aainstitutetext: Department of Physics, Institute of Theoretical Physics,
Beijing Normal University,
Beijing, 100875, Chinabbinstitutetext: Institute of Theoretical Physics,
Zhanjiang Normal University,
Zhanjiang, Guangdong, 524048, China

# Phase transitions, geometrothermodynamics and critical exponents of black
holes with conformal anomaly

Jie-Xiong Mo a,1 Wen-Biao Liu 111Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Conformal anomaly is an important concept which has various applications in
quantum field theory in curved space-time, string theory, black hole physics
and cosmology. Probing its influences in phase transitions of black holes is
of great physical importance. In this paper, we achieve this goal by
investigating the phase transitions of black holes with conformal anomaly in
canonical ensemble from different perspectives. Some interesting and novel
phase transition phenomena has been discovered. Firstly, we discuss the
behavior of the specific heat and the inverse of the isothermal
compressibility. It is shown that there are striking differences in Hawking
temperature and phase structure between black holes with conformal anomaly and
those without it. In the case with conformal anomaly, there exists local
minimum temperature corresponding to the phase transition point; Phase
transitions take place not only from an unstable large black hole to a locally
stable medium black hole but also from an unstable medium black hole to a
locally stable small black hole. Secondly, we probe in details the dependence
of phase transitions on the choice of parameters. The results show that black
holes with conformal anomaly have much richer phase structure than those
without it. There would be two, only one or no phase transition points
depending on the parameters we have chosen. The corresponding parameter region
are derived both numerically and graphically. Thirdly, geometrothermodynamics
are built up to examine the phase structure we have discovered. It is shown
that Legendre invariant thermodynamic scalar curvature diverges exactly where
the specific heat diverges. Furthermore, critical behaviors are investigated
by calculating the relevant critical exponents. And we proved that these
critical exponents satisfy the thermodynamic scaling laws, leading to the
conclusion that critical exponents and the scaling laws do not change even
when we consider conformal anomaly.

##  1 Introduction

Black hole thermodynamics has long been one of exciting and challenging
research fields ever since the pioneer research made by Bekenstein and Hawking
Bekenstein -Hawking1 . A variety of thermodynamic quantities of black holes
has been studied. In 1983, Hawking and Page Hawking2  discovered that pure
thermal radiation in AdS space becomes unstable above certain temperature and
collapses to form black holes. This is the well-known Hawking-Page phase
transition which describes the phase transition between the Schwarzschild AdS
black hole and the thermal AdS space. This phenomenon can be interpreted in
the AdS/CFT correspondence Maldacena9999  as the confinement/deconfinement
phase transition of gauge field Witten9999 . Since then, phase transitions of
black holes have been investigated from different perspectives. For recent
papers, see Sahay -Wenbiao1 .

One of the elegant approach is the thermodynamic geometry method, which was
first introduced by Weinhold Weinhold  and Ruppeiner Ruppeiner . Weinhold
proposed metric structure in the energy representation as
$g_{i,j}^{W}=\partial_{i}\partial_{j}M(U,N^{a})$ while Ruppeiner defined
metric structure as $g_{i,j}^{R}=-\partial_{i}\partial_{j}S(U,N^{a})$. These
metric structures are respectively the Hessian matrix of the internal energy
$U$ and the entropy $S$ with respect to the extensive thermodynamic variables
$N^{a}$. And Weinhold’s metrics were found to be conformally connected to
Ruppeiner’s metrics through the map $dS^{2}_{R}=\frac{dS^{2}_{W}}{T}$ Janyszek
. Ruppeiner’s metric has been applied to investigate various thermodynamics
systems for its profound physical meaning. For more details, see the nice
review Ruppeiner2 . Recently Quevedo et al. Quevedo2  presented a new
formalism called geometrothermodynamics, which allows us to derive Legendre
invariant metrics in the space of equilibrium states. Geometrothermodynamics
presents a unified geometry where the metric structure describes various types
of black hole thermodynamics Quevedo3 -Wenbiao2 .

Apart from the thermodynamic geometry, critical behavior also plays a crucial
role in the study of black hole phase transitions. The critical points of
phase transitions are characterized by the discontinuity of thermodynamic
quantities. So it is important to investigate the behavior in the neighborhood
of the critical point, especially the divergences of various thermodynamic
quantities. In classical thermodynamics, this goal is achieved by taking into
account a set of critical exponents, from which we can gain qualitative
insights into the critical behavior. These critical exponents are found to be
universal to a large extent (only depending on the dimensionality, symmetry
etc) and satisfy scaling laws, which can be attributed to scaling hypothesis.
Critical behavior of black holes accompanied with their critical exponents
have been investigated not only in asymptotically flat space time Davies
-Arcioni1  but also in the de Sitter and anti de Sitter space Muniain -Liu99 .

In this paper, we would like to focus our attention on the critical behavior
and geometrothermodynamics of static and spherically symmetric black holes
with conformal anomaly. As we know, conformal anomaly, an important concept
with a long history, has various applications in quantum field theory in
curved spaces, string theory, black hole physics and cosmology. So it is worth
probing its influences in phase transitions of black holes. Recently, Cai et
al. Cai9999  found a class of static and spherically symmetric black holes
with conformal anomaly, whose thermodynamic quantities were also investigated
in the same paper. It was found that there exists a logarithmic correction to
the well-known Bekenstein-Hawking area entropy. This discovery is quite
important in the sense that with this term one is able to compare black hole
entropy up to the sub-leading order, in the gravity side and in the
microscopic statistical interpretation side Cai9999 . Based on the metrics in
that paper, phase transitions of a spherically symmetric Schwarzschild black
hole have been investigated by taking into account the back reaction through
the conformal anomaly of matter fields recently Son9999 . It has been shown
that there exists an additional phase transition to the conventional Hawking-
Page phase transition. The entropy of these black holes has also been
investigated by using quantum tunneling approach Liran . Moreover, Ehrenfest
equation has been applied to investigate this class of black holes Chenghongbo
and it has been found that the phase transition is a second order one. Despite
of these achievements, there are still many issues left to be explored. Ref.
Son9999  mainly focus on the uncharged case . So it is natural to ask what
would happen to the charged black holes. Ref. Chenghongbo  concentrated their
efforts on the Ehrenfest equation in the grand canonical ensemble. So it is
worthwhile to study the phase transition in canonical ensemble. The dependence
of the phase structure on the parameter deserves to be further investigated.
One may also wonder whether the thermodynamic geometry and scaling laws still
works to reveal the phase structure and critical behavior when conformal
anomaly is taken into consideration. Motivated by these, we would like to
investigate the phase transition, geometrothermodynamics and critical
exponents in canonical ensemble.

The organization of our paper is as follows. In Sec. 2, the thermodynamics of
black holes with conformal anomaly will be briefly reviewed. In Sec. 3, phase
transitions will be investigated in details in canonical ensemble and some
interesting and novel phase transition phenomena will be disclosed. In Sec. 4,
geometrothermodynamics will be established to examine the phase structure we
find in Sec. 3. In Sec. 5, critical exponents will be calculated and the
scaling laws will be examined. In the end, conclusions will be drawn in Sec.
6.

##  2 A brief review of thermodynamics

The static and spherically symmetric black hole solution with conformal
anomaly has been proposed as Cai9999

$ds^{2}=f(r)dt^{2}-\frac{dr^{2}}{f(r)}-r^{2}(d\theta^{2}+sin^{2}\theta
d\varphi^{2}),$ (1)

where

$f(r)=1-\frac{r^{2}}{4\tilde{\alpha}}(1-\sqrt{1-\frac{16\tilde{\alpha}M}{r^{3}}+\frac{8\tilde{\alpha}Q^{2}}{r^{4}}}\,).$
(2)

The Newton constant $G$ has been set to one. Both $M$ and $Q$ are integration
constants. And the coefficient $\tilde{\alpha}$ is positive. The physical
meanings of $M$ and $Q$ were discussed in Ref. Cai9999 . $M$ is nothing but
the mass of the solution while $Q$ should be interpreted as $U(1)$ charge of
some conformal field theory.

When $M=Q=0$, the metric above reduces to

$ds^{2}=dt^{2}-dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta d\varphi^{2}),$ (3)

implying that the vacuum limit is the Minkowski space-time.

In the large $r$ limit, Eq.(2) becomes

$f(r)\approx 1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+O(r^{-2}),$ (4)

which behaves like the Reissner-Norström solution.

When $\tilde{\alpha}\rightarrow 0$, Eq.(2) reduces into

$f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}},$ (5)

Eqs.(1) and (5) consist of the metric of Reissner-Norström black hole.

Solving the equation $f(r)=0$, we can get the radius of black hole horizon
$r_{+}$, with which the mass of the black hole can be expressed as

$M=\frac{r_{+}}{2}+\frac{Q^{2}}{2r_{+}}-\frac{\tilde{\alpha}}{r_{+}}.$ (6)

The Hawking temperature can be derived as

$T=\frac{f^{\prime}(r_{+})}{4\pi}=\frac{r_{+}^{2}+2\tilde{\alpha}-Q^{2}}{4\pi
r_{+}(r_{+}^{2}-4\tilde{\alpha})}.$ (7)

The potential difference between the horizon and the infinity can be written
as

$\Phi=\frac{Q}{r_{+}}.$ (8)

The entropy was reviewed in Ref. Chenghongbo  as

$S=\pi r_{+}^{2}-4\pi\tilde{\alpha}ln{r_{+}^{2}}.$ (9)

##  3 Novel phase transition phenomena

In this section, we would like to investigate the phase transition of black
holes with conformal anomaly in canonical ensemble where the charge of the
black hole is fixed.

The corresponding specific heat can be calculated as

$C_{Q}=T(\frac{\partial S}{\partial
T})_{Q}=\frac{2\pi(r_{+}^{2}-4\tilde{\alpha})^{2}(Q^{2}-2\tilde{\alpha}-r_{+}^{2})}{r_{+}^{4}-8\tilde{\alpha}^{2}+10r_{+}^{2}\tilde{\alpha}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})}.$
(10)

Apparently, $C_{Q}$ may diverge when

$r_{+}^{4}-8\tilde{\alpha}^{2}+10r_{+}^{2}\tilde{\alpha}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})=0,$
(11)

which suggests a possible phase transition. However, the phase transition
point characterized by Eq.(11) is not apparent. To gain an intuitive
understanding, we plot Fig.1 using Eq.(10). To check whether the phase
transition point locates in the physical region, we also plot the Hawking
temperature using Eq.(7) in Fig.1. It is shown that the phase transition point
locates in the positive temperature region. From Fig.1 and Fig.1, we find that
there have been striking differences between the case $\tilde{\alpha}\neq 0$
and the case $\tilde{\alpha}=0$. In the case $Q=1,\tilde{\alpha}=0.1$, there
are two phase transition point while there is only one in the case
$\tilde{\alpha}=0$. The temperature in the case $\tilde{\alpha}=0$ increases
monotonically while there exists local minimum temperature in the case
$Q=1,\tilde{\alpha}=0.1$. Fig.1 can be divided into four phases. The first one
is thermodynamically stable ($C_{Q}>0$)with small radius. The second one is
unstable ($C_{Q}<0$)with meidium radius. The third one is thermodynamically
stable ($C_{Q}>0$)with medium radius while the fourth one is thermodynamically
unstable ($C_{Q}<0$)with large radius. So the phase transition take place not
only from an unstable large black hole to a locally stable medium black hole
but also from an unstable medium black hole to a locally stable small black
hole.

Figure 1: (a)$C_{Q}$ vs. $r_{+}$ for $Q=1,\tilde{\alpha}=0.1$ (b)$T$ vs.
$r_{+}$ for $Q=1,\tilde{\alpha}=0.1$

From Fig.1,we notice that the Hawking temperature has a local minimum value.
And the corresponding location can be derived through

$\frac{\partial T}{\partial
r_{+}}=-\frac{r_{+}^{4}-8\tilde{\alpha}^{2}+10r_{+}^{2}\tilde{\alpha}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})}{4\pi(r_{+}^{3}-4r_{+}\tilde{\alpha})^{2}}=0.$
(12)

It is quite interesting to note that the numerator of Eq.(12) is the same as
Eq.(11), which implies that the location which corresponds to the minimum
Hawking temperature also witnesses the existence of phase transition.

To probe the dependence of phase transition location on the choice of
parameter, we solve Eq.(11) and obtain the location of phase transition point
as

$r_{c}=\sqrt{\frac{3Q^{2}-10\tilde{\alpha}\pm\sqrt{132\tilde{\alpha}^{2}-76\tilde{\alpha}Q^{2}+9Q^{4}}}{2}}.$
(13)

With Eq.(13) at hand, we plot Fig.2 and Fig.2 which show the influence of
parameters $Q$ and $\tilde{\alpha}$ respectively.

Figure 2: (a)$r_{c}$ vs. $Q$ for $\tilde{\alpha}=0.1$ and $\tilde{\alpha}=0$
(b)$r_{c}$ vs. $\tilde{\alpha}$ for $Q=1$

It can be observed from Fig.2 and Fig.2 that black holes with conformal
anomaly have much richer phase structure than that without conformal anomaly.
When $\tilde{\alpha}=0$, the location of the phase transition $r_{c}$ is
proportional to the charge $Q$. However, the cases of black holes with
conformal anomaly are quite complicated. For $\tilde{\alpha}=0.1$, the curve
can be divided into three regions. Through numerical calculation, we find that
black holes have only one phase transition point when $Q\subset(0,0.4472)$.
When $0.4472<Q<0.7746$, there would be no phase transition at all. When
$Q>0.7746$, there exist two phase transition points, just as what we show in
Fig.1. And the distance between these two phase transition point becomes
larger with the increasing of $Q$. Fig.2 shows the case that the charge $Q$
has been fixed at one. We notice that there would be two phase transition
points when $0<\tilde{\alpha}<\frac{1}{6}$, which is consistent with Fig.1.
And the distance between these two phase transition point becomes narrower
with the increasing of $\tilde{\alpha}$. When
$\tilde{\alpha}\subset(\frac{1}{6},\frac{1}{2})$, there would be no phase
transition. When $\tilde{\alpha}>\frac{1}{2}$, there would be only one phase
transition point. To gain a three-dimensional understanding, we also include a
three dimensional figure of $C_{Q}$ in Fig.3 and Fig.3.

Apart from the specific heat, we would also like to investigate the behavior
of the inverse of the isothermal compressibility, which is defined as

Figure 3: (a)$C_{Q}$ vs. $Q$ and $r_{+}$ for $\tilde{\alpha}=0.1$ (b)$C_{Q}$
vs. $\tilde{\alpha}$ and $r_{+}$ for $Q=1$

$\kappa_{T}^{-1}=Q(\frac{\partial\Phi}{\partial Q})_{T}.$ (14)

Utilizing the thermodynamic identity relation

$(\frac{\partial\Phi}{\partial T})_{Q}(\frac{\partial T}{\partial
Q})_{\Phi}(\frac{\partial Q}{\partial\Phi})_{T}=-1,$ (15)

we obtain

$(\frac{\partial\Phi}{\partial Q})_{T}=-(\frac{\partial\Phi}{\partial
T})_{Q}(\frac{\partial T}{\partial Q})_{\Phi},$ (16)

where the second term on the right hand side can be calculated through

$(\frac{\partial T}{\partial Q})_{\Phi}=(\frac{\partial T}{\partial
r_{+}})_{Q}(\frac{\partial r_{+}}{\partial Q})_{\Phi}+(\frac{\partial
T}{\partial Q})_{r_{+}}.$ (17)

Utilizing Eqs.(7), (8), (14), (16), (17), we obtain the explicit form of
$\kappa_{T}^{-1}$ as

$\kappa_{T}^{-1}=\frac{Qr_{+}^{4}-Q^{3}r_{+}^{2}-4Q^{3}\tilde{\alpha}+10Qr_{+}^{2}\tilde{\alpha}-8Q\tilde{\alpha}^{2}}{r_{+}[r_{+}^{4}-8\tilde{\alpha}^{2}+10r_{+}^{2}\tilde{\alpha}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})]}.$
(18)

We show the behavior of $\kappa_{T}^{-1}$ in Fig.4. Comparing Fig.4 with
Fig.1, we find that the inverse of the isothermal compressibility
$\kappa_{T}^{-1}$ also diverges at the critical point.

Figure 4: The inverse of the isothermal compressibility $\kappa_{T}^{-1}$ vs.
$r_{+}$ for $Q=1,\tilde{\alpha}=0.1$

##  4 Geometrothermodynamics

According to geometrothermodynamics Quevedo2 , the $(2n+1)$-dimensional
thermodynamic phase space $\mathcal{T}$ can be coordinated by the set of
independent quantities {$\phi,E^{a},I^{a}$}, where $\phi$ corresponds to the
thermodynamic potential, and $E^{a},I^{a}$ are the extensive and intensive
thermodynamic variables respectively. The fundamental Gibbs 1- form defined on
$\mathcal{T}$ can then be written as $\Theta=d\phi-\delta_{ab}I^{a}dE^{b}$,
where $\delta_{ab}=diag(1,\cdots,1)$. Considering a non-degenerate Riemannian
metric $G$, a contact Riemannian manifold can be defined from the set
$(\mathcal{T},\Theta,G)$ if the condition $\Theta\wedge(\Theta)^{n}\neq 0$ is
satisfied. Utilizing a smooth map $\varphi:\varepsilon\rightarrow\mathcal{T}$,
i.e. $\varphi:(E^{a})\mapsto(\phi,E^{a},I^{a})$, a submanifold $\varepsilon$
called the space of thermodynamic equilibrium states can be induced.
Furthermore, a thermodynamic metric $g$ can be induced in the equilibrium
manifold $\varepsilon$ by the smooth map $\varphi$.

As proposed by Quevedo, the non-degenerate metric $G$ and the thermodynamic
metric $g$ can be written as follows Quevedo7

$G=(d\phi-\delta_{ab}I^{a}dE^{b})^{2}+(\delta_{ab}E^{a}I^{b})(\eta_{cd}dE^{c}dI^{d}),$
(19) $g=\varphi^{*}(G)=(E^{c}\frac{\partial\phi}{\partial
E^{c}})(\eta_{ab}\delta^{bc}\frac{\partial^{2}\phi}{\partial E^{c}\partial
E^{d}}dE^{a}dE^{d}),$ (20)

where $\eta_{ab}=diag(-1,\cdots,1)$.

To construct geometrothermodynamics of black holes with conformal anomaly in
canonical ensemble, we choose $M$ to be the thermodynamic potential and $S,Q$
to be the extensive variables. Then the corresponding thermodynamic phase
space is a 5-dimensional one coordinated by the set of independent
coordinates{$M,S,Q,T,\Phi$}. The fundamental Gibbs 1- form defined on
$\mathcal{T}$ can then be written as

$\Theta=dM-TdS-\Phi dQ.$ (21)

The non-degenerate metric $G$ from Eq.(19) can be changed into

$G=(dM-TdS-\Phi dQ)^{2}+(TS+\Phi Q)(-dSdT+dQd\Phi).$ (22)

Introducing the map

$\varphi:\\{S,Q\\}\mapsto\\{M(S,Q),S,Q,\frac{\partial M}{\partial
S},\frac{\partial M}{\partial Q}\\},$ (23)

the space of thermodynamic equilibrium states can be induced. According to
Eq.(19), the thermodynamic metric $g$ can be written as follows

$g=(S\frac{\partial M}{\partial S}+Q\frac{\partial M}{\partial
Q})(-\frac{\partial^{2}M}{\partial S^{2}}dS^{2}+\frac{\partial^{2}M}{\partial
Q^{2}}dQ^{2}).$ (24)

Utilizing Eqs.(6) and (9), we can easily calculate the relevant quantities in
Eq.(24) as

$\displaystyle\frac{\partial M}{\partial S}$
$\displaystyle=\frac{r_{+}^{2}+2\tilde{\alpha}-Q^{2}}{4\pi
r_{+}(r_{+}^{2}-4\tilde{\alpha})},$ (25) $\displaystyle\frac{\partial
M}{\partial Q}$ $\displaystyle=\frac{Q}{r_{+}},$ (26)
$\displaystyle\frac{\partial^{2}M}{\partial S^{2}}$
$\displaystyle=\frac{8\tilde{\alpha}^{2}-r_{+}^{4}-10r_{+}^{2}\tilde{\alpha}-Q^{2}(4\tilde{\alpha}-3r_{+}^{2})}{8\pi^{2}r_{+}(r_{+}^{2}-4\tilde{\alpha})^{3}},$
(27) $\displaystyle\frac{\partial^{2}M}{\partial Q^{2}}$
$\displaystyle=\frac{1}{r_{+}}.$ (28)

Comparing Eqs.(25),(26) with Eqs.(7),(8), we find

$\frac{\partial M}{\partial S}=T,\quad\frac{\partial M}{\partial Q}=\Phi,$
(29)

which proves the validity of the first law of black hole thermodynamics
$dM=TdS+\Phi dQ$. Substituting Eqs.(25)-(28) into Eq.(24), we can calculate
the component of the thermodynamic metric $g$ as

$\displaystyle g_{SS}=$
$\displaystyle\frac{1}{32\pi^{2}r_{+}^{2}(r_{+}^{2}-4\tilde{\alpha})^{4}}\times[r_{+}^{4}-8\tilde{\alpha}^{2}+10r_{+}^{2}\tilde{\alpha}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})]$
$\displaystyle\times[r_{+}^{4}+2r_{+}^{2}\tilde{\alpha}+Q^{2}(3r_{+}^{2}-16\tilde{\alpha})+8\tilde{\alpha}(Q^{2}-r_{+}^{2}-2\tilde{\alpha})ln{r_{+}}],$
(30) $\displaystyle g_{QQ}=$
$\displaystyle\frac{r_{+}^{4}-16Q^{2}\tilde{\alpha}+2r_{+}^{2}\tilde{\alpha}+3Q^{2}r_{+}^{2}+(8Q^{2}\tilde{\alpha}-8r_{+}^{2}\tilde{\alpha}-16\tilde{\alpha}^{2})ln{r_{+}}}{4r_{+}^{2}(r_{+}^{2}-4\tilde{\alpha})}.$
(31)

Utilizing Eqs.(30) and (31), we can obtain the Legendre invariant scalar
curvature as

$\mathfrak{R}_{Q}=\frac{A(x_{+},Q)}{B(x_{+},Q)},$ (32)

where

$\displaystyle B(x_{+},Q)=$
$\displaystyle[r_{+}^{4}+10r_{+}^{2}\tilde{\alpha}-8\tilde{\alpha}^{2}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})]^{2}$
$\displaystyle\times[r_{+}^{4}+Q^{2}(3r_{+}^{2}-16\tilde{\alpha})+2r_{+}^{2}\tilde{\alpha}+8\tilde{\alpha}(Q^{2}-r_{+}^{2}-2\tilde{\alpha})ln{r_{+}}]^{3}.$
(33)

Figure 5: Thermodynamic scalar curvature $R_{Q}$ vs. $r_{+}$ for
$Q=1,\tilde{\alpha}=0.1$

The numerator of the Legendre invariant scalar curvature is too lengthy to be
displayed here. From Eq.(33), we find that the Legendre invariant scalar
curvature shares the same factor
$r_{+}^{4}+10r_{+}^{2}\tilde{\alpha}-8\tilde{\alpha}^{2}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})$
with the specific heat $C_{Q}$ in its denominator , which implies that it
would diverge when
$r_{+}^{4}+10r_{+}^{2}\tilde{\alpha}-8\tilde{\alpha}^{2}+Q^{2}(4\tilde{\alpha}-3r_{+}^{2})=0$.
That is the exact condition that the phase transition point satisfies. To get
an intuitive sense on this issue, we plot Fig.5 showing the behavior of
thermodynamic scalar curvature $\mathfrak{R}_{Q}$. From Fig.5, we find that
thermodynamic scalar curvature $\mathfrak{R}_{Q}$ diverges at three locations.
Comparing Fig.5 with Fig.1, we find that the second diverging point which
corresponds to negative Hawking temperature does not have physical meaning.
Furthermore, the first and the third diverging points coincide exactly with
the phase transition point, which can be witnessed by comparing Fig.5 with
Fig.1. So we can safely draw the conclusion that the Legendre invariant metric
constructed in geometrothermodynamics correctly produces the behavior of the
thermodynamic interaction and phase transition structure of black holes with
conformal anomaly.

##  5 Critical exponents and scaling laws

In order to have a better understanding of the phase transition of black holes
with conformal anomaly, we would like to investigate their critical behavior
near the critical point by considering a set of critical exponents in this
section.

Before embarking on calculating critical exponents, we would like to reexpress
physical quantities near the critical point as

$\displaystyle r_{+}$ $\displaystyle=r_{c}(1+\Delta),$ (34) $\displaystyle
T(r_{+})$ $\displaystyle=T_{c}(1+\varepsilon),$ (35) $\displaystyle Q(r_{+})$
$\displaystyle=Q_{c}(1+\eta),$ (36)

where $|\Delta|\ll 1,|\varepsilon|\ll 1,|\eta|\ll 1$. Note that the footnote
”c” in this section denotes the value of the physical quantity (or the
expression) at the critical point. For example, $T_{c}$ corresponds to the
temperature at the critical point.

Critical exponent $\alpha$ is defined through

$C_{Q}\sim|T-T_{c}|^{-\alpha}.$ (37)

To obtain $T-T_{c}$, we would like to carry out Taylor expansion as below

$T(r_{+})=T_{c}+[(\frac{\partial T}{\partial
r_{+}})_{Q=Q_{c}}]_{r_{+}=r_{c}}(r_{+}-r_{c})+\frac{1}{2}[(\frac{\partial^{2}T}{\partial
r_{+}^{2}})_{Q=Q_{c}}]_{r_{+}=r_{c}}(r_{+}-r_{c})^{2}+higher\,order\,terms,$
(38)

from which we obtain

$\Delta=\frac{1}{r_{c}}\sqrt{\frac{2\varepsilon T_{c}}{D}},$ (39)

where

$D=[(\frac{\partial^{2}T}{\partial
r_{+}^{2}})_{Q=Q_{c}}]_{r_{+}=r_{c}}=\frac{r_{c}^{6}+24r_{c}^{4}\tilde{\alpha}-24r_{c}^{2}\tilde{\alpha}^{2}+32\tilde{\alpha}^{3}-2Q_{c}^{2}(3r_{c}^{4}-6r_{c}^{2}\tilde{\alpha}+8\tilde{\alpha}^{2})}{2\pi(r_{c}^{3}-4r_{c}\tilde{\alpha})^{3}}.$
(40)

In the above derivation, we have considered the fact that $C_{Q}$ diverges at
the critical point, making the second term at the right hand side of Eq.(38)
vanish. Substituting Eq.(34) into Eq.(10) and keeping only the linear terms in
its denominator, we obtain

$C_{Q}\simeq\frac{2\pi(r_{c}^{2}-4\tilde{\alpha})^{2}(Q_{c}^{2}-2\tilde{\alpha}-r_{c}^{2})}{\Delta(4r_{c}^{4}+20r_{c}^{2}\tilde{\alpha}-6Q_{c}^{2}r_{c}^{2})},$
(41)

which can be transformed via Eq.(39) into

$C_{Q}\simeq\frac{\pi\sqrt{2D}(r_{c}^{2}-4\tilde{\alpha})^{2}(Q_{c}^{2}-2\tilde{\alpha}-r_{c}^{2})}{(4r_{c}^{3}+20r_{c}\tilde{\alpha}-6Q_{c}^{2}r_{c})(T-T_{c})^{1/2}},$
(42)

Comparing Eq.(42) with Eq.(37), we can obtain $\alpha=1/2$.

Critical exponent $\beta$ is defined through the following relation when $Q$
is fixed,

$\Phi(r_{+})-\Phi(r_{c})\sim|T-T_{c}|^{\beta}.$ (43)

The above definition motivates us to carry out the Taylor expansion as

$\Phi(r_{+})=\Phi_{c}+[(\frac{\partial\Phi}{\partial
r_{+}})_{Q=Q_{c}}]_{r_{+}=r_{c}}(r_{+}-r_{c})+higher\,order\,terms,$ (44)

Utilizing Eq.(8), (44) and neglecting higher order terms of Eq.(43), we get

$\Phi(r_{+})-\Phi_{c}=[(\frac{\partial\Phi}{\partial
r_{+}})_{Q=Q_{c}}]_{r_{+}=r_{c}}\sqrt{\frac{2}{D}}(T-T_{c})^{1/2}=-\frac{Q_{c}}{r_{c}^{2}}\sqrt{\frac{2}{D}}(T-T_{c})^{1/2}.$
(45)

Comparing Eq.(43) with Eq.(45), we can obtain $\beta=1/2$.

Critical exponent $\gamma$ is defined through the following relation

$\kappa_{T}^{-1}\sim|T-T_{c}|^{-\gamma}.$ (46)

Substituting Eq.(34) and (39) into Eq.(18) and keeping only the linear term of
$\Delta$, we obtain

$\kappa_{T}^{-1}=\frac{\sqrt{D}(Q_{c}r_{c}^{4}-Q_{c}^{3}r_{c}^{2}-4Q_{c}^{3}\tilde{\alpha}+10Q_{c}r_{c}^{2}\tilde{\alpha}-8Q_{c}\tilde{\alpha}^{2})}{\sqrt{2}[5r_{c}^{4}-8\tilde{\alpha}^{2}+30r_{c}^{2}\tilde{\alpha}+Q_{c}^{2}(4\tilde{\alpha}-9r_{c}^{2})](T-T_{c})^{\frac{1}{2}}}.$
(47)

From Eq.(46) and (47), we find that $\gamma=1/2$

Critical exponent $\delta$ is defined for the fixed temperature $T_{c}$
through

$\Phi(r_{+})-\Phi(r_{c})\sim|Q-Q_{c}|^{1/\delta}.$ (48)

To obtain $Q-Q_{c}$, we would like to carry out Taylor expansion as

$Q(r_{+})=Q_{c}+[(\frac{\partial Q}{\partial
r_{+}})_{T}]_{r_{+}=r_{c}}(r_{+}-r_{c})+\frac{1}{2}[(\frac{\partial^{2}Q}{\partial
r_{+}^{2}})_{T}]_{r_{+}=r_{c}}(r_{+}-r_{c})^{2}+higher\,order\,terms,$ (49)

Utilizing the thermodynamic identity again, we get

$[(\frac{\partial Q}{\partial r_{+}})_{T}]_{r_{+}=r_{c}}=-[(\frac{\partial
T}{\partial r_{+}})_{Q}]_{r_{+}=r_{c}}[(\frac{\partial Q}{\partial
T})_{r_{+}}]_{r_{+}=r_{c}}=0.$ (50)

In the above derivation, we have taken into account the fact that $C_{Q}$
diverges at the critical point, making the first term at the right hand side
of Eq.(38) vanish. Substituting Eq.(34) and Eq.(36) into Eq.(49)and neglecting
the high order terms, we obtain

$\Delta=\sqrt{\frac{2Q_{c}\eta}{Er_{c}^{2}}},$ (51)

where

$E=[(\frac{\partial^{2}Q}{\partial
r_{+}^{2}})_{T}]_{r_{+}=r_{c}}=\frac{22r_{c}^{4}\tilde{\alpha}+32\tilde{\alpha}^{3}+16\tilde{\alpha}^{2}(r_{c}^{2}-Q_{c}^{2})-r_{c}^{4}(3Q_{c}^{2}+r_{c}^{2})}{2Q_{c}(r_{c}^{3}-4r_{c}\tilde{\alpha})^{2}}.$
(52)

Taylor expanding $\Phi$ near the critical point, we get

$\Phi(r_{+})=\Phi_{c}+[(\frac{\partial\Phi}{\partial
r_{+}})_{T}]_{r_{+}=r_{c}}(r_{+}-r_{c})+higher\,order\,terms,$ (53)

where the coefficient of the second term on the right hand side can be derived
as follows

$[(\frac{\partial\Phi}{\partial r_{+}})_{T}]_{r_{+}=r_{c}}=[(\frac{\partial
Q}{\partial r_{+}})_{T}]_{r_{+}=r_{c}}[(\frac{\partial\Phi}{\partial
Q})_{r_{+}}]_{r_{+}=r_{c}}+[(\frac{\partial\Phi}{\partial
r_{+}})_{Q}]_{r_{+}=r_{c}}=-\frac{Q_{c}}{r_{c}^{2}}.$ (54)

Utilizing Eq.(51), (53), (54),we get

$\Phi(r_{+})-\Phi_{c}\simeq-\frac{Q_{c}}{r_{c}^{2}}\sqrt{\frac{2(Q-Q_{c})}{E}},$
(55)

from which we can draw the conclusion that $\delta=2$.

Critical exponent $\varphi$ is defined through

$C_{Q}\sim|Q-Q_{c}|^{-\varphi}.$ (56)

Substituting Eq.(51) into Eq.(41), we obtain

$C_{Q}\simeq\frac{\pi
r_{c}\sqrt{2E}(r_{c}^{2}-4\tilde{\alpha})^{2}(Q_{c}^{2}-2\tilde{\alpha}-r_{c}^{2})}{\sqrt{Q-Q_{c}}(4r_{c}^{4}+20r_{c}^{2}\tilde{\alpha}-6Q_{c}^{2}r_{c}^{2})},$
(57)

Comparing Eq.(57) and (56), we find that $\varphi=1/2$.

Critical exponent $\psi$ is defined through

$S(r_{+})-S_{c}\sim|Q-Q_{c}|^{\psi}.$ (58)

Taylor expanding $S$ near the critical point, we obtain

$S(r_{+})=S_{c}+[(\frac{\partial S}{\partial
r_{+}})_{Q}]_{r_{+}=r_{c}}(r_{+}-r_{c})+higher\,order\,terms.$ (59)

Utilizing Eq.(9), (34), (51) and (59), we get

$S(r_{+})-S_{c}\simeq(2\pi
r_{c}-\frac{8\pi\tilde{\alpha}}{r_{c}})\sqrt{\frac{2(Q-Q_{c})}{E}},$ (60)

from which we obtain $\psi=1/2$.

Till now, we have finished the calculations of six critical exponents. They
are also equal to $1/2$ except $\delta=2$. Our results are in accordance with
those in classical thermodynamics. And it can be easily proved that critical
exponents we obtain in our paper satisfy the following thermodynamic scaling
laws

$\displaystyle\alpha+2\beta+\gamma=2,\,\alpha+\beta(\delta+1)=2,\,(2-\alpha)(\delta\psi-1)+1=(1-\alpha)\delta,$
$\displaystyle\gamma(\delta+1)=(2-\alpha)(\delta-1),\,\gamma=\beta(\delta-1),\,\varphi+2\psi-\delta^{-1}=1.$
(61)

##  6 Conclusions

The phase transition of black holes with conformal anomaly has been
investigated in canonical ensemble. Firstly, we calculate the relevant
thermodynamic quantities and discuss the behavior of the specific heat at
constant charge. We find that there have been striking differences between
black holes with conformal anomaly and those without conformal anomaly. In the
case $Q=1,\tilde{\alpha}=0.1$, there are two phase transition point while
there is only one in the case $\tilde{\alpha}=0$. The temperature in the case
$\tilde{\alpha}=0$ increases monotonically while there exists local minimum
temperature in the case $Q=1,\tilde{\alpha}=0.1$. This local minimum
temperature corresponds to the phase transition point. We also find that the
phase transitions of black holes with conformal anomaly take place not only
from an unstable large black hole to a locally stable medium black hole but
also from an unstable medium black hole to a locally stable small black hole.
We also study the behavior of the inverse of the isothermal compressibility
$\kappa_{T}^{-1}$ and find that $\kappa_{T}^{-1}$ also diverges at the
critical point.

Secondly, we probe the dependence of phase transitions on the choice of
parameters. The results show that black holes with conformal anomaly have much
richer phase structure than that without conformal anomaly. When
$\tilde{\alpha}=0$, the location of the phase transition $r_{c}$ is
proportional to the charge $Q$. By contrast, the case of black holes with
conformal anomaly is more complicated. For $\tilde{\alpha}=0.1$, the curve can
be divided into three regions. Through numerical calculation, we find that
black holes has only one phase transition point when $Q\subset(0,0.4472)$.
When $0.4472<Q<0.7746$, there would be no phase transition at all. When
$Q>0.7746$, there exist two phase transition points. And the distance between
these two phase transition points becomes larger with the increasing of $Q$.
In the case that the charge $Q$ has been fixed at one, we notice that there
would be two phase transition point when $0<\tilde{\alpha}<\frac{1}{6}$. And
the distance between these two phase transition points becomes narrower with
the increasing of $\tilde{\alpha}$. When
$\tilde{\alpha}\subset(\frac{1}{6},\frac{1}{2})$, there would be no phase
transition. When $\tilde{\alpha}>\frac{1}{2}$, there would be only one phase
transition point.

Thirdly, we build up geometrothermodynamics in canonical ensembles. We choose
$M$ to be the thermodynamic potential and build up both thermodynamic phase
space and the space of thermodynamic equilibrium states. We also calculate the
Legendre invariant thermodynamic scalar curvature and depict its behavior
graphically. It is shown that Legendre invariant thermodynamic scalar
curvature diverges exactly where the specific heat diverges. Based on this, we
can safely conclude that the Legendre invariant metrics constructed in
geometrothermodynamics can correctly produce the behavior of the thermodynamic
interaction and phase transition structure even when conformal anomaly is
taken into account.

Furthermore, we calculate the relevant critical exponents. They are also equal
to $1/2$ except $\delta=2$. Our results are in accordance with those of other
black holes. And it has been proved that critical exponents we obtain in our
paper satisfy the thermodynamic scaling laws. We conclude that the critical
exponents and the scaling laws do not change even when we consider conformal
anomaly. This may be attributed to the mean field theory.

###### Acknowledgements.

This research is supported by the National Natural Science Foundation of China
(Grant Nos.11235003, 11175019, 11178007). It is also supported by ”Thousand
Hundred Ten” project of Guangdong Province and Natural Science Foundation of
Zhanjiang Normal University under Grant No. QL1104.

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# Appropriateness of LLM-equipped Robotic Well-being Coach Language in the
Workplace: A Qualitative Evaluation

Micol Spitale1,2, Minja Axelsson1 and Hatice Gunes1 1 Department of Computer
Science and Technology, University of Cambridge, UK, 2 Department of
Electronic, Information, and Bio-engineering, Politecnico di Milano, Italy.
<EMAIL_ADDRESS>
###### Abstract

Robotic coaches have been recently investigated to promote mental well-being
in various contexts such as workplaces and homes. With the widespread use of
Large Language Models (LLMs), HRI researchers are called to consider language
appropriateness when using such generated language for robotic mental well-
being coaches in the real world. Therefore, this paper presents the first work
that investigated the language appropriateness of robot mental well-being
coach in the workplace. To this end, we conducted an empirical study that
involved 17 employees who interacted over 4 weeks with a robotic mental well-
being coach equipped with LLM-based capabilities. After the study, we
individually interviewed them and we conducted a focus group of 1.5 hours with
11 of them. The focus group consisted of: i) an ice-breaking activity, ii)
evaluation of robotic coach language appropriateness in various scenarios, and
iii) listing should _s_ and shouldn’t _s_ for designing appropriate robotic
coach language for mental well-being. From our qualitative evaluation, we
found that a language-appropriate robotic coach should (1) ask deep questions
which explore feelings of the coachees, rather than superficial questions, (2)
express and show emotional and empathic understanding of the context, and (3)
not make any assumptions without clarifying with follow-up questions to avoid
bias and stereotyping. These results can inform the design of language-
appropriate robotic coach to promote mental well-being in real-world contexts.

##  I INTRODUCTION

The last year has been characterised by a ground-breaking advancement in Large
Language Models (LLMs) that has revolutionised several research fields [1, 2],
including Human-Robot Interaction (HRI) research [3]. Robots have been
increasingly introduced in sensitive application contexts – such as mental
health [4, 5], therapy [6, 7], and elderly care [8]. In such sensitive
contexts, designing appropriate robot language is extremely important. As HRI
researchers, we are called to consider the appropriateness of using LLM-
generated language in robots deployed in the real world, and the social and
ethical implications of their language when interacting with people.

Within the HRI literature, robotic well-being coaches have been recently
investigated with the aim of promoting mental well-being in homes [9],
university accommodations[10], and workplaces [4, 11]. However, the
appropriateness of these robotic coaches’ language, generated using LLMs, have
not been analyzed. In the past, such appropriateness analyses have been
undertaken in other HRI applications such as robotic movement planning [12].
Past studies have shown how to design an appropriate prompt to guide
researchers in generating high-level motion planning [13, 14, 15]. For
example, Huang et al. [14] have used LLMs to explore the feasibility of
decomposing natural language expressions for high-level tasks (such as “make
breakfast”) into a designated set of executable steps (for instance, “open
fridge”); while Song et al. [13] proposed an effective way to use LLMs as
planners to execute complex tasks based on natural language instructions
within a visually perceived environment. Our work aims to extend such analysis
of appropriateness to the language robots use during HRI—specifically to the
design of language-appropriate robotic mental well-being coaches in real-world
settings.

This paper presents the first study that examines the appropriateness of
robotic mental well-being coach language in the workplace. To this end, we
conducted an empirical study with 17 employees who interacted with an LLM-
equipped robotic mental well-being coach over 4 weeks in their workplace [16].
After the study, we first interviewed them individually, and then we organised
a focus group that involved 11 of them. This focus group included: 1) an ice-
breaking activity, where participants were asked to describe the robotic coach
with five adjectives; 2) language appropriateness evaluation, where
participants were asked to assess the appropriateness of robotic coach
language in seven different scenarios; and 3) listing should _s_ and shouldn’t
_s_ , where participants were asked to list what a robotic coach should and
should not say and how it should behave when delivering well-being practices.

Our results show that a language-appropriate robotic well-being coach should
(1) ask deep questions which explore feelings of the coachees, (2) express and
show emotional and empathic understanding of the context, and (3) not make any
assumptions without clarifying with follow-up questions to avoid bias and
behaviours that enforce stereotypes. Our analysis and results contribute
towards understanding how to design language-appropriate robotic coaches to
promote mental well-being in adults in their workplace.

##  II RELATED WORK

###  II-A Appropriate LLM-equipped Agent Language

Designing embodied agent (e.g., robot, virtual agent) language that is
appropriate for a specific context can be a challenging task [17], especially
for sensitive application scenarios such as medicine [18], therapy [19],
healthcare [20, 21], just to mention a few. Given the widespread use of LLMs
in such applications, recent studies [18, 17] have started exploring the
challenges and opportunities that emerged from the adoption of LLMs for
language generation in these environments. For instance, Cabrera et al. [20]
conducted a literature review to discuss the bioethical dilemmas related to
the use of chatbots in the field of mental health, namely quality of care,
access and exclusion, responsibility and human supervision, and regulations
and policies for LLM-equipped use. Their review recommend that LLM-equipped
chatbots should be developed for mental health purposes, with tasks
complementary to the therapeutic care provided by human professionals, and
that their implementation should be properly regulated and should have a
strong ethical framework. Analogously, Cho et al. [19] have investigated the
LLM efficacy – in terms of empathy, communication skills, adaptability,
engagement, and the ability to establish a therapeutic alliance – in
interactive language therapy for high-functioning autistic adolescents. Their
results highlight the challenges of developing such therapeutic LLM-equipped
systems that cannot achieve the depth of personalization and emotional
understanding characteristic of human therapists and the importance of ethical
considerations in therapeutic contexts.

However, very few studies [22, 23] have investigated further how the LLM usage
in agents has impacted agent language and human perceptions toward the agent.
For example, Onorati et al. [23] proposed a robotic application to generate
verbal dialogues considering the user’s interests and preferences using LLM to
create the conversation. Their main goal was to make users perceive these
dialogues as interesting, to avoid disengagement during the interaction with
the robot. Their results show that participants positively engage and use the
robotic application. Brohan et al. [22] proposed a set of pre-trained skills
for a robot, which were used to constrain the model to propose natural
language actions that were both feasible and contextually appropriate to
provide high-level semantic knowledge about the real-world robotic task (e.g.,
use a vacuum cleaner).

This work presents the first evaluation of the appropriateness of the language
of an LLM-equipped robotic coach that delivered well-being coaching in a
workplace context.

###  II-B Robotic Coaches For Mental Well-being

Various studies have examined the use of robotic coaches to promote mental
well-being, e.g., [24, 25, 4, 10, 26, 27, 28]. Jibo robots were used in a
longitudinal study by Jeong et al. [29] to deliver positive psychology
interventions to students in home settings over the course of seven days. It
was found that participants gradually grew fond of the robot and experienced
improved well-being, improved mood, and readiness to change. Bodala et al.
[26] evaluated participants’ perceptions towards a human coach in comparison
to a teleoperated robotic coach across a five-week period. It was concluded
that both had favourable feedback, however, the human coach was preferred in
terms of likability, intelligence and level of animacy. Additionally, the
personality traits of the participants such as neuroticism and
conscientiousness were found to influence how they interacted with the robot.
Shi et al. [28] investigated the effect of physical embodiment and
personalisation on the user-perceived quality of text-to-speech (TTS) voices
for mindfulness. Their results showed that the user-personalized TTS voices
were able to perform nearly as well as human voices, indicating that user
personalisation could be a powerful approach to raise user perception of TTS
voice quality. When Abbasi et al. [5] looked into a robot’s potential to
evaluate mental well-being problems in children, they found that it was more
accurate at identifying likely anomalies than tests that relied on self- and
parent-reports. Recent studies have looked into the application of robotic
coaches in the workplace and public spaces. Spitale et al. [4, 11] conducted a
study involving employees of a tech company to interact with two different
forms of robotic coaches that delivered positive psychology exercises over 4
weeks. Their results showed that the robot form may impact the perception of
the coachees towards the robotic coach. Axelsson et al. [24] examined the use
of a robotic mindfulness coach at a public cafe over 4 weeks, finding that
participants thought the robot was useful as a guiding voice and a focal
point, but wanted the robot to be more responsive. However, these studies have
not examined the appropriateness of robotic coach language.

##  III METHODOLOGY

The ultimate goal of this work is to understand the appropriateness of the
language of a robotic coach for promoting mental well-being in the workplace.
This is done by undertaking an empirical study and then conducting individual
interviews and a focus group with employees who have interacted with the
robotic coach. This section describes the methodology of the study and the
qualitative approach by reporting the participant demographics, the empirical
study, the interview and focus group protocols, and the scenarios
investigated.

###  III-A Participants

We conducted an empirical study and individual semi-structured interviews with
17 employees (7 females, and 10 males, 4 of whom were 18-25 years old, 6 were
26-35 years old, 4 were 36-45 years old, and 3 were 46-55 years old). All
participants were employees of the Cambridge Consultants Inc. company, where
they had taken part in the empirical study (see Sec. III-B). We then asked
them whether they were willing to participate in a focus group without
specifying to them its ultimate goal. Eleven participants out of seventeen
agreed to attend the focus group: 6 females, and 5 males, 1 of whom were 18-25
years old, 4 were 26-35 years old, 3 were 36-45 years old, and 3 were 46-55
years old. All participants provided informed consent for their participation
and agreed on the usage of their data for scientific research. The focus group
design, the protocol, and the consent forms were approved by the Departmental
Ethics Committee of the University of Cambridge’s Department of Computer
Science & Technology.

###  III-B Study Protocol

Employees involved in this qualitative research took part in an empirical
study with an autonomous and adaptive robotic coach over four weeks at their
workplace, as described in detail in [16].

####  III-B1 Empirical Study

The robotic coach delivered four different positive psychology exercises (one
per week), namely savouring, gratitude, accomplishment and one door closes and
one door opens. The robot used a LLM-equipped framework for managing the
dialogue flow (i.e., we embedded in the robot OpenAI API for natural language
processing). The interaction between each employee and the robotic coach
lasted for about 10 minutes and included the following steps: (1) The robot
introduced itself and described the positive psychology practice (just in the
first session) and the exercise of the week (e.g., savouring exercise). (2)
The robot asked the coachee to think about a positive memory from the last
week and to share it with them. (3) The robot listened to the coachee’s
response. (4) The robot made a decision on the next step (summarise, ask for a
follow-up question, or start a new episode) based on an on-the-fly pre-trained
reinforcement learning model described in detail in [16]. (5) The robot
generated the action according to the decision made in (4) and listened to the
coachees’ response. (6) The robot repeated steps (4) and (5) for 8 turns. We
decided to fix the number of turns to 8 to ensure that the coaching practice
does not last more than 10 minutes. (7) The robot concluded the session by
thanking the employee and reminding them of the following week’s session. This
procedure was repeated for each session over the four weeks of the study.

####  III-B2 Interviews

At the end of the last week, we conducted individual semi-structured
interviews with all employees (17 in total) in a meeting room of the Cambridge
Consultants Inc. headquarters. We asked employees several questions, such as
“What was your overall impression of the robot?”, “Have you felt understood
and listened to?”, “Would you recommend this robot to a colleague or a
friend?” etc. We concluded the interview by debriefing the coachees, i.e.,
explaining the main goal of the empirical study, and answering their
questions.

####  III-B3 Focus Group

After two weeks, we conducted a 1.5-hour focus group with the employees online
via MS Teams. To facilitate the discussion, we asked participants to use the
online tool Miro111https://miro.com/. Two researchers were present during the
focus group: one of them was taking notes, and the other was leading the
discussion. The focus group included three activities, which were conducted
individually on the Miro Board. First (ice-breaking activity, 10 mins), the
researcher asked the employees to post on the Miro board five adjectives to
describe the robotic coach. This activity was chosen to better understand the
employees’ perceptions of the robotic coach as was previously done in [30].
Second, the researcher explained to the employees that the main goal of the
focus group was to better understand what robotic coaches should say to be
perceived as more appropriate to the context and the situation. Then, they
described the activity of scenario evaluations (40 mins). The employees were
presented with seven scenarios on the Miro board, and they were asked to write
for each scenario: how they felt about the scenario (“How do you feel?”), the
appropriateness score of the robotic coach language in that scenario (from
1-10), how would they modify the robot language (“Behavior modifications”),
and why they would modify what the robotic coach says (“Why”). Finally, the
researcher asked the participants to define a list of “should” _s_ and
“shouldn’t” _s_ (20 mins) of the robotic coach language, as depicted in Figure
2. At the end of each of the three activities, the researchers allocated
around 10 mins for group discussions with all the participating employees.

###  III-C Scenarios

In our empirical study, employees shared with the robotic coach various
positive episodes from their lives [16] (e.g., playing instruments, practicing
sports) that informed the definition of the workshop’s scenarios alongside
their grounding in Human-Computer Interaction (HCI) literature. While watching
the recordings of the study, we observed situations (i.e., what the coachee
shared and how the robotic coach followed up) that we used as scenarios in our
focus group, but modified for privacy reasons. As a result, we defined seven
scenarios that we encountered recurrently in our empirical study by framing
them from a Human-Computer Interaction (HCI) perspective [31]. HCI literature
[31] suggests that the research field has had three waves: (1) in the first
wave, the human is conceptualised as an object that functions following rigid
guidelines, and HCI focused on pragmatic solutions and objective measures
(e.g., time, task performance); (2) the second wave added subjectivity and
took into account the person in the interaction (besides the functionality of
the technology itself) and HCI focused on cognitive science and psychology
aspects (e.g., emotion, empathy) during the design of technologies for humans;
(3) in the third wave, researchers considered how the technology may reach our
everyday lives (homes, privacy etc.), by taking into account social-cultural
context (e.g., gender, culture). We applied these waves as the three drivers
for designing the seven scenarios as follows:

  1. 1.

Wave 1 (Objective Measures) - Efficiency and time pressure:

     * •

Scenario 1: You have just shared that you have been working out recently, and
you felt grateful for it. As follow up questions, the robot asks you the
questions “How many times a week are you working out?”, “When was the last
time you worked out?”.

     * •

Scenario 2: You have just shared one example you were grateful for during the
last week. After sharing that, the robot asked you to think about another
example you have been grateful for during the last week, giving you 30 seconds
to think about it.

  2. 2.

Wave 2 (Subjectivity) - Empathy and emotions:

     * •

Scenario 3: You have just shared that you have become an aunt and you have a
new nephew that you are very grateful for. The robot asked you “What makes you
grateful for the birth of your nephew?”.

     * •

Scenario 4: You have just shared that you have baked a cake as one of your
week’s accomplishments. The robot asked you the following question: “What are
the main ingredients for the cake you baked?”

     * •

Scenario 5: You have just shared that you have been by the beach last weekend
and you savoured the moment in which your feet touched the sand. The robot
said “So you just mentioned that you savoured the moment in which your feet
touched the sand, what were the senses activated in that moment? The smell,
the touch, the sound of the waves?”

  3. 3.

Wave 3 (Social-cultural Context) - Bias and stereotyping:

     * •

Scenario 6: You have just shared that you were grateful that your friend
texted you yesterday morning saying that she woke up very energised and she
left home after a couple of months of staying on the couch because of her
depression. The robot responds: “I’m really sorry that your friend was feeling
depressed. I’m glad to hear that now she is feeling better and that you were
able to support her. Regardless of all your effort, something in her mind
should change and you may feel very powerless. What qualities in you made you
able to support your friend?”

     * •

Scenario 7: You have just shared a great accomplishment of a friend of yours
at work who successfully delivered a very big project and you are really proud
of your friend. The robot assumes that your friend is male and asks you
follow-up questions referring to your friend as “he/him”.

##  IV FINDINGS

This section reports the qualitative evaluation of the findings from the
original study and the findings obtained from the interviews and workshop
activities approached via a HCI lens.

###  IV-A Interviews

Following the framework method for qualitative analysis [32] (i.e., using the
three HCI-inspired waves as an analysis framework), we summarise the interview
results as follows.

####  IV-A1 Wave 1: Efficiency and time pressure

Coachees noted that the robotic coach was focusing on the efficiency of their
actions rather than going deeper and exploring their feelings, and they also
felt pressured by the time constrains driven by the robotic coaches. P04 felt
that the interaction was like a “job interview” during the one door closes and
one opens exercise. She shared with the robotic coach a door that closed in
her work experience, and the robot kept asking her questions about her work
rather than focusing on positive things (e.g., it asked her about time
management). P04 also shared in her first week that she has practiced dance.
The robotic coach followed up with very practical question, e.g., technical
aspects of her dance practice, rather than asking her about how the dance made
her feel. Analogously, P03 was impressed with how the robot asked a relevant
follow-up question about the work task he had described to the robot. However,
he expressed disappointment in that the robotic coach asked him specific
details about his work task, rather than how his work accomplishment made him
feel. P17 shared with the robotic coach that he played some guitar, and the
robot asked several follow-up questions, such as, “How long have you been
practicing?”, “How many times a week do you practice?”, and implying through
these statements that the person should have practiced more. The participant
found those questions putting “pressure” on him, because he “hasn’t actually
practiced that much”. Regarding time pressure, P16 shared that the gratitude
exercise “put him on the spot to think of examples” of gratitude, and
analogously, P11 wished to have more time or at least not feel the pressure of
a time out (i.e., the robotic coach was giving the employees 30 seconds to
think about each experience before asking them to share).

####  IV-A2 Wave 2: Empathy and emotions

Employees also noted that the robotic coach was lacking empathic responses and
emotional understanding. Specifically, P15 highlighted that the robotic coach
made him “think of something I would not otherwise”, but without “really
making him have an emotional response to it”. P14 explicitly reported that the
robotic coach should be “more empathic” and also P07 thought that it was
“lacking compassion” and he didn’t feel “listened to”. Again, P06 found the
robotic coach not really going deeper in the conversation, but she thought
that this could be because the type of example she brought up were “not very
challenging”.

####  IV-A3 Wave 3: Bias and stereotyping

Finally, employees brought up concerns about bias and stereotyping in the
interviews. P10 highlighted that the robotic coach “asked really good follow-
up questions”, e.g., by validating her about the difficulty of dealing with
toddlers. She also mentioned that the robotic coach misheard her when she said
that she was giving a presentation and thought “Dave” gave the presentation.
She found this mistake to exemplify “gender inequality”, since she was trying
to share with the robotic coach an accomplishment that was hers.

###  IV-B Workshop: Ice-breaking Activity

Figure 1: Word cloud of the adjectives listed by the employees to describe the
robotic coach.

Figure 1 shows the word cloud of the adjectives listed by the employees to
describe the robotic coach they have interacted with. Five of the employees
used the word “calming” to describe the robotic coach because they found the
positive psychology exercises delivered by the robotic coach as a beneficial
practice to relax and stay calm. P01, P03 and P10 also pinpointed positive
aspects of the robotic coach’s personality by describing it as “friendly”,
“respectful” and “insightful”, and P01 also described its capabilities
positively using words like “responsive”, and P05 and P03 thought the robot
was “capable” and “adaptable”. However, P05, P08 and P14 perceived the robot
to be “slow”, “monotonous”, and “repetitive”, and P01 and P05 found the
robotic interaction to be “unnatural” and “rigid”.

These results show that employees have both positive and negative opinions of
the robot, by describing it as calming and insightful but questioning its
capability in terms of naturalness of the interaction and slow pace of the
conversation.

###  IV-C Workshop: Scenario Evaluation

Figure 2: List of should _s_ (green post-it on the left-side) and shouldn’t
_s_ (orange post-it on the right-side) identified by the employees regarding
the robotic coaches’ appropriate behaviours. Participants’ initials on the
post-its are anonymized using black boxes.

####  IV-C1 Wave 1: Efficiency and time pressure

Scenario 1 was interpreted very differently across employees. P03 and P05
thought that the follow up question (i.e., “how many times a week are you
working out?”, “when was the last time you worked out?”) of the robotic coach
was appropriate, and P05 felt that the robotic coach’s question can be “a
natural follow on in a conversation that could provide useful information”.
P06 believed that the response was appropriate however she suggested to
rephrase the follow-up question, e.g. “Why do you run?”, or “Is it something
that you started recently?”, or “How do you feel about it?”. On the other
hand, P04 and P02 found the follow-up question of the robotic coach very
inappropriate. P02 felt “nagged” and she wondered whether the robotic coach
“is trying to make [her] feel guilty for not doing more [exercise]”. Also, P04
noted that the robotic coach was not focusing on the gratitude aspect of the
exercise but only on facts. She suggested other follow-up questions such as
“What made you feel inspired to exercise?” or “Is this something you can plan
into your routine?”, to make the coachee reflect more why they are grateful
for the running experience. P07 believed that the robotic coach’s question
could have been appropriate depending on the familiarity between the robotic
coach and the coachee, and “whether the robot has already built rapport and
established a relationship”. He felt that for a first interaction, the
presented follow-up question may be perceived as challenging and
confrontational, while it may be more appropriate later in time.

All employees agreed on the inappropriateness of Scenario 2. They felt
“annoyed”, “not listened to”, “frustrated” and “doubtful”. This was because
they felt the robotic coach was dismissing their first answer and they felt
pressured about the timing. P07 thought that it is “artificial to wait for the
full 30 seconds if the participant is ready to speak”, and also P03 reported
that he felt it was stressful to be asked to come up with a new example in
only 30 seconds: “it can make it harder to come up with an example due to the
[time] pressure”. P06 suggested that the robotic coach can ask more follow-up
questions regarding the first example, and add asking about another experience
only as “optional”. P03 would like the robotic coach to “go deep/follow up in
the discussion for each example, rather than ask for many [examples]”.

####  IV-C2 Wave 2: Empathy and emotions

Seven employees found Scenario 3 appropriate in terms of the robotic coach’s
behaviour because they felt “listened to” and “engaged”. P01 reported that it
is very “nice to elaborate on and reinforce your positive feelings about [the]
example”. P04 suggested that the robotic coach could have also added something
like “family is often a cause of gratefulness” to show an understanding of the
context. Few of them believed that the robotic coach’s behaviour was
inappropriate. For example, P05 doubted that a “robot will truly to able to
understand it”, so he found inappropriate that a robotic coach asked such
personal questions. P13 also thought that the robotic coach could have dived
“into the reason why it might make you grateful”.

Four employees found the robotic coach’s behaviour in Scenario 4 completely
inappropriate and other two considered that behaviour barely appropriate
because it did not add any emotional or empathic value to the conversation.
P14 believed that asking a follow-up question about the cake’s ingredients has
“nothing to do with [the coachee]”, and suggested that talking about the
feelings related to the accomplishment of baking a cake would have been more
appropriate behaviour. Analogously, P13 reported that the robotic coach’s
question “has no value/benefit for the robot or the participant” and suggested
that the robotic coach could have asked questions more related to the emotions
felt during baking. P10 felt that the robotic coach was “dismissive” and found
its response to be a “robotic question”. P07 felt “validated” and “listened
to” but he found the question “pointless” because “talking about ingredients
is unlikely to elicit a response that leads to rapport/empathy”. In contrast,
P05, P04, and P02 found the robotic coach’s behaviour appropriate because they
felt that it was “interested in” what the employee baked.

Seven employees found the robotic coach’s behaviour in Scenario 5 very
appropriate. They felt “listened to” and “encouraged to think further about
how [the coachee] felt in that moment”. P04 found the robotic coach’s response
“almost perfect” because the robotic coach was asking to “return to the moment
and think about being there in more detail”. P02 also suggested that the
robotic coach could have asked to “explore how the sensations made [the
coachee] feel”. In contrast, three employees found the behaviour
inappropriate. For example, P14 felt rushed and he would have preferred that
the robotic coach gave him the “time to respond rather than give [the coachee]
options”.

####  IV-C3 Wave 3: Bias and ethics

All the employees found the robotic coach’s behaviour in Scenario 6 very
inappropriate. They felt “confused”, “awkward”, and “frustrated”. P03 noted
that the robotic coach was inappropriately making assumptions, and analogously
P07 noted that the robotic coach was “making an assumption about [the
coachee’s] emotional state when [the coachee] has simply described the facts
of the situation”. P02 highlighted that the robotic coach was a “bit
pessimistic” and assumed that “any progress the friend has made may be lost”,
and she suggested that instead the robotic coach should have considered the
qualities of the coachee and how they have been useful for the friend.
Analogously, P13 felt that the robotic coach’s answer was “not constructive”
and she suggested that the robotic coach should ask about what the coachee has
done to help their friend and give them support. She also highlighted that
this is a very sensitive subject and the robotic coach should be careful when
handling such situations that may be “close to” many families.

All the employees agreed that in Scenario 7 the robotic coach’s behaviour was
completely inappropriate because it assumed that the successful friend was a
male. They felt “frustrated” and “annoyed”. P04 was “pissed off” by the
robotic coach’s behaviour because she believed the robot was trained on models
that inherited social biases. She suggested that the robotic coach can be
trained to use gender neutral pronouns when the gender of the person is not
provided. Analogously P10 said that the robotic coach with such behaviour was
“perpetuating societal stereotypes”, and he suggested to keep gender
neutrality, and make no assumptions in this regard. To avoid making
assumptions, P03 suggested to ask more questions rather than fall into
stereotypical behaviour hypotheses.

###  IV-D Workshop: List of should _s_ and should not _s_

We asked the employees to identify how the robotic coach “should” and “should
not” behave while delivering positive psychology coaching sessions. Our
findings show that being able to make the participant “feel listened to” is
one of the main features employees believed the robotic coach should be
equipped with. P02 and P07 highlighted that the robotic coach should show cues
that suggest that it had heard the coachee when they were speaking. Employees
thought that the robotic coach should finish discussing their shared
experience by going “a bit more in depth when an answer is given”, and by
double-checking with the coachee if they have anything to add about the topic
before moving on to asking about another experience. Emotional understanding
and expression were other behaviours that employees identified as appropriate
for a robotic coach. P07 believed that the robotic coach needs to “demonstrate
that it should understand the emotional content of the participant’s answers”
and P03 suggested that it should have a “more expressive face”.

Employees highlighted that the robotic coach should not “make assumptions” and
“jump to conclusions too quickly”, to avoid introducing any type of biases. In
addition, the robotic coach should not repeat “verbatim” what the coachee has
said to avoid appearing mechnical, and should not interrupt them as that may
disrupt the coaching session.

##  V DISCUSSION & CONCLUSION

We collated the qualitative findings from interviews and the focus group, and
discuss these results as follows.

Reflecting the research trend of the first HCI wave, when technologies were
measured mostly in terms of objective metrics and performances [31], coachees
perceived that the robotic coach was too pragmatic and superficial and focused
on measurable accomplishments rather than on their feelings. Specifically, our
findings show that LLM-equipped robotic coach followed up with practical
questions and facts without focusing on the positive psychology aspects (e.g.,
asking why they were grateful for what they shared). Similar results were
found in [33] in which health professional interacted with ChatGPT and
identified limitations during mental health counselling such as rushing the
client, not addressing or evaluating safety concerns, putting clients at risk.
Again, [34] highlighted the risks of using LLMs in high education systems in
promoting superficial learning rather than deeper explanations. Future work
should investigate systematically how to avoid inappropriate responses
especially in delicate contexts such as mental well-being coaching.

Coachees also considered the LLM-equipped robotic coach not empathic and not
able to emotionally understand their experiences by highlighting the need of
putting the subjectivity and psychological aspects at the center of the
interaction as for the second HCI wave. Specifically, coachees felt engaged
and listened to in accordance to previous studies [33], but not understood
emotionally. This result is line with a previous work [35] that developed a
new way of prompt engineering ChatGPT to enhance the empathic and emotional
understanding capabilities of the chatbot.

In accordance with the third HCI wave that focused on the socio-cultural
importance, coachees pinpointed that the LLM-equipped robotic coach should not
make assumptions and make statements that reinforce social stereotyping.
Specifically, our results show that they perceived the robotic coach was
perpetuating social stereotypes and biases such as gender stereotypes. These
findings are in line with a previous study in which undergraduate students
interacted with ChatGPT in Arabic language, and ChatGPT generated data in the
context of counseling and mental health was not suitable for Arabic society,
customs, traditions, and culture [36]. This could have been attributed to the
bias in the training data as reported previously in a study that focused on
bias of ChatGPT in American society [37].

In summary, our findings suggest that a language-appropriate robotic coach
should:

  1. 1.

ask deep questions which explore feelings of the coachees,

  2. 2.

express and show emotional and empathic understanding of the context,

  3. 3.

not make any assumptions without clarifying with follow-up questions to avoid
bias and behaviours that enforce stereotypes.

We hope that these results can inform the design of language-appropriate
robotic coaches to promote mental well-being in various real-world contexts.

## ACKNOWLEDGMENT

We thank Cambridge Consultants Inc. and their employees for participating in
this study. Funding: M. Spitale and H. Gunes have been supported by the
EPSRC/UKRI under grant ref. EP/R030782/1 (ARoEQ) and EP/R511675/1. M. Spitale
is partially supported by PNRR-PE-AI FAIR project funded by the NextGeneration
EU program. M. Axelsson is funded by the Osk. Huttunen foundation and the
EPSRC under grant EP/T517847/1. Open Access: For open access purposes, the
authors have applied a Creative Commons Attribution (CC BY) licence to any
Author Accepted Manuscript version arising. Data access: Raw data related to
this publication cannot be openly released due to anonymity and privacy
issues.

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