# Variational Loop Vertex Expansion

Vasily<EMAIL_ADDRESS>
Université Paris-Saclay, CEA, List,
F-91120, Palaiseau, France

###### Abstract

Loop Vertex Expansion (LVE) was developed to construct QFT models with local
and non-local interactions. Using LVE, one can prove the analyticity in the
finite cardioid-like domain in the complex plain of the coupling constant of
the free energies and cumulants of various vector, matrix, or tensor-type
models. Here, applying the idea of choosing the initial approximation
depending on the coupling constant, we construct the analytic continuation of
the free energy of the quartic matrix model beyond the standard LVE cardioid
over the branch cut and for arbitrary large couplings.

##  1 Introduction

The present work stems from the constructive field theory method of the Loop
Vertex Expansion (LVE) and ideas of the variational perturbation theory. The
concept of LVE was first introduced in [1] as a constructive approach for
quartic matrix models aimed to provide bounds that are uniform in the size of
the matrix. In its original form, LVE combines an intermediate field
representation with replica fields and a forest formula [2, 3] to express the
free energy of the theory through a convergent sum over trees. Unlike
conventional constructive methods, this loop vertex expansion does not rely on
cluster expansions and does not entail conditions related to small/large field
considerations.

Like Feynman’s perturbative expansion, the LVE provides a straightforward way
to calculate connected quantities. In this method, the theory’s partition
function is represented as a sum over forests, and its logarithm is
essentially the same sum but constrained to connected forests – trees. This
property arises from the fact that the amplitudes factorize over the connected
components of the forest. The functional integrands associated with each
forest or tree exhibit absolute and _uniform_ convergence for all field
values. Together with the non-proliferation of trees (in comparison to Feynman
diagrams) this leads to the convergence of the LVE in the ’pacman’-like or
cardioid-like domains, see Fig. 1.

Figure 1: The left-hand side represents the typical LVE ’pacman’-type domain
of analyticity with the radius $r_{\alpha}$ shrinking with decreasing of the
angle $\alpha$. On the right, we present the cardioid domain, which can be
understood as a union of ’pacman’-type domains.

The convergence of the LVE implies the Borel summability of the standard
perturbation series [4], and the LVE directly computes the Borel sum.
Essentially, the loop vertex expansion performs an _explicit repacking_ of
infinitely many subsets of Feynman amplitude components, leading to a
convergent expansion rather than a divergent one [5].

In the context of combinatorial field theories involving matrices and tensors
[6, 7], appropriately rescaled to possess a non-trivial $N\to\infty$ limit [8,
9, 10, 11], the Borel summability achieved by the LVE is _uniform_ with
respect to the model’s size $N$, [1, 12, 13, 14]. The LVE method is extendable
to ordinary field theories with cutoffs, as discussed in [15]. An adapted
multiscale version, known as MLVE [16], incorporates renormalization
techniques [17, 18, 19, 20, 21], although it’s worth noting that models
developed so far are limited to the superrenormalizable type. The MLVE is
particularly effective in resumming renormalized series for non-local field
theories of matrix or tensorial types. Originally developed for the quartic
interactions, the LVE method was then generalized to higher-order interactions
under the name of the Loop Vertex Representation [22, 23, 24].

The main principle of the Variational Perturbation Theory (VPT) is the
utilization of the initial approximation chosen depending on the coupling
constant and/or on the order of the perturbative expansion, or optimized
depending on these parameters. Such ideas of the VPT were applied under
different names: Variational Perturbation Theory [25, 26], Optimized
Perturbation Theory [27], Convergent Perturbation Theory [28], Delta Expansion
[29], etc., – with the different levels of rigor to a various range of
physical problems. The applications of VPT include: energy levels of quantum
anharmonic oscillator [30, 31, 29]; critical indices in scalar quantum field
theories [32]; optimization of the QCD perturbative computations [27, 33];
computations in lattice models with real and scalar actions [34, 35, 36];
coefficients of the $\frac{1}{D}$ expansion in the vector model [37]. VPT was
used for constructing strong coupling expansions [38]. It was shown that VPT
(Delta Expansion) should be applicable in cases where the standard
perturbation theory is non-Borel summable [29]. It is worth noting that the
proof of the convergence of the VPT (Delta Expansion) for the double-well case
of the anharmonic oscillator is based on the analyticity of its energy levels
[39] in the area of the coupling constant Riemann surface, which is larger
than the one required for the Borel summability in the single-well potential
case.

In the current work, we unite the LVE method with the VPT idea of the initial
approximation depending on the coupling constant. Taking the quartic matrix
model as an example, we construct LVE with the modified initial approximation
depending on the coupling constant and prove the convergence of the
corresponding series for the free energy of the model for arbitrary large
coupling constants with the arguments $-\frac{3\pi}{2}<\phi<\frac{3\pi}{2}$.
The latter result is not optimal and might be improved up to
$-2\pi+\epsilon<\phi<2\pi-\epsilon$, where $\epsilon\geq\epsilon_{0}$,
$\epsilon_{0}>0$. This and the extension of the current results to cumulants
and a constructive version of the $1/N$ expansion are additional outcomes of
the method, left for further exploration. We also aim to investigate
connections with the resurgence theory in the spirit of the recently carried
out analysis of the vector model [40].

##  2 Statement of the result

To illustrate the main advantages of the Loop Vertex Expansion modified by the
initial approximation depending on the coupling, we study a quartic matrix
model. The partition function of this model is defined by

$\displaystyle{\cal Z}[\lambda,N]=\frac{1}{Z_{0}}\int
dM\exp\Big{\\{}-\operatorname{Tr}(MM^{\dagger})-\frac{\lambda}{2N}\operatorname{Tr}([MM^{\dagger}]^{2})\Big{\\}}\,,$
(1)

where $M$ are complex $N\times N$ matrices, and

$\displaystyle Z_{0}=\int
dM\exp\Big{\\{}-\operatorname{Tr}(MM^{\dagger}))\Big{\\}}\,.$ (2)

The measure $dM$ is given by

$dM=\pi^{N}\prod_{1\leq i,j\leq N}d\text{Re}(M_{ij})d\text{Im}(M_{ij})\,.$ (3)

The core object of our studies here is the free energy of the model defined,
as

$\displaystyle F[\lambda,N]=-\frac{1}{N^{2}}\log{\cal Z}[\lambda,N]\,.$ (4)

As was mentioned in the introduction the standard LVE tools allow one to prove
the analyticity of the free energy (4) or of the cumulants of the model in the
’pacman’-like or cardioid-like domains, as in Fig. 1. The sharpest result
regarding the model (1) was obtained by LVE in [12], among other results it
was shown there that the free energy (4) is analytic in the cardioid domain of
the coupling constant $\lambda$,

${\cal
C}=\Big{\\{}\lambda\in\mathbb{C}\,\Big{|}\arg\lambda=\phi\,,4|\lambda|<\cos^{2}\Big{(}{\frac{\phi}{2}}\Big{)}\Big{\\}}\,.$
(5)

The main result of the current paper, obtained by merging the ideas of the
variational perturbation theory and loop vertex expansion is given by the
following theorem.

######  Theorem 1.

For any $\lambda\in{\cal X}$, where ${\cal X}={\cal C}\cup{\cal Y}$, and
${\cal Y}$ is defined as a subset of the Riemann surface of $\sqrt{\lambda}$
with $\lambda\neq 0$ and $|\phi|<\frac{3\pi}{2}$, the free energy
$F[\lambda,N]$ of the model (1), is analytic in $\lambda$ uniformly in $N$.

##  3 New initial approximation and intermediate field representation

Hereafter we assume that $\lambda\neq 0$. The case of $\lambda=0$ is trivial
and the vicinity of $\lambda=0$ can be easily treated with the standard LVE
tools [12]. As a first step of our program, we change the Gaussian part of the
action which we treat as an unperturbed measure, in other words, we shift the
initial approximation. Then, introducing the intermediate field
representation, we obtain,

$\displaystyle{\cal Z}[\lambda,N]=\frac{1}{Z_{0}}\int dM\int
dA\,\exp\Big{\\{}-\frac{1}{2}\operatorname{Tr}(A^{2})-a\operatorname{Tr}(MM^{\dagger})$
$\displaystyle+\mathrm{i}\operatorname{Tr}\Big{(}A\big{[}\sqrt{\frac{\lambda}{N}}MM^{\dagger}+\frac{(1-a)\sqrt{N}}{\sqrt{\lambda}}\mathbb{1}\big{]}\Big{)}+\operatorname{Tr}\big{[}\frac{(1-a)^{2}N}{2\lambda}\mathbb{1}\big{]}\Big{\\}}\,,$
(6)

where $Re\,a>0$, $A$ field is realised by the $N\times N$ Hermitian matrices,
and the integral over $A$ is assumed to be normalized.

The integral over the initial degrees of freedom, matrices $M$ and
$M^{\dagger}$, is Gaussian, so it can be evaluated. The corresponding
covariance is given by
$\big{(}a-\mathrm{i}\sqrt{\frac{\lambda}{N}}A\big{)}\otimes 1$, and using that

$\displaystyle\det\left[\Big{(}a-\mathrm{i}\sqrt{\frac{\lambda}{N}}A\Big{)}\otimes
1\right]=a^{N}\exp\bigg{\\{}N\operatorname{Tr}\log\Big{(}1-\mathrm{i}\sqrt{\frac{\lambda}{a^{2}N}}A\Big{)}\bigg{\\}}\,,$
(7)

we obtain

$\displaystyle{\cal Z}[\lambda,N]$ $\displaystyle=$
$\displaystyle\frac{e^{\frac{N^{2}(1-a)^{2}}{2\lambda}}a^{N}}{\widetilde{Z}_{0}}\int
dA\,\exp\bigg{\\{}-\frac{1}{2}\operatorname{Tr}(A^{2})$ (8) $\displaystyle-$
$\displaystyle
N\operatorname{Tr}\log\left(1-\mathrm{i}\sqrt{\frac{\lambda}{a^{2}N}}A\right)-\mathrm{i}\operatorname{Tr}\bigg{(}A\frac{(1-a)\sqrt{N}}{\sqrt{\lambda}}\bigg{)}\bigg{\\}}\,.$

Then, we define the non-polynomial interaction, as

$\displaystyle{\cal
S}[\lambda,N,a](A)=\operatorname{Tr}\log\left(1-\mathrm{i}\sqrt{\frac{\lambda}{a^{2}N}}A\right)+\frac{\mathrm{i}}{\sqrt{N}}\operatorname{Tr}\bigg{(}A\frac{(1-a)}{\sqrt{\lambda}}\bigg{)}\,,$
(9)

and the normalization, as

$\displaystyle
K[\lambda,N,a]=\frac{e^{\frac{N^{2}(1-a)^{2}}{2\lambda}}a^{N}}{\widetilde{Z}_{0}}\,.$
(10)

Thus, the partition function can be written in the following form

$\displaystyle{\cal Z}[\lambda,N]=K[\lambda,N,a]\int
dA\,\exp\bigg{\\{}-\frac{1}{2}\operatorname{Tr}(A^{2})-N{\cal
S}[\lambda,N,a](A)\bigg{\\}}\,.$ (11)

Let us study the analytic properties of the partition function. We first have
the following bound.

######  Lemma 1.

Let $\frac{\lambda}{a^{2}}=\rho\mathrm{e}^{\mathrm{i}\theta}$ with $\rho>0$,
then:

$\Big{\|}\Big{(}1-\mathrm{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A\Big{)}^{-1}\Big{\|}\leq\frac{1}{\cos\frac{\theta}{2}}\,,$
(12)

where $\Big{\|}\cdot\Big{\|}$ stands for the operator norm.

###### Proof.

The first step is to rewrite the resolvent, as an integral,

$\Big{(}1-\mathrm{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A\Big{)}^{-1}=\frac{a\sqrt{N}}{\sqrt{\lambda}}\int_{0}^{\infty}d\alpha\,\exp\Big{\\{}-\alpha\frac{a\sqrt{N}}{\sqrt{\lambda}}+\alpha\mathrm{i}A\Big{\\}}\,.$
(13)

Using the latter representation, we immediately arrive at the bound for the
operator norm,

$\displaystyle\Big{\|}\Big{(}1-\mathrm{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A\Big{)}^{-1}\Big{\|}$
$\displaystyle\leq$
$\displaystyle\frac{|a|\sqrt{N}}{|\sqrt{\lambda}|}\int_{0}^{\infty}\exp\Big{\\{}-\alpha\text{Re}\big{(}\frac{a\sqrt{N}}{\sqrt{\lambda}}\big{)}\Big{\\}}\Big{\|}\exp\Big{\\{}\mathrm{i}\alpha
A\Big{\\}}\Big{\|}$ (14) $\displaystyle=$
$\displaystyle\frac{1}{\cos\frac{\theta}{2}}\,.$

∎

Hereafter, we write the parameter $a$ describing the initial approximation in
the following form,

$\displaystyle
a=x|\sqrt{\lambda}|e^{\mathrm{i}\psi}\,,\qquad-\frac{\pi}{2}<\psi<\frac{\pi}{2}\,,\qquad
x>0\,.$ (15)

Writing (11), as

${\cal Z}[\lambda,N]=K[\lambda,N,a]\int
dA\,\frac{\exp\bigg{\\{}-\frac{1}{2}\operatorname{Tr}(A^{2})-\mathrm{i}\operatorname{Tr}\bigg{(}A\frac{(1-a)\sqrt{N}}{\sqrt{\lambda}}\bigg{)}\bigg{\\}}}{\Big{[}\det\Big{(}1-\mathrm{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A\Big{)}\Big{]}^{N}}\,,$
(16)

and using lemma 1, one can show that this integral is convergent for
$\theta\in(-\pi,\pi)$, or equivalently for
$\lambda/a^{2}\in{\mathbb{C}}-{\mathbb{R}}^{-}$. Since the integrand is
analytic for $\theta\in(-\pi,\pi)$, and $\theta=\phi-2\psi$, we have the
following result.

######  Proposition 1.

${\cal Z}[N,\lambda]$ is analytic in $\lambda$ on the Riemann surface of the
square root with $-2\pi<\phi<2\pi$.

Note, that since ${\cal Z}[N,\lambda]$ may be zero for certain values of
$\lambda$, its analyticity does not imply the analyticity of the free energy.

##  4 The Loop Vertex Expansion

Using the representation (11) for the partition function, we expand the
interaction part of the exponent into the Taylor series, which is convergent
if $\lambda/a^{2}\in{\mathbb{C}}-{\mathbb{R}}^{-}$,

$\displaystyle{\cal
Z}=K[\lambda,N,a]\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\int
d\mu(A)\,\bigg{[}N{\cal S}[\lambda,N,a](A)\bigg{]}^{n}\,,$ (17)

where $d\mu(A)=dA\exp\\{-\frac{1}{2}\operatorname{Tr}(A^{2})\\}$ is the
normalized Gaussian measure for the Hermitian matrices $A$ (and we dropped the
arguments $\lambda$ and $N$ of ${\cal Z}$, and ${\cal S}[\lambda,N,a](A)$ we
will write ${\cal S}(A)$ in order to simplify the notation).

Applying the replica trick, we replace (at each order $n$) the integral over a
single matrix $A$ by the integral over a $n$-copies of $N\times N$ Hermitian
matrices $A=(A_{i})_{1\leq i\leq n}$. The measure of the Gaussian integral
over the replicated matrices $A$ is normalized with a degenerated covariance
$C_{ij}=1$. For any real positive symmetric matrix $C_{ij}$ the Gaussian
integral obeys

$\int
d\mu_{C}(A)\,A_{i|ab}A_{j|cd}=C_{ij}\,\delta_{ad}\delta_{bc}\,,~{}~{}~{}\int
d\mu_{C}(A)=1\,,$ (18)

where $A_{i|ab}$ corresponds to the matrix element of $A_{i}$ in the row $a$
and column $b$. The degenerated covariance in the Gaussian integral is
equivalent to insertion $n-1$ Dirac $\delta$-functions
$\delta(A_{1}-A_{2})\cdots\delta(A_{n-1}-A_{n})$. In Feynman diagrams, the
uniform covariance connects the various replicas together (with the
appropriate weights).

Then, the partition function is given by

$\displaystyle{\cal
Z}=K[\lambda,N,a]\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\int
d\mu_{C}(A)\prod_{i=1}^{n}\bigg{[}N{\cal S}(A_{i})\bigg{]}\,.$ (19)

To take the logarithm of the partition function, we are going to convert the
latter expansion into the sum over forests by applying the Bridges-Kennedy-
Abdessalam-Rivasseau [2, 3]. The first step to do it is to replace the
covariance $C_{ij}=1$ by $C_{ij}(x)=x_{ij}$, $x_{ij}=x_{ji}$ (which at the end
should be evaluated at $x_{ij}=1$) for $i\neq j$ and $C_{ii}(x)=1$. Then,

$\displaystyle{\cal Z}$ $\displaystyle=$ $\displaystyle
K[\lambda,N,a]\sum_{F\,\text{labeled
forest}}\frac{(-1)^{n}}{n!}\int_{0}^{1}\prod_{(i,j)\in
F}dt_{ij}\,\,\left(\prod_{(i,j)\in F}\frac{\partial}{\partial x_{ij}}\right)$
$\displaystyle\times$ $\displaystyle\bigg{\\{}\int
d\mu_{C(x)}(A)\prod_{i=1}^{n}\bigg{[}N{\cal
S}(A_{i})\bigg{]}\bigg{\\}}\bigg{|}_{x_{ij}=v^{F}_{ij}}\,,$

where $n$ is the number of vertices of the forest $F$, $i$ and $j$ label the
forest’s vertices, and there is a weakening parameter $t_{ij}$ per each edge
$(i,j)$ of the forest, and

$v^{F}_{ij}=\left\\{\begin{array}[]{ccl}\inf_{(k,l)\in{P}_{i\leftrightarrow
j}^{{F}}}t_{kl}&\text{if}&{P}_{i\leftrightarrow j}^{{F}}\,\text{exists}\\\
0&\text{if}&{P}_{i\leftrightarrow j}^{{F}}\,\text{does not
exist}\end{array}\right.\,,$ (21)

${P}_{i\leftrightarrow j}^{{F}}$ is the unique path in the forest $F$ joining
$i$ and $j$ (the infimum is taken to be $1$ if $i=j$).

Applying the following lemma, [12], we can take the logarithm.

######  Lemma 2.

Let ${\cal W}(T)$ be the weight of a tree $T$, not depending on the labels of
the tree vertices and the weight of a forest ${\cal W}(F)$ is defined to be
the product of the weights of its trees. Then, in the formal series sense, we
have

$\log\sum_{F\text{ labeled forests}}\frac{{\cal W}(F)}{|V(F)|!}=\sum_{T\text{
labeled trees}}\frac{{\cal W}(T)}{|V(T)|!}\,,$ (22)

where $|V(F)|$ and $|V(T)|$ are the number of vertices in $F$ and $T$
correspondingly.

Since the differentiation with respect to $x_{ij}$ and the Gaussian
integration factor over the trees in the forest $F$, we obtain

$\displaystyle\log{\cal Z}$ $\displaystyle=$ $\displaystyle\log
K[\lambda,N,a]+\sum_{T\,\text{labeled
trees}}\frac{(-1)^{n}}{n!}\int_{0}^{1}\prod_{(i,j)\in
T}dt_{ij}\,\left(\prod_{(i,j)\in T}\frac{\partial}{\partial x_{ij}}\right)$
(23) $\displaystyle\times$ $\displaystyle\bigg{\\{}\int
d\mu_{C(x)}(A)\prod_{i=1}^{n}\bigg{[}N{\cal
S}(A_{i})\bigg{]}\bigg{\\}}\bigg{|}_{v^{T}_{ij}}\,,$ (24) $\displaystyle
v^{T}_{ij}$ $\displaystyle=$ $\displaystyle\inf_{(k,l)\in{P}_{i\leftrightarrow
j}^{T}}t_{kl}\,.$ (25)

where ${P}_{i\leftrightarrow j}^{T}$ stands for the unique path joining
vertices $i$ and $j$ in the tree $T$. Expressing the Gaussian integral as a
differential operator,

$\displaystyle\int
d\mu_{C(x)}(A)F(A)=\left[e^{\frac{1}{2}\sum_{i,j}x_{ij}\operatorname{Tr}\left[\frac{\partial}{\partial
A_{i}}\frac{\partial}{\partial A_{j}}\right]}F(A)\right]_{A_{i}=0}\,,$ (26)

we see that

$\frac{\partial}{\partial x_{ij}}\bigg{(}\int
d\mu_{C(x)}(A)F(A)\bigg{)}=\frac{1}{2}\int
d\mu_{C(x)}(A)\,\operatorname{Tr}\left[\frac{\partial}{\partial
A_{i}}\frac{\partial}{\partial A_{j}}\right]F(A)\,.$ (27)

The latter differential operator acts on $i$ and $j$ vertices and connects
them by an edge.

The first derivative of the loop vertex (non-polynomial action ${\cal
S}(A_{i})$) is given by

$\displaystyle\frac{\partial}{\partial
A_{i|cd}}\bigg{[}\operatorname{Tr}\log\left(1-\mathrm{i}\sqrt{\frac{\lambda}{a^{2}N}}A\right)+\frac{\mathrm{i}}{\sqrt{N}}\operatorname{Tr}\bigg{(}A\frac{(1-a)}{\sqrt{\lambda}}\bigg{)}\bigg{]}=$
$\displaystyle\frac{\sqrt{\lambda}}{a\sqrt{N}}\bigg{(}1-\text{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A_{i}\Big{)}_{cd}^{-1}+\mathrm{i}\frac{(1-a)}{\sqrt{\lambda}\sqrt{N}}\mathbb{1}_{cd}\,.$
(28)

Only the first term of (28) is relevant for applying all further derivatives.
Therefore all other derivatives can be computed using the following recursive
relation,

$\frac{\partial}{\partial
A_{i|ab}}\frac{\sqrt{\lambda}}{a\sqrt{N}}\Big{(}1-\text{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A_{i}\Big{)}_{cd}^{-1}=\mathrm{i}\frac{\lambda}{a^{2}N}\Big{(}1-\text{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A_{i}\Big{)}_{ca}^{-1}\Big{(}1-\text{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A_{i}\Big{)}_{bd}^{-1}\,.$
(29)

Let $V(T)$ be the set of all vertices of the tree, and $E(T)$ be the set of
edges of the tree. We observe that in each tree $T$ with $|V(T)|>2$ there are
two types of vertices: internal vertices and leaves. If the vertex is
differentiated only once, it is a leaf, and it brings the contribution of the
form of (28). We say that leaf vertices have only one corner. Multiple
derivatives acting on the same vertex (corresponding to multiple edges hooked
to it) can act on either of the corners of the vertex, splitting it into two
corners and eliminating the constant term of (28), if it was present, see Fig.
2.

Figure 2: From left to right: a vertex without corners – corresponds to a
trivial tree, a vertex with only one corner – a leaf, vertex with two corners.

Each derivative brings an additional factor of $\frac{1}{\sqrt{N}}$, and
consequently, each edge of any tree comes with the factor $\frac{1}{N}$. In
the following we attribute factors $\frac{1}{N}$ to the edges, and define a
corner operator as

${\cal
C}=\begin{cases}\frac{\sqrt{\lambda}}{a}\Big{(}1-\text{i}\sqrt{\frac{\lambda}{a^{2}N}}A_{i}\Big{)}_{cd}^{-1}+\mathrm{i}\frac{(1-a)}{\sqrt{\lambda}}\mathbb{1}_{cd}\,,\text{
only one corner in the vertex}\\\ ~{}\\\
\frac{\sqrt{\lambda}}{a}\Big{(}1-\text{i}\sqrt{\frac{\lambda}{a^{2}N}}A_{i}\Big{)}_{cd}^{-1}\,.\text{
if there are more corners}\end{cases}\,,$ (30)

Therefore, the logarithm of ${\cal Z}$ can be expressed, as

$\displaystyle\log{\cal Z}=$ $\displaystyle\log
K[\lambda,N,a]+\sum_{T\,\atop\text{LVE tree}}{\cal A}_{T}[\lambda,N]\,,$ (31)
$\displaystyle{\cal A}_{T}[\lambda,N]=$
$\displaystyle\frac{N^{|V(T)|-|E(T)|}}{|V(T)|!}\int_{0}^{1}\prod_{e\in
E(T)}dt_{e}\,$ (32) $\displaystyle\times\int
d\mu_{C_{T}}(A)\,\operatorname{Tr}\Big{[}\mathop{\overrightarrow{\prod}}\limits_{c\in\partial
T\,\text{corner}}{\cal C}_{c}(i_{c})\Big{]}\,,$ (33)

where $i_{c}$ labels the vertex, to which the corner $c$ is attached, the
covariance $C_{T}$ is

$(C_{T})_{ij}=\inf_{(k,l)\in P^{T}_{i\leftrightarrow j}}t_{kl}$ (34)

and the infimum is taken to be $1$ if $i=j$.

##  5 Bounds

The expansion (33) contains three types of contributions, which should be
considered separately: a trivial tree with only one vertex, a tree with two
vertices, and all other trees. In the following we treat these cases step by
step.

###  5.1 Trivial tree

As always in the LVE formalism there is a trivial tree with only one vertex,
see Fig. 2. It requires a special treatment. The contribution of the trivial
tree is given by

$\displaystyle{\cal A}_{T_{1}}=\int
d\mu_{C(x)}(A)\bigg{[}N\operatorname{Tr}\log\left(1-\mathrm{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A\right)+\mathrm{i}\operatorname{Tr}\bigg{(}A\frac{(1-a)\sqrt{N}}{\sqrt{\lambda}}\bigg{)}\bigg{]}\,.$
(35)

The second term gives zero after the integration and the first can be bounded
by integrating by parts as

$\displaystyle{\cal A}_{T_{1}}$ $\displaystyle=$ $\displaystyle N\int
dA\,e^{-\frac{1}{2}\operatorname{Tr}\big{[}A^{2}\big{]}}\operatorname{Tr}_{i=j}\log(\mathbb{1}-\text{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A_{ij})$
(36) $\displaystyle=$ $\displaystyle N\int
dA\,e^{-\frac{1}{2}\operatorname{Tr}\big{[}A^{2}\big{]}}\operatorname{Tr}_{i=j}\int_{0}^{1}dt\,\frac{-\text{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}tA_{ik}}{(\mathbb{1}-\text{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A)_{kj}}$
$\displaystyle=$ $\displaystyle\int
d\mu(A)\,\operatorname{Tr}_{i=j}\int_{0}^{1}dt\,\frac{-\frac{\lambda}{a^{2}}t}{(\mathbb{1}-\text{i}\frac{\sqrt{\lambda}}{a\sqrt{N}}A)^{2}_{ij}}$

Since the integrand is analytic for $\theta\in(-\pi,\pi)$, and
$\theta=\phi-2\psi$, employing (15), we arrive at the following conclusion.

######  Lemma 3.

The amplitude of the trivial tree, ${\cal A}_{T_{1}}$, is analytic in
$\lambda$ and bounded on the Riemann surface of the square root with
$-2\pi<\phi<2\pi$.

###  5.2 Two-vertex tree

To derive the bound for the tree with two vertices, see Fig. 3, we start with
a general bound for the corner operators that are also useful for all other
trees. Using the lemma 1, we obtain

$\|{\cal
C}\|\leq\begin{cases}\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}+\Bigg{|}\frac{(1-a)}{\sqrt{\lambda}}\Bigg{|}\,,\text{
if there is only one corner in the vertex}\\\ ~{}\\\
\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}\,,\text{
if there are more corners}\end{cases}\,.$ (37)

Figure 3: Two-vertex tree.

Remembering the representation (15), for sufficiently large $x$, we can
simplify the one-corner bound in (37), requiring that

$\displaystyle\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}+\Bigg{|}\frac{(1-a)}{\sqrt{\lambda}}\Bigg{|}\leq\Bigg{|}\frac{3(1-a)}{2\sqrt{\lambda}}\Bigg{|}\,.$
(38)

It is not hard to see that the latter inequality is valid when

$\displaystyle x\geq x_{1}\,,\qquad
x_{1}=\frac{\cos\frac{\theta}{2}+\sqrt{\cos^{2}\frac{\theta}{2}+8|\lambda|\cos\frac{\theta}{2}}}{2|\sqrt{\lambda}|\cos\frac{\theta}{2}}\,.$
(39)

Note, as $-\pi<\theta<\pi$, we have that $\cos\frac{\theta}{2}>0$.

In the two-vertex tree, both vertices have only one corner, therefore, using
(38), taking into account that there are $2$ vertices, $1$ edge, and one
factor $N$ coming from the trace, we can bound the amplitude of this tree as

$\displaystyle|{\cal A}_{T_{2}}|\leq\frac{9N^{2}}{8}\int
d\mu_{C(x)}(A)\Bigg{|}\frac{(1-a)}{\sqrt{\lambda}}\Bigg{|}^{2}\,.$ (40)

Due to the analyticity of the integrand for $\theta\in(-\pi,\pi)$ and the
latter bound, we arrive at the following.

######  Lemma 4.

The amplitude of the trivial tree, ${\cal A}_{T_{2}}$, is analytic in
$\lambda$ and bounded on the Riemann surface of the square root with
$\lambda\neq 0$, $-2\pi<\phi<2\pi$.

###  5.3 Other trees

Obviously, the bound for the leaves is larger than for the internal vertices.
If we use it for all vertices, we can bound the amplitude of each tree, as

$\displaystyle\bigg{|}\operatorname{Tr}\Big{[}\mathop{\overrightarrow{\prod}}\limits_{c\in\partial
T\,\text{corner}}{\cal C}_{c}(i_{c})\Big{]}\bigg{|}$ $\displaystyle\leq
N\Bigg{(}\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}+\Bigg{|}\frac{(1-a)}{\sqrt{\lambda}}\Bigg{|}\Bigg{)}^{2|E(T)|}\,.$
(41)

However, the best results given by this bound are achieved when one takes
$a=1$, which corresponds to classic LVE results. It happens since it is
impossible to make simultaneously small both terms:
$\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}\text{
and }\Bigg{|}\frac{(1-a)}{\sqrt{\lambda}}\Bigg{|}$. To overcome this
difficulty, we observe that:

  * •

We can always make the term
$\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}$ as
small as needed, by varying the ratio $\frac{\sqrt{\lambda}}{a}$ (considering
large $x$).

  * •

The factor
$\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}+\Bigg{|}\frac{(1-a)}{\sqrt{\lambda}}\Bigg{|}$,
for which hereafter we will use the bound (38), comes only from the leave
vertices, and in a general tree there are not so many leaves, see Fig. 4.

Figure 4: 1) A tree with a maximal amount of leaves at the given order of the
loop vertex expansion. 2) A tree with a minimal amount of leaves at the given
order of the LVE. 3) Representation of an average LVE tree with not so many
leaves.

Being inspired by the latter observations, we first establish the bound for
the corner operators of a general tree.

######  Lemma 5.

If a tree has $n_{l}$ leaves, $n_{i}$ internal vertices, and
$n_{l}+n_{i}=|V(T)|>2$, its amplitude given by a trace of the product of the
corner operators is bounded by

$\displaystyle\bigg{|}\operatorname{Tr}\Big{[}\mathop{\overrightarrow{\prod}}\limits_{c\in\partial
T_{(n_{i},n_{l})}\,\text{corner}}{\cal C}_{c}(i_{c})\Big{]}\bigg{|}$
$\displaystyle\leq
N\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}^{2n_{i}-2}\Bigg{|}\frac{3(1-a)}{2a\cos\frac{\theta}{2}}\Bigg{|}^{n_{l}}\,.$
(42)

###### Proof.

We prove (42) by induction. For the base, it is enough to check (42) for the
tree with $n_{i}=1$ and $n_{l}=2$. According to (38), two leaves bring the
factor

$\displaystyle\Bigg{|}\frac{3(1-a)}{2\sqrt{\lambda}}\Bigg{|}^{2}\,,$ (43)

and two corner operators of the internal vertex give

$\displaystyle\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}^{2}\,.$
(44)

All together, taking into account the factor $N$ coming from the trace, we
have exactly (42) for $n_{i}=1$ and $n_{l}=2$.

Then, for the induction, we assume that (42) is valid for all trees with
$n_{v}$ vertices, $n_{l}+n_{i}=n_{v}$, and prove that it is then valid for all
trees with $(n_{v}+1)$ vertices. The trees with $(n_{v}+1)$ vertices can be
obtained from the trees with $n_{v}$ vertices by increasing the number of
leaves or number of internal vertices. In the first case, see Fig. 5, the
bound (42) gets an extra factor

$\displaystyle\Bigg{|}\frac{3(1-a)}{2\sqrt{\lambda}}\Bigg{|}\,,$ (45)

from the leaf, and an additional factor

$\displaystyle\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}$
(46)

from the new corner of the internal vertex, where the new leaf is attached.
Altogether, this gives (42) with the number of internal $n_{i}$ and number of
leaves $(n_{l}+1)$.

Figure 5: Possible ways to add a leaf to a tree.

The second case, see Fig. 6, when the number of vertices is augmented by
increasing the number of the internal vertices, can be interpreted as
attaching a new leaf to the already existing leaf – since the only way to
obtain an additional internal vertex is to convert a leaf into it. The
contribution of the original leaf will be preserved – by shifting to the
contribution of the new leaf. In addition, there will be a contribution from
two corner operators of the new internal vertex, the same as (44). All
together this gives (42) with the number of internal $(n_{i}+1)$ and number of
leaves $n_{l}$.

Figure 6: Possible ways to add an internal vertex to a tree.

∎

To prove the convergence of the series (33), we split the sum over the LVE
trees as

$\displaystyle\sum_{T\,\atop\text{LVE
tree}}=\sum_{2<|V(T)|<60}+\sum_{T_{\geq}}+\sum_{T_{<}}\,,$ (47)

where the first sum runs over all LVE trees with $2<|V(T)|<60$ number of
vertices, $T_{\geq}$ denotes the LVE trees which $\alpha|V(T)|$ leaves or more
and $|V(T)|\geq 60$ vertices, and $T_{<}$ stands for the LVE trees with less
than $\alpha|V(T)|$ leaves and $|V(T)|\geq 60$ vertices.

Let us now bound each sum of (47) separately. We start with the following
lemma.

######  Lemma 6.

For any $x$, satisfying

$\displaystyle x\geq x_{2}\,,\qquad
x_{2}=\frac{|\sqrt{\lambda}|+1}{|\sqrt{\lambda}|}\,,$ (48)

it is easy to see that

$\displaystyle\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}\leq\Bigg{|}\frac{(1-a)}{a\cos\frac{\theta}{2}}\Bigg{|}\leq\frac{\sqrt{2}}{\cos\frac{\theta}{2}}\,,$
(49)

and obviously

$\displaystyle\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}\leq\Bigg{|}\frac{3(1-a)}{2a\cos\frac{\theta}{2}}\Bigg{|}\leq\frac{3\sqrt{2}}{2\cos\frac{\theta}{2}}\,.$
(50)

###### Proof.

Recall that $\sqrt{\lambda}=e^{\text{i}\phi/2}|\sqrt{\lambda}|$, $\lambda\neq
0$, and that according to the representation (15),
$a=x|\sqrt{\lambda}|e^{\mathrm{i}\psi}$, $x>0$,
$-\frac{\pi}{2}<\psi<\frac{\pi}{2}$. Then, the first inequality in (50)
transforms to

$\displaystyle\frac{1}{x}\leq\Bigg{|}\frac{1-|\sqrt{\lambda}|xe^{\mathrm{i}\psi}}{|\sqrt{\lambda}|x}\Bigg{|}\,,$
(51)

and it is enough to satisfy it for $\psi=0$. Simplifying the absolute value
operation for $x\geq\frac{1}{|\sqrt{\lambda}|}$, we arrive at (48). The second
inequality in (50) rewrites, as

$\displaystyle\Bigg{|}\frac{(1-|\sqrt{\lambda}|xe^{\mathrm{i}\psi})}{|\sqrt{\lambda}|x\cos\frac{\theta}{2}}\Bigg{|}\leq\frac{\sqrt{2}}{\cos\frac{\theta}{2}}\,.$
(52)

Using the upper bound for the left-hand side of (52), we find that it is
enough to satisfy

$\displaystyle\Bigg{|}\frac{\sqrt{1+|\lambda|x^{2}}}{|\sqrt{\lambda}|x}\Bigg{|}\leq\sqrt{2}\,,$
(53)

what can be achieved for $x\geq\frac{1}{|\sqrt{\lambda}|}$. ∎

######  Lemma 7.

For $-\pi<\theta<\pi$, the sum of absolute values of trees amplitudes from the
finite set of trees in (47) is bounded by a constant,

$\displaystyle\sum_{2<|V(T)|<30}|{\cal A}_{T}|<const\,.$ (54)

###### Proof.

Follows from lemmas 5 and 6. ∎

Now we can estimate how many trees have more than $\alpha|V(T)|$ leaves (we
will be interested in $1>\alpha>1/2$).

######  Lemma 8.

The number of trees with $|V(T)|$ vertices and $\alpha|V(T)|$ or more leaves
with $1>\alpha>1/2$, $|T_{\geq}|$ is bounded by

$\displaystyle|T_{\geq}|$ $\displaystyle\leq$
$\displaystyle(|V(T)|-\left\lceil\alpha|V(T)|\right\rceil+1)\big{(}|V(T)|!\big{)}$
(55) $\displaystyle\times$ $\displaystyle
2^{|V(T)|-3}e^{\left\lceil\alpha|V(T)|\right\rceil}\bigg{(}\frac{1-\alpha}{\alpha}\bigg{)}^{\left\lceil\alpha|V(T)|\right\rceil}\,.$

###### Proof.

The number of labeled trees with $n$ vertices and $k$ leaves is given by

$\displaystyle{\cal N}(n,k)=\frac{n!}{k!}S(n-2,n-k)\,,$ (56)

where $S(n-2,n-k)$ is the Stirling number of the second type. We can bound
${\cal N}(n,k)$ using an upper bound for the Stirling’s numbers of the second
type,

$\displaystyle S(n-2,n-k)\leq\binom{n-3}{n-k-1}(n-k)^{k}\leq
2^{n-3}(n-k)^{k}\,,$ (57)

and a lower bound for the Gamma function

$\displaystyle k!\geq e^{-k}(k+1)^{k}\,.$ (58)

Thus, we have

$\displaystyle{\cal N}(n,k)\leq n!\frac{e^{k}}{k^{k}}2^{n-3}(n-k)^{k}\,.$ (59)

Applying it to $n=|V(T)|$, $k=\left\lceil\alpha|V(T)|\right\rceil$ – an
integer number larger or equal to $\alpha|V(T)|$, we obtain

$\displaystyle{\cal
N}(|V(T)|,\left\lceil\alpha|V(T)|\right\rceil)\leq\big{(}|V(T)|!\big{)}2^{|V(T)|-3}e^{\left\lceil\alpha|V(T)|\right\rceil}\bigg{(}\frac{1-\alpha}{\alpha}\bigg{)}^{\left\lceil\alpha|V(T)|\right\rceil}\,,$
(60)

where we have used an obvious relation

$\displaystyle\frac{|V(T)|-\left\lceil\alpha|V(T)|\right\rceil}{\left\lceil\alpha|V(T)|\right\rceil}\leq\frac{1-\alpha}{\alpha}\,.$
(61)

If we take $\frac{e}{1+e}<\alpha<1$, then $e\frac{1-\alpha}{\alpha}<1$, and it
decreases with increasing of $\alpha$ leading to decreasing of the bound (60).
Therefore for large values of $\alpha$ (for larger amounts of leaves), we will
have fewer trees. Consequently, we can bound the number of trees that have
$\alpha|V(T)|$ leaves or more just by multiplying the (60) by
$(|V(T)|-\left\lceil\alpha|V(T)|\right\rceil+1)$. ∎

Now we are ready to bound the sum of amplitudes of all trees $T_{\geq}$ with
$\alpha|V(T)|$ leaves or more.

######  Lemma 9.

For $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$ and $\alpha=\frac{59}{60}$, the sum
of absolute values of amplitudes of the trees $T_{\geq}$ is smaller than the
sum of an absolutely convergent series,

$\displaystyle\sum_{T_{\geq}}\big{|}{\cal
A}_{T_{\geq}}\big{|}\leq\frac{N^{2}e}{472}\sum_{v=60}^{\infty}(\frac{1}{60}v+1)\,\Bigg{(}\frac{7}{8}\Bigg{)}^{v}\,.$
(62)

###### Proof.

First of all, we chose $x\geq x_{3}$, $x_{3}=\max\\{x_{1},x_{2}\\}$. Then,
according to lemmas 5 and 6, the contributions of the corner operators of each
tree from $T_{\geq}$ can be bounded, as

$\displaystyle\bigg{|}\operatorname{Tr}\Big{[}\mathop{\overrightarrow{\prod}}\limits_{c\in\partial
T_{\geq}\,\text{corner}}{\cal C}_{c}(i_{c})\Big{]}\bigg{|}\leq
N\Bigg{(}\frac{3\sqrt{2}}{2\cos\frac{\theta}{2}}\Bigg{)}^{2n_{i}+n_{l}-2}\leq
N\Bigg{(}\frac{3\sqrt{2}}{2\cos\frac{\theta}{2}}\Bigg{)}^{2|V(T)|}\,.$ (63)

Taking into account the number of trees in $T_{\geq}$ at each order of LVE by
lemma 8, we obtain

$\displaystyle\sum_{T_{\geq}}\big{|}{\cal A}_{T_{\geq}}\big{|}$
$\displaystyle\leq$
$\displaystyle\frac{N^{2}}{8}\sum_{v=60}^{\infty}(v-\left\lceil\alpha
v\right\rceil+1)\,2^{v}e^{\left\lceil\alpha
v\right\rceil}\bigg{|}\frac{1-\alpha}{\alpha}\bigg{|}^{\left\lceil\alpha
v\right\rceil}\Bigg{(}\frac{3\sqrt{2}}{2\cos\frac{\theta}{2}}\Bigg{)}^{2v}$
(64) $\displaystyle\leq$
$\displaystyle\frac{N^{2}}{8}\sum_{v=60}^{\infty}(v-\alpha v+1)\,e^{\alpha
v+1}\bigg{|}\frac{1-\alpha}{\alpha}\bigg{|}^{\alpha
v+1}\Bigg{(}\frac{3}{\cos\frac{\theta}{2}}\Bigg{)}^{2v}\,.$

Now let us take $\alpha$ in such a way that

$\displaystyle\Bigg{(}\frac{3}{\cos\frac{\theta}{2}}\Bigg{)}^{2}\,e^{\alpha}\Bigg{|}\frac{1-\alpha}{\alpha}\Bigg{|}^{\alpha}<1\,.$
(65)

It is easy to see that, for any value of the $\cos\frac{\theta}{2}$, one can
choose such $\alpha$’s, sufficiently close to $\alpha=1$, that the inequality
(65) is satisfied. However, to handle our bounds we need to fix a certain
constant value of $\alpha$. For this, we need to additionally restrict values
of $\theta$. For simplicity, hereafter we consider
$-\frac{\pi}{2}<\theta<\frac{\pi}{2}$. Then, it is enough to find $\alpha$’s
satisfying

$\displaystyle
18\,e^{\alpha}\Bigg{|}\frac{1-\alpha}{\alpha}\Bigg{|}^{\alpha}<1\,.$ (66)

If, for instance, we take $\alpha=\frac{59}{60}$, the left-hand side of (66)
$\approx 0.873<\frac{7}{8}$ (note that $\frac{59}{60}>\frac{e}{1+e}$). This
completes the proof. ∎

######  Lemma 10.

For $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$ and $\alpha=\frac{59}{60}$, the sum
of absolute values of amplitudes of the trees $T_{<}$ is smaller than the sum
of an absolutely convergent series,

$\displaystyle\sum_{T_{<}}\big{|}{\cal A}_{T_{<}}\big{|}$ $\displaystyle\leq$
$\displaystyle
N^{2}\frac{2(x_{4}\cos\frac{\theta}{2})^{5}\sqrt{\lambda}}{3(x_{4}\sqrt{\lambda}-1)}\sum_{v=60}^{\infty}\frac{1}{v^{2}}\Bigg{(}\frac{1}{2}\Bigg{)}^{v}\,,$
(67)

where

$\displaystyle
x_{4}=\max\\{x3,\frac{2^{30}e^{30}}{\cos\frac{\theta}{2}}\Big{(}\frac{3\sqrt{2}}{2\cos\frac{\theta}{2}}\Big{)}^{59/30}\\}+1\,.$
(68)

###### Proof.

Recall that, for $x\geq x_{3}$,

$\displaystyle\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}<\Bigg{|}\frac{(1-a)}{a\cos\frac{\theta}{2}}\Bigg{|}<\frac{\sqrt{2}}{\cos\frac{\theta}{2}}\,.$
(69)

Taking this into account, we conclude that the corner operators of trees with
less than $\alpha|V(T)|$ leaves, with $\alpha=\frac{59}{60}$, are bounded by
the bounds for corner operators of trees from $T_{<}$ with the maximal
possible amount of leaves $(\left\lceil\frac{59}{60}|V(T)|\right\rceil-1)$.
Note that for any
$x>\max\\{x_{3},\frac{2^{30}e^{30}}{\cos\frac{\theta}{2}}\Big{(}\frac{3\sqrt{2}}{2\cos\frac{\theta}{2}}\Big{)}^{59/30}\\}$,
we have

$\displaystyle
e\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}^{\frac{1}{30}}\bigg{(}\frac{3\sqrt{2}}{2\cos\frac{\theta}{2}}\bigg{)}^{\frac{59}{60}}<\frac{1}{2}\,.$
(70)

Therefore, fixing $x_{4}$ as in (68), we obtain

$\displaystyle\bigg{|}\operatorname{Tr}\Big{[}\mathop{\overrightarrow{\prod}}\limits_{c\in\partial
T_{n_{l},n_{i}}\,\text{corner}}{\cal C}_{c}(i_{c})\Big{]}\bigg{|}\leq$
$\displaystyle
N\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}^{2|V(T)|-2\left\lceil\frac{59}{60}|V(T)|\right\rceil-2}\Bigg{|}\frac{3(1-a)}{2a\cos\frac{\theta}{2}}\Bigg{|}^{\left\lceil\frac{59}{60}|V(T)|\right\rceil-1}\leq$
$\displaystyle
N\Bigg{|}\frac{\sqrt{\lambda}}{a}\frac{1}{\cos\frac{\theta}{2}}\Bigg{|}^{\frac{1}{30}|V(T)|-4}\Bigg{|}\frac{3\sqrt{2}}{2\cos\frac{\theta}{2}}\Bigg{|}^{\frac{59}{60}|V(T)|}\leq$
$\displaystyle\frac{2(x_{4}\cos\frac{\theta}{2})^{5}\sqrt{\lambda}}{3(x_{4}\sqrt{\lambda}-1)}\frac{N}{e^{|V(T)|}}\Big{(}\frac{1}{2}\Big{)}^{|V(T)|}\,.$
(71)

The number of all LVE trees with $n$ vertices is given by $n^{n-2}$, this
amount can be employed as an upper bound for the number of trees $T_{<}$.
Using the lower bound for the factorial, we obtain

$\displaystyle|T_{<}|\leq\frac{n^{n-2}}{n!}\leq\frac{n^{n-2}}{\Big{(}\frac{n+1}{e}\Big{)}^{n}}\leq\frac{e^{n}}{n^{2}}\,.$
(72)

Combining (71) with (72), and (33), we obtain (67). ∎

Equations (33), (47), lemmas 3, 4, 7, 9, and 10 together with the theorem 1
from [12] give the proof of the theorem 1 of the current paper.

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††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Ionization-induced Long-lasting Orientation of Symmetric-top Molecules

Long Xu  AMOS and Department of Chemical and Biological Physics, The Weizmann
Institute of Science, Rehovot 7610001, Israel  Ilia Tutunnikov  AMOS and
Department of Chemical and Biological Physics, The Weizmann Institute of
Science, Rehovot 7610001, Israel  Yehiam Prior<EMAIL_ADDRESS>AMOS and Department of Chemical and Biological Physics, The Weizmann Institute
of Science, Rehovot 7610001, Israel  Ilya Sh. Averbukh
<EMAIL_ADDRESS>AMOS and Department of Chemical and Biological
Physics, The Weizmann Institute of Science, Rehovot 7610001, Israel

###### Abstract

We theoretically consider the phenomenon of field-free long-lasting
orientation of symmetric-top molecules ionized by two-color laser pulses. The
anisotropic ionization produces a significant long-lasting orientation of the
surviving neutral molecules. The degree of orientation increases with both the
pulse intensity and, counterintuitively, with the rotational temperature. The
orientation may be enhanced even further by using multiple delayed two-color
pulses. The long-lasting orientation may be probed by even harmonic generation
or by Coulomb-explosion-based methods. The effect may enable the study of
relaxation processes in dense molecular gases, and may be useful for molecular
guiding and trapping by inhomogeneous fields.

_Introduction_.—Field-free oriented molecules are essential in many studies,
such as ultrafast dynamic imaging, molecular tomography, and electron
diffraction, to name just a few. A much more comprehensive list of
applications may be found in a recent review by Koch et al. Koch  _et al._
(2019). Naturally, for practical applications, a sizable degree of orientation
is beneficial. One of the tools for inducing the molecular orientation is a
non-resonant two-color laser pulse consisting of the fundamental wave (FW) and
its second harmonic (SH). Using such fields, two different orientation
mechanisms have been identified and studied theoretically and experimentally
Kanai and Sakai (2001); De  _et al._ (2009); Oda  _et al._ (2010); Spanner
_et al._ (2012); Frumker  _et al._ (2012a); Znakovskaya  _et al._ (2014);
Kraus  _et al._ (2015). The first orientation mechanism, which is dominant at
low to moderate (non-ionizing) intensities, relies on the interaction of the
external fields with the molecular hyperpolarizability, which results in
asymmetric torques that orient the molecules along the polarization direction
of the SH field Kanai and Sakai (2001); De  _et al._ (2009); Oda  _et al._
(2010); Lin  _et al._ (2018); Xu  _et al._ (2021a). At high (ionizing)
intensities, the dominant orientation mechanism Spanner  _et al._ (2012) is
different – probability of ionization depends on the molecular orientation
with respect to the polarization direction of the asymmetric electric field of
the two-color pulse Spanner  _et al._ (2012); Frumker  _et al._ (2012a);
Znakovskaya  _et al._ (2014). As a result, immediately after the pulse, the
angular distribution of the surviving neutral molecules is asymmetric and has
a non-zero orientation on average. Note that for linear molecules, this
orientation disappears shortly after the excitation, but it periodically
reemerges due to the phenomenon of rotational quantum revivals Sh. Averbukh
and Perelman (1989); Robinett (2004).

Here, we theoretically investigate the ionization-induced orientation of
_symmetric-top molecules_ excited by intense two-color femtosecond laser
pulses. We demonstrate that in addition to the transient post-pulse
orientation, and unlike linear molecules, there also exists a significant
long-lasting orientation in these molecules. Long-lasting means that the
orientation exists not only at the revival times but between the revivals too.
In other words, the orientation signal has a non-zero baseline. Within the
idealized model of non-interacting rigid rotors used here, this orientation
lasts indefinitely. In practice, however, it will eventually be suppressed by
additional physical effects, e.g., by intermolecular collisions in gas cell
experiments. Related effects of long-lasting orientation have been recently
investigated in chiral Milner  _et al._ (2019); Tutunnikov  _et al._ (2020,
2021); Xu  _et al._ (2021a) and other non-linear Xu  _et al._ (2020, 2021b,
2021a) molecules excited by non-ionizing THz and laser pulses.

In what follows, we present our numerical analysis, outline our results on
significant long-lasting orientation, and discuss its dependence on intensity
and temperature. We conclude with a discussion of the experimental feasibility
of observing the predicted effect.

_Numerical methods_.—In our analysis, we simulate the rotational dynamics of
symmetric-top molecules within the rigid rotor approximation both classically
and quantum-mechanically. The symmetric-top molecules are excited by a two-
color laser pulse, consisting of the co-linearly polarized and phase-locked FW
field and its SH. The electric field is described by

$\mathbf{E}(t)=E_{0}f(t)[\cos(\omega t)+\varepsilon\cos(2\omega
t+\phi_{0})]\mathbf{e}_{Z},$ (1)

where $E_{0}$ and $\omega$ are the peak amplitude and the carrier frequency of
the FW, respectively. The laser pulse envelope is defined by $f(t)=\exp[-2\ln
2\,(t^{2}/\sigma^{2})]$, where $\sigma$ is the full width at half maximum
(FWHM) of the pulse intensity profile, and $\mathbf{e}_{Z}$ is a unit vector
along the laboratory $Z$ axis. Here, we set $\varepsilon=1$ and $\phi_{0}=0$.

The Hamiltonian describing molecular rotation driven by a two-color laser
pulse is given by $H(t)=H_{r}+H_{\mathrm{int}}$, where $H_{r}$ is the
rotational kinetic energy Hamiltonian and the interaction Hamiltonian is given
by

$H_{\mathrm{int}}=V_{\mathrm{pol}}+V_{\mathrm{hyp}}+V_{\mathrm{ion}}.$ (2)

The field-polarizability and field-hyperpolarizability interaction terms are
defined as Buckingham (2007)

$V_{\mathrm{pol}}=-\frac{1}{2}\sum_{i,j}\alpha_{ij}E_{i}E_{j},V_{\mathrm{hyp}}=-\frac{1}{6}\sum_{i,j,k}\beta_{ijk}E_{i}E_{j}E_{k},\\!\\!$
(3)

where $E_{i}$, $\alpha_{ij}$, and $\beta_{ijk}$ are the components of the
electric field vector, polarizability tensor, and hyperpolarizability tensor,
respectively.

Figure 1: (a) $\mathrm{CH_{3}F}$ molecule. Atoms are color-coded: black,
carbon; gray, hydrogen; green, fluorine. $\theta$ is the angle between the
molecular $a$ axis and the laboratory $Z$ axis, and $\chi$ represents the
rotation angle about the $a$ axis. (b) Structure factor $G(\theta,\chi)$ [see
Eqs. (7) and (8)] determining the angle dependence of the ionization rate.

The ionization depletion term $V_{\mathrm{ion}}$ is sensitive to molecular
orientation. For linear molecules, e.g., $\mathrm{HCl}$ Akagi  _et al._ (2009)
and $\mathrm{CO}$ Li  _et al._ (2011); Wu  _et al._ (2012); Spanner  _et al._
(2012), the ionization rate depends on the angle between the molecular axis
and the polarization direction. For symmetric-top molecules, e.g.,
$\mathrm{CH_{3}F}$ and $\mathrm{CH_{3}Br}$, belonging to the $C_{3v}$ point
group, the ionization rate depends on two angles, $\theta$ and $\chi$ (see
Fig. 1) Kraus  _et al._ (2015).

In this work we consider $\mathrm{CH_{3}F}$ as our example symmetric-top
molecule. The ionization process is modeled using a complex absorbing
potential

$V_{\mathrm{ion}}=-\frac{i}{2}\Gamma(\theta,\chi,t),$ (4)

where the ionization rate $\Gamma(\theta,\chi,t)$ is defined as Kraus  _et
al._ (2015) (within the weak-field asymptotic theory)

$\Gamma(\theta,\chi,t)=\begin{cases}W(t)|G(\theta,\chi)|^{2},&E(t)>0,\\\
W(t)|G(\pi-\theta,\pi+\chi)|^{2},&E(t)<0.\end{cases}$ (5)

Here, $E(t)=\mathbf{e}_{Z}\cdot\mathbf{E}(t)$, $W(t)$ is the field factor, and
$G(\theta,\chi)$ is the structure factor. The field factor is given by Kraus
_et al._ (2015)

$W(t)=\frac{\kappa}{2}\left(\frac{4\kappa^{2}}{|E(t)|}\right)^{2/\kappa-1}\exp\left[-\frac{2\kappa^{3}}{3|E(t)|}\right],$
(6)

where $\kappa=\sqrt{2I_{p}}$, and $I_{p}$ is the field-free energy of the
highest occupied molecular orbital (HOMO). We use the following model for the
structure factor

$G(\theta,\chi)=\left[\sin(\theta)+\frac{3}{2}\sin(2\theta)\right]G_{1}(\chi),$
(7)

where

$G_{1}(\chi)=\begin{cases}\,\,\,\,\sqrt{1+\sin(3\chi)},&0\leq\chi<7\pi/6,\\\
-\sqrt{1+\sin(3\chi)},&7\pi/6\leq\chi<11\pi/6,\\\
\,\,\,\,\sqrt{1+\sin(3\chi)},&11\pi/6\leq\chi<2\pi.\end{cases}\\!\\!\\!$ (8)

$G(\theta,\chi)$ defined in Eqs. (7) and (8) closely approximates the
structure factor of field-dressed HOMO of $\mathrm{CH_{3}F}$ with the largest
dipole moment Kraus  _et al._ (2015) (the orbital from which the strong field
ionization preferentially occurs). The definition in Eq. (5) accounts for the
oscillations of the laser electric field along the $Z$ axis. We quantify the
degree of orientation of surviving neutral molecules using the thermally
averaged quantum expectation value of $\cos(\theta)$, $\braket{\cos(\theta)}$.
Further details on the quantum simulations can be found in Xu  _et al._
(2021b); Tutunnikov  _et al._ (also see Sec. I of the Supplemental Material
Sup ).

In classical simulations, we use the Monte Carlo approach to simulate the
behavior of a classical ensemble consisting of $N=10^{7}$ sample molecules. A
detailed description can be found in Tutunnikov  _et al._ (2019); Xu  _et al._
(2021b) (also see Sec. I of the Supplemental Material Sup ). Following the
ionization depletion, the classical degree of orientation is given by

$\displaystyle\braket{\cos(\theta)}(t)=\sum\limits_{n=1}^{N}\rho(\theta_{n},\chi_{n},t)\cos(\theta_{n}),$
(9)

where the relative weight (non-ionized fraction) of the $n$-th molecule is

$\displaystyle\rho(\theta_{n},\chi_{n},t)=N_{\mathrm{neu}}^{-1}\exp\left[-\int_{0}^{t}\Gamma(\theta_{n},\chi_{n},t^{\prime})\,dt^{\prime}\right],$
(10)

and the total number of surviving neutral molecules is

$\displaystyle
N_{\mathrm{neu}}=\sum\limits_{n=1}^{N}\exp\left[-\int_{0}^{t}\Gamma(\theta_{n},\chi_{n},t^{\prime})\,dt^{\prime}\right],$
(11)

Here, $\theta_{n}$ and $\chi_{n}$ are the time-dependent angles of the $n$-th
molecule. The population of surviving neutral molecules is defined as
$N_{\mathrm{neu}}/N$.

_Results_.—Molecular parameters of $\mathrm{CH_{3}F}$ are provided in Sec. II
of the Supplemental Material Sup . Figure 2 shows the calculated, classically
and quantum-mechanically, time-dependent orientation factor of neutral
molecules following a single two-color pulse applied at $t=0$. The parameters
used in this calculation are: the rotational temperature is
$T=300\,\mathrm{K}$, the laser wavelengths are 800 nm (FW) and 400 nm (SH),
the peak intensity is $7\times 10^{13}\,\mathrm{W/cm^{2}}$, and
$\sigma=20\,\mathrm{fs}$, see Eq. (1).

The three panels of Fig. 2 show the orientation factor obtained for various
combinations of interaction terms [see Eq. (2)]. All cases include the field-
polarizability interaction, $V_{\mathrm{pol}}\propto\cos^{2}(\theta)$ which is
a symmetric function of $\theta$ (about $\theta=\pi/2$). A torque-kick by such
a potential results in molecular alignment only (for review, see Stapelfeldt
and Seideman (2003)). The two other terms, $V_{\mathrm{hyp}}$ and
$V_{\mathrm{ion}}$ are asymmetric functions of $\theta$ and thus induce
molecular orientation. All three panels depict the immediate response of
$\braket{\cos(\theta)}$ to the laser excitation near $t=0$. This transient
orientation effect is similar to the one studied in linear molecules excited
by two-color laser pulses De  _et al._ (2009); Oda  _et al._ (2010); Wu and
Zeng (2010); Mun and Sakai (2018); Mun  _et al._ (2019); Mun and Kim (2020);
Mellado-Alcedo  _et al._ (2020); Wang and Henriksen (2020). At room
temperature and field parameters used here, the transient molecular
orientation induced by the field-hyperpolarizability interaction alone [Fig.
2(a)] is negligible compared to the orientation resulting from the ionization
depletion [Fig. 2(b)]. Accordingly, the curves in Fig. 2(b) and Fig. 2(c) [in
which all the interaction terms are included, see Eq. (2)] are almost
indistinguishable. These results are consistent with previous results reported
for linear molecules Spanner  _et al._ (2012); Znakovskaya  _et al._ (2014),
where it was shown that at high (ionizing) intensities, the orientation
mechanism of ionization depletion dominates.

Figure 2: Time-dependent orientation factor at $T=300\,\mathrm{K}$ for
different orientation mechanisms calculated classically and quantum-
mechanically. Here the field intensity is $I_{0}=7\times
10^{13}\,\mathrm{W/cm^{2}}$ and the pulse duration is
$\sigma=20\,\mathrm{fs}$. About 40% of neutral molecules survive after the
ionization.

In this work, we focus on the long-term orientation existing in symmetric-top
molecules. Figures 2(b) and 2(c) show that under the stated conditions and for
these molecules, following the transient orientation the degree of orientation
doesn’t return to zero but persists at a constant value till the first revival
and beyond. _This effect of ionization-induced long-lasting orientation
doesn’t exist in linear molecules, and it is the main result of this Letter_.
Note that the quantum and classical results agree well on the short time
scale, during the initial transient response, and predict the same degree of
long-lasting orientation on the long time scale, suggesting that the long-
lasting orientation stems from a classical origin. Long-lasting orientation
has been previously observed in chiral Milner  _et al._ (2019); Tutunnikov
_et al._ (2020, 2021); Xu  _et al._ (2021a) and studied in other non-linear
molecules Xu  _et al._ (2020, 2021b, 2021a) excited by non-ionizing THz and
laser pulses. In these cases, the orientation mechanisms, including the
interactions with the polarizability and hyperpolarizability, were considered.
Here, the ionization depletion mechanism of ionization depletion gives rise to
unprecedented degrees of long-lasting orientation at room temperature. The
degree of long-lasting orientation ($\approx-0.033$), as shown in Figs. 2(b)
and 2(c), is an order of magnitude higher than values reported in previous
studies.

_Mechanism_.—Next, we discuss the mechanism behind this large ionization-
induced long-lasting orientation. Under field-free condition, the symmetry
axis (dipole) of symmetric-top molecules precesses around the (conserved,
space-fixed) vector of angular momentum, whereas linear molecules rotate in a
plane perpendicular to the angular momentum Landau and Lifshitz (1976). It is
this precession that is the source of the long-lasting orientation in
symmetric-top molecules. The degree of long-lasting orientation (with respect
to the $Z$ axis) of a single symmetric-top molecule is given by the
combination of three quantities $L_{a}L_{Z}/L^{2}$ Xu  _et al._ (2020, 2021b)
(see also Sec. III of the Supplemental Material Sup ), where $L_{a}$ and
$L_{Z}$ are the projections of the angular momentum along the molecular
symmetry axis [the molecular $a$ axis in Fig. 1(a)] and the laboratory $Z$
axis, respectively, and $L$ is the magnitude of the angular momentum. Note
that in the presence of $Z$-polarized pulses, $L_{a}$ and $L_{Z}$ are
conserved quantities.

Initially, before the laser pulse, the molecules are isotropically distributed
in space, namely there is an equal number of molecules having positive (along
the $Z$ axis) and negative (against the $Z$ axis) long-lasting orientations.
Therefore, the ensemble-averaged long-lasting orientation
$\braket{L_{a}L_{Z}/L^{2}}$ vanishes Xu  _et al._ (2021b). The co-linearly
polarized two-color laser pulse preferentially ionizes molecules oriented more
or less along its polarization axis (with $\theta<\pi/2$) [see Fig. 1(b)].
Thus, this selective ionization depletion breaks the symmetry of the molecular
ensemble, generating a non-zero long-lasting orientation of the surviving (not
ionized) neutral molecules, as shown in Figs. 2(b) and 2(c).

For symmetric-top molecules, within the model used here, the long-lasting
orientation is permanent. In experiments, however, this orientation will be
gradually destroyed by other physical effects such as intermolecular
collisions, centrifugal distortion, and radiation emission caused by rapidly
rotating molecular permanent dipole moments. Furthermore, while the
centrifugal distortion is known to lead to the decay of the revivals’ peaks
due to the dephasing of the rotational states Damari  _et al._ (2016);
Babilotte  _et al._ (2016), the average dipole remains almost unchanged (see
Xu  _et al._ (2020)). The radiative emission gradually decreases the
rotational energy Damari  _et al._ (2016); Babilotte  _et al._ (2016), but the
relative energy loss during a single revival is negligible for a rarefied
molecular gas.

_Intensity dependence_.—One of the ways to enhance the degree of the
ionization-induced long-lasting orientation is by increasing the laser pulse
energy – peak intensity and/or pulse duration, or investing the higher energy
in several pulses. Figure 3 shows the classically calculated long-lasting
orientation factor, $\braket{\cos(\theta)}_{p}$, and the population of
surviving neutral molecules as functions of the laser intensity for single or
multiple delayed pulses. The value of $\braket{\cos(\theta)}_{p}$ is taken
after the pulse(s), when the orientation factor reaches a constant value. As
expected, with the increasing input energy, the long-lasting orientation
factor (in absolute value) initially grows [for $I_{0}<7\times
10^{13}\,\mathrm{W/cm^{2}}$, see Fig. 3(a)], while the population of surviving
neutral molecules decreases monotonically [see Fig. 3(b)].

After reaching its maximum value, the long-lasting orientation decreases with
increasing laser intensity. The reason for this decrease is the structure
factor $G(\theta,\chi)$ shown in Fig. 1(b). Due to the anisotropy of the
ionization process, molecules at $\theta\approx 0,\,0.6\pi,\,\pi$ have a
relatively low ionization rate. Therefore, when the ionization saturates at
high laser intensities, only these molecules survive, but these molecules have
a relatively low contribution to the long-lasting orientation. The combination
of increased ionization yield, but counterproductive selectivity of low-
contributing molecules limits the usefulness of increasing the laser intensity
beyond a certain point.

Figure 3: Classically calculated (a) long-lasting orientation and (b)
population of neutral molecules after the pulse(s) as functions of the laser
intensity. The time delay between pulses is 0.5 ps. Here $T=300\,\mathrm{K}$
and $\sigma=20\,\mathrm{fs}$. Both orientation mechanisms are taken into
account.

A way to avoid the limits imposed on high intensities is to use multiple
pulses instead of a single strong pulse. Consider the application of several
delayed two-color pulses. Figure 3(a) shows that the maximum long-lasting
orientation is about $-0.033$ for one pulse, $-0.069$ for two pulses, and
$-0.107$ for three pulses. Here, the time delay between each pulse is set to
0.5 ps, a relatively long time delay which allows the molecules to rotate
between the pulses. This way, after each pulse additional neutral molecules
rotate to an orientation more favorable for ionization by the next pulse. This
approach overcomes the ionization saturation limit that exists in the case of
a single pulse, and is similar to the use of multiple pulses for achieving
enhanced alignment while avoiding ionization Leibscher  _et al._ (2003, 2004).
Naturally, adding more pulses results in a progressively lower population of
surviving neutral molecules. Nevertheless, for a fixed population, a higher
long-lasting orientation is achieved by applying multiple delayed pulses. For
example, for $N_{\mathrm{neu}}/N\approx 40\%$, the long-lasting orientation is
about $-0.033$ for one pulse, $-0.049$ for two pulses, and $-0.054$ for three
pulses. _This observation is an additional main result of this Letter._
Optimization of the time delay may allow further enhancement of the long-
lasting orientation.

_Temperature dependence_.—Next, we consider the temperature dependence of the
long-lasting orientation as depicted in Fig. 4. At $T=0$, the long-lasting
orientation is zero for all three curves shown in Fig. 4(a), as a result of
$L_{a}=L_{Z}=0$ (these are conserved quantities). There is an optimal
temperature ($T<1\,\mathrm{K}$ in this example) at which the long-lasting
orientation induced by the field-hyperpolarizability interaction (together
with the field-polarizability interaction) reaches the maximum ($\approx
0.006$). Above the optimal temperature, the long-lasting orientation decays
with increasing temperature Xu  _et al._ (2021b). In sharp contrast, the
ionization-induced long-lasting orientation factor (in absolute value)
increases monotonically with the temperature, which is another principal
finding of this work.

Figure 4: Classically calculated (a) long-lasting orientation factor and (b)
population of neutral molecules as functions of temperature for different
orientation mechanisms. The laser parameters are the same as in Fig. 2:
$I_{0}=7\times 10^{13}\,\mathrm{W/cm^{2}}$, $\sigma=20\,\mathrm{fs}$. The
population for the orientation mechanism of hyperpolarizability remains 1.

As mentioned above, the long-lasting orientation is given by
$L_{a}L_{Z}/L_{f}^{2}$, where $L_{f}=|\mathbf{L}_{f}|$ is the magnitude of the
angular momentum after the pulse. $L_{Z}$ and $L_{a}$ are conserved in the
case of excitation by linearly polarized laser pulses and thus are functions
of temperature only. The short two-color pulse has a two-fold effect: (i) it
(almost) instantaneously ionizes the molecules at particular angles (the
effect described by $V_{\mathrm{ion}}$), and (ii) it changes the molecular
angular momentum, $\mathbf{L}_{f}=\mathbf{L}_{i}+\delta\mathbf{L}$, where
$\mathbf{L}_{i}$ is the initial angular momentum, and $\delta\mathbf{L}$ is
the change caused by $V_{\mathrm{pol}}$. Around the room temperature (in the
examples considered here), the effect of $V_{\mathrm{pol}}$ becomes
negligible, i.e., $L_{a}L_{Z}/L_{f}^{2}\approx L_{a}L_{Z}/L_{i}^{2}$, and the
degree of long-lasting orientation reaches a constant value determined by
$V_{\mathrm{ion}}$. At lower temperatures, $V_{\mathrm{pol}}$ has a
detrimental effect by increasing the total angular momentum, such that
$L_{a}L_{Z}/L_{f}^{2}<L_{a}L_{Z}/L_{i}^{2}$ effectively lowers the long-
lasting orientation.

At temperatures above room temperature, the ionization can no longer be
considered instantaneous because the molecules rotate fast enough to change
their orientation during the pulse. Accordingly, more molecules get ionized,
as seen from the populations in Fig. 4(b). The higher the ionization yield,
the higher asymmetry of the molecular ensemble, manifesting in the higher
long-lasting orientation.

_Conclusions_.—We have theoretically demonstrated a sizable ionization-induced
long-lasting orientation of symmetric-top molecules excited by two-color laser
pulses. The mechanism leading to this observation is the selective ionization
of the polar molecules at particular angles with respect to the laser’s
electric field, and the ability of symmetric-top molecules to precess around
the fixed angular momentum vector. We show that using a proper sequence of
delayed two-color pulses allows for enhancing the long-lasting orientation of
neutral molecules without drastic depletion of their population. Due to the
required precession, the enhanced orientation favors high temperature (room
temperature and above in our examples). The long-lasting orientation may be
measured with the help of Coulomb explosion Znakovskaya  _et al._ (2014), or
by observing second (or higher order) harmonic generation in the gas phase,
which is sensitive to the lack of inversion symmetry Frumker  _et al._ (2012b,
c); Kraus  _et al._ (2012). Long-lasting orientation can pave the way to the
study of relaxation processes in dense molecular gases that are not otherwise
accessible to direct probing (see analogous applications of persistent
alignment Hartmann and Boulet (2012); Vieillard  _et al._ (2013)). In
addition, molecular focusing, guiding, and trapping by inhomogeneous fields
rely on molecular orientation Gershnabel and Sh. Averbukh (2011a, b); Küpper
_et al._ (2012), and their observation may be facilitated by this long-lasting
orientation effect.

###### Acknowledgements.

L.X. is a recipient of the Sir Charles Clore Postdoctoral Fellowship. This
research was made possible in part by the historic generosity of the Harold
Perlman Family.

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