# Evaporation of dynamical horizon with the Hawking temperature in the
K-essence emergent Vaidya spacetime

Bivash Majumder<EMAIL_ADDRESS>Department of Mathematics, Prabhat
Kumar College, Contai, Purba Medinipur-721404, India
Department of Mathematics, Raiganj University, Raiganj-733134, India  Goutam
Manna<EMAIL_ADDRESS>Department of Physics, Prabhat Kumar
College, Contai, Purba Medinipur-721404, India  Mrinal Manna
<EMAIL_ADDRESS>Department of Physics, Ramakrishna Mission
Vivekananda Educational and Research Institute, Belur Math, Howrah-711202,
India

###### Abstract

We observe that how the mass is falling because of the Hawking Radiation in
the K-essence Vaidya Schwarzschild spacetime by using the dynamical horizon
equation and evaluate the hawking temperature using the tunnelling formalism.
Here we consider the K-essence action to be of the Dirac-Born-Infeld variety,
the physical spacetime to be a general static spherically symmetric black hole
and also we restrict our self that the K-essence scalar field to be a function
solely of the advanced or the retarded time. Manna et. al. have established
the connection between the K-essence emergent gravity scenario and
generalizations of Vaidya spacetime. Using the modified definition of the
dynamical horizon by Sawayama, we conclude that the results are different from
the usual Vaidya spacetime geometry.

Emergent gravity, K-essence, Vaidya spacetimes

###### pacs:

04.20.-q; 04.20.Jb; 04.70.-s

##  I Introduction

It is generally accepted that a black hole is originated by gravitational
collapse and it produces thermal radiation as it were very hot with its
temperature is proportional to the surface gravity haw1 ; haw2 ; haw3 ; haw4 ;
haw5 ; haw6 ; haw7 ; haw8 ; haw9 ; haw10 . This thermal radiation causes a
slow decrease of mass of the black hole by assuming that the spacetime is to
be static or stationary haw1  provided the radiated energy is so small
compared to the mass energy of the black hole. For sufficiently large
radiation, it can be improved via Einstein equation. In this connection, a
non-static solution of the Einstein’s field equations for spheres of fluids
radiating energy has been derived by Vaidya vai1 ; vai2 . Also Vaidya
established the nonstatic analogs of Schwarzschild’s interior solution in vai3
; vai4  and solved the problem of gravitational collapse with radiation in
vai5 . This solution satisfies the physical feature of allowing a positive
definite value of the density of collapsing matter and gives the total
luminosity of the object as it is observed by a stationary observer at
infinity to be zero when the object collapses to the Schwarzschild’s
singularity. It is relevant to believe that the Vaidya spacetime vai1 ; vai2 ;
vai3 ; vai4 ; vai5  is a non-stationary Schwarzschild spacetime. Corresponding
to the gravitational collapse of a null fluid, the generalized Vaidya
spacetime was developed by Husain husain  and Wang et. al. wang . In this
context, recently the K-essence generalizations of Vaidya spacetime has been
established by Manna et. al. gm1 ; gm2  in which time dependence of the metric
comes from the kinetic energy ($\phi_{v}^{2}$) of the K-essence scalar field
($\phi$). In parikh ; mitra1 ; mitra2 ; mitra3 ; mitra4 , authors has
discussed about hawking radiation haw1 ; haw2 ; haw3 ; haw4 ; haw5 ; haw6 ;
haw7 ; haw8 ; haw9 ; haw10  using tunnelling mechanism and this processes in
semi-classical quantum mechanics can be described by the method of complex
path analysis which is proposed in padma1 ; padma2 . In mann , Kerner and Mann
have established that the Hawking temperature is independent of the angular
part of the spacetime in general. The Hawking radiation of Vaidya or Vaidya-
Bonner spacetime is deeply discussed in kuroda ; siahaan ; tang ; wanglin .

In the theory of K-essence babi1 ; babi2 ; babi3 ; babi4 ; babi5 ; scherrer1 ;
scherrer2 , the scalar field is dominated by the kinetic energy over potential
energy. The non-trivial dynamical solutions of the K-essence equation of
motion with non-canonical kinetic terms spontaneously breaks Lorentz
invariance and also it changes the metric for the perturbations around these
solutions. The perturbation propagates with the metric in the so called
emergent or analogue curved spacetime. Manna et. al. gm3 ; gm4 ; gm5 ; gm6
have developed simplest form of K-essence emergent gravity metric
$\bar{G}_{\mu\nu}$ by considering the Dirac-Born-Infeld (DBI) model born1 ;
born2 ; born3 . This form is not conformally equivalent to the usual
gravitational metric $g_{\mu\nu}$. The lagrangian is non-canonical and it does
not depend explicitly on the field itself in the theory of K-essence field.
The general form of the Lagrangian for the K-essence model is:
$L=-V(\phi)F(X)$ where
$X=\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$.

Across the dynamical horizons, the expression of fluxes of energy and angular
momentum which are carried by gravitational wave and also the equation which
describe how the dynamical horizon radius changes, derived in ashtekar1 ;
ashtekar2 ; ashtekar3 ; ashtekar4 ; hayward ; badri . The definition of
dynamical horizon is: A smooth, three-dimensional, space-like submanifold $H$
in a space-time $\cal{M}$ is said to be a dynamical horizon if it can be
foliated by a family of closed $2-$surfaces such that, on each leaf $S$, the
expansion $\Theta_{(l)}$ of one null normal $l^{a}$ vanishes and the expansion
$\Theta_{(n)}$ of the other null normal $n^{a}$ is strictly negative. And the
modified definition proposed by Sawayama sawayama  is: A smooth, three-
dimensional, spacelike or timelike submanifold $H$ in a space-time is said to
be a dynamical horizon if it is foliated by a preferred family of 2-spheres
such that, on each leaf $S$, the expansion $\Theta_{(l)}$ of a null normal
$l^{a}$ vanishes and the expansion $\Theta_{(n)}$ of the other null normal
$n^{a}$ is strictly negative. Following ashtekar3 , in the concept of world
tubes , if the marginally trapped tubes (MTT) is

  1. 1.

spacelike, then it is called a dynamical horizon (DH) and under some
conditions that it provides a quasi-local representation of an evolving black
hole.

  2. 2.

timelike, then the causal curves can transverse it in both inward and outward
directions, where it does not represent the surface of a black hole in any
useful sense, it is called a timelike membrane (TLM).

  3. 3.

null, then it describes a quasi-local description of a black hole in
equilibrium and is called an isolated horizon (IH).

In this work, using dynamical horizon equation, we have studied the Hawking
effect by using dynamical horizon equation in the K-essence emergent Vaidya
spacetime based on Sawayama sawayama  and tunneling formalism parikh ; mitra1
; mitra2 ; mitra3 ; mitra4 ; padma1 ; padma2 ; kuroda ; siahaan ; gm3 ; gm4 ;
gm5 .

The paper is organized as follows: In section 2, we have briefly discussed the
K-essence emergent geometry and corresponding K-essence Vaidya spacetime. We
describe the dynamical horizons by considering Schwarzschild black hole as a
background for the K-essence emergent Vaidya spacetime in section 3. In the
next section, we have discussed in detail of the dynamical horizon equation
for the K-essence Vaidya Schwarzschild spacetime. Also, we have discussed the
corresponding Hawking radiation using dynamical horizon equation and tunneling
mechanism. The Last section is our conclusion.

##  II Brief review of K-essence and K-essence-Vaidya Geometry

###  II.1 K-essence Geometry

The K-essence scalar field $\phi$ minimally coupled to the background
gravitational metric $g_{\mu\nu}$ has action babi1 -babi5

$\displaystyle S_{k}[\phi,g_{\mu\nu}]=\int d^{4}x{\sqrt{-}g}L(X,\phi)$ (1)

where $X={1\over 2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ and the energy-
momentum tensor is

$\displaystyle T_{\mu\nu}\equiv{2\over\sqrt{-g}}{\delta S_{k}\over\delta
g^{\mu\nu}}=L_{X}\nabla_{\mu}\phi\nabla_{\nu}\phi-g_{\mu\nu}L$ (2)

$L_{\mathrm{X}}={dL\over dX},~{}~{}L_{\mathrm{X}X}={d^{2}L\over
dX^{2}},~{}~{}L_{\mathrm{\phi}}={dL\over d\phi}$ and $\nabla_{\mu}$ is the
covariant derivative defined with respect to the gravitational metric
$g_{\mu\nu}$. The scalar field equation of motion is

$\displaystyle-{1\over\sqrt{-g}}{\delta
S_{k}\over\delta\phi}=\tilde{G}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi+2XL_{X\phi}-L_{\phi}=0$
(3)

where

$\displaystyle\tilde{G}^{\mu\nu}\equiv
L_{X}g^{\mu\nu}+L_{XX}\nabla^{\mu}\phi\nabla^{\nu}\phi$ (4)

and $1+{2XL_{XX}\over L_{X}}>0$.

Using the conformal transformations $G^{\mu\nu}\equiv{c_{s}\over
L_{x}^{2}}\tilde{G}^{\mu\nu}$ and $\bar{G}_{\mu\nu}\equiv{c_{s}\over
L_{X}}G_{\mu\nu}$, with $c_{s}^{2}(X,\phi)\equiv{(1+2X{L_{XX}\over
L_{X}})^{-1}}$ we have gm3 ; gm4 ; gm5

$\displaystyle\bar{G}_{\mu\nu}={g_{\mu\nu}-{{L_{XX}}\over{L_{X}+2XL_{XX}}}\nabla_{\mu}\phi\nabla_{\nu}\phi}$
(5)

Here one must always have $L_{X}\neq 0$ for $c_{s}^{2}$ to be positive
definite and only then equations $(1)-(4)$ will be physically meaningful.

If $L$ is not an explicit function of $\phi$ then the equation of motion (3)
reduces to;

$\displaystyle-{1\over\sqrt{-g}}{\delta
S_{k}\over\delta\phi}=\bar{G}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=0$ (6)

Note that for non-trivial spacetime configurations of $\phi$, the emergent
metric $G_{\mu\nu}$ is, in general, not conformally equivalent to
$g_{\mu\nu}$. So $\phi$ has properties different from canonical scalar fields,
with the local causal structure also different from those defined with
$g_{\mu\nu}$. We consider the DBI type Lagrangian as gm3 ; gm4 ; gm5 ; born1 ;
born2 ; born3 ,

$\displaystyle L(X,\phi)=1-V(\phi)\sqrt{1-2X}$ (7)

for $V(\phi)=V=constant$ and $kinetic~{}energy~{}of~{}\phi>>V$
i.e.$(\dot{\phi})^{2}>>V$. This is a typical for the K-essence fields where
the kinetic energy dominates over the potential energy. Then
$c_{s}^{2}(X,\phi)=1-2X$. For scalar fields
$\nabla_{\mu}\phi=\partial_{\mu}\phi$. Thus (5) becomes

$\displaystyle\bar{G}_{\mu\nu}=g_{\mu\nu}-\partial_{\mu}\phi\partial_{\nu}\phi$
(8)

The corresponding geodesic equation for the K-essence theory in terms of the
new Christoffel connections $\bar{\Gamma}$ is gm3 ; gm4 ; gm5

$\displaystyle\frac{d^{2}x^{\alpha}}{d\lambda^{2}}+\bar{\Gamma}^{\alpha}_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}=0$
(9)

where $\lambda$ is an affine parameter and

$\displaystyle\bar{\Gamma}^{\alpha}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\Gamma^{\alpha}_{\mu\nu}+\Big{(}1-2X\Big{)}^{-1/2}\bar{G}^{\alpha\gamma}\Big{[}\bar{G}_{\mu\gamma}\partial_{\nu}\Big{(}1-2X\Big{)}^{1/2}$
(10) $\displaystyle+$
$\displaystyle\bar{G}_{\nu\gamma}\partial_{\mu}\Big{(}1-2X\Big{)}^{1/2}-\bar{G}_{\mu\nu}\partial_{\gamma}\Big{(}1-2X\Big{)}^{1/2}\Big{]}$
$\displaystyle=$
$\displaystyle\Gamma^{\alpha}_{\mu\nu}-\frac{1}{2(1-2X)}\Big{[}\delta^{\alpha}_{\mu}\partial_{\nu}X+\delta^{\alpha}_{\mu}\partial_{\nu}X\Big{]}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

###  II.2 K-essence-Vaidya Geometry

Considering a general spherically symmetric static (black hole) Eddington-
Finkelstein line element gm1

$\displaystyle ds^{2}=f(r)dv^{2}-2\epsilon dvdr-r^{2}d\Omega^{2}$ (11)

where $d\Omega^{2}=d\theta^{2}+sin^{2}\theta d\Phi^{2}$. When $\epsilon=+1$,
the null coordinate $v$ represents the Eddington advanced time (outgoing),
while when $\epsilon=-1$, it represents the Eddington retarded time
(incoming).

From (8) the emergent spacetime is described by the line element

$\displaystyle dS^{2}=ds^{2}-\partial_{\mu}\phi\partial_{\nu}\phi
dx^{\mu}dx^{\nu}.$ (12)

Considering the scalar field $\phi(x)=\phi(v)$, so that the emergent spacetime
line element is

$\displaystyle dS^{2}=\Big{[}f(r)-\phi_{v}^{2}\Big{]}dv^{2}-2\epsilon
dvdr-r^{2}d\Omega^{2}$ (13)

where $\phi_{v}=\frac{\partial\phi}{\partial v}$. It is noted that the
assumption on $\phi$ actually violates local Lorentz invariance, since in
general, spherical symmetry would only require that $\phi(x)=\phi(v,r)$. But
in the K-essence theory, the dynamical solutions are spontaneously break
Lorentz invariance. So, in this context, our choice of form of the K-essence
scalar field is physically permissible.

Now comparing the emergent spacetime (13) with the metric husain ; wang  of
the generalized Vaidya spacetimes corresponding to gravitational collapse of a
null fluid (take $\epsilon=+1$)

$\displaystyle
dS_{V}^{2}=\left(1-\frac{2m(v,r)}{r}\right)dv^{2}-2dvdr-r^{2}d\Omega^{2}$ (14)

where the mass function is

$\displaystyle m(v,r)=\frac{1}{2}r\Big{[}1+\phi_{v}^{2}-f(r)\Big{]}.$ (15)

These forms (13) or (14) of metrics are satisfying all the required energy
conditions haw-ellis  (weak, strong, dominant) for generalizations of the
K-essence emergent spacetime with the Vaidya spacetime provided
$\phi_{v}\phi_{vv}>0~{};~{}1+\phi_{v}^{2}>f+rf_{r}~{};~{}2f_{r}+rf_{rr}>0$
which have established by Manna et. al. gm1 .

##  III Dynamical horizons

Following Ashtekar and Krishnan ashtekar2  and Sawayama sawayama , we can
discuss behavior of the dynamical horizon of the K-essence emergent Vaidya
spacetime. The generalized Vaidya spacetime (13) or (14) in the presence of
kinetic energy of the K-essence scalar field $\phi_{v}^{2}$ can be written as

$\displaystyle dS^{2}=F(v,r)dv^{2}-2dvdr-r^{2}d\Omega^{2}$ (16)

where $v^{a}$ is a null vector. Following Sawayama sawayama , we can define

$\displaystyle a=\frac{dr}{dr^{*}}$ (17)

where $r^{*}$ is tortoise coordinate defined as $v=t+r^{*}$. There are two
null vectors,

$\displaystyle l^{a}=\left[\begin{array}[]{c}l^{t}\\\ l^{r^{*}}\\\
l^{\theta}\\\ l^{\Phi}\end{array}\right]=\left[\begin{array}[]{c}a^{-1}\\\
-a^{-1}\\\ 0\\\ 0\end{array}\right]$ (26)

corresponding to the null vector $v^{a}$ , and the other is

$\displaystyle n^{a}=\left[\begin{array}[]{c}n^{t}\\\ n^{r^{*}}\\\
n^{\theta}\\\ n^{\Phi}\end{array}\right]=\left[\begin{array}[]{c}a^{-1}\\\
\frac{F}{F-2a}a^{-1}\\\ 0\\\ 0\end{array}\right].$ (35)

The expansions $\Theta_{(l)}$ and $\Theta_{(n)}$ of the two null vectors
$l^{a}$ and $n^{a}$ are sawayama

$\displaystyle\Theta_{(l)}=\frac{1}{r}(2F-a)$ (36)

and

$\displaystyle\Theta_{(n)}=\frac{1}{ar}\left(\frac{-2F^{2}+aF-2a^{2}}{-F+2a}\right).$
(37)

We can see that as $\Theta_{(l)}=0\Rightarrow 2F-a=0$, and the other null
expansion $\Theta_{(n)}$ is strictly negative which are the required
conditions for the dynamical horizon ashtekar1 ; ashtekar2 ; sawayama . So the
horizons of our case are dynamical. Note that, based on Ashtekar et. al.
ashtekar1 ; ashtekar2 ; ashtekar3 ; ashtekar4 ; hayward , Manna et. al. gm1
also have established the possibility of the dynamical horizon for the
K-essence emergent Vaidya spacetime.

###  III.1 Schwarzschild black hole as background:

In this case $f(r)=1-\frac{2M}{r}$, then (16) become

$\displaystyle
dS^{2}=\Big{(}1-\frac{2M}{r}-\phi_{v}^{2}\Big{)}dv^{2}-2dvdr-r^{2}d\Omega^{2}$
(38)

with

$\displaystyle
F=\Big{(}1-\frac{2M}{r}-\phi_{v}^{2}\Big{)}\equiv\Big{(}1-\frac{2m(v,r)}{r}\Big{)}$
(39)

and the mass function become

$\displaystyle m(v,r)=M+\frac{r}{2}\phi_{v}^{2}.$ (40)

where the tortoise coordinate

$\displaystyle r^{*}=\frac{1}{N}\Big{[}r+B~{}ln(r-B)\Big{]}$ (41)

where $N\equiv N(v)=(1-\phi_{v}^{2})$, $B=2M^{\prime}(v)=\frac{2M}{N}$ with
$M^{\prime}(v)=\frac{M}{1-\phi_{v}^{2}}$. Note that in the above spacetime
(38) always $\phi_{v}^{2}<1$. If $\phi_{v}^{2}>1$, the signature of this
spacetime will be ill-defined. Also the condition $\phi_{v}^{2}\neq 0$ holds
good instead of $\phi_{v}^{2}=0$, which leads to non-applicability of the
K-essence theory. Now differentiating the above equation (41) with respect to
$r^{*}$ we get

$\displaystyle
a=F\Big{[}1-\frac{1}{N}\Big{(}\frac{dB}{dv}~{}ln(r-B)-\frac{B}{r-B}\frac{dB}{dv}\Big{)}$
$\displaystyle+\frac{1}{N^{2}}\Big{(}r+B~{}ln(r-B)\frac{dN}{dv}\Big{)}\Big{]}$
(42)

Now we solve $\Theta_{(l)}=0$, to determine the dynamical horizon radius as

$\displaystyle
2F-a=2F-F\Big{[}1-\frac{1}{N}\Big{(}\frac{dB}{dv}~{}ln(r-B)-\frac{B}{r-B}\frac{dB}{dv}\Big{)}$
$\displaystyle+\frac{1}{N^{2}}\Big{(}r+B~{}ln(r-B)\frac{dN}{dv}\Big{)}\Big{]}=0~{}~{}~{}~{}~{}~{}~{}$
(43)

Which gives two values of $r$, one is

$\displaystyle r_{D}=2M^{\prime}(v)=\frac{2M}{1-\phi_{v}^{2}}=\frac{2M}{N}$
(44)

by solving $F=0$ and the other can be found by solving
$1+\frac{d}{dv}\Big{[}\frac{B}{N}~{}ln(r_{D}-B)\Big{]}-\frac{r_{D}}{N^{2}}\frac{dN}{dv}=0$
gives

$\displaystyle(r_{D}-B)^{B}e^{r_{D}}=e^{-vN}$ (45)

This value of $r_{D}$ can be written in terms of the Wright Omega function
($\omega$) corless1  as

$\displaystyle
r_{D}=B\Big{[}1+\omega(Z)\Big{]}=\frac{2M}{N}\Big{[}1+\omega(Z)\Big{]}$ (46)

with $Z=-[1+ln(B)+vC]$, $C=\frac{N^{2}}{2M}=\frac{(1-\phi_{v}^{2})^{2}}{2M}$.
It is to be noted that this dynamical radius is outside $r_{D}=\frac{2M}{N}$

The Wright Omega function ($\omega$) is a single-valued function, defined in
terms of the Multi-valued Lambert W function corless2  as
$\omega(Z)=W_{\mathcal{K}(Z)}(e^{Z})$ where
$\mathcal{K}(Z)(=[\frac{(Im(Z)-\pi)}{2\pi}])$ is the unwinding number of $Z$.
The sign of this unwinding number is such that $ln(e^{Z})=Z+2\pi
i\mathcal{K}(Z)$ which is opposite to the sign used in corless3 . The
algebraic properties corless1  of the Wright Omega function ($\omega$) are

$\displaystyle\frac{d\omega}{dZ}$ $\displaystyle=$
$\displaystyle\frac{\omega}{1+\omega}$ (47) $\displaystyle\int{\omega^{n}dZ}$
$\displaystyle=$
$\displaystyle\Bigg{\\{}\begin{split}\frac{\omega^{n+1}-1}{n+1}+\frac{\omega^{n}}{n}~{}~{}if~{}n\neq-1\\\
ln~{}\omega-\frac{1}{\omega}~{}~{}~{}~{}~{}~{}~{}~{}~{}if~{}n=-1\end{split}$
(48)

with the analytic property $Z=\omega+ln~{}\omega$.

Here mention that if we consider the usual Vaidya spacetime without K-essence
scalar field $\phi$ i.e., $m(v,r)\equiv M(v)$ and $\phi_{v}^{2}=0$, then from
the above equation (45) we can get back to the Sawayama’s sawayama  result

$\displaystyle r_{D}=2M(v)\left[1+e^{-v/2M(v)}\right].$ (49)

##  IV DYNAMICAL HORIZON EQUATION

From (16), we can construct the Ricci scalar ($\bar{R}$) and the Ricci tensors
($\bar{R}_{\mu\nu}$) of the K-essence emergent Vaidya spacetime as

$\displaystyle\bar{R}_{vv}=\frac{1}{r}\partial_{v}F-\frac{F}{2}\partial_{r}^{2}F-\frac{F}{r}\partial_{r}F~{}:~{}$
$\displaystyle\bar{R}_{rv}=\bar{R}_{vr}=\frac{1}{2}\partial_{r}^{2}F+\frac{1}{r}\partial_{r}F~{}:~{}\bar{R}_{rr}=0~{};~{}$
$\displaystyle\bar{R}_{\theta\theta}=F+r\partial_{r}F-1~{};~{}\bar{R}_{\Phi\Phi}=sin^{2}\theta\bar{R}_{\theta\theta}~{};~{}$
$\displaystyle\bar{R}=-[\partial_{r}^{2}F+\frac{4}{r}\partial_{r}F+\frac{2}{r^{2}}(F-1)]$
(50)

Using these values of the Ricci scalar and the Ricci tensors (50), we can
construct the components of the energy momentum tensor for the K-essence
emergent Vaidya spacetime through the “emergent” Einstein’s equation
$\bar{R}_{\mu\nu}-\frac{1}{2}\bar{G}_{\mu\nu}\bar{R}=8\pi\bar{T}_{\mu\nu}$,
taking the gravitational constant $G=1$, as

$\displaystyle 8\pi\bar{T}_{vv}$ $\displaystyle=$
$\displaystyle\frac{1}{r}\partial_{v}F+\frac{1}{r}F\partial_{r}F+\frac{F}{r^{2}}(F-1),$
(51) $\displaystyle 8\pi\bar{T}_{vr}$ $\displaystyle=$
$\displaystyle-[\frac{1}{r}\partial_{r}F+\frac{1}{r^{2}}(F-1)],$ (52)
$\displaystyle 8\pi\bar{T}_{rr}$ $\displaystyle=$ $\displaystyle 0,$ (53)
$\displaystyle 8\pi\bar{T}_{\theta\theta}$ $\displaystyle=$
$\displaystyle-[\frac{r^{2}}{2}\partial_{r}^{2}F+r\partial_{r}F]~{};~{}8\pi\bar{T}_{\Phi\Phi}=sin^{2}\theta\bar{T}_{\theta\theta}.$
(54)

Now, for the Schwarzschild background case, the components of the energy-
momentum tensors for the spherically symmetric K-essence emergent
Schwarzschild Vaidya spacetime (38) are

$\displaystyle 8\pi\bar{T}_{vv}$ $\displaystyle=$
$\displaystyle-\frac{2}{r^{2}}[\partial_{v}m+F\partial_{r}m]$ (55)
$\displaystyle=$
$\displaystyle-\left[\frac{2\phi_{v}\phi_{vv}}{r}+\frac{\phi_{v}^{2}}{r^{2}}(1-\frac{2M}{r}-\phi_{v}^{2})\right]$
$\displaystyle 8\pi\bar{T}_{vr}$ $\displaystyle=$
$\displaystyle\frac{2}{r^{2}}\partial_{r}m=\frac{\phi_{v}^{2}}{r^{2}}$
$\displaystyle~{}or~{}8\pi\bar{T}_{vr^{*}}$ $\displaystyle=$
$\displaystyle\frac{2a}{r^{2}}\partial_{r}m=a\frac{\phi_{v}^{2}}{r^{2}}$ (56)
$\displaystyle 8\pi\bar{T}_{rr}$ $\displaystyle=$ $\displaystyle
8\pi\bar{T}_{r^{*}r^{*}}=0$ (57)

Following Sawayama sawayama , we can derive the energy-momentum tensor
$\bar{T}_{\hat{t}l}$ for the integration of the dynamical horizon equation
exactly stated in ashtekar1 ; ashtekar2 ; sawayama  as

$\displaystyle\frac{1}{2G}(R_{2}-R_{1})=\int_{\Delta
H}T_{ab}\hat{\tau}^{a}\xi_{(R)}^{b}d^{3}V$ $\displaystyle+\frac{1}{16\pi
G}\int_{\Delta H}N_{R}[|\sigma|^{2}+2|\zeta|^{2}]d^{3}V$ (58)

where $R_{2}$, $R_{1}$ are the radii of the dynamical horizon, $T_{ab}$ is the
stress-energy tensor, $|\sigma|^{2}=\sigma_{ab}\sigma^{ab}$,
$|\zeta|^{2}=\zeta_{a}\zeta^{a}$, $\sigma_{ab}$ is the shear,
$\zeta^{a}=\tilde{q}^{ab}\hat{r}^{c}\nabla_{c}l_{b}$, with the two-dimensional
metric $\tilde{q}^{ab}$, and $\xi_{(R)}^{a}=N_{R}l^{a}$ with $N_{R}=|\partial
R|$, where $R$ is the radius of the dynamical horizon. This is the dynamical
horizon equation that tells us how the horizon radius changes due to the
matter flow, shear, and expansion. In the spherically symmetric case, the
second term of the right-hand side of the dynamical equation vanishes.

At first, we can write $\bar{T}_{tl}$ in terms of $\bar{T}_{vv}$ and
$\bar{T}_{vr^{*}}$ as

$\displaystyle\bar{T}_{tl}=\bar{T}_{vv}-\bar{T}_{vr^{*}}=-\frac{1}{4\pi
r^{2}}\frac{5}{2}\partial_{v}m$ $\displaystyle=-(\frac{1}{4\pi
r^{2}})\frac{5}{2}r\phi_{v}\phi_{vv}$ (59)

Considering $\hat{t}^{a}$ being the unit vector in the direction of $t^{a}$,
then we have

$\displaystyle\bar{T}_{\hat{t}l}=-\frac{1}{4\pi
r^{2}}\frac{5}{2}(\partial_{v}m)F^{-1}$ (60)

with $F$ is defined in (39). At the horizon, the expression of
$\bar{T}_{\hat{t}l}$ is $\bar{T}_{\hat{t}l}\Big{|}_{r=r_{D}}=\frac{1}{4\pi
r_{D}^{2}}\frac{5r_{D}}{4}(\partial_{v}N)F^{-1}$ where we have used $m\equiv
m(v,r)=M+\frac{r}{2}(1-N)$.

To evaluate, the dynamical horizon integration (58) in terms of the Wright
$\omega$ function, at first, we derive some terms as follows: Using equation
(46), we get

$\displaystyle\frac{dr_{D}}{dv}=\Bigg{[}-\frac{2M}{N^{2}}\Big{(}1+\omega(Z)\Big{)}+\frac{2M}{N}~{}\frac{\omega(Z)}{1+\omega(Z)}\times$
$\displaystyle\Big{(}\frac{1}{N}-\frac{Nv}{M}\Big{)}\Bigg{]}\partial_{v}N-\frac{N~{}\omega(Z)}{1+\omega(Z)}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(61)

At the horizon, we obtain, from the above equation (61)

$\displaystyle\partial_{v}N=\frac{N~{}\omega(Z)}{1+\omega(Z)}~{}\Bigg{[}-\frac{2M}{N^{2}}\Big{(}1+\omega(Z)\Big{)}$
$\displaystyle+\frac{2M}{N}~{}\frac{\omega(Z)}{1+\omega(Z)}~{}\Big{(}\frac{1}{N}-\frac{Nv}{M}\Big{)}\Bigg{]}^{-1}$
(62)

and from (46)

$\displaystyle\frac{dr_{D}}{dN}=\Big{(}\frac{N}{\partial_{v}N}\Big{)}\frac{\omega(Z)}{1+\omega(Z)}.$
(63)

Now, re-write the equation (39) in terms of Write $\omega$ function as

$\displaystyle F=\frac{N~{}\omega(Z)}{1+\omega(Z)}$ (64)

and from the equation (40)

$\displaystyle\partial_{v}m=-\Big{(}\frac{r_{D}}{2}\Big{)}\partial_{v}N.$ (65)

Changing the order of integration $r_{D}$ to $N$ in dynamical horizon
integration (58), we have

$\displaystyle\frac{1}{2}\left[\frac{2M}{N}(1+\omega(Z))\right]\Biggr{|}_{N_{1}}^{N_{2}}=\frac{5}{4}\int_{N_{1}}^{N_{2}}\frac{2M}{N}(1+\omega(Z))dN$
(66)

since in the above calculation, the functions $\partial_{v}N$ and $F^{-1}$
with fixed $r_{D}$ are used only in the integration.

Now, using the equations (60) to (65) and inserting the above equation (66) in
the equation (58) and taking the limit $N_{2}\rightarrow N_{1}=N$, we have

$\displaystyle-\frac{M}{N^{2}}\Big{(}1+\omega(Z)\Big{)}+\frac{M}{N}\frac{\omega(Z)}{(1+\omega(Z))}\Big{(}\frac{1}{N}-\frac{Nv}{M}\Big{)}$
$\displaystyle-\frac{5}{2}\frac{M}{N}\Big{(}1+\omega(Z)\Big{)}=0~{}~{}~{}~{}~{}$
(67)

which is the dynamical horizon equation for the spherically symmetric
K-essence emergent Schwarzschild Vaidya spacetime. This dynamical horizon
equation in the presence of the kinetic energy of the K-essence scalar field
is far different from the usual Vaidya dynamical horizon equation in sawayama
.

##  V HAWKING RADIATION IN THE K-ESSENCE SCHWARZSCHILD VAIDYA SPACETIME

Now we discuss about the Hawking radiation haw1 ; haw2 ; haw3 ; haw4 ; haw5 ;
haw6 ; haw7 ; haw8 ; haw9 ; haw10 . To solve this problem, we consider two
ideas, which are (1) to use the dynamical horizon equation (58), and (2) to
use the K-essence Schwarzschild Vaidya metric (38) using tunneling mechanism
parikh ; mitra1 ; mitra2 ; mitra3 ; mitra4 ; padma1 ; padma2 ; kuroda ;
siahaan .

###  V.1 Using the dynamical horizon equation:

Considering the result of Candelas cand , for the matter on the dynamical
horizon, which assumes that spacetime is almost static and is valid near the
horizon sawayama , from (38), $r\sim 2m(\equiv\frac{2M}{N})$

$\displaystyle\bar{T}_{tl}$ $\displaystyle=$
$\displaystyle\frac{-1}{2\pi^{2}(1-\frac{2m}{r})}\int_{0}^{\infty}\frac{w^{3}dw}{e^{8\pi
mw}-1}$ (68) $\displaystyle=$
$\displaystyle\frac{-1}{2m^{4}\pi^{2}c(1-\frac{2m}{r})}$

with $c=15\times 8^{4}=61440$, where we have used the following value of the
integration
$\int_{0}^{\infty}\frac{w^{3}dw}{e^{bw}-1}=\frac{\pi^{4}}{15b^{4}}$. The
matter energy of the equation (68) is negative near $r\sim 2m$, which means
that in the dynamical horizon equation, the K-essence Schwarzschild Vaidya
black hole absorbs negative energy i.e., the black hole radius decreases. Note
that, the K-essence Schwarzschild Vaidya mass
$m(\equiv\frac{M}{1-\phi_{v}^{2}})$, at near the horizon, is greater than the
Schwarzschild mass as $\phi_{v}^{2}<1$. Also, the dynamic behavior of the mass
function $m(v,r)$ is carried by the kinetic energy ($\phi_{v}^{2}$) of the
K-essence scalar field which is defined by the equation (40). Here we use the
dynamical horizon equation, since we need only information about the matter
near the horizon, without solving the full Einstein equation with the
backreaction. Now following sawayama , we can get

$\displaystyle\bar{T}_{\hat{t}l}=\frac{1}{2m^{4}\pi^{2}c(1-\frac{2m}{r})}.$
(69)

Again, by changing the order of integration of the right hand side of the
dynamical horizon equation (58) from $r_{D}$ to $N$, we get

$\displaystyle\int_{r_{1}}^{r_{2}}4\pi r_{D}^{2}\bar{T}_{\hat{t}l}dr_{D}$ (70)
$\displaystyle=$ $\displaystyle
b\int_{N_{1}}^{N_{2}}\Big{[}\frac{2M}{N}(1+\omega(Z))\Big{]}^{2}\frac{(1+\omega(Z))}{(N\omega(Z))}\times$
$\displaystyle\Big{[}M+\frac{M}{N}(1+\omega(Z))(1-N)\Big{]}^{-4}\Big{(}\frac{dr_{D}}{dN}\Big{)}dN$
$\displaystyle=$ $\displaystyle
b\int_{N_{1}}^{N_{2}}\Big{[}\frac{2M}{N}(1+\omega(Z))\Big{]}^{2}\times$
$\displaystyle\Big{[}M+\frac{M}{N}(1+\omega(Z))(1-N)\Big{]}^{-4}\Big{(}\frac{1}{\partial_{v}N}\Big{)}dN$

where $b=\frac{2}{\pi c}$ is a constant and
$\frac{dr_{D}}{dN}=\frac{N\omega(Z)}{1+\omega(Z)}\times\frac{1}{\partial_{v}N}$.

Since in our case, the time dependence of the K-essence Schwarzschild Vaidya
metric is carrying $\phi_{v}^{2}$ i.e., $N(=1-\phi_{v}^{2})$. Now inserting
(70) into right hand side of the dynamical horizon equation (58), we obtain

$\displaystyle\frac{1}{2}\Big{[}\frac{2M}{N}\Big{(}1+\omega(Z)\Big{)}\Big{]}\Biggr{|}_{N_{1}}^{N_{2}}=b\int_{N_{1}}^{N_{2}}\Big{[}\frac{2M}{N}\Big{(}1+\omega(Z)\Big{)}\Big{]}^{2}\times$
(71)
$\displaystyle\Big{[}M+\frac{M}{N}\Big{(}1+\omega(Z)\Big{)}(1-N)\Big{]}^{-4}\Big{(}\frac{1}{\partial_{v}N}\Big{)}dN$
$\displaystyle+$
$\displaystyle\frac{5}{4}\int_{N_{1}}^{N_{2}}\frac{2M}{N}\Big{(}1+\omega(Z)\Big{)}dN$

where
$r_{D}=B\Big{[}1+\omega(Z)\Big{]}=\frac{2M}{N}\Big{[}1+\omega(Z)\Big{]}$. Now
taking the limit $N_{2}\rightarrow N_{1}=N$ and $\omega(Z)\neq 0$, the
dynamical horizon equation (71) becomes

$\displaystyle
M^{2}\Bigg{[}1+\Big{(}1-\frac{1}{N}\Big{)}\Big{(}1+\omega(Z)\Big{)}\Bigg{]}^{4}$
(72) $\displaystyle=$
$\displaystyle\frac{4b(1+\omega(Z))^{3}}{N^{3}\omega(Z)}\Bigg{[}\frac{1}{2}-\frac{5(1+\omega(Z))}{4}\times$
$\displaystyle\Bigg{(}-\frac{1+\omega(Z)}{N}+\frac{\omega(Z)}{1+\omega(Z)}\Big{(}\frac{1}{N}-\frac{Nv}{M}\Big{)}\Bigg{)}^{-1}\Bigg{]}^{-1}$

provided $N\neq 0$ and $M\neq 0$. From this equation (72), we can say that $N$
is a function of $v$, therefore $\phi_{v}^{2}$ is a function of $v$ since
$\phi_{v}^{2}=1-N$ . This equation is much more different from Sawayama’s
sawayama  transcendental equation (37) because the equation (72) consists the
wright omega function $\omega(Z)$ ( where $Z=-[1+ln(B)+vC]$ ) in a extensive
way and analytically there is no possibility to find $N$.

But in this section, our main objective is to find the behavior of the mass of
the black hole $m(v,r)$ with $v$ for fixed $M$. In sawayama , from the
equation (37) one can find the behaviour of $M(v)$ directly with the help of
numerical methods but here first we have to solve the equation (72) for $N$
numerically for a particular value of $M$ provided $M\neq 0$. So that we can
find the value of $\phi_{v}^{2}$ for that particular value of $M$ by using the
expression $\phi_{v}^{2}=1-N$. And then finally we are able to see the
behavior of the mass $m(v,r)$ with $v$ using the relation (40).

Now if we solve the equation (72) for $M=1$, $M=5$ and $M=10$ separately the
numerical solutions of $\phi_{v}^{2}$ always decreases with increasing $v$ as
shown in the left column of the figure (1). Now substituting these values of
$\phi_{v}^{2}$ for $M=1$ and $r=10$ in (40) we find that the mass how the
black hole $m(v,r)$ is decreasing but not tends to zero as shown in the first
figure of the right column of (1). Similar behaviour of $m(v,r)$ can be found
if we solve the equations (72) and (40) for M=5, r=100 and M=10, r=50 as shown
in the remaining two figures.

Now analyzing this decreasing behaviour of $\phi_{v}^{2}$ with $v$ we can
conclude that the mass of the blackhole in the K-essence Schwarzschild Vaidya
spacetime decreases. In limiting case, $m(v,r)\rightarrow M$ as
$\phi_{v}^{2}\rightarrow 0^{+}$ when $r$ and $M$ are fixed or
$m(v,r)\rightarrow M$ as $r\rightarrow 0$ when $\phi_{v}^{2}$ and $M$ are
fixed and so there is no possibility of full evaporation since $M\neq 0$ which
is quite different from Sawayama’s result sawayama  where mass of the black
hole always vanishes.

Figure 1: (Color online) Figures in the left column show the numerical
solutions of $\phi^{2}_{v}$ for M=1, M=5, and M=10. Figures in the right
column show the numerical solutions of $m(v,r)$ for (M=1, r=10) , (M=5, r=100)
and (M=10, r=50).

###  V.2 Using the tunneling formalism:

To calculate the Hawking temperature using tunnelling formalism parikh ;
mitra1 ; mitra2 ; mitra3 ; mitra4 ; padma1 ; padma2 ; kuroda ; siahaan ; gm3 ;
gm4 ; gm5 , consider a massless particle in a black hole (38) background is
described by the Klein-Gordon equation

$\displaystyle\hbar^{2}\left(-\bar{G}\right)^{-1/2}\partial_{\mu}\left(\bar{G}^{\mu\nu}\left(-\bar{G}\right)^{1/2}\partial{\nu}\Psi\right)$
(73)

where $\Psi$ can be taken in the form

$\displaystyle\Psi=\textit{exp}\left(\frac{i}{\hbar}S+...\right)$ (74)

to obtain the leading order in $\hbar$ the Hamilton-Jacobi equation is

$\displaystyle\bar{G}^{\mu\nu}\partial_{\mu}S\partial_{\nu}S=0$ (75)

considering $S$ is independent of $\theta$ and $\Phi$. Therefore, we have,

$\displaystyle
2\partial_{v}S\partial_{r}S+\Big{(}1-\frac{2M}{r}-\phi_{v}^{2}\Big{)}\Big{(}\partial_{r}S\Big{)}^{2}=0$
(76)

Now, let us choose the action $S$ in the following form siahaan

$\displaystyle S(v,r)=-\int_{0}^{v}E(v^{\prime})dv^{\prime}+S_{0}(v,r)$ (77)

so that

$\displaystyle\partial_{v}S=-E(v)+\partial_{v}S_{0};$ $\displaystyle
and~{}~{}\partial_{r}S=\partial_{r}S_{0}$ (78)

Since $S_{0}$ is dependent on $v$ and $r$, so we can write,

$\displaystyle\frac{dS_{0}}{dr}=\partial_{r}S_{0}+\frac{dv}{dr}\partial_{v}S_{0}$
$\displaystyle=\partial_{r}S_{0}+\frac{2}{F}\partial_{v}S_{0}$ (79)

where we have used $\frac{dv}{dr}=\frac{2}{F}$, $F$ is defined in equation
(39). Now applying the equations (78) and (79) in equation (76), we get

$\displaystyle F\frac{dS_{0}}{dr}=2E(v)$ (80)

since $\partial_{r}S_{0}\neq 0$. Therefore, we can obtain the solution of
$S_{0}$ as

$\displaystyle S_{0}=2E(v)\int\frac{dr}{F}\equiv
2E(v)\int\frac{dr}{(1-\frac{2M}{r}-\phi_{v}^{2})}$
$\displaystyle=\frac{2E(v)}{N}\int\frac{r~{}dr}{r-2M/N}=2\pi
i\frac{4ME(v)}{N^{2}}$ (81)

with $N=1-\phi_{v}^{2}$. Here, we have used the Cauchy-integral formula since
in the integral, $r$ is analytic inside and on any simple closed contour $C$
which is taken in the positive sense and $\frac{2M}{N}$ is a point interior to
$C$. Therefore, (77) become,

$\displaystyle S(v,r)=-\int_{0}^{v}E(v^{\prime})dv^{\prime}+2\pi
i\frac{4ME(v)}{N^{2}}$ (82)

So the wave function for the outgoing (and ingoing) massless particle can be
read as

$\displaystyle\Psi_{out}(v,r)=exp\Big{[}\frac{i}{\hbar}\Big{(}-\int_{0}^{v}E(v^{\prime})dv^{\prime}+\pi
i\frac{4ME(v)}{N^{2}}\Big{)}\Big{]}$ (83)
$\displaystyle\Psi_{in}(v,r)=exp\Big{[}\frac{i}{\hbar}\Big{(}-\int_{0}^{v}E(v^{\prime})dv^{\prime}-\pi
i\frac{4ME(v)}{N^{2}}\Big{)}\Big{]}$ . (84)

The tuneling rate for the outgoing particle is

$\displaystyle\Gamma\sim e^{-2\textit{Im}S}\sim e^{-2\frac{4\pi
ME(v)}{N^{2}}}=e^{-\frac{E(v)}{K_{B}T}}$ (85)

where $K_{B}$ is Boltzman Constant. Therefore, the Hawking temperature is

$\displaystyle T_{H}=\frac{1}{8\pi K_{B}}\frac{N^{2}}{M}=\frac{1}{8\pi
K_{B}}\frac{(1-\phi_{v}^{2})^{2}}{M}$ (86)

If we consider $\phi_{v}^{2}=0$ and $m(v,r)=M(v)$, then we can get back the
usual Hawking temperature for the Vaidya spacetime tang .

##  VI Conclusion

The connection between the K-essence geometry and the Vaidya spacetime have
been established by Manna et. al. gm1 , which generates a new spacetime namely
”the K-essence emergent Vaidya spacetime”. We have derived the dynamical
horizon equation (67) for the spherically symmetric K-essence emergent
Schwarzschild Vaidya spacetime which is far different from the Sawayama’s
sawayama  result i.e. the usual Vaidya dynamical horizon equation. We have
investigated the Hawking radiation in the K-essence emergent Vaidya spacetime
according to the modified definition of dynamical horizon sawayama . Our study
of Hawking radiation using the dynamical horizon equation gives the
transcendental equation (72) which is much more complicated than the
Sawayama’s sawayama  transcendental equation (38). This transcendental
equation (72) in the presence of the wright omega function corless1 , can be
solved numerically not analytically. The numerical solutions for different
fixed values of $M$ and $r$, $\phi_{v}^{2}$ and $m(v,r)$ always decreases with
increasing $v$ but not tends to zero since $M\neq 0$. Analytically we have
shown that the black hole mass $m(v,r)$ in the K-essence emergent
Schwarzschild Vaidya spacetime always decreases but not fully vanishes by
using dynamical horizon equation as $\phi_{v}^{2}\rightarrow 0^{+}$ . The main
difference between Vaidya spaecetime and K-essence emergent Schwarzschild
Vaidya spacetime is that the mass always vanishes in the former case but not
in later case. Though in both cases we can see that the decreasing behaviour
of the mass of a blackhole. And finally we have evaluated the Hawking
temperature $T_{H}=\frac{1}{8\pi K_{B}}\frac{(1-\phi_{v}^{2})^{2}}{M}$ using
tunneling mechanism, again which is different from the usual Vaidya case.

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# Dramatic Changes in the Electronic Structure Upon Transition to the
Collapsed Tetragonal Phase in CaFe2As2

R. S. Dhaka  Rui Jiang  S. Ran  S. L. Bud’ko  P. C. Canfield  Division of
Materials Science and Engineering, The Ames Laboratory, U.S. DOE  Department
of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA  Milan
Tomic̀  Roser Valentí  Institut für Theoretische Physik, Goethe-Universität
Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany  Yongbin
Lee  Division of Materials Science and Engineering, The Ames Laboratory, U.S.
DOE  B. N. Harmon  Adam Kaminski  Division of Materials Science and
Engineering, The Ames Laboratory, U.S. DOE  Department of Physics and
Astronomy, Iowa State University, Ames, Iowa 50011, USA

###### Abstract

We use angle-resolved photoemission spectroscopy (ARPES) and density
functional theory (DFT) calculations to study the electronic structure of
CaFe2As2 in previously unexplored collapsed tetragonal (CT) phase. This
unusual phase of the iron arsenic high temperature superconductors was hard to
measure as it exists only under pressure. By inducing internal strain, via the
post growth, thermal treatment of the single crystals, we were able to
stabilize the CT phase at ambient-pressure. We find significant differences in
the Fermi surface topology and band dispersion data from the more common
orthorhombic-antiferromagnetic or tetragonal-paramagnetic phases, consistent
with electronic structure calculations. The top of the hole bands sinks below
the Fermi level, which destroys the nesting present in parent phases. The
absence of nesting in this phase along with apparent loss of Fe magnetic
moment, are now clearly experimentally correlated with the lack of
superconductivity in this phase.

###### pacs:

74.25.Jb,74.62.Dh,74.70.-b,79.60.-i

The AFe2As2 (A = Ca, Sr, Ba) materials have become one of the key systems for
the study of Fe-based high temperature superconductivity Kamihara08 ;
Takahashi08 ; XHChen08 ; TYChen08 ; GFChen08 ; Rotter08 ; Cruz08 ; SefatPRL08
; Ni08 ; Canfield09 . Detailed substitution and pressure studies on these
systems have revealed that superconductivity is intimately linked to the
magnetic state of iron and can be turned on or off by manipulating the nature
of the order, the fluctuations or even the existence of Fe-magnetism Rotter08
; Ni08 ; Cruz08 ; SefatPRL08 ; Canfield09 ; Johnston10 ; CanfieldRev10 ;
Hirschfeld ; LeePRB09 ; AlirezaJPCM08 . Moreover, in contrast to cuprate
superconductors, the Fe-based superconductors are multiorbital systems and
orbital degrees of freedom have been also proposed to be important for the
understanding of the structural, magnetic and superconducting properties of
these materials Krueger ; Lee1 ; Lee2 . At ambient pressure CaFe2As2 manifests
a strongly first order, coupled structural / magnetic phase transition at 170
K that is exceptionally pressure sensitive with a remarkably large pressure
derivative Ni08 ; Canfield09 . For pressures as low as 0.4 GPa another
dramatic, first order phase transition to a non-moment bearing collapsed
tetragonal phase Ni08  is stabilized near 100 K and rapidly increases in
temperature for higher applied pressures. Whereas the electronic properties of
CaFe2As2 in the ambient pressure phases LiuPRL09 ; KondoPRB10  were found to
be similar to the other parent compounds of 122 family ShimojimaPRL10 ;
YangPRL09 ; SatoPRL09 ; FinkPRB09 ; JensenPRB11 ; GuodongPRB09 ; ZhouPRB10 ;
YiPRB09 , the electronic behavior of the CT phase is so far largely unexplored
due to the need for at least 0.4 GPa of external pressure.

Post growth annealing/quenching of CaFe2As2 samples grown from excess FeAs can
be used to control the degree of internal strain due to Fe-As precipitates
associated with a small width of formation RanPRB11 ; RanPRB12 ; GatiPRB12 .
One of the important findings of these works is that the non-magnetic,
collapsed tetragonal phase can be stabilized in an ambient pressure sample by
using internal strain. Key spectroscopic tools, such as ARPES and STM, which
normally cannot be combined with pressures even at the 0.1 GPa level, can now
be brought to play in order to understand the electronic properties of the
collapsed tetragonal phase.

In this Letter we present ARPES measurements of the electronic properties of
CaFe2As2 in the CT phase and compare them with the orthorhombic and (non-
collapsed) tetragonal phases. Our data demonstrate that the band dispersion
and FS topology change significantly with the transition to the CT phase, in
agreement with our electronic structure calculations. In particular the tops
of the bands associated with Fe dxy and dxz+dyz states at the center of the
Brillouin zone move from above $E_{\rm F}$ to just below $E_{\rm F}$,
eliminating the hole pockets. Whereas there are significant changes elsewhere
(to maintain electron count), loss of the nesting between the hole and
electron sheets brought about by the shifts we observe are of key importance
for magnetic and superconducting states in the low pressure phase DhakaPRL13 .
Our findings provide a peek into a phase that was previously accessible only
under external pressure and provide further evidence of an intimate relation
between the nature and nesting of the Fermi surface and magnetism in the iron
arsenic high temperature superconductors.

Figure 1: (a) Schematic phase diagram for CaFe2As2 system based on data in
Ref. RanPRB11 . Blue, green and red circles mark the annealing condition of
samples used in this study. The hatched area marks the boundary between two
distinct phases. (b-d) band dispersion along cuts #1 and #2 for the three
samples measured at T=15 K and $k_{z}=$15.3$\pi/c$. Figure 2: The band
structure of CaFe2As2 in three different phases measured at $k_{z}=$13.5
$\pi/c$. (a) Fermi surface map (b) band dispersion along cut #1 (marked in
panel a) (c) band dispersion along cut #2 measured at 190 K in
tetragonal/paramagnetic phase for a sample that was post-growth annealed at
400∘C. (d–f) the same plots for the same sample measured at 15K in
orthorhombic/antiferromagnetic phase. Arrow in panel (d) points to small
pockets associated with AFM order, while arrows in panels (e,f) point to band
splitting caused by AFM order. (g–i) the same plots for a sample that was
post-growth annealed at 960∘C measured at 110 K again in
tetragonal/paramagnetic phase. (k–m) the same plots for the “960C” sample, but
measured at T=15 K in the CT phase. (n) EDCs at the corner of the zone,
k=(1,1). The location of the peak marks the energy of the bottom of the
electron pocket. (o) MDCs at the center of the zone - bottom two curves and
corner of the zone - top two curves. Figure 3: Results of the DFT calculation
of the band structure in CT at $k_{z}=$15.3$\pi/c$ corresponding to ARPES data
shown on Fig. 1. (a) Left column shows band dispersion at the zone center, and
right column corresponds to zone corner. (b) Calculated band dispersion along
kz direction. Dashed lines mark the $k_{z}$ values of data in Figs. 1 and 2.
Colors denote orbital contributions.

Single crystals of CaFe2As2 were grown from excess FeAs by slowly cooling a
melt of CaFe4As4 from 1180∘C to 960∘C and then decanting off the excess
liquid, essentially quenching the samples from 960∘C to room temperature,
which is referred as “960C” sample RanPRB11 . Upon cooling this sample becomes
nonmagnetic and has CT unit cell with significantly reduced $c$/$a$ ratio for
T$\lesssim$100 K. This is remarkably similar to what has been observed for Sn
grown CaFe2As2 samples under hydrostatic pressure (0.6 GPa)RanPRB11 . Post
growth thermal treatments of these samples involve annealing at temperatures
ranging from 350 to 800∘C for 24 hours, and subsequently quenching them from
this annealing temperature to room temperature. Here, we have measured three
samples identified by their annealing/quenching temperatures: “400C”, “700C”
and “960C”. As shown in Fig. 1a, these three samples cover the salient parts
of the phase diagram: $\sim$170 K transition to low-temperature
AF/orthorhombic (“400C”); $\sim$120K transition to AF/orthorhombic (“700C”),
and $\sim$100 K transition to non-magnetic CT phase (“960C”). At ambient
conditions, the properties of the last two samples are remarkably similar to
ones where hydrostatic pressure is applied to Sn grown samples of 0.35 and
0.675 GPa Canfield09 ; RanPRB11 . ARPES measurements have been performed (in
grazing incidence geometry shown in Fig. 1(v) of Ref. DhakaPRL11, ) at
beamline 10.0.1 of the Advanced Light Source (ALS) using a Scienta R4000
electron analyzer. The measurements at Ames Laboratory were acquired using a
Scienta SES2002 electron analyzer and a GammaData helium ultraviolet lamp. The
samples were mounted on an Al pin using UHV compatible epoxy and in situ
cleaved perpendicular to the $c$-axis, yielding single layer surfaces in the
$a$-$b$ plane. All ARPES data were collected in ultrahigh vacuum with
pressures better than 4$\times$10-11 torr. The energy and angular resolutions
were set at $\approx 15$ meV and $\approx 0.3^{\circ}$, respectively. The
Fermi level ($E_{\rm F}$) of the samples was referenced by measuring for a Au
film evaporated in-situ onto the sample holder. High symmetry points were
defined the same way as in Refs. LiuNP10, . The measurements carried out on
several samples yielded similar results for the band dispersion and FSs. Ab-
initio calculations Tomic1 ; Tomic2  based on density functional theory as
implemented in the Vienna ab initio simulations package (VASP) Kresse
combined with the Fast Inertial Relaxation Engine (FIRE) algorithm FIRE  were
performed at finite pressure, both for compressive and tensile stress. A
collapsed tetragonal phase was obtained in the calculations only for
isotropically applied pressures as well as uniaxial pressures along the
c-axis.

The evolution of the band structure with annealing/quenching temperature,
which is equivalent to applying pressure, for $k_{z}=15.3~{}\pi/c$ is shown in
Figure 1. Because there is a significant decrease in the c-axis lattice
parameter in the CT phase, we used different photon energies in order to
compare data at similar values of $k_{z}$. This momentum plane was selected
for best signal to background ratio and clarity of the band dispersion in data
(matrix elements). This momentum plane is accessed using 45 eV photons for
orthorhombic/tetragonal phases and 62 eV for the CT phase. Panels b–d show the
band dispersion along two cuts (marked in Fig. 2a) for “400C”, “700C” and
“960C” samples, respectively. The first two samples remain in
orthorhombic/antiferromagnetic phase at low temperatures. At this value of
$k_{z}$ and T=15 K, the matrix elements and relatively low scattering allow
observation of two well separated hole bands in “400C” sample. These bands
broaden in “700C” sample, most likely due to some degree of inhomogenous
stress being present. The “960C” sample at low temperature is in the CT phase,
which is equivalent to application of 0.7 GPa of hydrostatic pressure to
stress free, Sn flux grown samplesCanfield09 ; RanPRB11 ; RanPRB12 . The band
structure in this phase is very different indeed as illustrated in panel (d).
The hole pockets sink below the Fermi energy while the electron band moves to
higher binding energy and becomes more dispersive and sharper. The vanishing
of large portions of the hole Fermi sheets in CT phase resembles to a degree
situation in its structural analog - CaFe2P2ColdeaPRL09 .

In Figure 2 we plot the Fermi surface and band dispersion in all three phases
that occur in the CaFe2As2. The tetragonal/paramagnetic phase can be accessed
at higher temperatures in both “400C” and “960C” samples (see phase diagram in
Fig. 1a). The data in orthorhombic and tetragonal phases was measured using 35
eV, which corresponds to $k_{z}=13.5~{}\pi/c$. The same value of $k_{z}$ is
accessed in CT phase when using 45 eV photons. The Fermi surface maps for
these phases are shown in panels (a) and (g) and the band dispersions along
two symmetry cuts are shown in panels (b, c) and (h, i) for “400C” and “960C”
samples, respectively. The electronic structure of the tetragonal phase
consists of two hole pockets at the center of the zone and two electron
pockets at the corners of the zone. Due to increased scattering at high
temperatures the pairs of bands appear as a single, broader streak of
intensity. These high temperature data are very similar to data measured in
parent compounds in the 122 family at ambient pressures LiuPRL09 ; DhakaPRL11
; VilmercatiPRB09 . The “400C” sample has similar properties to samples at
ambient pressure RanPRB11  e. g. grown using Sn flux that are stress free (see
phase diagram in panel 1a). Upon cooling the “400C” sample enters the well
documented orhtorhombic/antiferromagnetic phase. This is evident from data in
Figs. 2(d,e,f), where complicated Fermi surface reconstruction (straight,
nested sections of the hole pockets and sharp, high intensity points in
electron pockets) and splitting of the bands at high energy is present. This
is fully consistent with previous data in the AFM state LiuPRB09 ; YangPRL09 ;
ZhangPRL09 ; KondoPRB10 ; LiuPRL09 ; YiPRB09  and signifies that “400C”
samples are indeed ambient/low pressure equivalent.

Figure 4: FS maps of CaFe2As2 “960C” sample measured with (a) 35, (b) 45, (c)
49 and (d) 62 eV photon energies at T${}_{\rm S}\approx$15 K. (e) intensity
plots close to the center of the zone along cut#1 for photon energies from 30
eV to 80eV. (f) EDCs at the center of the zone for photon energies from 30 eV
to 80 eV, plotted every 1 eV. Thick (black) lines mark photon energies of 45
eV and 62 eV used in Figs. 1 and 2.

As previously mentioned, the “960C” sample shows a transition from the high
temperature tetragonal phase (Fig. 2g-i) to the CT phase (Fig. 2 k-m) upon
cooling. Overall, the band dispersion in the CT phase at this $k_{z}$ plane
(panels l and m) is very similar to data in Fig. 1. The appearance of brighter
spots at different (from data in Fig. 1d) momentum points for both hole and
electron bands signifies modifications in the matrix elements due to different
photon energies used. Overall the electronic properties in the CT phase agree
qualitatively with the recently reported theoretical calculations Tomic1 ;
Tomic2 ; SannaPRB12 . Perhaps the most striking characteristic of the CT phase
is the sinking of the hole pockets under the $E_{\rm F}$ and associated with
this lack of the Fermi surface nesting. We quantify the key changes in the
band dispersion by plotting the Energy Distribution Curves (EDCs) and Momentum
Distribution Curves (MDC) at the high symmetry points in panels (n) and (o).
In the CT phase, the bottom of the electron pocket moves to higher binding
energy from $\sim$70 meV to $\sim$90 meV, as shown in panel (n). At the same
time the diameter of the electron pocket shrinks slightly and the MDC peaks
become sharper (top two curves in panel (o)). The MDC close to zone center
have two peaks in the tetragonal phase, which signifies presence of a finite
size hole pocket. Upon transition to the CT phase only weaker, single peak
centered at the (0,0) is present (lower curve in panel (o)) that is consistent
with an absence of the hole pocket at this $k_{z}$ plane and the top of this
band being located below the chemical potential.

Results of DFT calculations of the band structure across the orthorhombic/CT
phase transition are shown in Fig. 3. We observe that simulations under
hydrostatic pressure mimic ARPES data most closely, when renormalization by a
factor of five is employed correlations . In particular, we find close
correspondence between the observed Fermi surface behavior of ”700C” and
”960C” structures and P/Pc=0.95, and P/Pc=1.02 hydrostatic pressure
structures, in the immediate vicinity of the Orthorhombic-to-CT phase
transition at Pc. At the transition, the dxy, dyz and dxz bands abruptly drop
below the Fermi level around the $\Gamma$-point, causing hole cylinders
parallel to kz-axis to quickly disappear completely within the collapsed
tetragonal phase, as observed in the ARPES measurements. These changes in the
electronic structure show clearly that the phase transition tetragonal/CT and
orthorhombic/CT cannot be explained through a rigid band shift, as has been
also discussed in Co-doped Ca122 Damascelli . In panel (b) we plot the
calculated band dispersion in the CT phase along $z$-direction at the center
of the zone. Except for a very small crossing at $E_{\rm F}$ by one of the
bands around the $k_{z}=14~{}\pi/c$, most of the bands are located below the
chemical potential.

To examine the electronic properties along the $z$-direction and ensure that
the differences we reported in the CT phase do not arise due to some
unforeseen complications related to $k_{z}$ dispersion, we measured the FS and
band structure using wide range of photon energies. The FS maps for selected
photon energies are shown in Figs. 4 (a–d). There is very little variation in
overall shape of the Fermi surface. The intensity of the inner and outer
electron pockets changes with photon energy due to matrix elements, but their
separation remains almost constant. Intensity plots along cut #1 in CT phase
measured using several photon energies are shown in Fig. 4e. The top of the
band at the center of the zone appears to be consistently located below the
chemical potential and the crossing predicted by the calculations (see Fig.
3b) is not visible in the data directly. This could be due to matrix elements,
or overwhelming intensity from the broad bands located below $E_{\rm F}$. In
Fig. 4f we plot EDC curves at the zone center as a function of photon energy
with 1 eV step. In addition to a sharp peak just below $E_{\rm F}$, a second
peak at higher binding energy can be observed for range of $k_{z}$ values
centered at $\sim$12.5 $\pi/c$ and $\sim$16 $\pi/c$. These structures most
likely correspond to overlap of dxz and d${}_{z^{2}}$ bands shown in Fig. 3b.
Based on DFT calculation, one would expect two broader, main peaks at binding
energies of -50 meV and -150 meV, while those peaks are observed in the data
at -18 meV and -68 meV. This discrepancy can be attributed to correlation
effects which are not properly accounted for in DFT calculations. These
effects can lead to strong band renormalizations Kotliar ; Valenti .The Fermi
crossing around the $k_{z}=14~{}\pi/c$ is not directly seen in the data, most
likely due to relatively weak intensity of this band.

In conclusion, our results reveal the electronic structure of CT phase in
unsubstituted CaFe2As2. Our data demonstrates that the FS and band dispersion
in CT phase is significantly different from ambient pressure phases and in
qualitative agreement with theoretical calculations for this phase. The hole
pockets do not extend over appreciable range of the $k_{z}$ values and the
destruction of nesting is accompanied by the vanishing of Fe magnetic moments
and order. The resulting lack of magnetic fluctuations could be responsible
for the absence of superconductivity in this phase.

We thank Sung-Kwan Mo for instrumentation support at the ALS. This work was
supported by the U.S. Department of Energy (DOE), Office of Science, Basic
Energy Sciences, Materials Science and Engineering Division. Ames Laboratory
is operated for the U.S. DOE by Iowa State University under contract # DE-
AC02-07CH11358 (sample growth, ARPES measurements and data analysis). MT and
RV thank the Deutsche Forschungsgemeinschaft for funding through grant SPP
1458 (DFT calculations). The Advanced Light Source is supported by the
Director, Office of Science, Office of Basic Energy Sciences, of the U.S.
Department of Energy under Contract No. DE-AC02-05CH11231.

Note added. After submission of this manuscript, we became aware of study
electronic structure of CT phase in chemically substituted Ca(Fe1-xRhx)2As2 by
Tsubota et al. Tsubota .

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