0$,
we can define for every $k\in\\{1,\cdots,K\\}$

$\text{max}_{k}:=\max\\{j:\quad\text{max}_{k-1}+1\leq j\leq
n,\,\,\,\bar{m}_{j}=\bar{m}_{\text{max}_{k-1}+1}\\}.$ (57)

To wit: for any $k\in\\{1,\cdots,K\\}$ the index $\text{max}_{k}$ is the last
one for which the corresponding entry in ${\bf\bar{m}}$ assumes the $k$-th
higher value in the whole vector: we will refer to the largest sections of the
vector ${\bf\bar{m}}$ where all entries are equal as blocks. By construction,
$j=\text{max}_{k}$ for some $k$ iff $\bar{m}_{j}<\bar{m}_{j+1}$.
As in the case $1)$ above, if $\bar{m}_{n}=1$ then

$D_{n}\geq\boldsymbol{\Phi}[G]-\frac{1}{\bar{m}_{n}}H(G\mid\mu)-\boldsymbol{\Phi}[\nu]+\frac{1}{\bar{m}_{n}}H(\nu\mid\mu);\\\
$ (58)

holds with equality. If $\bar{m}_{n}<1$ then, considering the real function
$m\in(0,\bar{m}_{\text{max}_{1}+1}]\to g_{\text{max}_{1}}(m)$ defined by (48)
for $j\equiv\text{max}_{1}$, to wit

$g_{\text{max}_{1}}(m)=\begin{cases}P(\bar{q},m,\dots,m)&\text{if}\,\,\max_{1}=n\,,\\\
P(\bar{q},m,\dots,m,\bar{m}_{\text{max}_{1}+1},\dots,\bar{m}_{n})&\text{if}\,\,\max_{1}<n\,;\end{cases}$

we notice that it must have a minimum point in
$m=\bar{m}_{1}\in(0,\bar{m}_{max_{1}+1})$, specifically for
$m\neq\bar{m}_{\text{max}_{1}+1}$ (recall that we have set
$\bar{m}_{n+1}\equiv 1$). This implies that $\bar{m}_{1}$ must be a critical
point of $g_{\text{max}_{1}}$, i.e. that
$\sum\limits_{k=1}^{\text{max}_{1}}\partial_{m_{k}}P(\bar{q},{\bf\bar{m}})=0$,
which implies

$H(G^{(\text{max}_{1})}\mid\mu^{(\text{max}_{1})})=\Theta\log
2\sum\limits_{k=1}^{\text{max}_{1}}\gamma_{k}.$ (59)

If now $K=1$, i.e. if ${\bf m}$ is made of one block and $\text{max}_{1}=n$,
the latter is exactly $H(G\mid\mu)=\Theta\log 2$. If instead $K\geq 2$, i.e.
if $\bar{\bf m}$ is made of at least two blocks and $\text{max}_{1}<n$, we
easily get, as above, that
$\sum_{k=\text{max}_{1}+1}^{\text{max}_{2}}\partial_{m_{k}}P(\bar{q},\bar{\bf
m})=0$, which implies

$\sum_{k=\text{max}_{1}+1}^{\text{max}_{2}}{\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{k-1}(H^{\,\bar{q},{\bf\bar{m}}}_{k})=\Theta\log
2\sum_{k=\text{max}_{1}+1}^{\text{max}_{2}}\gamma_{k}.$ (60)

Through the chain-rule, (59) and (60), we get

$\displaystyle H(G\mid\mu)=$ $\displaystyle
H(G^{(\text{max}_{1})}\mid\mu^{(\text{max}_{1})})+\sum_{k=\text{max}_{1}+1}^{n}{\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{k-1}(H^{\,\bar{q},{\bf\bar{m}}}_{k})$
(61) $\displaystyle=$ $\displaystyle\Theta\log
2\sum\limits_{k=1}^{\text{max}_{1}}\gamma_{k}+\Theta\log
2\sum\limits_{k=\text{max}_{1}+1}^{\text{max}_{2}}\gamma_{k}+\sum_{k=\text{max}_{2}+1}^{n}{\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{k-1}(H^{\,\bar{q},{\bf\bar{m}}}_{k}).$

It should now be clear to the reader that by repeating the same reasoning $K$
times, since $\text{max}_{K}\equiv n$, we arrive at

$\displaystyle H(G\mid\mu)=$ $\displaystyle\Theta\log
2\sum\limits_{j=0}^{K}\sum\limits_{k=\text{max}_{j}+1}^{\text{max}_{j+1}}\gamma_{k}=\Theta\log
2.$ (62)

Specifically, as before, the inequality (58) holds also if $\bar{m}_{n}<1$.
From (50) and some elementary manipulations one finds

$\displaystyle H(G\mid\mu)$
$\displaystyle=H(G^{(n-1)}\mid\mu^{(n-1)})+{\sf{E}}^{\,\bar{q},{\bf\bar{m}}}_{n-1}(H^{\,\bar{q},{\bf\bar{m}}}_{n})$
(63) $\displaystyle=H(G^{(n-1)}\mid\mu^{(n-1)})+\bar{m}_{n}\int
P_{n}\,dG-\bar{m}_{n}\int P_{n-1}\,dG^{(n-1)}$
$\displaystyle=H(G^{(n-1)}\mid\mu^{(n-1)})+\bar{m}_{n}\Phi^{\prime}(\bar{q})\int\varphi\,dG+\bar{m}_{n}\int
f\,dG-\bar{m}_{n}\int P_{n-1}\,dG^{(n-1)}.$

Through the latter, we get

$\displaystyle\frac{1}{\bar{m}_{n}}H(G\mid\mu)=$
$\displaystyle\frac{1}{\bar{m}_{n}}H(G^{(n-1)}\mid\mu^{(n-1)})+\Phi^{\prime}(\bar{q})\int\varphi
dG+\int fdG-\int P_{n-1}dG^{(n-1)}$ (64)

Plugging the latter into (58), and recalling the explicit expression (17) for
$\boldsymbol{\Phi}$, we find

$\displaystyle D_{n}$ $\displaystyle\geq\Phi\left(\int\varphi
dG\right)-\Phi\left(\int\varphi
d\nu\right)-\Phi^{\prime}(\bar{q})\int\varphi\,dG-\int fd\nu\,+$ (65)
$\displaystyle\qquad\qquad+\frac{1}{\bar{m}_{n}}H(\nu\mid\mu)-\frac{1}{\bar{m}_{n}}H(G^{(n-1)}\mid\mu^{(n-1)})+\int
P_{n-1}\,dG^{(n-1)}$
$\displaystyle\geq\frac{1}{\bar{m}_{n}}H(\nu\mid\mu)-\Phi^{\prime}(\bar{q})\int\varphi
d\nu-\int fd\nu-\frac{1}{\bar{m}_{n}}H(G^{(n-1)}\mid\mu^{(n-1)})+\int
P_{n-1}\,dG^{(n-1)}$

the second inequality by concavity of $\Phi$, using that
$\bar{q}=\int\varphi\,dG$.
Recall now that a chain-rule analogous to (50) holds for every measure
$\nu\in\mathcal{K}$. Specifically, if $(X_{1},\dots,X_{n})$ is a random vector
distributed according to some $\nu\in\mathcal{M}_{1}^{+}(\mathbb{R}^{n})$ and
${\bf x}_{n-1}=(x_{1},\dots,x_{n-1})\in\mathbb{R}^{n-1}$, indicating with
$K_{\nu}^{({\bf x}_{n-1})}\in\mathcal{M}_{1}^{+}(\mathbb{R})$ the regular
conditional probability distribution of $X_{n}$ given
$X_{1}=x_{1},\dots,X_{n-1}=x_{n-1}$ the following holds

$\displaystyle H(\nu\mid\mu)$
$\displaystyle=H(\nu^{(n-1)}\mid\mu^{(n-1)})+\int\,\,H(K_{\nu}^{({\bf
x}_{n-1})}\mid\mu_{n})\,\,\nu^{(n-1)}(d{\bf x}_{n-1}).$

Using this formula we get

$\displaystyle
H(\nu\mid\mu)-\int\,\,\bar{m}_{n}\left[\Phi^{\prime}(\bar{q})\varphi({\bf
x})+f({\bf x})\right]\,\,\nu(d{\bf x})+\bar{m}_{n}\int P_{n-1}({\bf
x}_{n-1})\,\,\nu^{(n-1)}(d{\bf x}_{n-1})$ (66)
$\displaystyle=H(\nu^{(n-1)}\mid\mu^{(n-1)})+\int\,\,H(K_{\nu}^{({\bf
x}_{n-1})}\mid\mu_{n})\nu^{(n-1)}(d{\bf x}_{n-1})\,+$
$\displaystyle\quad-\int\left\\{\int\bar{m}_{n}\left[\Phi^{\prime}(\bar{q})\varphi({\bf
x})+f({\bf x})\right]K_{\nu}^{({\bf x}_{n-1})}(dx_{n})-\bar{m}_{n}P_{n-1}({\bf
x}_{n-1})\right\\}\,\,\nu^{(n-1)}(d{\bf x}_{n-1})$
$\displaystyle=H(\nu^{(n-1)}\mid\mu^{(n-1)})+\int\,\,H(K_{\nu}^{({\bf
x}_{n-1})}\mid\mu_{n})\nu^{(n-1)}(d{\bf x}_{n-1})\,+$
$\displaystyle\quad-\int\left\\{\int\bar{m}_{n}\left[\Phi^{\prime}(\bar{q})\varphi({\bf
x})+f({\bf x})\right]K_{\nu}^{({\bf x}_{n-1})}(dx_{n})-\log\int
e^{\bar{m}_{n}P_{n}({\bf x})}\mu_{n}(dx_{n})\right\\}\nu^{(n-1)}(d{\bf
x}_{n-1}).$

Now, the Donsker and Varadhan variational formula for relative entropies
implies that for every fixed ${\bf x}_{n-1}\in\mathbb{R}^{n-1}$

$\displaystyle H(K_{\nu}^{({\bf x}_{n-1})}\mid\mu_{n})$
$\displaystyle\geq\int\bar{m}_{n}\left[\Phi^{\prime}(\bar{q})\varphi({\bf
x})+f({\bf x})\right]K_{\nu}^{({\bf x}_{n-1})}(dx_{n})-\log\int
e^{\bar{m}_{n}\left[\Phi^{\prime}(\bar{q})\varphi({\bf x})+f({\bf
x})\right]}\mu_{n}(dx_{n})$

which, as $P_{n}({\bf x})=\Phi^{\prime}(\bar{q})\varphi({\bf x})+f({\bf x})$,
implies through (66) that

$\displaystyle\frac{1}{\bar{m}_{n}}H(\nu\mid\mu)-\int\,\,\left[\Phi^{\prime}(\bar{q})\varphi+f\right]\,\,d\nu\geq-\int
P_{n-1}\,\,d\nu^{(n-1)}+\frac{1}{\bar{m}_{n}}H(\nu^{(n-1)}\mid\mu^{(n-1)}).$
(67)

Plugging the latter into (65) we find exactly $D_{n}\geq D_{n-1}$.
It is now enough to notice that the exact same argument (with $\Phi$ the
identity function) implies $D_{k}\geq D_{k-1}$ for every
$k\in\\{1,\cdots,n-1\\}$.
As we proved that $D_{1}\geq 0$, the claim (52) follows and the proof of the
Lemma is concluded. ∎

It follows from Lemma 4.1 that the unique solution to the Boltzmann-Gibbs
principle must be given by $G^{\bar{q},\bar{\bf m}}$, where $(\bar{q},\bar{\bf
m})$ is a minimum point for $P$ on $[\inf\varphi,\sup\varphi]\times\Delta$. As
one easily checks that
$\boldsymbol{\Phi}[G]-H(G\mid\mu)=\boldsymbol{\Phi}[G^{\bar{q},\bar{\bf
m}}]-H(G^{\bar{q},\bar{\bf m}}\mid\mu)=P(\bar{q},\bar{\bf m})$, Theorem 2.2
follows from Theorem 2.1 together with Lemma 4.1.

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# Hybrid single-pair charge-2 Weyl semimetals

P. Zhou<EMAIL_ADDRESS>Hunan Provincial Key laboratory of Thin Film
Materials and Devices, School of Material Sciences and Engineering, Xiangtan
University, Xiangtan 411105, China  Y. Z. Hu  Hunan Provincial Key laboratory
of Thin Film Materials and Devices, School of Material Sciences and
Engineering, Xiangtan University, Xiangtan 411105, China  B. R. Pan  Key
Laboratory of Low-dimensional Materials and Application Technology, School of
Material Sciences and Engineering, Xiangtan University, Xiangtan 411105, China
F. F. Huang  Key Laboratory of Low-dimensional Materials and Application
Technology, School of Material Sciences and Engineering, Xiangtan University,
Xiangtan 411105, China  W. Q. Li  Key Laboratory of Low-dimensional Materials
and Application Technology, School of Material Sciences and Engineering,
Xiangtan University, Xiangtan 411105, China  Z. S. Ma  Hunan Provincial Key
laboratory of Thin Film Materials and Devices, School of Material Sciences and
Engineering, Xiangtan University, Xiangtan 411105, China  L. Z. Sun
<EMAIL_ADDRESS>Hunan Provincial Key laboratory of Thin Film Materials and
Devices, School of Material Sciences and Engineering, Xiangtan University,
Xiangtan 411105, China

###### Abstract

Intuitively, the dispersion characteristics of Weyl nodes with opposite
charges in single-pair charge-2 Weyl semimetals are the same, quadratic or
linear. We theoretically predicted that single-pair hybrid charge-2 Weyl
semimetals (the nodes with opposite charges show quadratic Weyl and linear
charge-2 Dirac characteristics, respectively) can be protected by specific
nonsymmorphic symmetries in spinless systems. Moreover, the symmetries force
the pair of Weyl points locate at the center and corners of the first
Brillouin zone (FBZ), respectively. Consequently, nontrivial surface states
run through the entire FBZ of the system fascinating for future experimental
detection and device applications. The hybrid phase is further verified with
the help of first-principles calculations for the phonon states in realistic
material of Na2Zn2O3. The new phase will not only broaden the understanding of
the Weyl semimetals, but also provide an interesting platform to investigate
the interaction between the two types of Weyl fermions with different
dispersions.

In high-energy physics, Weyl fermionsHirschberger et al. (2016); Wan et al.
(2011); Lv et al. (2015); Xu et al. (2015); Xiao et al. (2015); Yan and Felser
(2017) are relativistic massless particles that are restricted by the Poincare
symmetry. The low-energy quasiparticle excitations of Weyl fermions in
condensed matters protected by crystalline symmetries are alternative
platforms to study the novel physics of the elementary particlesFang et al.
(2012); Weng et al. (2015); Dai (2016); Armitage et al. (2018). The Weyl
fermions are monopoles of Berry curvature in the Brillouin zone (BZ) and the
charges associated with the monopoles are known as their chiralityLiu et al.
(2017); Nielsen and Ninomiya (1981a); Fang et al. (2003); Udagawa and
Bergholtz (2016). In crystalline materials, the Bloch function enriches the
chiral charges to be $\pm$1Meng et al. (2019); Borisenko et al. (2019);
Soluyanov et al. (2015); Yao et al. (2019); Xie et al. (2019); Xia et al.
(2019); Nie et al. (2022), $\pm$2Xu et al. (2011); Zhang et al. (2018); Miao
et al. (2018); Ding et al. (2022); Li et al. (2021); Zhong et al. (2022);
Sreeparvathy et al. (2022), $\pm$3Tsirkin et al. (2017); Tang et al. (2017),
and even $\pm$4Liu et al. (2021); Wang et al. (2022a); Chang et al. (2017);
Luo et al. (2022), relying on the degree of degeneracy of the Weyl points and
band dispersion around the Weyl nodes. In comparison with the conventional
nodesYu et al. (2016); Sun et al. (2015, 2016); Chang et al. (2016); Bulmash
et al. (2014) with the charges of $\pm$1, the nodes with higher chargeZhang et
al. (2020); Yu et al. (2022); Cui et al. (2021); Huang et al. (2015); Zhang
and Niu (2015) are fascinating since the chiral charges determine the number
of chiral edge states. Consequently, the Weyl points with higher monopole
charges will result in intensified Fermi arcs. The anomalous Hall
effectSteiner et al. (2017); Higo et al. (2018); Burkov (2014), as well as the
effects linked to the chiral anomaly such as negative magnetic resistanceSon
and Spivak (2013); Huang et al. (2015), will be strengthened with the increase
in the monopole charge. Furthermore, the intrinsic anisotropy of quadratic or
cubic dispersion of higher charge Weyl cones will result in novel correlation
phenomena. Therefore, Weyl semimetals with higher charge receive great
attention, recently.

Figure 1: Four types of single-pair Weyl nodes. (a) Single-pair ”pure” nodes
with a chiral charge of $\pm$1 and linear dispersion in all directionsZhang et
al. (2019); Nie et al. (2022). (b) Single-pair ”pure” nodes with a chiral
charge of $\pm$2Wang et al. (2022b); Li et al. (2022). There are a quadratic
dispersion in specific plane and linear dispersion in the perpendicular
orientations to the plane. (c) Single-pair ”pure” nodes with a chiral charge
of $\pm$4Wang et al. (2022a). The nodes show a $\textbf{\emph{k}}^{3}$-type
dispersion splitting along one direction and quadratic dispersion splitting
along other directions. (d) Single-pair ”hybrid” nodes, including a double-
Weyl point (twofold degenerate point) with quadratic dispersion and a charge-2
Dirac point (fourfold-degenerate point) with linear dispersion, respectively.

As for the studies on Weyl semimetals, one of the critical obstacles is that
there are usually too many pairs of Weyl points around the Fermi level in
condensed matters, for example 12 pairs of Weyl nodes in TaAsWeng et al.
(2015); Lv et al. (2015); Dai (2016); Xu et al. (2016), which will complicate
their experimental detections and device applications. Thus, to find Weyl
semimetals with fewer pairs of Weyl nodes arouses much interest. According to
the Nielsen-Ninomiya no-go theoremNielsen and Ninomiya (1981b, c), the total
chirality in the entire three-dimensional BZ of a Weyl semimetal must be zero.
Therefore, theoretically, the minimum number of Weyl nodes in a condensed
matter is single-pair. This kind of system will be helpful for us to capture
the main physical phenomenon and the associated topological features through
simplifying the theoretical analysis and related experimental results. Very
recently, several Weyl semimetals with single-pair Weyl nodes have been
reportedXu et al. (2011); Zhang et al. (2019); Wang et al. (2019); Nie et al.
(2022); Wang et al. (2022b, a); Li et al. (2022). The chiral charges of them
can be $\pm$1, $\pm$2, and $\pm$4\. Up to now, the two Weyl nodes with
opposite charges in single-pair Weyl semimetals are all the same dispersion
characteristics, as shown in Fig. 1(a)-1(c), we call them “pure” type single-
pair Weyl semimetals. As for charge-2 single-pair Weyl semimetals, considering
half-filling and excluding spin-1 Weyl node, there are two kinds of typical
band dispersion for charge-2 Weyl nodes: quadratic Weyl and charge-2 Dirac
nodesBradlyn et al. (2016); Geilhufe et al. (2017); Huang et al. (2016); Liu
et al. (2022); Li et al. (2021). The quadratic Weyl node (also called double-
Weyl node) is twofold degenerate with quadratic dispersion along specific high
symmetry lines, as shown in Fig. 1(b). Whereas the charge-2 Dirac node is
composed of degenerate Weyl nodes with same chiral charges. Until now, single-
pair charge $\pm$2 Weyl semimetals previously reported are all pure type, as
shown in Fig. 1(b). An interesting issue is that can the symmetries of
condensed matter enforce the single-pair ”hybrid” type charge-2 Weyl
semimetal? Namely, the nodes with opposite charges show quadratic Weyl and
linear charge-2 Dirac characteristics, respectively, as shown in Fig. 1(d).
Such new phase will not only broaden the understanding of the Weyl semimetals,
but also provide an interesting platform to investigate the interaction
between the two kinds of Weyl fermions with different dispersions.
In present letter, using the theory of elementary band representationBradlyn
et al. (2017); Cano et al. (2018) (EBR) and considering the symmetries of the
Bloch states at high symmetry point and their combinatorics between different
wavevectorsSlager et al. (2013); Kruthoff et al. (2017), we theoretically
predicted that hybrid single-pair charge-2 Weyl semimetal can be protected by
specific nonsymmorphic space groups, such as $P4_{1}2_{1}2$ and
$P4_{3}2_{1}2$, in spinless systems. Moreover, the symmetries enforce the pair
of Weyl points locate at the center and corners of the FBZ, respectively, that
any two-dimensional plane in the reciprocal space (except for those containing
the Weyl points) is topologically nontrivial with nonzero Chern number (1 in
$k_{z}$ $>$ 0 and -1 in $k_{z}$ $<$ 0 in FBZ). Consequently, nontrivial
surface states run through the entire surface FBZ of the system due to the
unconventional Chern number distribution. Inspired by the theoretical results,
we further verified the new charge-$\pm$2 Weyl phase in the optical phonon
states of realistic material Na2Zn2O3 with the help of first-principles
calculations, which is fascinating for future experimental detection and
device applications.

Figure 2: Four potential band structures forming double-Weyl node along high-
symmetry line $\Gamma$-Z, where +1, -1, $+i$, and $-i$ are the eigenvalues of
the operation ${C}_{4}$. The time-reversal symmetry results in the degeneracy
of the eigenstates at the $\Gamma$ point and the degeneracy point located at
the Fermi level if half-filling is taken into account. The purple dots show
the conventional Weyl points with linear dispersion in all directions, while
the green points show the double-Weyl nodes with quadratic dispersion in the
kz = 0 plane and linear dispersion along $\Gamma$-Z. Figure 3: (a) The crystal
structure adopted in the TB model. The bonds with sapphire and purple denote
the nearest-neighbor hopping and next nearest-neighbor hopping terms,
respectively. The fractional coordinates of the four sites in the unit cell
are (0.40, 0.40, 0.00), (0.90, 0.10, 0.25), (0.60, 0.60, 0.50), and (0.10,
0.90, 0.75), respectively. (b) Energy band structure from TB model and the
$\pm$ 2 are the Chern numbers of the Weyl Points. (c) and (d) are the
distribution of charge-2 Weyl points in our model and conventional charge-2
Weyl semimetalXu et al. (2011), respectively. The red and green balls indicate
the Weyl points with Chern number +2, and -2, respectively. (e) Phase diagram
of the TB model with different $t_{0}$/$t_{1}$. The QAHI phases with Chern
number +1 [QAHI(1)] and -1 [QAHI(-1)] are shaded in green, and red,
respectively. The white area indicates the normal insulator (NI).

In present work, we focus on the three-dimensional (3D) Weyl semimetal in
class AI (spinless and time-reversal symmetric) with additional crystalline
symmetries. The single-valued representations can be used to characterize the
symmetry features of this class of systems. Concretely, they can be used to
describe spinless electrons, phonons of crystalline materials or some
artificial systems. By examining the little cogroup irreducible
representations (irreps) of all 230 space groups, we find the single-pair of
hybrid charge-2 Weyl nodes can be realized in the space groups of
$P4_{1}2_{1}2$ and $P4_{3}2_{1}2$, and the details will be discussed below.
Considering the similar characteristics, we concentrate on the space group
$P4_{3}2_{1}2$ (No. 96). Under time-reversal symmetry, there are two EBRs in
this space group and they can be induced from the Wyckoff position 4a with the
local site point group $C_{2}$. At the $A$ point, only two conjugated two-
order little cogroup irreps exist: $A_{1}\oplus A_{2}$, indicating that the
bands at this point must be fourfold-degenerate. The fourfold degeneracy can
be derived from the anticommutation between S2x and S2y, as well as their
commutation relation with time-reversal operator $T$. A k$\cdot$p model
analysis reveals that the dispersion is linear along all directions (see
details in Supplemental materialssup ). Therefore, a double Weyl node will
form at the high symmetry point. $\Gamma$-Z is a $C_{4}$ rotation invariant
path and the states on this path can be written as the eigenstates of $C_{4}$.
As mentioned above, the set of EBRs can be induced from Wyckoff position 4a.
Considering the eigenvalues at $\Gamma$ and Z as listed in Tab. S1 as well as
the compatibility relations, there are 12 types of energy band connections as
shown in Fig. 2 and Fig. S1. For all the cases, Weyl points are formed under
half-filling condition. However, only four types of energy band connections
can produce charge-2 Weyl node, as shown in Fig. 2. Among them, only two cases
as shown in Figs. 2(a) and 2(b) can fulfill the single-pair charge-2 Weyl
semimetals. Considering the similarity of the two charge-2 Weyl nodes, we take
the case as shown in Fig. 2(a) as an example. At high symmetry point $\Gamma$
with the wave vector $q_{\Gamma}=(0,0,0)$, the Hamiltonian is constrained by
the point group $D_{4h}$. Considering the time-reversal invariant of $\Gamma$
point, the double-degenerate states must have the $C_{4}$ eigenvalues
$\pm{i}$, then the $u_{c}$/$u_{v}$ = -1, which indicates the bands is
quadratic in $q_{x,y}$ with the monopole charge $\pm$2Fang et al. (2012). If
the charge-2 Dirac point at $A$ point and quadratic Weyl point around the
$\Gamma$ point share the same ”occupation”, the system will feature hybrid
single-pair Weyl nodes.

Figure 4: (a) The crystal structure of Na2Zn2O3, where the yellow, gray and
red spheres represent the Na, Zn and O atoms, respectively. (b) The bulk BZ
and the projected surface BZ for the (001) and (010) surfaces, respectively.
The charge-2 points with Chern number -2, and +2 are schematically shown by
the green and red spheres, respectively. (c) Calculated phonon dispersion and
DOS. The zoom in phonon spectra in the range of 14.9 to 18.0 THz. (d) The
Chern numbers of such nontrivial phonon branches are illustrated around the
$\Gamma$ and A points.

To further confirm and investigate the general properties of a system with a
single-pair of hybrid charge-2 nodes, we construct a four-site TB model as
shown in Fig. 3(a). In the model, a tetragonal lattice with four sites per
unit cell is adopted. The four sites locate at the Wyckoff 4a of the space
group $P4_{3}2_{1}2$. The fourfold screw symmetry S4z := ($x$, $y$, $z$)
$\rightarrow$ ($y+\frac{1}{2}$, $-x+\frac{1}{2}$, $z+\frac{1}{4}$) makes the
atoms going from 1 $\rightarrow$ 2 $\rightarrow$ 3 $\rightarrow$ 4, as shown
in Fig. 3(a). The Hamiltonian with only nearest-neighbor and next-nearest-
neighbor hopping terms can be written as

$\displaystyle H=$ $\displaystyle\sum_{\left\langle
i,j\right\rangle}t_{0}(a_{i}^{\dagger}c_{j}+b_{i}^{\dagger}d_{j})$ (1)
$\displaystyle+\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}t_{1}(a_{i}^{\dagger}b_{j}+a_{i}^{\dagger}d_{j}+b_{i}^{\dagger}c_{j}+c_{i}^{\dagger}d_{j})+H.c.,$

where $a^{\dagger}(a)$, $b^{\dagger}(b)$, $c^{\dagger}(c)$, and
$d^{\dagger}(d)$ are the creation (annihilation) operators on sites 1 to 4,
respectively. $t_{0}$ and $t_{1}$ are the nearest-neighbor and next nearest-
neighbor hopping parameters, respectively. The band structures derived from
the TB model with $t_{0}$ = 0.4 and $t_{1}=1.0$ (the evolution of the band
structures according to the parameters will be discussed below) are shown in
Fig. 2(b). The results indicate that, at half-filling, the energy bands host a
doubly-degenerate point at the $\Gamma$ point with linear dispersion along
$\Gamma$-Z high symmetry line and quadratic along $\Gamma$-X high symmetry
line, whereas a linear fourfold point forms at the A point. The chiral charge
or Chern numbers of them are -2, and +2, respectively. The symmetry enforced
distribution of the two types of charge-2 Weyl points is shown in Fig. 3(c).
Generally, for a Weyl semimetal, quantum anomalous Hall insulator (QAHI) phase
is restricted in the space of BZ between the two Weyl pointsXu et al. (2011).
For example, as for charge-2 semimetals, the Chern number of QAHI phase
between the Weyl points is 2, denoted as QAHI(2) as shown in Fig. 3(d). The
results of the hybrid single-pair Weyl semimetal show that the QAHI phase
distribute throughout the whole 3D BZ with Chern numbers of +1 [QAHI(1)] and
-1 [QAHI(-1)] for $k_{z}>0$ and kz $<$ 0, respectively, as shown in Fig. 3(c).
The result is similar to the case of higher-order Weyl semimetalsWang et al.
(2020), the different Chern numbers on either sides of $k_{z}$ = 0 plane are
caused by the quadratic Weyl point at the $\Gamma$. Consequently, nontrivial
surface states run through the entire surface FBZ of the system due to the
unconventional Chern number distribution, which will be favorite for
experimental detection and device applications.
To investigate the appearance condition of the hybrid single-pair Weyl
semimetal phase, the phase diagram in function of $t_{0}/t_{1}$ ratio is
examined and the results are shown in Fig. 3(e). The ratio of $t_{0}/t_{1}$ is
restricted in the range of 0.0-3.5 because there is no new phase out of the
range. The results reveal that the semimetal with hybrid nodes can only be
found in the range of 0.0-1.0. At the phase transition point $t_{0}/t_{1}$ $=$
1, the eigenvalues of the lowest three states are degenerate at the $\Gamma$
point as shown in Fig. S2(a). When $t_{0}/t_{1}$ $>$ 1, the degenerate bands
at the $\Gamma$ point gap out and the charge-2 quadratic Weyl node around the
$\Gamma$ point disappears. However, two Weyl points with charge -1 arise on
the $\Gamma$-Z as shown in Fig. S2(b) corresponding to the situation of Fig.
S1(c). There are three Weyl points in the systems, two of them with charge of
-1 and one with charge of +2. The green and red lines in Fig. 3(e) show where
the Weyl points with charge of -1 are, and they separate the trivial and
nontrivial phases. It is significant to note that along with the evolution of
the ratio, the bands at the A point remains fourfold degenerate with the Chern
number of +2. In the range of the ratios, the first BZ is divided into three
sections as shown in Fig. S3(b): the kz planes in the white zone have the
Chern number of 0, which corresponds to normal insulator, whereas the other
two zones are quantum anomalous Hall with Chern number of +1 or -1,
respectively. However, only when the $t_{0}/t_{1}$ $<$ 1, the quantum
anomalous Hall effect origins from the hybrid charge-2 semimetal phase.
The above discussions mainly focus on the space group of $P4_{1}2_{1}2$.
Similar conclusions can be produced in the space group $P4_{3}2_{1}2$.
Moreover, the nontrivial hybrid single-pair charge-2 Weyl semimetal phase
discussed above is not restricted to specific system. Generally, the new phase
can be realized in spinless electronic materials, phononic or photonic
crystals, acoustic metamaterials. In present work, we take phonons in a
crystalline material as example to illustrate the realization of the
nontrivial topological phase in realistic materials with the help of first-
principles method.

Figure 5: The surface LDOS of Na2Zn2O3 in the (001) surface (a) and (010)
surface (b). (c) The phonon surface arcs at 16.63 THz projected on the semi-
infinite (001) surface. The green and red dots indicate the projections of
Weyl points with $\mathcal{C}=+2$ and $\mathcal{C}=-2$, respectively. (d) The
double-helicoid surface states are described by the Weierstrass elliptic
function.

To concrete the hybrid single-pair charge-2 semimetal phase in term of
realistic materials, through simply searching for the materials belong to
above two nonsymmorphic space group ( $P4_{1}2_{1}2$ and $P4_{3}2_{1}2$), we
find that the single-pair hybrid charge-2 semimetal states can be realized in
the optic phonons of 3D Na2Zn2O3. The material has been experimentally
synthesized via solid state reaction method since 1996Trinschek and Jansen
(1996). As shown in Fig. 4(a), it crystallizes in a tetragonal Bravais lattice
with the space group $P4_{3}2_{1}2$. The optimized lattice constants are a = b
= 6.174 $\AA$ and c = 9.397 $\AA$, which agree well with the experimental
valuesTrinschek and Jansen (1996). The primitive cell has two inequivalent O
atoms that occupy the Wyckoff positions 4a and 4b, respectively. Based on
density functional perturbation theoryGonze and Lee (1997), we calculate the
phonon spectraTogo and Tanaka (2015) and density of states (DOS) of Na2Zn2O3
with the Vienna ab initio simulation package (VASP)Kresse and Furthmüller
(1996), the results are shown in Fig. 4(c). There are three different groups
of isolated 4-band high-frequency (larger than 14 THz) optic modes and they
are mainly contributed by the vibrations of O atoms. The zoom in phonon
dispersions in the range of 14.9 to 18.0 THz, as shown in Fig. 4(d), indicate
that a fourfold-degenerate point and a quadratic double-degenerate point
present at the A and ${\Gamma}$ points, respectively, forming a single-pair of
hybrid charge-2 nodes. The two bands forming the node around the $\Gamma$
point show quadratic dispersion in $k_{z}$=0 plane and linear dispersion in
other orientations, which is consistent with the above theoretical analysis.
The Chern numbers of the Weyl points obtained with Wilson-loop methodYu et al.
(2011); Wu et al. (2018) are denoted in Figs. 4(d), where green and red dots
denote the Weyl points with Chern number of -2 and +2, respectively. Fig. 4(b)
illustrates the distribution of the quadratic Weyl node $(\mathcal{C}=-2)$ at
${\Gamma}$ point (in the center of the BZ) and the charge-2 Dirac points
$(\mathcal{C}=+2)$ at A point (at the corner of the BZ). The result is
analogous to the theoretical conclusion above as shown in schematic Fig. 3(c).
As mentioned above, one of the intriguing properties of the hybrid single-pair
charge-2 semimetal phase is that the nontrivial surface states run through the
entire projected FBZ of the system due to the unconventional Chern number
distribution. To illustrate the topological characteristics throughout the
whole BZ of Na2Zn2O3, we compute the Chern number of horizontal two-
dimensional planes on both sides of the Weyl point at the $\Gamma$ along
$k_{z}$ orientation. The results show that the Chern numbers are 1 and -1 for
$k_{z}>0$ and $k_{z}<0$, respectively, which shares the same nontrivial
topological states with the case of Fig. 3(c). To further confirm the
topological phonon properties of Na2Zn2O3, we calculate the surface local
density of statesSancho et al. (1985, 1984); Wu et al. (2018) (LDOS) on the
(001) and (010) surfaces spanning the frequency range of 15 to 18 THz, the
results are shown in Figs. 5(a) and 5(b). There are surface states connect the
projected charge-2 nodes on the (001) and (010) surfaces further confirming
the nontrivial properties of the single-pair charge-2 Weyl semimetal Na2Zn2O3.
The isofrequency surface for the frequency of 16.63 THz on the (001) surface
exhibits chiral surface arcs rotating around the Weyl points, as shown in Fig.
5(c), that is the signature of nontrivial property for the material (the
circles around the $\bar{M}$ derive from the energy difference between the two
opposite charge Weyl points). The global picture of the surface statesFang et
al. (2016) also can be described by the Weierstrass elliptic functionZhang et
al. (2018) and they are composed with two surface sheets wind around the two
charge-2 Weyl points, as shown in Fig. 5(d). The surface states are periodic
in complex plane and have two singular points at (0, 0) and ($\pi$, $\pi$),
which correspond to the positions of chiral charge +2 and -2, respectively.
Using the theory of EBR, we propose that hybrid single-pair charge-2 Weyl
semimetal can be forced in spinless systems with specific nonsymmorphic space
groups of $P4_{1}2_{1}2$ and $P4_{3}2_{1}2$. Such new phase will provide an
interesting platform to investigate the interaction between the two types of
Weyl fermions with different dispersions. Moreover, the symmetry enforced Weyl
node distribution produce nontrivial double-helicoid surface states that
connect the projected nodes and run through the entire surface FBZ of the
system, which is fascinating for future experimental detection and device
applications. Although, in present work, we only take the simplified hybrid
single-pair charge-2 optical phonon states in realistic material of Na2Zn2O3
as an example, the novel phase will appear in the spinless electronic
materials, phononic or photonic crystals, acoustic metamaterials as long as
the nonsymmorphic symmetries are fulfilled. As for Na2Zn2O3, its surface
phonon states can be investigated experimentally using high resolution
electron energy loss spectroscopyZhu et al. (2015), helium scatteringDelaire
et al. (2015), or THz spectroscopyBaxter and Guglietta (2011). Moreover, the
topological surface phonon states will improve electron-phonon interactions
when the material form interface with other material that will benefit
interfacial superconductivity.
This work is supported by the National Natural Science Foundation of China
(Grant No. 11804287, 11574260), Hunan Provincial Natural Science Foundation of
China (2019JJ50577, 2021JJ30686), and Hunan Provincial Innovation Foundation
for Postgraduate (CX20190479).

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