#  Fluctuating field series: towards calculations of correlated systems with
high accuracy

Ya. S. Lyakhova  Russian Quantum Center, Moscow 121205, Russia  National
Research Nuclear University MEPhI, Moscow 115409, Russia  S. D. Semenov
Russian Quantum Center, Moscow 121205, Russia  Moscow Institute of Physics and
Technology, Dolgoprudny, 141701, Russia  A. I. Lichtenstein  Institut für
Theoretische Physik, Universität Hamburg, Notkestraße 9, 22607 Hamburg,
Germany  A. N. Rubtsov<EMAIL_ADDRESS>Russian Quantum Center, Moscow 121205,
Russia  Lomonosov Moscow State University, Moscow 119991, Russia

###### Abstract

We introduce regular series expansion for weakly- and moderately-correlated
fermionic systems, based on Fluctuating Local Field approach. The method
relies on the explicit account of leading fluctuating mode(s) and is therefore
suitable for medium-sized lattices. It can be also used as a solver for the
cluster approximations for infinite-size lattices. We introduce classical
fluctuating field coupled to fermionic collective mode(s). This way,
fluctuations in selected modes are treated in a non-perturbative way. Other
degrees of freedom are accounted for the diagram expansion performed at each
value of the fluctuating field. The method is benchmarked for the $U/t=1$ and
$U/t=2$ Hubbard lattices at half-filling. Results for susceptibility in the
antiferromagnetic channel along with the single particle density of states are
compared with the numerically exact quantum Monte Carlo data. Calculations up
to the third order of the series expansion are performed and show a uniform
convergence to the reference result for the susceptibility. This convergence
is observed well below the DMFT Neél temperature and makes our practically
simple method applicable in a much wider temperature range than DMFT-based
diagrammatic schemes.

††preprint: APS/123-QED

##  I Introduction

An accurate description of fluctuation phenomena in correlated materials
remains one of challenging problems in condensed matter theory, in particular
when it comes to correlated systems [1, 2]. Despite the clear importance of
numerical modelling of the phase transition phenomena, modern theoretical
tools do not always provide their predictive description. A two-dimensional
Hubbard model is a particularly known example of a system being hard for a
quantitatively accurate modelling. This model is of a great importance because
it is believed to describe the electronic properties of a number of intriguing
systems: High-Tc cuprates [3], twisted bilayer graphene [4] or ultracold atoms
[5].

The challenges encountered when modeling the Hubbard systems arise from at
least three sources. First, when Hubbard on-site repulsion $U$ is compatible
with bandwidth, the Hubbard model exhibits strong correlation effects, mostly
local on sites and short-ranged. Second, collective fluctuations of different
nature may be present in the system. The role of fluctuation phenomena is
greatly enhanced in a two-dimensional or layered systems, compared to the bulk
materials. Third, a large number of possible ordered phases with very similar
free energy have to be considered for the doped Hubbard model. One can mention
antiferromagnetically-ordered, striped, superconducting, spin wave, charge
density wave and other phases. Theoretical description of all these mutually
influencing phenomena requires highly accurate numerical methods capable of
taking into account different energy and length scales. Various approaches,
mostly based on the pure mean-field or mean-field-like schemes, aim to
establish reliable information about the phase diagram of the Hubbard system.
In this cases parental method is usually extended. One of frequently used
approaches here is the perturbative extension. GW method built on the basis of
the Density Functional Theory [6, 7], Dual Fermions [8, 9], TRILEX [10] and
D-TRILEX [11] represent different implementations of this approach. Other ways
to increase the accuracy of scheme is to use hybrid methods or various cluster
extensions [12]. A birds eye look to the existing schemes is given in the
review [13]. A particular family of diagram methods constructed around the
Dynamical Mean Field Theory (DMFT) is surveyed in [14]. State of the art
results for the Hubbard model obtained by different computational schemes are
outlined in [15].

The paper [16] gives an important survey of the results obtained by different
extensions of the DMFT for a particular case of the half-filled Hubbard model
with $U=2t$, where $t$ is the hoping amplitude between the neighboring sites.
This case is much simpler than a generic Hubbard model with arbitrary
parameters. First, half-filling ensures that the model shows fluctuations and
ordering solely in the antiferromagnetic (AF) channel, so one should not deal
with an interplay between different modes and candidate states. Second, the
system at $U=2t$ is moderately correlated. It could be expected that its
description is simpler than the strongly correlated case of $U\gtrsim 4t$. The
results can be summarized as follows. The parental approximation, DMFT,
predicts a Neél transition at the inverse temperature $\beta\approx 12$. This
is an artifact, because Mermin-Wagner theorem forbids formation of the
antiferromagnetic ordering in a planar system at finite temperature. The
extended schemes indeed get rid of this transition. Even more, a very
reasonable accuracy is seen when calculating both single-particle (the self-
energy) and two-particle (the AF susceptibility and correlation length)
properties, but only for $\beta\lesssim 15$. We conclude that the domain of
applicability of the known DMFT extensions just slightly exceeds the thermal
range defined by the Neél temperature of the parental approximation.

Recently, there was presented a novel approach to the description of
correlated systems, called Fluctuating Local Field (FLF) [17]. This method
starts with the exact transformation of the partition function, introducing an
ensemble of systems at different values of artificial fluctuating field. In
practice FLF is based on a parental scheme, e.g. stationary mean field theory
[18, 19, 20, 21] or DMFT [22]. The scheme has proved its effectiveness for
weakly correlated systems, getting rid of the above-mentioned artifacts and
providing good accuracy at low calculational cost.

Here, we construct a perturbation theory based on the FLF approach. To
demonstrate our method, we work with a simple example of the 2D Hubbard system
at half-filling operating under assumption that there is a single AF
fluctuating mode, that is, the system under consideration is assumed not large
in size. Moreover, we limit ourselves with weakly- and moderately-correlated
systems, which allows us to work with the formal expansion w.r.t. the order of
Hubbard interaction $U$. However, it was noticed in [19] that the FLF
transformation leads to appearance of the effective long-range interaction in
this system which competes with the on-site repulsion and thus effectively
form a new small parameter. The main result of the present paper is that the
FLF expansion converges regularly within a wide temperature range, much larger
than the domain of applicability of usual weak-coupling expansion. This opens
a pathway to future extension of the method to strongly correlated systems by
constructing the FLF series on top of the DMFT result.

We start with the formulation of FLF formalism in Sec. II, and derivation of
the perturbation series for the FLF free energy in Sec. III. It allows us to
calculate the single-particle and two-particles observables of the Hubbard
system in Sec. IV and V. Limiting ourselves with the single-mode assumption,
we extend our approach to large systems by constructing the FLF cluster
extension in the spirit of Cluster DMFT (CDMFT) in Sec. VI. Having conducted
all the mentioned steps for weakly correlated regime $U=1$, we move to more
detailed consideration of possible divergences and examine the $U=2$ case,
which is more illustrative here. We provide a regularization procedure,
smoothing the disconvergence of the perturbation series in moderately
correlated regime in Sec. VII. Comparison of the results obtained within the
FLF perturbation approach with the DMFT and numerically exact Quantum Monte-
Carlo (QMC) [23] data demonstrates high accuracy for Green’s function, local
spectral density function and the susceptibility. We discuss the results of
our work and provide the outlook in Sec. VIII.

##  II System and formalism

We consider the two-dimensional Hubbard system with $N$ sites at half-filling.
It can be described by the following Hamiltonian:

$\displaystyle H=$ $\displaystyle-t\sum_{\langle
jj^{\prime}\rangle,\sigma}\left(c^{\dagger}_{j,\sigma}c_{j^{\prime},\sigma}+c^{\dagger}_{j^{\prime},\sigma}c_{j,\sigma}\right)+$
(1)
$\displaystyle+U\sum_{j}\left(n_{j\uparrow}-\frac{1}{2}\right)\left(n_{j\downarrow}-\frac{1}{2}\right)-\sum_{j}h^{z}\left(n_{j\uparrow}-n_{j\downarrow}\right).$

Here, the first term describes the nearest-neighbour pair $\langle
jj^{\prime}\rangle$ hoppings with the amplitude $t$, the second term describes
the on-site Coulomb repulsion, and the third term couples the magnetic moment
of the lattice to the local external magnetic field $\bm{h}_{j}$ along the
$z$-axis. Moving forward, we will discuss only the case of
antiferromagnetically (AF) ordered field
$\bm{h}_{j}=\bm{h}e^{i\bm{K}\bm{r}_{j}}$ with $\bm{K}=\left(\pi,\pi\right)$
and the real space vector $\bm{r}_{j}$.

The partition function of the Hubbard system can be expressed symbolically in
the path integral formalism as:

$Z=\int
e^{-\left[\mathcal{S}^{0}_{h}+\mathcal{S}_{U}\right]}\mathcal{D}[c^{\dagger},c],$
(2)

where $\mathcal{S}_{h}^{0}$ is the action of non-interacting system in
external field $\bm{h}$, and $\mathcal{S}_{U}$ is the Hubbard interaction term
action. We conclude the full action:

$\mathcal{S}_{h}[c^{\dagger},c]=\underbrace{-c^{\dagger}G_{0}^{-1}c-\beta
Nh^{l}s^{l}}_{\text{{\hbox{\mathcal{S}^0_h}}}}+\underbrace{U\theta}_{\text{{\hbox{\mathcal{S}_U}}}},$
(3)

where $G_{0}$ is a bare Green’s function, $\beta$ is reverse temperature, and

$\displaystyle\mathcal{S}_{U}=U\theta$
$\displaystyle=U\int_{0}^{\beta}\sum_{j}\left(c_{j\uparrow}^{\dagger}(\tau)c_{j\uparrow}(\tau)-\frac{1}{2}\right)\left(c_{j\downarrow}^{\dagger}(\tau)c_{j\downarrow}(\tau)-\frac{1}{2}\right)d\tau$
(4)

is an abbreviation of the on-site interaction in the action formalism.
$h^{l}s^{l}$ represents the scalar product between external field $h^{l}$ and
collective AF static spin variable:

$s^{l}=\frac{1}{\beta
N}\sum_{j\sigma\sigma^{\prime}}\int_{0}^{\beta}c_{j\sigma\tau}^{\dagger}\sigma^{l}_{\sigma\sigma^{\prime}}c_{j\sigma^{\prime}\tau}e^{i\bm{K}\bm{r}_{j}}d\tau,$
(5)

where $\sigma^{l}_{\sigma\sigma^{\prime}}$ is the $l-$Pauli matrix, and the
integration is w.r.t. to the imaginary time $\tau$.

###  II.1 Fluctuating local field formalism

The Fluctuating Local Field (FLF) formalism can be developed as follows: one
should define the instability channel and the corresponding order parameter;
couple the artificially introduced field to this order parameter; and let this
field fluctuate in the controlled way. Formally, the FLF approach starts with
the exact transformation of the partition function:

$Z=\left(\frac{\beta N}{2\pi\lambda}\right)^{3/2}\iint
e^{-\mathcal{S}_{h}-\frac{\beta
N}{2\lambda}\left(\bm{\nu}-\bm{h}-\lambda\bm{s}\right)^{2}}\mathcal{D}\left[c^{\dagger},c\right]d^{3}\nu.$
(6)

Here we introduced the fluctuating field $\bm{\nu}$, which is a classical
field coupled to the AF spin variable $\bm{s}$, as we consider the AF
instability channel. Parameter $\lambda$ can be considered as a gauge
parameter of transformation (6). We will use this freedom of choice later. In
what follows, we will omit the pre-integral coefficient for the sake of
shortness, as it does not affect the physics.

Expanding the brackets in (6) leads to the following expression:

$\displaystyle Z$ $\displaystyle=\iint
e^{-\mathcal{S}_{\nu}^{\prime}}e^{-\frac{\beta
N}{2\lambda}\left(\bm{\nu}-\bm{h}\right)^{2}}\mathcal{D}\left[c^{\dagger},c\right]d^{3}\nu=$
$\displaystyle=\int\mathcal{Z}_{\nu}e^{-\frac{\beta
N}{2\lambda}\left(\bm{\nu}-\bm{h}\right)^{2}}d^{3}\nu,$ (7)

where the new action $\mathcal{S}_{\nu}^{\prime}$ consists of two parts:

$\mathcal{S}_{\nu}^{\prime}[c^{\dagger},c]=\underbrace{-c^{\dagger}G_{0}^{-1}c-\beta
N\nu^{l}s^{l}}_{\text{{\hbox{\mathcal{S}^0_{\nu}}}}}+\underbrace{U\theta+\frac{\beta
N\lambda}{2}s^{l}s^{l}}_{\text{{\hbox{\mathcal{S}_{int}^{\prime}}}}}.$ (8)

Here $\mathcal{S}^{0}_{\nu}$ is a bare fermionic action polarized by the
external field $\bm{\nu}$, and $\mathcal{Z}_{\nu}$ is the partition function
of the system with action $\mathcal{S}^{\prime}_{\nu}$ at fixed $\bm{\nu}$.
The new effective interaction action $\mathcal{S}_{int}^{\prime}$ now involves
two components, the Hubbard on-site interaction $U\theta$, which tends to the
AF order, and the effective long-range ferromagnetic interaction $\frac{\beta
N\lambda}{2}s^{l}s^{l}$. This spin-spin interaction weakens the AF
fluctuations, thus indeed allowing us to treat them as approximately gaussian.
Full range of fluctuations is then recovered via the integration over
$\bm{\nu}$.

Eq. (II.1) describes the partition function of the Hubbard system $Z$ as the
average over the gaussian ensemble of systems at fixed value of fluctuating
field $\bm{\nu}$ with the partition function $\mathcal{Z}_{\nu}$. Practical
implementation of this approach demands that we approximate
$\mathcal{Z}_{\nu}$ in some way. The simple and still effective method is to
use the mean-field-like approximation for this. Such an approach was applied
for this task for instance in [18] and [22], where the authors developed the
FLF method on the basis of stationary and dynamical mean field theories
respectively. In this work we propose to exploit the effective small parameter
$\mathcal{S}^{\prime}_{int}$ and obtain $\mathcal{Z}_{\nu}$ perturbatively.
Once this is done we will be able to calculate physical observables
$\mathcal{O}$ as an average over the FLF ensemble

$\left\langle\mathcal{O}\right\rangle_{FLF}=Z^{-1}\int\mathcal{O}(\bm{\nu})\mathcal{Z}_{\nu}e^{-\frac{\beta
N}{2\lambda}\left(\bm{\nu}-\bm{h}\right)^{2}}d^{3}\nu.$ (9)

In the particular case of $\bm{h}\to 0$ and for observables
$\tilde{\mathcal{O}}$, which are spherically symmetric w.r.t. $\bm{\nu}$, this
expression can be reduced to 1D-integrals

$\left\langle\tilde{\mathcal{O}}\right\rangle_{FLF}=\frac{4\pi\int_{0}^{+\infty}\tilde{\mathcal{O}}(\nu)\mathcal{Z}_{\nu}e^{-\frac{\beta
N}{2\lambda}\nu^{2}}\nu^{2}d\nu}{4\pi\int_{0}^{+\infty}\mathcal{Z}_{\nu}e^{-\frac{\beta
N}{2\lambda}\nu^{2}}\nu^{2}d\nu}.$ (10)

##  III Perturbation theory series

In this section we develop the perturbation theory series, assuming that
$\mathcal{S}^{\prime}_{int}$ is a small parameter. As we will see later (Eq.
(19)), the parameter $\lambda$ is proportional to the Hubbard interaction
constant $U$. Hence, the perturbation expansion is constructed w.r.t. orders
of $U$. We will derive the corrections to the quantities of non-interacting
zeroth approximation at first. Once this is done, we will find the conditions
ensuring the validity of this approach. We start with the expansion of the
partition function $\mathcal{Z}_{\nu}$:

$\mathcal{Z}_{\nu}=\int\left[1-\left(U\theta+\frac{\beta
N\lambda}{2}s^{l}s^{l}\right)+\frac{1}{2}\left(U^{2}\theta\theta+\beta
NU\lambda\theta s^{l}s^{l}+\frac{(\beta
N\lambda)^{2}}{4}s^{l}s^{l}s^{l^{\prime}}s^{l^{\prime}}\right)+\dots\right]e^{-\mathcal{S}_{0}^{\nu}}\mathcal{D}\left[c^{\dagger},c\right].$
(11)

Here and in the following, summation over the repeated indices is assumed. Now
we may note that

$s^{m}s^{m}e^{-\mathcal{S}^{\nu}_{0}}=\frac{1}{(\beta
N)^{2}}\frac{\partial^{2}}{\partial\nu^{m}\partial\nu^{m}}e^{-\mathcal{S}^{\nu}_{0}}=\frac{1}{(\beta
N)^{2}}\Delta_{\nu}e^{-\mathcal{S}^{\nu}_{0}}.$ (12)

Let us denote $\mathcal{Z}^{0}_{\nu}=\int
e^{-\mathcal{S}_{\nu}^{0}}\mathcal{D}\left[c^{\dagger},c\right]$, which is the
partition function of non-interacting system at fixed value of $\bm{\nu}$
(compare with (II.1)). The average over this ensemble reads

$\langle\dots\rangle_{0}=\left(\mathcal{Z}^{0}_{\nu}\right)^{-1}\int\dots
e^{-\mathcal{S}_{\nu}^{0}}\mathcal{D}\left[c^{\dagger},c\right].$ (13)

It allows us to rewrite (11) in the form

$\mathcal{Z}_{\nu}=\left[1-\left(U\langle\theta\rangle_{0}+\frac{\lambda}{2\beta
N}\Delta_{\nu}\right)+\frac{1}{2}\left(U^{2}\langle\theta\theta\rangle_{0}+\frac{U\lambda}{\beta
N}\Delta_{\nu}\langle\theta\rangle_{0}+\frac{\lambda^{2}}{(2\beta
N)^{2}}\Delta_{\nu}\Delta_{\nu}\right)+\dots\right]\mathcal{Z}^{0}_{\nu}.$
(14)

It is useful to express this result in terms of free energy. Taking the
definition $\mathcal{Z}_{\nu}=\text{exp}\left(-\beta NF_{\nu}\right)$, we can
derive the perturbation series

$F_{\nu}\approx F_{\nu}^{(n)}=F^{0}_{\nu}+\delta F^{1}_{\nu}+\delta
F^{2}_{\nu}+\dots+\delta F^{n}_{\nu},$ (15)

where $F^{0}_{\nu}$ is the free energy of non-interacting system, and
$F_{\nu}^{(n)}$ is the $n$th order approximation of exact free energy,
including up to the $n$th order corrections $\delta F^{n}_{\nu}$. In the
particular case of negligible external magnetic field $\bm{h}\to 0$ we have:

$\delta
F^{1}_{\nu}=U\Theta_{1}+\frac{\lambda}{2}(F^{0}_{\nu})^{{}^{\prime}2}-\frac{\lambda}{2\beta
N}\left[\frac{2}{\nu}(F^{0}_{\nu})^{\prime}+(F^{0}_{\nu})^{{}^{\prime\prime}}\right],$
(16) $\displaystyle\delta F^{2}_{\nu}=-$
$\displaystyle\frac{U^{2}}{2}\Theta_{2}+\lambda
U(F^{0}_{\nu})^{\prime}\Theta_{1}^{\prime}-\frac{\lambda U}{2\beta
N}\left[\frac{2}{\nu}\Theta_{1}^{\prime}+\Theta_{1}^{{}^{\prime\prime}}\right]$
$\displaystyle+\frac{\lambda^{2}}{2}(F^{0}_{\nu})^{{}^{\prime}2}(F^{0}_{\nu})^{{}^{\prime\prime}}+\frac{\lambda^{2}}{8\beta
N}\left[\frac{4}{\nu^{2}}(F^{0}_{\nu})^{{}^{\prime}2}-2(F^{0}_{\nu})^{{}^{\prime\prime}2}\right.$
$\displaystyle\left.-\frac{8}{\nu}(F^{0}_{\nu})^{\prime}(F^{0}_{\nu})^{{}^{\prime\prime}}-4(F^{0}_{\nu})^{\prime}(F^{0}_{\nu})^{{}^{\prime\prime\prime}}\right]$
$\displaystyle+\frac{\lambda^{2}}{8(\beta
N)^{2}}\left[\frac{4}{\nu}(F^{0}_{\nu})^{{}^{\prime\prime\prime}}+(F^{0}_{\nu})^{(IV)}\right].$
(17)

Here we used the notation $\beta N\Theta_{1}=\langle\theta\rangle_{0}$, $\beta
N\Theta_{2}=\langle\theta\theta\rangle_{0}-\langle\theta\rangle_{0}\langle\theta\rangle_{0}$.
The derivatives are taken w.r.t. the magnitude of fluctuating field
$\bm{\nu}$. The terms are grouped according to their extensity/intensity, i.e.
their dependence on $\beta N$. See Appendix for the derivation of the third
order corrections to free energy.

Note that the trick with derivatives in (12) is not the only way to derive
perturbation series. The same result can be obtained using the Feynman
diagrams technique. According to the linked-cluster theorem [24], the free
energy is determined by only connected diagrams, which simplifies the task
significantly. For instance, there are only three connected diagrams of the
first order (see Fig. 2a). The local one, which is proportional to $U$ and is
denoted with a bold dot, represents the on-site Hubbard term. And the
effective long-range interaction proportional to $\lambda$ is expressed
through diagrams with curved lines. The first two diagrams correspond to the
first two terms in (16). They are both built of local propagators, which means
they represent extensive quantities. Similar reasoning can be applied to the
higher order corrections. The second order diagrams are shown in Figs. 2b-d.
For the third order diagrams see Appendix.

Up to this point we conducted the derivation assuming that
$\mathcal{S}^{\prime}_{int}\to 0$. Now we can figure out whether this
assumption is valid. To do this we consider the first correction to the free
energy (16). We write down explicitly what the derivatives of the free energy
$F_{\nu}^{0}$ mean:

$\displaystyle F_{\nu}^{0}$ $\displaystyle=-\frac{1}{\beta N}\ln\int
e^{c^{\dagger}G_{0}^{-1}c-\beta
N\nu^{l}S^{l}}\mathcal{D}\left[c^{\dagger},c\right],$
$\displaystyle(F_{\nu}^{0})^{\prime}$ $\displaystyle=\left\langle
s^{z}\right\rangle_{0},\quad(F_{\nu}^{0})^{{}^{\prime\prime}}=\beta
N\left(\left\langle s^{z}\right\rangle_{0}^{2}-\left\langle
s^{z}s^{z}\right\rangle_{0}\right).$ (18)

Consider now the first two terms of (16). One can see that

$\left\langle\theta\right\rangle_{0}\equiv\left\langle\left(n_{j\uparrow}-\frac{1}{2}\right)\left(n_{j\downarrow}-\frac{1}{2}\right)\right\rangle_{0}=-\frac{1}{2}\left\langle
s^{z}\right\rangle_{0}^{2},$ (19)

and thus, these two terms cancel each other at $\lambda_{0}=U/2$. The last,
intensive term of $\delta F_{\nu}^{1}$ corresponds to the Hartree-Fock
approximation. The same result for $\lambda$ can be obtained if we consider
the MF as the saddle-point approximation of the FLF ensemble (9). See [22] for
details. We keep $\lambda_{0}=U/2$ for all of the following numerical
calculations.

At this point it is worthwhile to mark the domain of applicability of our
scheme. The hierarchy of diagrams in powers of $(\beta N)^{-1}$ indicates the
limits on the lattice size and the temperature. Namely, the FLF perturbative
approach is applicable for medium-sized lattices at moderate temperatures,
such that it possesses collective excitations and at the same time the physics
is still defined by a single collective mode.

Next, we will move to the practical implementation of our method. In this work
we limited ourselves with the 3rd order corrections to keep the calculations
simple and fast, as is provided by the FLF. Specifically, the 4th order
diagrams include four-leg vertices, which requires comparatively heavy
calculations. Fortunately, the 3rd order already provides us with high
accuracy results. The numerical results for the free energy corrections up to
the 3rd order are presented in the Fig. 1. One can see that the 3rd order
approximation is very close to the 2nd order one, demonstrating convergence.
Overall, it leads us to the conclusion that introduction of the effective
long-range interaction indeed allows us to develop the convergent perturbation
series. Finally, it enables us to calculate the physical observables (9) and
(10) up to the needed order of accuracy $n$ using that

$\mathcal{Z}_{\nu}\approx\mathcal{Z}^{(n)}_{\nu}=e^{-\beta NF^{(n)}_{\nu}}.$
(20)

In what follows we will refer to the $n$-order approximation of the FLF
ensemble as FLF-$n$.

Figure 1: FLF free energy approximation $F^{(n)}_{\nu}$ for $n=0,\,1,\,2,\,3$
as a function of the fluctuating field magnitude $\nu$. The parameters of the
Hubbard system are $U/t=1$, $\beta=10$, the size of the lattice is $6\times
6$.

a)

b)

c)

d)

Figure 2: First order (a) and second order (b, c, d) diagrams. Bold dots
denote local Hubbard interaction $U$ and curved lines denote effective long-
range FLF interaction $\lambda$. Solid lines with arrows mean free fermionic
propagators. Gray diagrams cancel each other at $\lambda=U/2$.

##  IV Single-particle quantities

Figure 3:  Imaginary and real parts of the Green’s functions
$\mathcal{G}_{\bm{k}}(iw)$ in $\bm{k}$-space near Fermi surface. Comparison of
the numerical results obtained within the 3rd order FLF approximation with the
QMC results. The parameters of the Hubbard system are $U/t=1$, $\beta=10$, the
lattice size is $6\times 6$.

The FLF ensemble (9) provides us with a powerful tool to investigate the
properties of many-particle systems without complex techniques. Here we start
with calculation of single-particle quantities. The temperature Green’s
function in the frequency domain can be defined as

$\mathcal{G}_{\bm{k}}(iw_{n})=\int_{0}^{\beta}\mathcal{G}_{\bm{k}}(\tau)e^{iw_{n}\tau}d\tau,$
(21)

where

$\mathcal{G}_{\bm{k}}(\tau)=-{\cal Z}^{-1}\int
c_{\bm{k}}(\tau)c_{\bm{k}}^{{\dagger}}(0)e^{-[\mathcal{S}^{0}_{h}+\mathcal{S}_{U}]}\mathcal{D}\left[c^{\dagger},c\right].$
(22)

Here $\bm{k}$ denotes a quasimomentum vector, and
$iw_{n}=i{\pi\left(2n+1\right)}{\beta^{-1}}$ is the $n$th fermionic Matsubara
frequency.

According to the FLF approach we can approximate (21) with the expectation
value over the ensemble of non-interacting systems perturbed by new
interaction term $\mathcal{S}_{int}^{\prime}$ (8), which is a small parameter
for $\lambda=\frac{U}{2}$ as it was shown in the previous section. Since
perturbations are already taken into account by (20), we approximate the exact
Green’s function $\mathcal{G}(iw_{n})$ by the bare Green’s function
$G_{k}^{\nu}(iw_{n})$ averaged over the FLF ensemble

$\mathcal{G}_{\bm{k}}(iw_{n})\approx\langle
G_{\bm{k}}^{\nu}(iw_{n})\rangle_{FLF}=\left\langle\frac{1}{iw_{n}-e_{\bm{k}}^{\nu}}\right\rangle_{FLF}.$
(23)

Here $e_{\bm{k}}^{\nu}$ is the energy of the non-interacting system in the
external field $\bm{\nu}$.

Figure 4:  Imaginary and real parts of the Green’s functions
$\mathcal{G}_{\bm{r}}(iw)$ in $\bm{r}$-space. Comparison of the numerical
results obtained within the 3rd order FLF approximation with the QMC results.
The parameters of the Hubbard system are $U/t=1$, $\beta=10$, the lattice size
is $6\times 6$.

In Fig. 3 and Fig. 4 we represent the numerical results for the Green’s
function $\mathcal{G}(iw)$ in $\bm{k}$-space and in $\bm{r}$-space
respectively, calculated in the 3rd order FLF approximation. The comparison
with the numerically exact Quantum Monte-Carlo (QMC) results shows that the
FLF Green’s functions perfectly matches with numerically exact ones at
moderately low temperature $\beta=10$. This allows us to obtain reliable
experimentally observable quantities without complex computational techniques.
For instance, to obtain the local density of states $\mathcal{A}_{\bm{k}}(w)$
in the QMC approach one would have to deal with analytic continuation of the
QMC data. To do the same in the FLF scheme we calculate the local Green’s
function in the real frequencies domain $\mathcal{G}_{\bm{k}}\left(w\right)$
by the formal substitution $iw_{n}\rightarrow w+i0$:

$\mathcal{A}_{\bm{k}}(w)=-\frac{1}{\pi}\Im{\mathcal{G}_{\bm{k}}(w)}\approx-\frac{1}{\pi}\Im{\left\langle\frac{1}{w-e_{\bm{k}}^{\nu}}\right\rangle_{FLF}}.$
(24)

The numerical results for the local density of states
$\mathcal{A}(w)=\sum_{\mathbb{k}}\mathcal{A}_{\mathbb{k}}(w)$ in the FLF
approximation are shown in the Fig. 5. However, approximation of full Green’s
function by bare one (23) makes the accuracy incomparable with QMC data for
such quantities as self-energy for low temperatures.

Figure 5: Local density of states
$\mathcal{A}(w)=\sum_{k}{\mathcal{A}_{k}(w)}$ calculated in the 3rd order FLF
approximation. The parameters of the Hubbard system are $U/t=1$, $\beta=10$,
the size of the lattice is $6\times 6$.

##  V Susceptibility in the antiferromagnetic channel

In order to further explore the power of the FLF ensemble we consider
susceptibility $\chi$ in the spin-channel. As we already mentioned, the half-
filled Hubbard model possesses fluctuations in the AF channel. Mean-field-like
schemes predict unphysical transition to the AF-ordered state at finite
temperature, which manifests as the infinite susceptibility to the external AF
magnetic field $\bm{h}$. The possible cause of this effect is that MF schemes
freeze all the fluctuations, which actually break the ordered states. If this
assumption is valid, taking into account AF fluctuations in the FLF approach
would cure this artifact.

The magnetic susceptibility can be defined as

$\chi=\left.\frac{1}{\beta N}\partial^{2}_{\bm{h}}\ln Z\right|_{\bm{h}\to 0}.$
(25)

Using the expression (II.1) for the partition function of the system and the
definition of average over the FLF ensemble (9) we obtain:

$\chi=\frac{\beta
N}{3\lambda^{2}}\langle\nu^{2}\rangle_{FLF}-\frac{1}{\lambda}.$ (26)

The $1/3$ coefficient appeared due to the spatial isotropy.

The numerical results are presented in terms of the Curie constant
$C=\chi/\beta$, and are shown in the Fig. 6. We considered the 2D Hubbard
lattice with $6\times 6$ sites at $U=t=1$. We compared the results taking into
account from $0$th to $3$rd order corrections with the DMFT and with the QMC
results. One can see that FLF-3, DMFT and QMC methods all lead to coinciding
results for comparatively high temperatures $\beta<5$. At $\beta\approx 9$
though the DMFT predicts transition to the ordered phase, which is unphysical.
FLF scheme describes the physics around this temperature qualitatively
correct. Quantitative correctness of the FLF results can be concluded from the
comparison with the QMC data. Though statistical noise leads to some errorbar
of the QMC points, one can see that the FLF-3 results are nearly within the
margin of error of the numerically exact QMC method.

Figure 6: Curie constant $C=\chi/\beta$ as a function of the inverse
temperature $\beta$. Comparison of the numerical results obtained within
different orders of FLF approximation with the DMFT and QMC results. The
parameters of the Hubbard system are $U/t=1$, the size of the lattice is
$6\times 6$.

##  VI Cluster extension

Assuming that fluctuations belong to a single mode we limit the available
range of systems with small clusters. However, the FLF approach could still be
applied to larger systems in the spirit similar to cluster DMFT extensions. In
this section we develop cellular FLF (c-FLF) method in the CDMFT-like manner
[25].

We start with the definition of the self-energy

$\Sigma_{\bm{k}}(z_{n})=z-e_{\bm{k}}-\mathcal{G}_{\bm{k}}(z_{n})^{-1},$ (27)

where $z$ is a real $z_{n}=w_{n}+i0$ or imaginary frequency $z_{n}=iw_{n}$. We
make use of the coarse-graining and decompose the momentum vector in two parts
$\bm{k}=\tilde{\bm{k}}+\delta\bm{k}$, where $\tilde{\bm{k}}$ is the coarsened
vector defined on the finite size cluster and $\delta\bm{k}$ is the residual
vector. Using (23) as the approximation for the cluster Green’s function we
obtain the cluster self-energy

$\Sigma_{\tilde{{}_{\bm{k}}}}(z)=z-e_{\tilde{{}_{\bm{k}}}}-\left\langle\left(z-e_{\tilde{{}_{\bm{k}}}}^{\nu}\right)^{-1}\right\rangle^{-1}_{FLF}.$
(28)

Assuming now that the essential interaction physics is captured by the cluster
self-energy $\Sigma_{\bm{k}}(z)\approx\Sigma_{\tilde{{}_{\bm{k}}}}(z)$ we go
back to (27) and finally obtain the lattice Green’s function

$\mathcal{G}_{\bm{k}}(z)=\frac{1}{z-e_{\bm{k}}-\Sigma_{\bm{\tilde{k}}}(z)}.$
(29)

Figure 7: Local density of states $\mathcal{A}(w)$ calculated within the c-FLF
approach with clusters of different sizes $N_{x}\times N_{y}$. The Hubbard
system parameters are $U/t=1$, $\beta=10$.

In Fig. 7 we show the results for the local density of states $\mathcal{A}(w)$
for infinite lattice calculated in the c-FLF approach with clusters of
different sizes. All the considered cluster sizes lead to almost the same
results indicating the convergence. In the Fig. 7 we may observe a pseudogap,
which can be due to two reasons. The first one is the finite size of clusters.
One can see that the pseudogap indeed reduces with the growth of the cluster
size. The second one may take origin in the approximation of the exact Green’s
function $\mathcal{G}_{\bm{k}}(z)$ by the ensemble of non-interacting ones
(23). It seems reasonable that this artifact could be suppressed by a more
sophisticated approximation for $\mathcal{G}_{\bm{k}}(z)$, although this point
in the Hubbard phase diagram is very close to the metal-insulator transition
point calculated by the diagrammatic Monte Carlo [26].

##  VII Convergence issues and relation to Landau free energy

In the previous sections we considered the numerical results obtained in the
FLF approach based on the perturbation series for free energy (15). We noticed
that this rather simple-to-implement method leads us to good accuracy for
comparatively high temperatures ($\beta\geq 10$) and for weakly-medium
correlated systems $U/t<2$. In this section we address the question, to which
extent the proposed FLF perturbation scheme is convergent. To this aim, let’s
consider the applicability of the FLF scheme to systems at lower temperatures
and/or in stronger correlated regime. In Fig. 8 we demonstrate the 0th and the
3rd order FLF free energy approximation as the function of fluctuating field
magnitude $\nu$ for $U/t=2$. One can see that the discrepancy between these
two curves for $\nu\to 0$, seeming harmless for $U/t=1$ (see Fig. 1), seeds
the divergence for $U/t=2$ in this region. It makes us unable to use the FLF
perturbative approach as is for this regime. In this section we develop the
regularization procedure based on the idea of Landau free energy flow which
would allow us to extend the range of the method applicability.

Figure 8: Comparison of the 0th $F^{(0)}$ and 3rd $F^{(3)}$ order FLF free
energies with the regularized ansatz function $\mathcal{F}_{\bm{a}}$. Color
map indicates the difference $\Delta F$ between the 0th and the 3rd order FLF
free energies. The Hubbard system parameters are $U/t=2$, $\beta=10$, the
lattice size is $4\times 4$.

Let us consider the half-filled Hubbard cluster in the view of
phenomenological Landau theory of phase transitions. This way, we can describe
the system in terms of the order parameter, which is the collective AF spin
$S^{l}$ (5) in our case. Then we can represent the exact partition function
(2) as the double integral

$Z=\int
d^{3}\tilde{s}\int\displaylimits_{\bm{s}[c^{\dagger},c]=\tilde{\bm{s}}}e^{-\mathcal{S}_{h}[c^{\dagger},c;\bm{s}]}\mathcal{D}\left[c^{\dagger},c\right]$
(30)

of $e^{-\mathcal{S}_{h}}$ over Grassman variables $c^{\dagger},c$ at fixed
values of the order parameter $\tilde{\bm{s}}$. It leads us to the definition
of the Landau free energy $F_{L}(\tilde{\bm{s}})$:

$Z=\int e^{-\beta NF_{L}\left(\tilde{\bm{s}}\right)}d^{3}\tilde{s}.$ (31)

Reformulating the FLF transformation (6) in this spirit we obtain

$Z=\iint e^{-\beta NF_{L}\left(\tilde{\bm{s}}\right)}e^{-\frac{\beta
N\lambda}{2}\left(\tilde{\bm{s}}-\tilde{\bm{s}}^{\prime}\right)^{2}}d^{3}\tilde{s}d^{3}\tilde{s}^{\prime},$
(32)

where $\tilde{\bm{s}}^{\prime}=\bm{\nu\lambda}$, and we omitted the dependence
on $\bm{h}$ for the sake of shortness. Integrating this over the Grassman
variables at some given value of $\lambda$, we can rewrite this as

$Z=\int e^{-\beta N\mathcal{F}(\tilde{\bm{s}},\lambda)}d^{3}\tilde{s},$ (33)

where $\mathcal{F}(\tilde{\bm{s}},\lambda)$ is a double variable function. At
this point one may notice that expressions (31)–(33) formally describe the
solution of the heat flow Cauchy problem

$\begin{cases}\left(\partial_{\lambda^{-1}}-\Delta_{\tilde{s}}\right)e^{-\beta
N\mathcal{F}(\tilde{\bm{s}},\lambda)}=0,\\\ \left.e^{-\beta
N\mathcal{F}(\tilde{\bm{s}},\lambda)}\right|_{\lambda^{-1}\to 0}=e^{-\beta
NF_{L}(\tilde{\bm{s}})},\end{cases}$ (34)

obtained by a convolution (33) of the fundamental solution
$\text{exp}\left(-\beta N\lambda\tilde{\bm{s}}^{2}/2\right)$ with the initial
condition of (34). Here $\lambda^{-1}$ plays the role of time variable, and
$(2\beta N)^{-1}$ can be considered as the thermal conductivity.

This observation allows us to derive the regularization procedure based on the
flow of free energy. Our aim is to find some function
$\mathcal{F}(\tilde{\bm{s}},\lambda)$ that best suits the 3rd order FLF free
energy $F^{(3)}$, but has a more regular behavior at $\nu\to 0$. The backward
Cauchy problem (34) is ill-conditioned, so we will not solve it directly.
Instead, we set the initial function to be an ansatz
$\mathcal{F}_{\bm{a}}(\tilde{\bm{s}})$ depending on vector of parameters
$\bm{a}$, and let it evolve through the convolution (31). Since the physically
meaningful range of the order parameter is $\tilde{s}\leq 1$, we impose an
additional condition

$\left.e^{-\beta
N\mathcal{F}(\tilde{\bm{S}},\lambda)}\right|_{\tilde{S}>1}=0.$ (35)

The particular value of $\bm{a}$ would be the one at which
$\mathcal{F}_{\bm{a}}(\tilde{\bm{S}},\lambda_{0})$ is the closest to the 3rd
order FLF Landau free energy $F^{(3)}(\tilde{\bm{S}},\lambda_{0})$ at
$\lambda_{0}$. As a measure of similarity, we use the loss function

$\mathcal{L}_{\alpha}=\int\frac{\left|\mathcal{F}_{\bm{a}}(\tilde{\bm{s}},\lambda_{0})-F^{(3)}(\tilde{\bm{s}},\lambda_{0})\right|^{2}}{\alpha(\Delta
F)^{2}+1}d^{3}\tilde{s}\rightarrow min,$ (36)

where

$\Delta
F=\max\limits_{\tilde{s}>\tilde{s}^{\prime}}\left|F^{(3)}(\tilde{s}^{\prime})-F^{0}(\tilde{s}^{\prime})\right|$
(37)

is the disconvergence between the 0th and the 3rd order FLF free energy, and
$\alpha$ is the regularization constant. The denominator of this loss function
(36) is constructed so that it reduces the contribution of the dominant
disconvergence region. In particular, we choose the value of the $\alpha$
parameter such that $\alpha(\Delta F)^{2}\approx 1$. Once the values of
$\bm{a}$ are found, we will be able to use $\mathcal{F}_{\bm{a}}$ the same way
as the FLF free energy to calculate observables (9).

In the particular case of negligible external magnetic field $\bm{h}\to 0$ the
Hubbard system possesses spherical symmetry, and we can use the finite even-
order polynomial ansatz

$\mathcal{F}_{\mathbb{a}}(\tilde{\bm{s}})=a_{0}+a_{2}\tilde{s}^{2}+...+a_{2p}\tilde{s}^{2p},\quad
p\in\mathbb{N}.$ (38)

We present the results for the
$\mathcal{F}_{\bm{a}}(\nu)=\mathcal{F}_{\bm{a}}(\tilde{\bm{s}}/\lambda)$
obtained for $4\times 4$ Hubbard lattice at $U/t=2$ and $\beta=10$ in Fig. 8.
In this case, we set $\alpha=10^{6}$, and we limited ourselves with the order
of polynomial $p=5$. Nevertheless, we should mention that as the temperature
decreases, the behavior of the Landau free energy $F_{L}$ becomes more
complex, and the order of the polynomial has to be increased. We also note
that the problem posed is not convex, and therefore has many local minima. To
estimate the global one, we used the SHGO method [27] implemented in SciPy
python package [28].

(a) Local Green’s function

(b) Local density of states

Figure 9: Comparison of the numerical results obtained within the 3rd order
FLF approximation with the regularized approximation. Imaginary part of the
local Green’s function (a) and the local density of states (b). The parameters
of the Hubbard system are $U/t=2$, $\beta=10$, the lattice size is $4\times
4$.

One can see that the evolved ansatz function almost coincides with the 3rd
order FLF free energy $F^{(3)}$ for $\nu>0.5$, and the divergence for $\nu\to
0$ is significantly smoother.

As an example of how this regularization manifests in observables, we provide
the numerical results for the local Green’s function and local spectral
function $\mathcal{A}(w)$ in Fig. 9. Comparison of the pure 3rd order FLF
approximation with the regularized approximation demonstrates that the second
one makes local Green’s function for small imaginary frequencies $iw$ more
precise (see Fig. 9(a)). What is more noteworthy, regularization approach
eliminates nonphysical pseudogap in the local spectral function in Fig. 9(b).

##  VIII Conclusion and outlook

In this work we derived the extended version of the Fluctuating Local Field
method using the perturbation theory approach. We considered the half-filled
2D Hubbard lattices, which are known to possess collective spin fluctuations,
as the particular example. The FLF scheme, originally based on the exact
transformation of the partition function, introduces an auxiliary classical
fluctuating field coupled to the collective order parameter of the system in
the leading fluctuation channel. For the system under consideration it leads
to the emergence of an effective long-range interaction $\lambda$ competing
with the local interaction $U$. We show that for $\lambda_{0}=U/2$ these two
interactions form a small parameter, enabling us to construct a regular series
expansion w.r.t. $U$.

In order to investigate the efficiency of the proposed FLF perturbation scheme
we calculated several physical quantities, namely Green’s function, local
density of states, and magnetic susceptibility. Comparison with much more
sophisticated DMFT method, and with the numerically exact QMC method shows
that the FLF perturbation approach leads to good accuracy preserving the
comparative simplicity of the implementation.

To obtain physically consistent results, one should keep in mind the range of
applicability of the proposed FLF scheme. First of all, as the series
expansion is constructed w.r.t. the powers of $U$, it better suits weak-
coupling regime. Nevertheless, the FLF perturbation series allows us to go
beyond the limitations of usual perturbation schemes and leads to good
accuracy even for $U/t=1$. And the proposed regularization procedure, based on
the flow of the Landau free energy, even extends the range of applicability to
$U/t\lesssim 2$, which already falls within the range of moderately correlated
regime. Handling lattices with larger $U/t$ would require an FLF scheme
constructed around DMFT, which will be in the scope of our future works.

Secondly, number of the lattice sizes should be large enough to form a
collective degree of freedom. At the same time, the lattice should not be too
large, as we expect that fluctuations belong to a single mode. Similar
argumentation holds for the temperature range. Formally it is reflected in a
diagram hierarchy in powers of $(\beta N)^{-1}$. Nevertheless, we show that
the limitation on the size of the lattice can be overcome using the cluster
extension of the FLF perturbation scheme.

Regardless of the fact that we focused on the single mode fluctuations in the
spin channel in this work, the flexibility of the FLF approach actually allows
one to describe the physics of the systems with other types of fluctuations,
both in single mode and multi mode regimes. Describing the physics of systems
with competing fluctuation channels will be in the scope of our future work,
as well as the FLF perturbation extension around DMFT.

## Acknowledgements

Ya.S.L., S.D.S., and A.N.R. acknowledge the support from the Russian Quantum
Technologies Roadmap.

*

##  Appendix A Third order corrections

This section is dedicated to the derivation of the third order corrections to
the free energy. We start from (11) and write explicitly the third term of
partition function expansion $\delta\mathcal{Z}^{3}_{\nu}$ coming from
$e^{-\mathcal{S}^{\prime}_{int}}$:

$\delta\mathcal{Z}^{3}_{\nu}=-\frac{1}{6}\int\left[U^{3}\theta\theta\theta+\frac{3\beta
NU^{2}\lambda}{2}S^{l}S^{l}\theta\theta+\frac{3(\beta
N)^{2}U\lambda^{2}}{4}S^{l}S^{l}S^{m}S^{m}\theta+\frac{(\beta
N\lambda)^{3}}{8}S^{l}S^{l}S^{m}S^{m}S^{p}S^{p}\right]e^{-\mathcal{S}_{0}^{\nu}}\mathcal{D}\left[c^{\dagger},c\right].$
(39)

Using the trick (12), we can rewrite this in the following form:

$\delta\mathcal{Z}^{3}_{\nu}=\left[-\frac{U^{3}}{6}\langle\theta\theta\theta\rangle_{0}-\frac{\lambda
U^{2}}{4\beta
N}\Delta_{\nu}\langle\theta\theta\rangle_{0}-\frac{\lambda^{2}U}{8(\beta
N)^{2}}\Delta_{\nu}\Delta_{\nu}\langle\theta\rangle_{0}-\frac{\lambda^{3}}{48(\beta
N)^{3}}\Delta_{\nu}\Delta_{\nu}\Delta_{\nu}\right]\mathcal{Z}_{\nu}^{0}.$ (40)

Finally, for the third order correction to the free energy $\delta
F^{3}_{\nu}$ we have

$\displaystyle\delta F^{3}_{\nu}$
$\displaystyle=\frac{U^{3}}{6}\Theta_{3}+\frac{\lambda
U^{2}}{2}\left(-\Theta_{2}^{\prime}(F_{\nu}^{0})^{\prime}+\Theta_{1}^{{}^{\prime}2}\right)+\frac{\lambda
U^{2}}{2\beta
N}\left(\frac{\Theta_{2}^{\prime}}{\nu}+\frac{\Theta_{2}^{{}^{\prime\prime}}}{2}\right)+$
(41)
$\displaystyle+\frac{\lambda^{2}U}{2}\left(\Theta_{1}^{{}^{\prime\prime}}(F_{\nu}^{0})^{{}^{\prime}2}+2\Theta_{1}^{\prime}(F_{\nu}^{0})^{\prime}(F_{\nu}^{0})^{{}^{\prime\prime}}\right)+$
$\displaystyle+\frac{\lambda^{2}U}{2\beta
N}\left(-\frac{2\Theta_{1}^{\prime}(F_{\nu}^{0})^{{}^{\prime\prime}}}{\nu}-\Theta_{1}^{{}^{\prime\prime}}(F_{\nu}^{0})^{{}^{\prime\prime}}+\frac{2\Theta_{1}^{\prime}(F_{\nu}^{0})^{\prime}}{\nu^{2}}-\frac{2\Theta_{1}^{{}^{\prime\prime}}(F_{\nu}^{0})^{\prime}}{\nu}-\Theta_{1}^{{}^{\prime\prime\prime}}(F_{\nu}^{0})^{\prime}-\Theta_{1}^{\prime}(F_{\nu}^{0})^{{}^{\prime\prime\prime}}\right)+$
$\displaystyle+\frac{\lambda^{2}U}{2(\beta
N)^{2}}\left(\frac{\Theta_{1}^{(IV)}}{4}+\frac{\Theta_{1}^{{}^{\prime\prime\prime}}}{\nu}\right)+$
$\displaystyle+\frac{\lambda^{3}}{2}\left((F_{\nu}^{0})^{{}^{\prime}2}(F_{\nu}^{0})^{{}^{\prime\prime}2}+\frac{(F_{\nu}^{0})^{{}^{\prime}3}(F_{\nu}^{0})^{{}^{\prime\prime\prime}}}{3}\right)+$
$\displaystyle+\frac{\lambda^{3}}{2\beta
N}\left(-\frac{(F_{\nu}^{0})^{{}^{\prime\prime}3}}{3}-\frac{2(F_{\nu}^{0})^{\prime}(F_{\nu}^{0})^{{}^{\prime\prime}2}}{\nu}-\frac{2(F_{\nu}^{0})^{{}^{\prime}3}}{3\nu^{3}}+\frac{(F_{\nu}^{0})^{{}^{\prime}2}(F_{\nu}^{0})^{{}^{\prime\prime\prime}}}{\nu}+\right.$
$\displaystyle\left.+\frac{2(F_{\nu}^{0})^{{}^{\prime}2}(F_{\nu}^{0})^{{}^{\prime\prime}}}{\nu^{2}}-\frac{2(F_{\nu}^{0})^{{}^{\prime}2}(F_{\nu}^{0})^{{}^{\prime\prime\prime}}}{\nu}-\frac{(F_{\nu}^{0})^{{}^{\prime}2}(F_{\nu}^{0})^{(IV)}}{2}-2(F_{\nu}^{0})^{\prime}(F_{\nu}^{0})^{{}^{\prime\prime}}(F_{\nu}^{0})^{{}^{\prime\prime\prime}}\right)+$
$\displaystyle+\frac{\lambda^{3}}{2(\beta
N)^{2}}\left(\frac{(F_{\nu}^{0})^{\prime}(F_{\nu}^{0})^{(V)}}{4}+\frac{5}{12}(F_{\nu}^{0})^{{}^{\prime\prime\prime}2}-\frac{(F_{\nu}^{0})^{\prime}(F_{\nu}^{0})^{{}^{\prime\prime\prime}}}{\nu^{2}}+\frac{2(F_{\nu}^{0})^{{}^{\prime\prime}}(F_{\nu}^{0})^{{}^{\prime\prime\prime}}}{\nu}+\right.$
$\displaystyle+\left.\frac{(F_{\nu}^{0})^{\prime}(F_{\nu}^{0})^{(IV)}}{\nu}+\frac{(F_{\nu}^{0})^{{}^{\prime\prime}}(F_{\nu}^{0})^{(IV)}}{2}\right)+$
$\displaystyle+\frac{\lambda^{3}}{2(\beta
N)^{3}}\left(-\frac{(F_{\nu}^{0})^{(V)}}{4\nu}-\frac{(F_{\nu}^{0})^{(VI)}}{24}\right).$

Here $\beta
N\Theta_{3}=2\left\langle\theta\right\rangle_{0}\left\langle\theta\right\rangle_{0}\left\langle\theta\right\rangle_{0}-3\left\langle\theta\right\rangle_{0}\left\langle\theta\theta\right\rangle_{0}+\left\langle\theta\theta\theta\right\rangle_{0}$,
and the terms are grouped according to their extensity/intensity.

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# Exploring the Effects of AI-assisted Emotional Support Processes in Online
Mental Health Community

Donghoon Shin  Seoul National UniversitySeoulKorea<EMAIL_ADDRESS>,  Subeen
Park  Seoul National UniversitySeoulKorea<EMAIL_ADDRESS>,  Esther Hehsun
Kim  Seoul National UniversitySeoulKorea<EMAIL_ADDRESS>,  Soomin Kim  Seoul
National UniversitySeoulKorea<EMAIL_ADDRESS>,  Jinwook Seo  Seoul
National UniversitySeoulKorea<EMAIL_ADDRESS>and  Hwajung Hong
KAISTDaejeonKorea<EMAIL_ADDRESS>
(2022)

###### Abstract.

Social support in online mental health communities (OMHCs) is an effective and
accessible way of managing mental wellbeing. In this process, sharing
emotional supports is considered crucial to the thriving social supports in
OMHCs, yet often difficult for both seekers and providers. To support
empathetic interactions, we design an AI-infused workflow that allows users to
write emotional supporting messages to other users’ posts based on the
elicitation of the seeker’s emotion and contextual keywords from writing.
Based on a preliminary user study (N = 10), we identified that the system
helped seekers to clarify emotion and describe text concretely while writing a
post. Providers could also learn how to react empathetically to the post.
Based on these results, we suggest design implications for our proposed
system.

online mental health community, AI-infused system, emotional support, peer
support

††copyright: acmlicensed††copyright: acmcopyright††journalyear:
2022††conference: CHI Conference on Human Factors in Computing Systems
Extended Abstracts; April 29-May 5, 2022; New Orleans, LA, USA††booktitle: CHI
Conference on Human Factors in Computing Systems Extended Abstracts (CHI ’22
Extended Abstracts), April 29-May 5, 2022, New Orleans, LA, USA††price:
15.00††doi: 10.1145/3491101.3519854††isbn: 978-1-4503-9156-6/22/04††ccs:
Human-centered computing Collaborative and social computing Figure 1.
Keyscreens of our system we designed and examined. Once the seeker writes down
the post, the system automatically detects (a)-① emotion and (a)-② contextual
keywords, where the user can select among them or newly add one. (b) By
filtering posts using the metadata from (a), providers can explore posts among
the list of posts. Once clicking the post, providers can start reading it and
(c)-③ commenting empathetically based on three phrase recommendations for
emotional reaction, followed by (c)-④ the prompts asking users to leave
interpretation and exploration of the post. The interface elements were
translated from Korean

##  1\. Introduction

Online mental health communities (OMHCs) have become a prevalent medium of
promoting mental wellness through collaborative interactions among people
(Sharma and De Choudhury, 2018). For instance, users in multiple mental health
sub-Reddits benefit from actively sharing their challenging experiences and
gaining feedback from their peers online (Sharma and De Choudhury, 2018). By
posting their experiences and concerns and reacting to them, individuals in
OMHCs can actively share emotional and informational support with people with
similar conditions (Cutrona and Suhr, 1994).

Here, ensuring active emotional support-sharing processes among users is
particularly crucial to maintaining thriving OMHCs (Elliott et al., 2011), yet
often considered challenging (Prescott et al., 2019). For providers, it is
often burdensome and overwhelming to be emotionally engaged (Highet et al.,
2004; Barney et al., 2009), leading to their dropouts (de Alva et al., 2015).
Plus, since most of the existing OMHCs are text-based (Hoermann et al., 2017),
providers have difficulty in converting their empathetic thoughts in a text-
based form (Gibson et al., 2016; Pérez-Rosas et al., 2017). Similarly, it is
equally important, yet difficult, for seekers to ensure that they disclose
their experiences concretely to help providers better understand seeker’s
experiences and react emotionally (Altman and Taylor, 1973; Yang et al.,
2019).

In line with such concerns, previous studies have devised theoretical guidance
to support individuals to learn how to react emotionally in text-based
settings (Sharma et al., 2020). On top of these, recent studies have begun to
feature the role of AI in supporting emotional reaction processes. For
example, Peng et al. designed MepBots, a writing assistance that aids
empathetic reaction processes (Peng et al., 2020). Still, such an approach is
limited in that the quality of the seeker’s posts (e.g., concreteness) is also
a crucial factor that affects the emotional supports (Li et al., 2016; Yang et
al., 2019). Thus, rather than merely supporting the empathy-providing process,
understanding the holistic workflow - from writing the post (seeker’s side) to
navigating posts and reacting to such seekers (provider’s side) - is
important.

Thus, we aim to explore the feasibility of AI in augmenting the overall
workflow of empathetic communication by considering both seeker and provider
sides in OMHCs. Specifically, we designed and evaluated an AI-infused mental
health community app, which supports the scaffolded interactions (writing
posts, exploring posts, and reacting to posts) specialized to facilitate
empathetic communication in OMHCs by detecting the emotion and contextual
keywords and recommending appropriate triggers/prompts to assist provider’s
reaction. From the preliminary user study with 10 participants, we identified
the feasibility and areas of improvement for enhancing emotional support
processes in AI-infused OMHCs.

##  2\. Related Work

###  2.1. Online Mental Health Community

Online mental health communities (OMHCs) are the prevalent medium of
exchanging social support with peers experiencing similar challenges,
providing high accessibility and anonymity (O’Leary et al., 2017; Sharma and
De Choudhury, 2018; Peng et al., 2021; Li et al., 2016; Naslund et al., 2016).
To date, several studies have explored the dynamics of peer supports in OMHCs
or mental health-related discussions, such as relationship between user
satisfaction and their community knowledge (Peng et al., 2021), cross-cultural
variances of OMHCs (Pendse et al., 2019), and motivations and practices of
mental health discussions in Twitch (Uttarapong et al., 2022).

For such OMHCs to thrive and individuals to improve mental health (e.g.,
reducing depression/anxiety), it is known that active sharing (e.g., sharing
experiences, expressing emotions, and restructuring maladaptive cognition) are
particularly crucial (Lepore et al., 2002; Kushner and Sharma, 2020; Morris et
al., 2015; Pfeiffer et al., 2011; Pruksachatkun et al., 2019; Radcliffe et
al., 2007). Specifically, previous works suggested that people were more
satisfied when their received support matches the needed support (Wolff et
al., 2013; Vlahovic et al., 2014), in the form of informational or emotional
support. These are known to increase both members’ commitment and satisfaction
when they sought and received the matched support (Yang et al., 2017; Peng et
al., 2021). Here, emotional supports are particularly known to lead support-
seeker to express more satisfaction when receiving emotional support from the
reaction (Peng et al., 2021). Thus, to guide long-lasting and thriving OMHC
environments, it is crucial to ensure emotional supports between seekers and
providers.

However, sharing emotional supports in OMHCs is especially challenging for
both support seekers and providers (Gibson et al., 2016; Pérez-Rosas et al.,
2017). For instance, providers often find it difficult to convert their
empathetic thoughts in a text-based form (Gibson et al., 2016; Pérez-Rosas et
al., 2017). Similarly, it is difficult for seekers to disclose their
experiences concretely, which helps providers better understand their
experiences and react emotionally (Altman and Taylor, 1973; Yang et al.,
2019). As such, researchers have begun to devise systems that support the
writing process of seekers and providers (Wu et al., 2019; Peng et al., 2020).
However, they were limited in that the system (i) assessed posts based on the
quality, which may discourage and negatively affect users (Wu et al., 2019),
or (ii) only focused on the provider’s side (Peng et al., 2020).

###  2.2. AI-mediated Communication

Advances in artificial intelligence (AI) and machine learning (ML) have
transformed the paradigm of how people communicate and collaborate, and
changed the pattern of forming interpersonal relationships. While traditional
computer-mediated communication (CMC) focuses on the interactions in which
communication is transmitted and mediated via technologies, AI-Mediated
Communication (AI-MC) assumes that communication is modified, amplified, and
even generated by a computational agent (Hancock et al., 2020). For example,
recommendation algorithms used in social networking online dating applications
determine who we associate and converse with.

Various methods have been proposed to enhance interpersonal relationships
using AI and ML technology. The most representative case would be recommending
appropriate responses. Human-AI hybrid conversational system was developed for
reply suggestions (Gao and Jiang, 2021). Robertson et al. (Robertson et al.,
2021) proposed a system that classifies problematic email reply suggestions by
analyzing email content (e.g., semantic, tonal coherence) and social context
(e.g., communication dynamics). Furthermore, an AI-infused chatbot agent was
proposed to recommend common interests between the strangers based on their
social media data (Shin et al., 2021).

In line with previous studies of using the assists of AI to enhance
interpersonal communication, this work premises that technology could support
the mental health community by augmenting interpersonal relationships between
community members (Gui et al., 2017). Specifically, we design and develop a
system that aids emotional support workflow in OMHCs by harnessing AI to (i)
detect/recommend emotion/contextual keywords for seekers and (ii) recommend
triggers/scaffolds of empathetic emotional reactions for providers. Based on
the preliminary user study with 10 users, we discuss enhancements on AI-
infused emotional support workflow across seekers and providers.

##  3\. Prototype Design

###  3.1. Overall Concept and Workflow

The goal of our system is to support empathetic peer-support workflow in
OMHCs. Here, in addition to identifying the emotion of the seeker’s post, it
is also important to ensure that providers understand the overall contextual
information (e.g., place, reason, subject in conflict) to provide more
detailed and empathetic response (Pfeil and Zaphiris, 2007). Thus, we decided
to let users elicit and leverage (i) emotion and (ii) contextual information
through their exchange, with the aid of AI.

###  3.2. System Design

In OMHCs, supports between providers and seekers are often shared as a form of
post, a text-based unit that basically contains content written by the seeker
and its comments from the providers. Specifically, this process can be
conceptualized with the following interactions: (i) seekers post their
emotional contexts, (ii) providers explore such posts in the community, and
they (iii) react to the post (Gui et al., 2017; Peng et al., 2020; Sharma and
De Choudhury, 2018; Peng et al., 2021). Thus, we designed our proposed system
by following three interactions:

####  3.2.1. Writing a post

First, the user is asked to freely describe their personal experiences that
required emotional support (Figure 1-(a)). Once the user enters text, the
system automatically detects the emotion (①) and contextual keywords (②) from
the text, which are considered as key information used for psychological
therapy assessment (Ehrenreich et al., 2007). Specifically, the system
recommends (i) one most likely representative emotion and (ii) up to three
most salient contextual keywords. In this process, to ensure user agency, the
user may also enter their own customized keywords.

For recommending the representative emotion, pre-trained model classifies
texts in one among six emotions (anger, sadness, happiness, surprise, fear,
and distress). Specifically, this taxonomy follows six basic emotions
suggested by Ekman (Ekman, 1999), except for ‘disgust’ altered by ‘distress’
to cover as many topics in OMHCs as possible. Since we deployed model
structure to be generalizable to other emotion sets as well, we believe our
emotion categorization can easily be altered to other sets of emotions.
Detailed training processes and the usage of models are described in Section
3.3.

####  3.2.2. Reacting to the post

To assist support providers to react empathetically, we decided to use EPITOME
(Sharma et al., 2020), a text-based empathic reaction framework. Devised
through the collaboration with psychologists and synthesizing existing
literature, EPITOME is conceptualized as follows: (i) emotional reaction
(expressing emotional phrases of their own), (ii) interpretation (delivering
understandings based on seekers’ writing), and (iii) exploration (exploring
information not stated in the post). In this process, by leveraging emotion
detected and saved from the writing process, we decided to support (i)
emotional reaction as a form of trigger, with the rest of the processes guided
by the prompts (Figure 1-(c)).

Specifically, we designed our system to recommend providers with three
representative phrases to start their reaction with (trigger; ③). To collect
the phrases suitable for users to start with, we first recruited a mental
health counselor from the university, who has more than 2 years of counseling
experience. Then, we asked them to recommend three short phrases of emotional
reaction targeted to each emotion, along with two phrases that can be used for
every negative emotion. After gathering all the responses, we designed the
system to randomly choose and present two among three targeted phrases based
on the detected emotion of the post, and one between two generalized emotions.

Once the user initiates writing a comment by selecting one of the recommended
phrases (emotional reaction), they can start entering their own feedback
regarding the post based on the prompts offered (④). Here, the system offers
prompts that guide users to (i) interpret the situation with their own words
(interpretation) and then (ii) throw a question about the post (exploration),
consecutively. Each parenthesis above indicates the element of EPITOME (Sharma
et al., 2020), the empathic reaction framework we decided to utilize.

####  3.2.3. Exploring posts

In our interface, posts are sorted by time by default. Here, filtering tags
located on top of the interface can be used to filter posts including specific
emotions or contexts. Each of the emotion/context tag collections consists of
tags collected from entire posts and once tags are selected, the system only
shows posts containing more than one selected tag until entire tags are
deselected.

###  3.3. Model Selection

####  3.3.1. Emotion detection

For building an emotion classification model, we utilized ELECTRA (Clark et
al., 2020), a state-of-the-art language training model, with fine-tuning
targeted to the main language of our interface (Korean) (Kim, 2020). Then, we
trained using Korean emotional dialogue corpus (Hub, 2020), a dataset used for
the training of the model that includes 270K transcribed sentences of
emotional dialogues. Among various emotions available for the training, data
that fall under six basic emotions that we previously defined were used to
train the model. As such, we obtained 4.6K sentences and split them into
approximately $8:1$ ratio for training and validation purposes. Then, we
trained the model with a learning rate of 1E-5 for 3 epochs with the batch
size of $N=16$, where we gained the validation set accuracy of 0.7083.

####  3.3.2. Keyword extraction

For the contextual keyword extraction, we used KRWordRank (Kim et al., 2014),
a WordRank-based unsupervised Korean word extraction method. Once the keyword
extraction begins, the system triggers the WordRank-based algorithm (Mihalcea
and Tarau, 2004) by default to extract the maximum of 3 keywords from the
posting. In this process, if the algorithm fails to extract the keyword, the
system performs lemmatization using Okt (Ryu, 2018), a Korean text processor
specialized in community postings. For both WordRank and lemmatization, we
predefined unnecessary stopwords (e.g. today, me) and have those tasks drop
stopwords in order to enhance the quality of recommendations.

Our system was developed as a mobile app targeted to iOS environment. The
server for our detection models was deployed on AWS EC2, and user responses
are programmed to be saved on Google Firebase with an anonymous ID.

##  4\. Preliminary User Study

The goal of this study is to identify the feasibility and possible
enhancements of our proposed AI-assisted emotional support workflow. As such,
we ran a preliminary user study with 10 participants using the system we
designed.

###  4.1. Methods

####  4.1.1. Participants

From two major college online communities in South Korea, we recruited
participants who self-reported that they often required emotional support and
share their emotions in an online medium such as OMHCs or social media. Since
we designed and deployed our system targeted to mobile settings, we screened
users with the self-reported daily usage of smartphones of at least 4 hours on
average for ensuring their familiarity with mobile settings. In addition,
people who are currently in the process of mental therapy were excluded to
avoid conflict with it. As such, 10 participants were recruited
(M${}_{\text{age}}$ = 22.6, SD${}_{\text{age}}$ = 1.84; 6 female).

####  4.1.2. Study procedure

In order to compare our system with the existing mental health community
settings, we designed a control interface, whose interface is identical except
for the recommendation functionalities offered while users write, explore, and
comment on the posts. Then, we ran a within-subject design study where 5
participants were assigned to use control interface and our interface
consecutively, and reversed order for the 5 rest participants, to control the
order effect. The overall procedure is illustrated in Figure 2.

Figure 2. Research procedure of our study. 10 participants were randomly
assigned to two conditions and used our/control interfaces, followed by the
survey and interview sessions. Once the study for 10 participants are all
complete, their responses are passed to mental health counselor to rate their
quality

First, we briefly introduced the research objectives and process, and the
participants were given a link that guides them to install our prototype
application. In each condition, the participants were requested to (i) write
two new posts about anxious situations they often faced in daily life, and
(ii) write reactions to two of others’ posts. For users to follow (ii), we
showed participants with pre-defined posts by crawling existing posts from
Korean OMHC after removing all the sensitive information. None of the
participants’ data was reused for the purpose of another participant’s study.

Each participant was then asked to complete the survey (5-point Likert scale;
1 = strongly disagree, 5 = strongly agree) regarding (i) the ease of using the
interface for both our and control system and (ii) satisfaction with the
assist of AI for each interaction. Then, we ran a semi-structured interview
session to collect their overall experiences of using our system and its
future enhancements. Each interview response was transcribed and later open-
coded.

Once the study procedures for every participant were complete, each of the
participants was compensated with 15,000 KRW (12 USD equivalent). Then, we
passed their response data to university mental health counselor and had them
evaluate posts and comments based on the following metric on a 5-point Likert
scale, respectively: (i) how much does the post draw emotionally supportive
reaction? (ii) how much does the comment show empathy toward the post? In this
process, to mitigate bias stemming from the interface design, we had the
counselor evaluate them only by viewing texts without showing any interface
element.

####  4.1.3. Privacy, ethics, and disclosure

Our system asks users to take note of their emotional contexts, which are
later passed to the mental health counselor for the purpose of evaluation.
Thus, we considered it particularly important to strictly ensure their privacy
during the study. As such, we formulated and followed the following
precautionary steps: (i) we had our study approved by the university IRB and
strictly followed the approved procedure, (ii) all the privacy-sensitive
information (e.g., name, affiliation) in transcription and user-provided data
was masked once collected.

###  4.2. Results

Figure 3. Survey of our study. (a) Users find it less burdensome to use our
interface compared to the control interface, except for the posting. (b) AI-
assisted posting (i.e., keyword recommendation), reacting (i.e., showing
triggers/prompts to assist reaction), and exploring posts (filtering posts
with keywords collected from posts) had the average satisfaction score of 4.0
or above

####  4.2.1. Writing a post

Our system induces users to view the detected emotion and contextual keywords
and lets them iterate writing based on them. Regarding such an approach,
participants offered several positive feedback:

  * •

Clarifying emotion and circumstance one faced: First of all, participants
responded that our system helped them become aware of the emotion and
circumstances they face, before writing down the post: “When I was pretending
as if I was emotionally stable, the system elicited my emotion through leading
to iterate over, which led me to better figure out myself.” (P10)

  * •

Help express their thoughts as a form of text: In addition to clarifying the
emotion and circumstance they faced, such iterative processes of getting
feedback from AI and re-writing texts were also reported to help them the
process of converting such thoughts into writing. Since our system detects and
presents the emotion/contextual keywords and keeps updating once the post is
edited, they reported that they kept making their posts more machine-
detectably concrete: “The system led me to keep in mind that I should show
emotion and contexts in my post, leading to writing in a more readable and
structured way.” (P7)

However, analysis from the mental health counselor shows that such
concreteness was not directed to the increase of the level of inducing
empathetic reaction ($t=-0.1787$, $p=0.43$). Thus, further design
consideration is required to lead the concreteness of the post to inducing an
empathetic reaction.

  * •

Emotional reliance toward the system: Interestingly, some participants
reported that they could emotionally rely on the system. In other words, by
viewing that our system actively suggests the emotion/contextual keywords as
well as the way of reaction, they felt that the system listened to and would
help them in terms of their emotional challenges. They anticipated that this
would make them feel relieved in real-life emotional challenges: “Seeing the
recommendations, I felt like the system would solve the emotional issue I
faced.” (P4)

On top of the potential of our post-writing interface, we also identified
several limitations of our system as follows:

  * •

Increased awareness of others: One participant showed the concern of being
more aware of other people. Since our interaction unit is emotion and
contextual keywords curated the seeker, the participant reported they became
more aware of the others while writing the post: “I wrote carefully because I
thought other people will look at my tags and consume them.” (P1)

  * •

Reluctance toward a feeling of being ‘diagnosed’ by AI: In addition to the
increased awareness toward others, one user pointed out the feeling of being
‘diagnosed’ by the emotion suggested by our system. Specifically, by having
their emotion detected by the model, the system may require the user to become
unexpectedly sober in terms of their emotion: “When writing a post,
unintentional ‘diagnosis’ might be made through processes such as emotional
judgment, so I thought the room for own deliberation had been decreased.”
(P10)

####  4.2.2. Reacting to the post

From the analysis of mental health counselor, comments from our system
($M=4.55$, $SD=0.51$) presented significant improvement in terms of empathy
compared to comments from control ($M=3.25$, $SD=0.92$; $t=-4.4316$,
$p<0.0001$). In addition to possible attributes that might have led to such
enhancement, we describe several benefits that our interface gave to the
users:

  * •

Opportunity to learn how to empathize with others: Along with the comment
trigger for emotional reaction, our system provides prompts that induce
providers to include interpretation and exploration of the post, which results
in the lowered burden of commenting. This acted as a ‘learning tool’ which
helped people, who originally had difficulty, learn how to react
empathetically: “I had been unfamiliar with how to react to such harsh
feelings, but using this system, I could learn it through the trigger and
guidance of the system.” (P2)

  * •

Fostering a healthy community: Considering the vulnerability of users in
OHMCs, it is crucial to prevent harmful feedback or comments to keep OMHCs
safe (Chancellor et al., 2016). In this study, participants responded that the
AI-assisted reaction process of our system has the potential of inducing a
healthy feedback process and maintaining the community safer: “By viewing how
the technology (triggers/prompts for reaction) offers candidates for
neutralized expressions, I thought that people in (online mental health)
communities may reduce their interpersonal conflicts.” (P5)

  * •

Improved understanding towards posts: Participants reported that explicitly
revealed emotional/contextual information enabled them to easily relate to a
post and also helped deepen their understanding about the post: “Starting by
understanding overall contexts of the post, I could better understand and
empathize the post.” (P7)

At the same time, we could also gain several challenges and enhancements
regarding the reaction process:

  * •

Limiting the style of overall supports: Although the commenting process of our
system has elicited a positive range of responses for most of the
participants, one participant (P9) expressed concern about the process of
‘regularizing’ reactions. They worried if our reaction-support method would
make the overall comments in a too empathetic way, leading to reduce benefits
from the reaction with opposite views or emotions (e.g., energetic): “When I
have a hard time, I sometimes want to meet a lively person to get better (…) I
was worried if most of the reactions (in our interface) would overflow only
with the empathy toward sadness.” (P9)

  * •

Burden of comment-writing process: The survey result for the AI-assisted
commenting system was shown positive. Still, two of the participants (P7, P9)
reported that the interactions patterns might be burdensome in terms of their
long-term usage since it requires users to follow the prompts: “Overall load
of clicking one of the triggers and following prompts was pretty high for me.”
(P7)

####  4.2.3. Exploring posts

Lastly, participants showed unanimity on the efficiency of the filtering
system in terms of exploring posts. Taking into account that the tag-based
filtering system is commonly found in online communities, such positive
feedback implies the feasibility of integrating our AI-driven keyword
suggestions into the existing filtering system in OMHCs: “I usually look for
articles that have similar situations/emotions to me (…) it was convenient
curating the posts that have similar emotions like mine.” (P9)

##  5\. Discussion & Limitation

Throughout the study, we could identify the feasibility of AI-assisted writing
processes in supporting interpersonal interactions in OMHCs, as well as its
feasibility of forming a healthier community environment. Specifically,
leveraging AI-driven emotion/contextual keyword elicitation was reported to
induce seekers to clarify expression for AI to better understand, ultimately
assisting the writing process to be more concrete. However, from the expert
analysis, we realized that such concreteness did not necessarily lead to
emotional support. Thus, further design iteration to connect from such
concreteness to inducing empathetic reaction would be required.

Furthermore, our system was reported to assist the support providers by
assisting them to learn how to react empathetically. Considering that such an
empathetic reaction is a key to thriving OMHCs (Kushner and Sharma, 2020), we
could see the feasibility of extending our work to long-term deployment
settings. However, considering that some participants worried if our community
setting might only be filled with monotonous, emotionally empathetic
reactions, it is would also be beneficial to diversify the type of reactions
(e.g., supporting energetic reactions) to enrich the community environment.

Still, our study presents several limitations. First, we conducted our study
only with 10 participants. Thus, additional participants may be required in
terms of generalizability. Second, this study was run in a lab-based setting,
where participants were asked to follow designated actions. To collect a more
lively experience of users in OMHCs aided by the AI-assisted writing process,
a deployment study in the wild might be needed.

##  6\. Conclusion

In this work, we presented an AI-infused writing workflow in OMHCs that
penetrates from the seeker’s posting to the provider’s reaction. Leveraging AI
models, our system recommends emotional/contextual information from postings
and provides extra cues that help people to provide more empathetic responses
based on their understandings toward seeker’s personal experiences. Through
the preliminary user study, we could identify the feasibility of AI-assisted
emotional support processes in fostering healthy OMHCs, as well as their
enhancements.

###### Acknowledgements.

This work has been led by the first author in the midst of his graduate school
application season, and it would have been impossible to finalize this work
without the assistance from many people on his applications. Thus, we would
like to genuinely thank Jaeyoon Song, Wesley Deng, Katelyn Morrison, Joanne
Ma, Soomin Kim, and many others for their invaluable help on Donghoon’s road
to his graduate studies. This work was supported by SNU Undergraduate Research
Program through the Faculty of Liberal Education, Seoul National University
(2021-05).

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