Kasner Metrics and Very Special Geometry

W. A. Sabra
Physics Department, American University of Beirut
Lebanon

Abstract

> We consider general charged Kasner-like solutions for the theory of five-
> dimensional supergravity coupled to Abelian vector multiplets in arbitrary
> space-time signature. These solutions, depending on the choice of
> coordinates, can be thought of as generalisations of Melvin/Rosen
> cosmologies, flux-branes and domain walls.

##  1 Introduction

In recent years an active area of research has been the classification and
analysis of supersymmetric gravitational backgrounds in supergravity theories
with various space-time dimensions and signatures. The present work deals with
non-supersymmetric cosmological and static solutions of the theories of five-
dimensional supergravities coupled to vector multiplets in arbitrary space-
time signature [1]. In these supergravity theories, the dynamics of the
bosonic fields can be described in terms of the so called very special
geometry [2]. Early analysis of supersymmetric solutions, with vanishing
fermionic fields, was based on starting with a specific ansatz for the space-
time metric and fixing it alongside the bosonic fields by requiring the
existence of some Killing spinors (see for example [3]). A systematic
classification of solutions admitting Killing spinors for minimal and general
five-dimensional supergravity theory was first considered in [4]. Later many
interesting solutions were found including the novel black ring solutions with
an event horizon of topology $S^{1}\times S^{2}$[5]. Five-dimensional
supergravity theories with vector multiplets in all possible space-time
signatures were recently considered in [6, 7]. A rigorous analysis of
supersymmetry algebras in all space-time signatures was given in [8]. A
generalisation of the results of [4] related to time-like solutions to all
five-dimensional supergravity was considered in [9]

In addition to the supergravity models based on symmetric scalar manifolds, a
large class of supergravities with Lorentzian signature can be obtained as
compactifications of eleven-dimensional supergravity [10] on a Calabi-Yau
3-folds [11]. Similarly, five-dimensional supergravity theories in various
space-time signatures can be obtained via Calabi-Yau 3-folds compactification
of the exotic eleven-dimensional supergravity theories constructed by Hull
[12].

It is our purpose in the present work to study Kasner-like cosmological and
static solutions to five-dimensional supergravity theories. The analysis of
classical time-dependent gravitational solutions in string theory and
supergravities is important to investigate the role that these theories can
play in gravitational physics and cosmology.

The plan of this paper is as follows. In the next section we briefly review
the vacuum Kasner metrics in four and five dimensions and their extensions to
solutions of Einstein-Maxwell theory. In section three, we present some basic
properties of the ungauged five-dimensional $N=2$ supergravity theory and the
very special geometry underlying its structure as well as the equations of
motion for the metric, gauge and scalar fields of the theory. We present two
classes of solutions with some explicit examples in section four where we also
include a summary of our results.

##  2 Solutions of Einstein-Maxwell is 4 and 5 dimensions

The original vacuum four-dimensional Euclidean Kasner solution depends only on
one variable and is given by [13]

$ds^{2}=x_{1}^{2a_{1}}dx_{1}^{2}+x_{1}^{2a_{2}}dx_{2}^{2}+x_{1}^{2a_{3}}dx_{3}^{2}+x_{1}^{2a_{4}}dx_{4}^{2}\text{
},$ (2.1)

where the constants $a_{i}$, known as the Kasner exponents, satisfy two
conditions

$a_{2}+a_{3}+a_{4}=1+a_{1},\text{ \ \ \ \
}a_{2}^{2}+a_{3}^{2}+a_{4}^{2}=\left(1+a_{1}\right)^{2}.$

The Kasner metric is actually valid for all space-time signatures [14] and
with a change of coordinates can take the form

$ds^{2}=\epsilon_{0}d\tau^{2}+\left(\epsilon_{1}\tau^{2a}dx^{2}+\epsilon_{2}\tau^{2b}dy^{2}+\epsilon_{3}\tau^{2c}dz^{2}\right)$
(2.2)

where $\epsilon_{i}$ take the values $\pm 1$ and the constants satisfy the
conditions

$a+b+c=a^{2}+b^{2}+c^{2}=1.$ (2.3)

Kasner metric is related to the metrics found by Weyl [15], Levi-Civita [16]
and Wilson [17] and was also rediscovered by many authors [18]. For solutions
with Lorentzian signature, we can obtain the vacuum cosmological solutions

$ds^{2}=-d\tau^{2}+\tau^{2a}dx^{2}+\tau^{2b}dy^{2}+\tau^{2c}dz^{2},$ (2.4)

as well as the anisotropic vacuum domain wall solutions

$ds^{2}=-r^{2a}dt^{2}+\left(dr^{2}+r^{2b}dy^{2}+r^{2c}dz^{2}\right).$ (2.5)

The vacuum Kasner metric can be promoted to solutions with a non-trivial gauge
field [20, 19]. For Einstein-Maxwell theory with the action

$S=\int d^{4}x\sqrt{\left|g\right|}\left(R-\frac{1}{4}\,F^{2}\right),$ (2.6)

one obtains the solutions

$ds^{2}=\left(1+\frac{\epsilon_{3}Q^{2}}{16c^{2}}\tau^{2c}\right)^{2}\left(\epsilon_{0}d\tau^{2}+\epsilon_{1}\mathrm{\tau}^{2a}dx^{2}+\epsilon_{2}\mathrm{\tau}^{2b}dy^{2}\right)+\left(1+\frac{\epsilon_{3}Q^{2}}{16c^{2}}\tau^{2c}\right)^{-2}\epsilon_{3}\mathrm{\tau}^{2c}dz^{2},$

with the gauge field-strength two-form given by

$F=Q\mathrm{\tau}^{2c-1}\left(1+\frac{\epsilon_{3}Q^{2}}{16c^{2}}\tau^{2c}\right)^{-2}d\tau\wedge
dz.$ (2.7)

The dual magnetic Lorentzian cosmological solutions, with $\tau$ being a time-
like coordinate and $F=Qdx_{1}\wedge dx_{2}$ [19] are equivalent to the family
of anisotropic cosmologies of Rosen [21].

In five dimensional gravity with arbitrary space-time signature, the Kasner
metrics take the form

$ds^{2}=\epsilon_{0}d\tau^{2}+\mathop{\displaystyle\sum}\limits_{i=1}^{4}\epsilon_{i}\tau^{2a_{i}}dx_{i}^{2}$
(2.8)

with the conditions

$\mathop{\displaystyle\sum}\limits_{i=1}^{4}a_{i}=\mathop{\displaystyle\sum}\limits_{i=1}^{4}a_{i}^{2}=1.$
(2.9)

For Einstein-Maxwell action in five dimensions

$S=\int d^{5}x\sqrt{\left|g\right|}\left(R-\frac{\epsilon}{4}\,\,F^{2}\right)$
(2.10)

with $\epsilon=\pm 1,$ two classes of solutions can be found [20]. One is
given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left(c_{1}+\frac{\epsilon\epsilon_{4}Q^{2}\,}{12a_{4}^{2}c_{1}}\tau^{2a_{4}}\right)\left(\epsilon_{0}d\tau^{2}+\mathop{\displaystyle\sum}\limits_{i=1}^{3}\epsilon_{i}\tau^{2a_{i}}dx_{i}^{2}\right)+\epsilon_{4}\left(c_{1}+\frac{\epsilon\epsilon_{4}Q^{2}\,}{12a_{4}^{2}c_{1}}\tau^{2a_{4}}\right)^{-2}\tau^{2a_{4}}dx_{4}^{2}\text{
},$ $\displaystyle F$ $\displaystyle=$ $\displaystyle
Q\,\tau^{2a_{4}-1}\left(c_{1}+\frac{\epsilon\epsilon_{4}Q^{2}\,}{12a_{4}^{2}c_{1}}\tau^{2a_{4}}\right)^{-2}d\tau\wedge
dx_{4},$

and the other class of solutions is for $F=Pdx_{1}\wedge dx_{2},$ with
constant $P$ and is given by

$ds^{2}=e^{2U}\left(\epsilon_{0}d\tau^{2}+\epsilon_{1}\tau^{2a_{1}}dx_{1}^{2}+\epsilon_{2}\tau^{2a_{2}}dx_{2}^{2}\right)+e^{-U}\left(\epsilon_{3}\tau^{2a_{3}}dx_{3}^{2}+\epsilon_{4}\tau^{2a_{4}}dx_{4}^{2}\right)\text{
},$ (2.11)

with

$e^{U}=\left(c_{1}-\frac{\epsilon\epsilon_{0}\epsilon_{1}\epsilon_{2}P^{2}}{12\left(a_{3}+a_{4}\right)^{2}c_{1}}\tau^{2\left(a_{3}+a_{4}\right)}\right).$
(2.12)

##  3 $5D$ $N=2$ Supergravity and Very Special Geometry

Ignoring the hypermultiplets, the Lagrangian of the bosonic fields of all the
five-dimensional supergravity can be given by [1, 7]

$\mathcal{L}_{5}=\sqrt{|g|}\left(R-\frac{\epsilon}{2}G_{IJ}F_{\mu\nu}^{I}F^{\mu\nu
J}-g_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}-\frac{1}{24}\epsilon^{\mu\nu\rho\sigma\tau}C_{IJK}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\tau}^{K}\right).$
(3.1)

The indices $I,$ $J,$ $K$ take values $0,1,...,n,$ where $n$ is the number of
the scalar fields $\phi^{i}.$ The constants $C_{IJK}$ are symmetric on all
indices. Note that for supergravity theories originating from Calabi-Yau
compactification, $C_{IJK}$ are the intersection numbers. The bosonic sector
of the theory can be described in terms of the so called very special geometry
[2]. We define the special coordinates $X^{I}=X^{I}(\phi)$ satisfying

$X^{I}X_{I}=1,\qquad\mathcal{V=}{\frac{1}{6}}C_{IJK}X^{I}X^{J}X^{K}=1,$ (3.2)

where, $X_{I}$, the dual coordinate, is defined by

$X_{I}={\frac{1}{6}}C_{IJK}X^{J}X^{K}.$ (3.3)

The gauge coupling metric is derived from the prepotential $\mathcal{V}$ and
is given by

$G_{IJ}=-{\frac{1}{2}}{\frac{\partial}{\partial
X^{I}}}{\frac{\partial}{\partial
X^{J}}}(\ln\mathcal{V})|_{\mathcal{V}=1}={\frac{9}{2}}X_{I}X_{J}-{\frac{1}{2}}C_{IJK}X^{K}.$

Moreover, the scalar metric satisfies

$g_{ij}=G_{IJ}\partial_{i}X^{I}\partial_{j}X^{J}|_{\mathcal{V}=1}\text{ },$
(3.4)

where we have $\partial_{i}={\frac{\partial}{\partial\phi^{i}}.}$ As such, we
have the relation
$g_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}=G_{IJ}\partial_{\mu}X^{I}\partial^{\mu}X^{J}.$
We also have the very special geometry relations

$G_{IJ}X^{J}={\frac{3}{2}}X_{I}\,,\qquad dX_{I}\,=-\frac{2}{3}G_{IJ}dX^{J}.$
(3.5)

The Einstein, Maxwell and scalar field equations of motion derived from (3.1)
can be written as

$\displaystyle
R_{\mu\nu}-G_{IJ}\partial_{\mu}X^{I}\partial_{\nu}X^{J}-\epsilon{G}_{IJ}\left(F_{\mu\alpha}F_{\nu}{}^{\alpha}-{\frac{1}{6}}g_{\mu\nu}F^{2}\right)$
$\displaystyle=$ $\displaystyle 0,$ (3.6)
$\displaystyle\nabla_{\mu}\left(\epsilon
G_{IJ}F^{I\mu\nu}\right)+\frac{1}{16}\epsilon^{\nu\lambda\sigma\mu\tau}C_{IJK}F_{\lambda\sigma}^{J}F_{\mu\tau}^{K}$
$\displaystyle=$ $\displaystyle 0,$ (3.7)
$\displaystyle\sqrt{|g|}\partial_{i}G_{IJ}\left(\frac{\epsilon}{2}F_{\mu\nu}^{I}F^{\mu\nu
J}+\partial_{\mu}X^{I}\partial^{\mu}X^{J}\right)-2\partial_{\mu}\left(\sqrt{|g|}G_{IJ}\partial^{\mu}X^{J}\right)\partial_{i}X^{I}$
$\displaystyle=$ $\displaystyle 0.$ (3.8)

##  4 General Solutions and Examples

To solve for the equations of motion (3.6)-(3.8), we start with the following
metric

$ds^{2}=e^{2U}\left(\epsilon_{0}d\tau^{2}+\epsilon_{1}\mathrm{\tau}^{2a_{1}}dx^{2}+\epsilon_{2}\mathrm{\tau}^{2a_{2}}dy^{2}+\epsilon_{3}\mathrm{\tau}^{2a_{3}}dz^{2}\right)+\epsilon_{4}e^{-4U}\mathrm{\tau}^{2a_{4}}dw^{2}$
(4.1)

where the constants $a_{i},$ satisfying the Kasner conditions (2.9) and $U$ is
a function of $\tau$ only. For the gauge fields, we consider solutions with
only $F_{\tau w}^{I}$ non-vanishing and functions of $\tau,$ the equations of
motion (3.7) then reduce to

$\partial_{\tau}\left(\tau e^{2U}G_{IJ}F^{J\tau w}\right)=0$

and thus imply the solution

$G_{IJ}F^{J\tau w}=e^{-2U}\frac{q_{I}}{\tau}.$ (4.2)

where $q_{I}$ being constants. Substituting the value of the gauge field-
strength into the Einstein equations of motion (3.6), we obtain

$\displaystyle\ddot{U}+2\dot{U}^{2}+\left(1-2a_{4}\right)\frac{\dot{U}}{\tau}$
$\displaystyle=$ $\displaystyle-\frac{1}{3}G_{IJ}\dot{X}^{I}\dot{X}^{J},$
(4.3) $\displaystyle 3\left(\ddot{U}+\frac{\dot{U}}{\tau}\right)$
$\displaystyle=$
$\displaystyle\epsilon\epsilon_{4}\tau^{2a_{4}-2}e^{-4U}G^{IJ}q_{I}q_{J}\text{
},$ (4.4)

here the dot symbol indicates differentiation with respect to the coordinate
$\tau.$ In order to solve these equations, we write the dual coordinate in the
form

$X_{I}(\tau)=\frac{1}{3}e^{-2U}f_{I}(\tau),$ (4.5)

then with the help of the relations of very special geometry, the following
equations can be derived

$\displaystyle\dot{X}_{I}$ $\displaystyle=$
$\displaystyle-2\dot{U}X_{I}+\frac{1}{3}e^{-2U}\dot{f}_{I}\text{ },$
$\displaystyle\dot{U}$ $\displaystyle=$
$\displaystyle\frac{1}{6}e^{-2U}\dot{f}_{I}X^{I},$ $\displaystyle\ddot{U}$
$\displaystyle=$
$\displaystyle\frac{1}{6}e^{-2U}\left(X^{I}\ddot{f}_{I}-\frac{1}{2}e^{-2U}G^{IJ}\dot{f}_{I}\dot{f}_{J}\right),$
$\displaystyle G_{IJ}\dot{X}^{I}\dot{X}^{J}$ $\displaystyle=$
$\displaystyle-3\ddot{U}-6\dot{U}^{2}+\frac{1}{2}e^{-2U}X^{I}\ddot{f}_{I}\text{
},$ (4.6)

which when substituted in the equations (4.3) and (4.4) give

$\displaystyle
X^{I}\left[\ddot{f}_{I}+\left(1-2a_{4}\right)\frac{\dot{f}_{I}}{\tau}\right]$
$\displaystyle=$ $\displaystyle 0,$ (4.7) $\displaystyle
e^{-2U}X^{I}\left(\ddot{f}_{I}+\frac{\dot{f}_{I}}{\tau}\right)-e^{-4U}G^{IJ}\left(\frac{1}{2}\dot{f}_{I}\dot{f}_{J}+2\epsilon\epsilon_{4}q_{I}q_{J}\tau^{2a_{4}-2}\right)$
$\displaystyle=$ $\displaystyle 0.$ (4.8)

Equation (4.7) admits the solution

$f_{I}=\left(A_{I}+B_{I}\mathrm{\tau}^{2a_{4}}\right)$ (4.9)

with constant $A_{I}$ and $B_{I}.$ Equation (4.8) then implies the conditions

$G^{IJ}\left(A_{J}B_{I}a_{4}^{2}-\epsilon\epsilon_{4}q_{I}q_{J}\right)=0.$
(4.10)

The scalar equation of motion (3.8) for our solutions reduces to

$\partial_{i}G^{IJ}\left({\frac{1}{2}}\dot{f}_{I}\dot{f}_{J}-\frac{1}{2}f_{J}\left(\frac{\dot{f}_{I}}{\tau}+\ddot{f}_{I}\right)+{2\epsilon_{4}\epsilon}q_{I}q_{J}\mathrm{\tau}^{2a_{4}-2}\right)=0$
(4.11)

which implies the conditions

$\partial_{i}G^{IJ}\left(A_{J}B_{I}a_{4}^{2}-\epsilon\epsilon_{4}q_{I}q_{J}\right)=0.$
(4.12)

Note that (3.2) implies the relation

$e^{4U}=\frac{1}{6}G^{IJ}f_{I}f_{J}.$ (4.13)

We consider a second class of of solutions represented by the metric

$ds^{2}=e^{2U}\left(\epsilon_{0}d\tau^{2}+\epsilon_{1}\mathrm{\tau}^{2a_{1}}dx^{2}+\epsilon_{2}\mathrm{\tau}^{2a_{2}}dy^{2}\right)+e^{-U}\left(\epsilon_{3}\mathrm{\tau}^{2a_{3}}dz^{2}+\epsilon_{4}\mathrm{\tau}^{2a_{4}}dw^{2}\right)$
(4.14)

and the gauge field-strength two-form $F^{I}=p^{I}dx\wedge dy$ with constant
$p^{I}.$ For the metric (4.14), one obtains for the Einstein equations of
motion

$\displaystyle\ddot{U}+\left(1-2l\right)\frac{\dot{U}}{\tau}+\dot{U}^{2}$
$\displaystyle=$ $\displaystyle-\frac{2}{3}G_{IJ}\dot{X}^{I}\dot{X}^{J},$
(4.15) $\displaystyle\ddot{U}+\frac{\dot{U}}{\tau}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\epsilon\epsilon_{0}\epsilon_{1}\epsilon_{2}e^{-2U}\tau^{-2s}{G}_{IJ}p^{I}p^{J}$
(4.16)

where we have $l=a_{3}+a_{4}$ and $s=a_{1}+a_{2}.$ To proceed in finding
solutions, we set

$X^{I}(\tau)=e^{-U}h^{I}(\tau),$ (4.17)

then using the relations of very special geometry, we obtain the following
relations

$\displaystyle\dot{U}$ $\displaystyle=$ $\displaystyle
e^{-U}X_{I}\dot{h}^{I},$ $\displaystyle\ddot{U}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}e^{-2U}G_{IJ}\dot{h}^{I}\dot{h}^{J}+e^{-U}X_{I}\ddot{h}^{I},$
$\displaystyle G_{IJ}\dot{X}^{I}\dot{X}^{J}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}\dot{U}^{2}+e^{-2U}G_{IJ}\dot{h}^{I}\dot{h}^{J}.$
(4.18)

Using (4.18), equations (4.15) and (4.16) then give

$\displaystyle
X_{I}\left[\ddot{h}^{I}+\left(1-2l\right)\frac{\dot{h}^{I}}{\tau}\right]$
$\displaystyle=$ $\displaystyle 0,$ (4.19) $\displaystyle
G_{IJ}\left[\dot{h}^{I}\dot{h}^{J}-h^{J}\left(\ddot{h}^{I}+\frac{\dot{h}^{I}}{\tau}\right)-\epsilon\epsilon_{0}\epsilon_{1}\epsilon_{2}\tau^{-2s}p^{I}p^{J}\right]$
$\displaystyle=$ $\displaystyle 0,$ (4.20)

which can be solved by

$h^{I}=C^{I}+D^{I}\tau^{2l},$ (4.21)

with the conditions

$G_{IJ}\left(4l^{2}C^{J}D^{I}+\epsilon\epsilon_{0}\epsilon_{1}\epsilon_{2}p^{I}p^{J}\right)=0.$
(4.22)

The scalar equation of motion also implies the conditions

$\partial_{i}G_{IJ}\left(4l^{2}C^{J}D^{I}+\epsilon\epsilon_{0}\epsilon_{1}\epsilon_{2}p^{I}p^{J}\right)=0.$
(4.23)

Moreover, the condition (3.2) for our solutions implies

$e^{3U}={\frac{1}{6}}C_{IJK}h^{I}h^{J}h^{K}.$ (4.24)

In what follows, some explicit solutions are constructed. We start with the
supergravity models where the scalar manifold is a symmetric space. For such
theories, the prepotential $\mathcal{V}$ defined in (3.2) is related to the
norm forms of degree three Euclidean Jordan algebras. For non-simple Jordan
algebras, the corresponding symmetric scalar manifolds $\mathcal{M}$ are given
by

$\mathcal{M}=SO(1,1)\times\frac{SO(n-1,1)}{SO(n-1)}$ (4.25)

and the prepotential ${\mathcal{V}}$ factorizes into a linear times a
quadratic form in $(n-1)$ scalars and takes the form

${\mathcal{V}}={\frac{1}{2}}X^{1}\left(\eta_{ab}X^{a}X^{b}\right),\quad
a,b=2,\dots,n.$ (4.26)

For these models, the following relations hold

$X^{I}={\frac{9}{2}}C^{IJK}X_{J}X_{K}\text{ },\text{ \ \ \
}G^{IJ}=2X^{I}X^{J}-6C^{IJK}X_{K}\text{ },$ (4.27)

where $C^{IJK}$ is defined by

$C^{IJK}=\delta^{II^{\prime}}\delta^{JJ^{\prime}}\delta^{KK^{\prime}}C_{I^{\prime}J^{\prime}K^{\prime}}\,.$
(4.28)

Explicitly, the components of $G_{IJ}$ and its inverse are given by

$\displaystyle G_{11}$ $\displaystyle=$
$\displaystyle{\frac{9}{2}}\left(X_{1}\right)^{2},\text{ \ \ \
}G_{1a}=0,\text{ \ \ \
}G_{ab}=\frac{9}{2}X_{a}X_{b}-\frac{1}{2}\eta_{ab}X^{1},$ $\displaystyle
G^{11}$ $\displaystyle=$ $\displaystyle 2\left(X^{1}\right)^{2},\text{ \ \ \ \
\ }G^{1a}=0,\text{ \ \ \ }G^{ab}=2X^{a}X^{b}-6\eta^{ab}X_{1}.$ (4.29)

For the first class of solutions in (4.1), the non-vanishing gauge fields and
scalars are given by

$\displaystyle F^{1\tau w}$ $\displaystyle=$ $\displaystyle
e^{-2U}G^{11}\frac{q_{1}}{\tau},\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$
$\displaystyle F^{a\tau w}$ $\displaystyle=$ $\displaystyle
e^{-2U}G^{ab}\frac{q_{b}}{\tau},$ $\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{3}e^{-2U}\left(A_{1}+B_{1}\mathrm{\tau}^{2a_{4}}\right),\text{
\ \ \ }$ $\displaystyle X_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{3}e^{-2U}\left(A_{a}+B_{a}\mathrm{\tau}^{2a_{4}}\right),$
$\displaystyle\vskip 12.0pt plus 4.0pt minus 4.0ptX^{1}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}\eta^{ab}e^{-4U}\left(A_{a}+B_{a}\mathrm{\tau}^{2a_{4}}\right)\left(A_{b}+B_{b}\mathrm{\tau}^{2a_{4}}\right),\text{
\ }$ $\displaystyle X^{a}$ $\displaystyle=$
$\displaystyle\eta^{ab}e^{-4U}\left(A_{1}+B_{1}\mathrm{\tau}^{2a_{4}}\right)\left(A_{b}+B_{b}\mathrm{\tau}^{2a_{4}}\right).$
(4.30)

Note that for these models we have

$X^{1}X_{1}=\frac{1}{3},\text{ \ \ \ }X^{a}X_{a}=\frac{2}{3},$

thus we obtain for our solution

$e^{6U}={\frac{1}{2}}\eta^{ab}\left(A_{a}+B_{a}\mathrm{\tau}^{2a_{4}}\right)\left(A_{b}+B_{b}\mathrm{\tau}^{2a_{4}}\right)\left(A_{1}+B_{1}\mathrm{\tau}^{2a_{4}}\right).$
(4.31)

The conditions (4.10) and (4.12) are satisfied by allowing for two independent
charges $q_{1}$ and $q_{a}=Q,$ and $A_{a}=B_{a}=A,$ with

$\left(q_{1}\right)^{2}=\epsilon\epsilon_{4}A_{1}B_{1}a_{4}^{2},\text{ \ \
}Q^{2}=\epsilon\epsilon_{4}A^{2}a_{4}^{2}.$ (4.32)

Similarly one can construct solutions with two independent charges for the
second class of metrics given in (4.14).

Next we consider the $U(1)^{3}$ supergravity with the prepotential
$\mathcal{V}=X^{1}X^{2}X^{3}=1$ and
$G_{IJ}=\frac{1}{2\left(X^{I}\right)^{2}}\delta_{IJ}.$ We first consider the
first class of solutions given in (4.1). Using (4.5), we obtain

$\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle{\frac{1}{3}}X^{2}X^{3}=\frac{1}{3}e^{-2U}H_{1},\text{ \ \ \ \ \
}$ (4.33) $\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle{\frac{1}{3}}X^{1}X^{3}=\frac{1}{3}e^{-2U}H_{2},\text{ \ \ \ \ \
\ }$ (4.34) $\displaystyle X_{3}$ $\displaystyle=$
$\displaystyle{\frac{1}{3}}X^{1}X^{2}=\frac{1}{3}e^{-2U}H_{3},$ (4.35)

where $H_{I}=\left(a_{I}+b_{I}\mathrm{\tau}^{2a_{4}}\right).$ The special
coordinates and the metric are given by

$X^{I}=\frac{\left(H_{1}H_{2}H_{3}\right)^{1/3}}{H_{I}},\text{ \ \ }$ (4.36)

and

$ds^{2}=\left(H_{1}H_{2}H_{3}\right)^{1/3}\left(\epsilon_{0}d\tau^{2}+\epsilon_{1}\mathrm{\tau}^{2a_{1}}dx^{2}+\epsilon_{2}\mathrm{\tau}^{2a_{2}}dy^{2}+\epsilon_{3}\mathrm{\tau}^{2a_{3}}dz^{2}\right)+\epsilon_{4}\left(H_{1}H_{2}H_{3}\right)^{-2/3}\mathrm{\tau}^{2a_{4}}dw^{2}.$
(4.37)

Using (4.2), we obtain for the gauge fields (with no summation over the index
$I$)

$F^{I\tau w}=\frac{2q_{I}}{\tau H_{I}^{2}}\left(H_{1}H_{2}H_{3}\right)^{1/3}$
(4.38)

with the three independent charges given by
$q_{I}^{2}=\epsilon\epsilon_{4}a_{I}b_{I}a_{4}^{2}.$

Using the general class of solutions given in (4.14), we obtain for the
$U(1)^{3}$ supergravity the solutions given by

$ds^{2}=\left(H_{1}H_{2}H_{3}\right)^{2/3}\left(\epsilon_{0}d\tau^{2}+\epsilon_{1}\mathrm{\tau}^{2a_{1}}dx^{2}+\epsilon_{2}\mathrm{\tau}^{2a_{2}}dy^{2}\right)+\left(H_{1}H_{2}H_{3}\right)^{-1/3}\left(\epsilon_{3}\mathrm{\tau}^{2a_{3}}dz^{2}+\epsilon_{4}\mathrm{\tau}^{2a_{4}}dw^{2}\right)$
(4.39)

and

$F^{I}=q^{I}dx\wedge dy,\text{ \ \ \ \ \
}X^{I}=\frac{H^{I}}{\left(H_{1}H_{2}H_{3}\right)^{1/3}},\text{ }$ (4.40)

where

$H^{I}=A^{I}+B^{I}\tau^{2l}$

and the three independent charges are given by

$\left(q^{I}\right)^{2}=-4\epsilon\epsilon_{0}\epsilon_{1}\epsilon_{2}l^{2}A^{I}B^{I}.$
(4.41)

As as special case, we can obtain from (4.37) a Lorentzian solution for which
all Kasner exponents vanish except for $a_{4}.$ This takes the form

$ds^{2}=\left(H_{1}H_{2}H_{3}\right)^{1/3}\left(-d\tau^{2}+dx^{2}+dy^{2}+dz^{2}\right)+\left(H_{1}H_{2}H_{3}\right)^{-2/3}\tau^{2}dw^{2},$
(4.42)

where the gauge fields and scalars are as in (4.36) and (4.38) and we choose

$H_{I}=\left(1+\epsilon q_{I}^{2}\tau^{2}\right).$ (4.43)

This solution can be thought of as a Melvin cosmology with non-trivial scalar
fields and three independent electric charges. Another special solution can be
obtained by analytic continuation or by simply setting
$\epsilon_{0}=-\epsilon_{1}=\epsilon_{2}=\epsilon_{3}=\epsilon_{4}=a_{4}=1$ in
(4.37). This can be given (after relabelling of the coordinates) by

$ds^{2}=\left(H_{1}H_{2}H_{3}\right)^{1/3}\left(-dt^{2}+dr^{2}+dy^{2}+dz^{2}\right)+\left(H_{1}H_{2}H_{3}\right)^{-2/3}r^{2}dw^{2}$
(4.44)

with

$\displaystyle H_{I}$ $\displaystyle=$ $\displaystyle\left(1+\epsilon
q_{I}^{2}r^{2}\right),$ $\displaystyle X^{I}$ $\displaystyle=$
$\displaystyle\frac{\left(H_{1}H_{2}H_{3}\right)^{1/3}}{H_{I}},\text{ \ \ \ \
}F^{Irw}=\frac{2q_{I}}{rH_{I}^{2}}\left(H_{1}H_{2}H_{3}\right)^{1/3}.$ (4.45)

This solution can be thought of an anisotropic Melvin domain wall with non-
trivial gauge and scalar fields. Solutions with cyclic $w$ correspond to five
dimensional generalisations of Melvin fluxtubes [22]. One can also obtain from
(4.39), magnetically charged cosmological solutions represented by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left(H_{1}H_{2}H_{3}\right)^{2/3}\left(-d\tau^{2}+dx^{2}+dy^{2}\right)+\left(H_{1}H_{2}H_{3}\right)^{-1/3}\left(dz^{2}+\mathrm{\tau}^{2}dw^{2}\right)$
$\displaystyle H^{I}$ $\displaystyle=$ $\displaystyle
1+\epsilon\frac{\left(q^{I}\right)^{2}}{4}\tau^{2},$ (4.46)

with bosonic fields as given in (4.40). Static solutions can also be obtained
from (4.46) by analytic continuation.

It is known that any solution of the five-dimensional $U(1)^{3}$ supergravity
can be uplifted to a solution of eleven-dimensional supergravity and a
solution of type IIB supergravity. The eleven-dimensional solutions take the
form [5]

$ds_{11}^{2}=ds_{5}^{2}+X^{1}\left(dz_{1}^{2}+dz_{2}^{2}\right)+X^{2}\left(dz_{3}^{2}+dz_{4}^{2}\right)+X^{3}\left(dz_{5}^{2}+dz_{6}^{2}\right)$
(4.47)

where $ds_{5}^{2}$ represents the five-dimensional metric and the coordinates
$z^{i}$, $i=1,..,6$, parametrize $T^{6}.$ The type IIB solution is given by

$ds_{10}^{2}=\left(X^{3}\right)^{1/2}ds_{5}^{2}+\left(X^{3}\right)^{-3/2}\left(dz+A^{3}\right)^{2}+X^{1}\left(X^{3}\right)^{1/2}\left(dz_{1}^{2}+dz_{2}^{2}+dz_{3}^{2}+dz_{4}^{2}\right)$
(4.48)

with the dilaton $\phi$ and the Ramond-Ramond 3-form field strength $F_{(3)}$
given by

$e^{2\phi}=\frac{X^{1}}{X^{2}},\text{ \ \ \
}F_{(3)}=\left(X^{1}\right)^{-2}\ast_{5}F^{1}+F^{2}\wedge\left(dz+A^{3}\right),$
(4.49)

with $\ast_{5}$ being the Hodge dual with respect to $ds_{5}^{2}.$

In conclusion, time-dependent and static gravitational solutions for the
theory of five-dimensional supergravity with non-trivial scalar and gauge
fields in various space-time dimensions have been considered. As in the
original metric constructed by Kasner, our solutions depend only on one
parameter which we have denoted by $\tau.$ Two classes of solutions were found
and some explicit solutions with two and three charges were constructed.
Depending on the choice of coordinates, the resulting solutions can be thought
of as generalizations of the electric Melvin or magnetic Rosen cosmologies,
flux-branes and domain walls. Our results can be generalised to supergravity
with various space-time dimensions. Another interesting direction for further
investigation is to consider solutions of gauged supergravity theories which
have a cosmological constant or a scalar potential. Those solutions will have
potential applications to (A)dS/CFT correspondence. We hope to report on this
in our future work.

Acknowledgements:

This work is supported in part by the National Science Foundation under grant
number PHY-1620505.

## References

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# Non-Hermitian guided modes and exceptional points using loss-free negative-
index materials

Li-Ting Wu1, Xin-Zhe Zhang2, Ru-Zhi Luo2, and Jing Chen2,3
<EMAIL_ADDRESS>$1$ School of Information and Communication Engineering,
Nanjing Institute of Technology, Nanjing 211167, China
$2$ MOE Key Laboratory of Weak-Light Nonlinear Photonics, School of Physics,
Nankai University, Tianjin 300071, China
$3$ Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan, Shanxi 030006, China

###### Abstract

We analyze the guided modes in coupled waveguides made of negative-index
materials without gain or loss. We show that it supports non-Hermitian
phenomenon on the existence of guided mode versus geometric parameters of the
structure. The non-Hermitian effect is different from parity-time
($\mathcal{PT}$) symmetry, and can be explained by a simple coupled-mode
theory with an anti-$\mathcal{PT}$ symmetry. The existence of exceptional
points and slow-light effect are discussed. This work highlights the potential
of loss-free negative-index materials in the study of non-Hermitian optics.

123

###### pacs:

123

##  I Introduction

Recent years the non-Hermitian physics, especially parity-time
($\mathcal{PT}$) symmetry R01 ; R02 ; R03 ; R04 ; R05 ; R06 ; R07 , has
attracted much attention in the society of optics, because it provides an
additional degree of freedom in manipulating the dynamics of optical waves.
Nevertheless, the requirement on proper strength of gain and loss effects
hinders the realistic applications of $\mathcal{PT}$ symmetric optics. The
advances of other categories of non-Hermitian optics, e.g. anti-$\mathcal{PT}$
symmetry R08 ; R09 ; R10 ; R11 , might overcome this drawback because gain and
loss are not strictly required.

The simplest configuration in studying $\mathcal{PT}$ symmetric optics is two
parallel waveguides (WGs). Such a kind configuration can be readily studied by
using Maxwell’s equations, and it supports some intriguing optical effects
especially the stopped light at exceptional points (EPs) R12 ; R13 ; R14 ; R15
that separate the conserved and broken $\mathcal{PT}$ phases R16 ; R17 ; R18 ;
R19 ; R20 . However, for an optical wave propagating inside a WG, it contains
multiple important physical parameters especially the wavevector $k$
charactering the propagation of phase front and the Poynting vector $S$
presenting the propagation of energy flux. They carry different informations
about the wave, and might have different directions of propagation R21 ; R22 .
The extreme situation is that in a negative-index material (NIM) with
simultaneous negative magnetic permeability ($\mu_{\text{NIM}}<0$) and
electric permittivity ($\epsilon_{\text{NIM}}<0$), where the direction of $k$
is opposite to that of $S$ R23 ; R24 ; R25 ; R26 ; R27 . However, non-
Hermitian feature of coupled WGs containing NIM has not been extensively
discussed. Only recently do we notice that Mealy etc. al. R28  have showed
that EP of degeneracy can be obtained in two coupled WGs by a proper coupling
of forward and backward waves, where the backward waves are generated by a
proper designed grating.

In this article, we check the optical waves in coupled WGs made of NIM and
positive-index material (PIM). A PIM has positive magnetic permeability and
electric permittivity, and covers all the dielectric materials in nature. We
show that the guided modes in this system also support non-Hermitian feature,
even when all the constituent media are loss-free. At a given angular
frequency and WG thicknesses, the wavevector $k$ of the eigenmodes and the
associated eigenvectors vary with the distance $a$ between the two WGs, and
they coalesce at a critical value of $a_{c}$ below which the wavevector $k$
becomes complex. This phase transition point is an EP. A simple non-Hermitian
coupled-mode theory is utilized to explain our results, which implies that the
field dynamics induced by NIM belongs to anti-$\mathcal{PT}$ symmetry.
Features at EPs including the slow-light effect are discussed.This study
highlights the important novelties of NIM, especially its great potential in
the study of non-Hermitian optics by bypassing many restrictions of
$\mathcal{PT}$ symmetry.

##  II Simulation and analysis

Let us consider the simple structure shown in Fig. 1. It contains two straight
WGs made of lossless media surrounded by air. One WG is PIM, and we assume
that it is a dielectric of $\epsilon_{\text{PIM}}=4$ and $\mu_{\text{PIM}}=1$.
The other WG is a NIM that has been discussed by various authors R29 ; R30 ;
R31 . The well documented structure of NIM with
$\epsilon_{\text{NIM}}=1-\omega_{p}^{2}/\omega^{2}$ and
$\mu_{\text{NIM}}=1-F\omega^{2}/(\omega^{2}-\omega_{0}^{2})$ can be utilized
here, where $\omega_{p}=2\pi\times 10$ GHz, $\omega_{0}=2\pi\times 4$ GHz, and
$F=0.56$ R29 ; R30 ; R31 . An effective NIM is achieved between 4 GHz and 6
GHz.

Because NIM requires an intrinsic dispersion of
$\partial(\epsilon_{\text{NIM}}\omega)/\partial\omega>0$ and
$\partial(\mu_{\text{NIM}}\omega)/\partial\omega>0$ so as to give a positive
energy density R21 ; R22 , in our below analysis we would keep the angular
frequency $\omega$ a constant, and test the variation of the wavevectors $k$
of the eigenmodes versus a geometric parameter of the coupled WGs. Here we set
$\omega=2\pi\times 5$ GHz (a free-space wavelength of 6cm), and the dispersion
gives $\epsilon_{\text{NIM}}=-3$ and $\mu_{\text{NIM}}=-0.556$. The index of
refraction equals
$n_{\text{NIM}}=\sqrt{\epsilon_{\text{NIM}}}\sqrt{\mu_{\text{NIM}}}=-1.291$.
Thickness $b$ of each WG is 4 cm. The distance $a$ between the two WGs is
chosen as the variable.

Figure 1: Schematic of the configuration under investigation, which contains
two straight WGs made of lossless media. One medium is a dielectric with
$\epsilon_{\text{PIM}}>0$ and $\mu_{\text{PIM}}=1$, while the other one is a
NIM with $\epsilon_{\text{NIM}}=-3$ and $\mu_{\text{NIM}}=-0.556$.

Assuming the direction of the wavevector $k$ is $z$, the dispersion and
distribution of the eigenmodes can be numerically found by using Maxwell’s
equations and boundary conditions R14 ; R21 ; R22 . For example, considering
the transverse-electric (TE) polarized eigenmodes with non-vanishing $E_{y}$
component

$E_{y}=e^{-jkz+j\omega
t}\left\\{\begin{array}[]{cc}E_{1}e^{-\beta(x-a/2-b)},&x>a/2+b\\\
E_{2}e^{-j\alpha_{\text{NIM}}(x-a/2)}+E_{3}e^{+j\alpha_{\text{NIM}}(x-a/2)},&a/2+b>x>a/2\\\
E_{4}e^{+\beta x}+E_{5}e^{-\beta x},&a/2>x>-a/2\\\
E_{6}e^{-j\alpha_{\text{PIM}}(x+a/2)}+E_{7}e^{+j\alpha_{\text{PIM}}(x+a/2)},&-a/2>x>-a/2-b\\\
E_{8}e^{+\beta(x+a/2+b)},&x<-a/2-b\\\ \end{array}\right.$ (1)

with

$k^{2}-\beta^{2}=\omega^{2}/c^{2},$ (2)
$k^{2}+\alpha_{m}^{2}=\epsilon_{m}\mu_{m}\omega^{2}/c^{2},$ (3)

where the subscript $m$ stands for NIM and PIM. The associated magnetic fields
can be found from Eq. (1) using $\nabla\times\vec{E}=-\partial\vec{B}/\partial
t$. By applying the electromagnetic boundary conditions, and defining

$F_{m}=\frac{\beta+j\alpha_{m}/\mu_{m}}{\beta-j\alpha_{m}/\mu_{m}},$ (4)
$\Upsilon_{m}=\frac{F_{m}\exp(j2\alpha_{m}b)-F_{m}^{-1}}{\exp(j2\alpha_{m}b)-1},$
(5)

it is readily to show that the eigensolutions $k$ are given by

$\Upsilon_{\text{NIM}}\Upsilon_{\text{PIM}}-\exp(-2\beta a)=0.$ (6)

By substituting the eigensolutions $k$ back into Eq. (1) we can calculate the
distributions of fields. Formula for transverse-magnetic (TM) polarized
eigenmodes can be developed similarly.

Figure 2: (a) Variation of $k$ versus $a$. (b) and (d) are the distributions
of field $E_{y}$ and Poynting vector $S_{z}$ at the two dispersion points in
(a) where $a=3$ cm. (c) shows the results in a PIM-PIM structure. Note that
$\text{Im}(E_{y})$ is zero in (b) and (d), and the $y$-coordinate of
$\text{Im}(k)$ in (a) is shown at the right side, which is properly chosen so
that the origins of the curves $\text{Im}(k)$ overlap with EP.

Equation (6) can be numerically solved, for example, by calculating the left-
hand side of it at different $k$ values to find zeroes. A standard result of
$k$ versus $a$ is shown in Fig. 2(a). We can see when the WG distance $a$ is
large enough so that the two WGs are weakly coupled, two separated eigenmodes
can be achieved. Each eigenmode represents a localized mode in a single WG.
From the distributions of field $E_{y}$ and Poynting vector $S_{z}$ shown in
Figs. 2(b) and 2(d) we can conclude that the eigenmode at
$k_{\text{NIM}}=1.328$ cm-1 is supported by the NIM WG, where the direction of
energy flux is opposite to that of $k$. The other one at
$k_{\text{PIM}}=1.186$ cm-1 is localized at the PIM WG. From the number of
nodes ($E_{y}=0$) inside each WG we can see the field supported by the planar
PIM WG is a TE2 mode, and that in the NIM WG is a TE1 one R30 . Note that the
wavevector $k$ is real here, and we have assumed that the expansion
coefficient $E_{1}$ in Eq. (1) is real. As a result, the imaginary component
$\text{Im}(E_{y})$ is zero and is not shown in Figs. 2(b) and 2(d). The
coefficient $\exp(-jkz+j\omega t)$ in Eq. (1) is neglected when plotting the
distribution of fields.

Figure 2(a) displays a unique phenomenon that is absent in ordinary dielectric
WGs. When $a$ decreases, the modes in the two WGs can couple together to form
hybrid eigenmodes. However, here the $k$ values of the eigenmodes approach to
each other. When $a$ is smaller than a critical distance $a_{c}$, the
eigenmodes coalesce. Below $a_{c}$ only complex $k$ exists (brown line and
dots in Fig. 2(a)), which can be found by searching for the solutions of Eq.
(6) in the space formed by $\text{Re}(k)$ and $\text{Im}(k)$. This phenomenon
is in sharp contrast with the ordinary belief about coupled PIM-PIM WGs, that
when $a$ decreases, the coupling between the two WGs would become stronger and
the split in $k$ should increase rather than decrease. To illustrate this
difference, we show an example about $k$ versus $a$ in a PIM-PIM configuration
in Fig. 2(c), where NIM in the structure of Fig. 1 is replaced by a conjugate
PIM with $\epsilon_{\text{PIM}}=3$ and $\mu_{\text{PIM}}=0.556$. We can see in
this PIM-PIM configuration all the solutions of $k$ are real, and the two
branches of $k$ do not coalesce.

Figure 3: Distributions of fields $E_{y}$ at some different points in the $k$
versus $a$ curves. They confirm that the transition point at $a_{c}$ is an EP.
Note that $\text{Im}(E_{y})$ is no longer zero when $a<a_{c}$ where
$\text{Im}(k)\neq 0$.

The coalescence of dispersion at a critical value of $a_{c}$ shown in Fig. 2
resembles EPs in $\mathcal{PT}$ symmetry very much R12 ; R13 ; R14 ; R15 ; R16
; R17 ; R18 ; R19 ; R20 . We check the eigenmodes around this point (see Fig.
3), and find that when approaching this point from $a>a_{c}$, the field
distributions of the two eigenmodes become more and more similar with each
other. So at this critical point not only the eigensolutions $k$ but also the
associated eigenvectors coalesce simultaneously. This point is not a diabolic
point in Hermitian system, but a standard EP R16 ; R17 ; R18 . The region of
$a>a_{c}$ ($a<a_{c}$), where the eigensolutions $k$ are real (complex),
possesses an exact (broken) non-Hermitian phase.

Figure 3 also displays the distributions of fields when $a<a_{c}$. In this
phase-broken region, at each given $a$ value we can generally find two complex
solutions of $k$, which are conjugate to each other. The complex solutions of
$k$ render complex values of $\beta$ and $\alpha_{m}$ given by Eqs. (2) and
(3). Consequently, $\text{Im}(E_{y})$ is no longer zero (see the insets of
Fig. 3).

To check how the curves of $k$ versus $a$ shown in Fig. 2(a) vary when
parameters of the coupled WGs change, we repeat the calculations at different
$\epsilon_{\text{PIM}}$ values. It would change the $k_{\text{PIM}}$ value of
the PIM WG. One can also achieve the same effect by changing the thickness of
it. As for the NIM WG, we would keep all the parameters constant so that
$k_{\text{NIM}}$ does not vary in this article.

Figure 4 displays the results when $\epsilon_{\text{PIM}}$ increases linearly
from 4.3 to 4.6. The increased $\epsilon_{\text{PIM}}$ would push the
$k_{\text{PIM}}$ value in the PIM WG close to or even pass $k_{\text{NIM}}$ in
the NIM WG. From Fig. 4 we can see with $\epsilon_{\text{PIM}}$ increasing,
the loop of $k$ versus $a$ would shrink, and eventually disappear around
$\epsilon_{\text{PIM}}=4.5$. When $\epsilon_{\text{PIM}}$ further increases
from 4.5, the curve appears again and becomes larger. Since
$k_{\text{NIM}}=1.328$ cm-1 does not change, the scenario of
$\epsilon_{\text{PIM}}=4.5$ in fact is around the critical point where the
degeneracy of $k_{\text{PIM}}=k_{\text{NIM}}$ takes place.

Figure 4: Variation of $k$ versus $a$ at different $\epsilon_{\text{PIM}}$
values, from (a) $\epsilon_{\text{PIM}}=4.3$ to (d)
$\epsilon_{\text{PIM}}=4.6$.

##  III A non-Hermitian coupled-mode theory

In the above simulation all the media in the coupled WGs are loss-free.
Nevertheless, the coalescence of eigenmodes at a critical value of $a_{c}$
hints that this loss-free system can possess non-Hermitian phenomenon. Here we
could utilize a non-Hermitian coupled-mode theory R21  in the form of
$M\Psi=k_{\pm}\Psi$ to explain the results, which is

$\left[\begin{array}[]{cc}k_{\text{NIM}}&-\gamma\\\ \gamma&k_{\text{PIM}}\\\
\end{array}\right]\left[\begin{array}[]{cc}\psi_{\text{NIM}}\\\
\psi_{\text{PIM}}\\\
\end{array}\right]=k_{\pm}\left[\begin{array}[]{cc}\psi_{\text{NIM}}\\\
\psi_{\text{PIM}}\\\ \end{array}\right].$ (7)

Here $k_{\text{NIM}}$ and $k_{\text{PIM}}$ are the wavevectors of guided modes
in separated NIM and PIM WGs, respectively, at the given angular frequency
$\omega$. Both $k_{\text{NIM}}$ and $k_{\text{PIM}}$ are real. Parameter
$\gamma$ is also real and characterizes the mutual coupling between the two
WGs. The basic vectors $\psi_{\text{NIM}}$ and $\psi_{\text{PIM}}$ represent
field components of the guided modes in characterizing the eigenvectors.
Evidently, Eq. (7) is non-Hermitian because the off-diagonal elements are not
conjugates, i.e. $M_{12}\neq M_{21}^{*}$. The choice of
$M_{21}=-M_{12}=\gamma$ can be explained by simultaneously considering two
aspects. One is that the total energy should be conserved in this loss-free
system R28 ; R32 . The other is that the propagating directions of energy in
the NIM and PIM WGs are opposite to each other, so the coupling between them
is contradirectional R28 ; R32 .

To see whether Eq. (7) can explain the main features in our former analysis,
we can first solve it. Defining

$k_{\text{NIM}}=k_{0}+\Delta,$ (8) $k_{\text{PIM}}=k_{0}-\Delta,$ (9)

where $k_{0}$ is the mean wavevector, and $\Delta$ is the detuning, the eigen-
solutions $k_{\pm}$ are

$k_{\pm}=k_{0}\pm\sqrt{\Delta^{2}-\gamma^{2}}.$ (10)

By using Eq. (10) we can fit the curves $k$ versus $a$ given by Eq. (6) and
find how the magnitude of $\gamma$ varies with the WG distance $a$. Results of
$\epsilon_{\text{PIM}}=4$ is shown in Fig. 5. When performing the best
fitting, we firstly choose proper values of $k_{\text{NIM}}$ and
$k_{\text{PIM}}$, which are $1.328$ cm-1 and $1.186$ cm-1, respectively, for
the results shown in Fig. 5. The values of $k_{\text{NIM}}$ and
$k_{\text{PIM}}$, as well as those of $k_{0}=1.257$ cm-1 and $\Delta=0.071$
cm-1, are kept as constants. Then, based on the $k_{\pm}$ values obtained from
Eq. (6) we can use Eq. (10) to find the values of $\gamma$. Because the two
solutions of Eq. (6) might give different values of $\gamma$, we would only
keep the averaged one, which is then substituted back into Eq. (10) to test
the discrepancy from Eq. (6). Similar fitting process is also performed in the
phase-broken region of complex $k$ values.

From Fig. 5(a) we can see the prediction from Eq. (10) fits well with those of
Eq. (6). The deviation between $\text{Re}(k)$ in the phase-broken region might
be fixed by assuming a spatial dependence of $k_{\text{NIM}}$ and
$k_{\text{PIM}}$ on $a$, which would not be discussed here. It is interesting
to emphasize that the magnitude of $\gamma$ exponentially decreases with
increasing $a$, and can be approximately expressed by

$\gamma=\gamma_{0}\exp(-a/L_{c})$ (11)

where $\gamma_{0}=0.18$ cm-1. The decay length $L_{c}$ equals 1.45 cm, very
close to the value of 1.438 cm given by
$\beta^{-1}=(k_{0}^{2}-\omega^{2}/c^{2})^{-1/2}$. Evidently, when $a$ is
smaller than $a_{c}$, the mutual coupling strength $|\gamma|$ is stronger than
$|\Delta|$, then the phase of the coupled WGs is spontaneously broken and
gives complex $k$ values. When $a$ is larger than $a_{c}$, $|\gamma|$ becomes
smaller than $|\Delta|$, and the phase is conserved. Position $a_{c}$ is the
critical phase transition point of EP where $|\gamma|=|\Delta|$. When the two
WGs are far away from each other, $\gamma$ is zero, and $k_{\pm}=k_{\text{NIM,
PIM}}$. Now the eigenvectors are $\Psi=[1,0]^{T}$ and $\Psi=[0,1]^{T}$, and
the fields are localized in separated WGs.

Figure 5: Variations of (a) $k$ and (b) $\gamma$ versus the WG distance $a$ by
using Eq. (10) to fit the results from Eq. (6). Green line in (b) is an
exponential fit of $\gamma$ using Eq. (11).

Furthermore, the detuning $\Delta$ determines the size of the loop in the $k$
versus $a$ spectrum. When $k_{\text{NIM}}=k_{\text{PIM}}$, i.e. the
eigensolutions in the NIM and PIM WGs are degenerated, the phase is always
spontaneously broken because now $\Delta=0$ and any nonzero $\gamma$ would
give complex $k_{\pm}$ values. It is the case shown in Fig. 4(c) at
$\epsilon_{\text{PIM}}=4.5$. Away from this degeneracy scenario, $|\Delta|$
increases and the loop becomes more and more great, which can be observed in
other plots in Fig. 4.

Now, we can pay attention to the EP where

$\Delta=\pm\gamma$ (12)

that gives $k_{\pm}=k_{0}$. The eigenfuntion of the eigenmode is

$\left[\begin{array}[]{cc}\psi_{\text{NIM}}\\\ \psi_{\text{PIM}}\\\
\end{array}\right]_{\text{EP}}=\left[\begin{array}[]{cc}1\\\
\text{sign}(\Delta/\gamma)\\\ \end{array}\right].$ (13)

where the function sign($x$) equals 1 ($-1$) when $x>0$ $(x<0)$. In other
words, at this EP the fields in the two WGs are in-phase (a phase difference
of zero) or out-phase (a phase difference of $\pi$) with each other. The exact
phase difference is determined by the sign of $\Delta/\gamma$. Generally
$\gamma$ is determined by the overlap integral of fields in the gap between
the two WGs and its sign is fixed R14 ; R15 ; R28 , but
$\text{sign}(\Delta/\gamma)$ can be tuned by changing the resonant conditions
of $k_{\text{NIM, PIM}}$ in the two WGs, e.g. by changing the index of
refraction inside or the thicknesses of WGs. As a consequence, the eigenvetcor
at EP is tunable.

Note that here we could not make any comment on the amplitudes of fields in
the two WGs because the refractive indexes in them are not required to be
equal. It is in sharp contrast with $\mathcal{PT}$ symmetric WGs that
generally only symmetric configurations are considered R10 ; R11 ; R12 . This
drawback hinders us to reveal more unique features about EP. However, the
phase-jump, as a signature of $\text{sign}(\Delta/\gamma)$ at EP, is an
observable, e.g. by choosing two sets of $\epsilon_{\text{PIM}}$ values so
that in one case $\Delta>0$ while in the other case $\Delta<0$, and then
analyzing the field patterns inside the structure. The results about the
distributions of $E_{y}$ at EPs when $\epsilon_{\text{PIM}}=4.4(\Delta>0)$ and
$\epsilon_{\text{PIM}}=4.6(\Delta<0)$ are shown in Fig. 6. From the two plots
we can see the fields in the NIM WG are almost identical, but the fields in
the PIM WG are flipped with respect to each other. It is then evident that the
phase between the two WGs are shifted by $\pi$ in the two scenarios.
Furthermore, associated with the $\pi$-phase jump there exists a node inside
the gap between the two WGs (red arrow in Fig. 6(b)), which implies that in
this scenario $\psi_{\text{NIM}}=-\psi_{\text{PIM}}$. From the fact of
$\Delta<0$ in Fig. 6(b) we can also conclude that $\gamma>0$.

Figure 6: Distributions of fields $E_{y}$ at EPs when (a)
$\epsilon_{\text{PIM}}=4.4(\Delta>0)$, and (b)
$\epsilon_{\text{PIM}}=4.6(\Delta<0)$, respectively. Red arrow represents the
node in (b), which is associated with the $\pi$-phase jump at EP when $\Delta$
varies its sign.

The group velocities $v_{g}$ of the guided modes can also be analyzed.
However, since here the angular frequency $\omega$ is kept constant, $v_{g}$
cannot be found from $\partial\omega/\partial k$. Here we calculate $v_{g}$ by
using the Poynting vector $S_{z}$ and the energy density $W$ via
$v_{g}=S_{z}/W$. When calculating the energy density $W$ we have adapted the
formula of
$\epsilon_{0}\partial(\epsilon_{\text{NIM}}\omega)/\partial\omega|E|^{2}+\mu_{0}\partial(\mu_{\text{NIM}}\omega)/\partial\omega|H|^{2}$
R21 ; R22 . At 5 GHz, the utilized dispersion gives
$\partial(\epsilon_{\text{NIM}}\omega)/\partial\omega=5$ and
$\partial(\mu_{\text{NIM}}\omega)/\partial\omega=4.975$.

Figure 7: Group velocity $v_{g}$ versus $a$ when $\epsilon_{\text{PIM}}=4.0$.
At EP $v_{g}$ is zero.

The curves of $v_{g}$ versus $a$ at $\epsilon_{\text{PIM}}=4.0$ are shown in
Fig. 7. As expected, the values of $v_{g}$ are limited in the region defined
by $v_{g}=-0.2c$ of NIM (it is negative because the energy flux is backward
propagating) and $v_{g}=0.5c$ of PIM, where $c$ is the speed of light. When
the field is mainly localized in the NIM WG, $v_{g}$ is negative. Otherwise
$v_{g}$ is positive. As $a$ decreases, the split between $k$ also decreases,
and $v_{g}$ of the two branches approach to each other. At EP the group
velocity is zero, which can be explained by the balanced positive energy flux
in PIM (including the surrounded air) and negative energy flux in NIM,
respectively.

Note that the stopped light at EP demonstrated here is different from that in
$\mathcal{PT}$ symmetric WGs R12 ; R13 ; R14 . Here the stopped light is
associated with the negative flux in NIM, and the whole structure is still
Hermitian because no loss or gain is presented. On the contrary, the stopped
light in $\mathcal{PT}$-symmetric WGs with spatial distributions of gain and
loss cannot be explained by the propagating of energy flux associated with
real-valued fields R14 , but is the consequence of zero $c$-product R12 ; R33
of the non-Hermitian system. The $c$-product can be applied when the two WGs
are not only geometric but also electromagnetic (except for the loss/gain
effect) identical so that $\psi$ in the eigenvector can be unambiguously
defined R12 ; R33 . Here the two WGs are made of different media, possess
different indexes of refraction, and can have different thicknesses, so the
applicability of $c$-product needs further discussion.

##  IV Discussion

Above we have shown that non-Hermitian optical effects can be observed in
loss-free coupled WGs made of NIMs. Here we would like to make further
comments on the category of the non-Hermitian effect. From Eq. (7) we can see
it can be classified into the anti-$\mathcal{PT}$ symmetric one because the
diagonal elements of the $2\times 2$ matrix $M$ satisfy
$\text{Im}(M_{11})=\text{Im}(M_{22})$ and $M_{12}=-M_{21}^{*}$. It should be
emphasized that when utilizing Eq. (7) we have assumed that the basic vectors
$\psi_{\text{NIM}}$ and $\psi_{\text{PIM}}$ represent field components of the
guided modes that characterize the wave functions. The set of analogues of the
basic vectors $\psi_{\text{NIM}}=E_{y}^{\text{NIM}}$ and
$\psi_{\text{PIM}}=E_{y}^{\text{PIM}}$ is adopted in our analysis. However,
since the two WGs are not identical with each other, the choice of the basis
vectors is arbitrary. For example, we can also utilize another set of
analogues by adding an additional phase of $\pi/2$ to the basic vector
$\psi_{\text{PIM}}$. Now, Eq. (7) becomes

$\left[\begin{array}[]{cc}k_{\text{NIM}}&j\gamma\\\ j\gamma&k_{\text{PIM}}\\\
\end{array}\right]\left[\begin{array}[]{cc}\psi_{\text{NIM}}\\\
j\psi_{\text{PIM}}\\\
\end{array}\right]=k_{\pm}\left[\begin{array}[]{cc}\psi_{\text{NIM}}\\\
j\psi_{\text{PIM}}\\\ \end{array}\right].$ (14)

Equation (14) is just the standard anti-$\mathcal{PT}$ symmetric operator that
has been discussed in many literatures R08 ; R09 ; R10 ; R11 . It can also
explain all the features on the dispersion and distributions of fields. As for
which one is preferred in studying NIM, we still prefer Eq. (7) because the
node shown in Fig. 6(b) can be more intuitively explained.

The importance of this article is that it proves we can access non-Hermitian
optics without using gain or loss. It also shows that the negative energy flux
in NIM has many realistic impacts on future applications, especially by
considering the fact that all the media are loss-free. Experimental
investigation of the non-Hermitian effect discussed here can utilize
documented NIM design R26 ; R27 , and transfer to other frequency regimes by
simply rescaling the geometric parameters of the coupled-WG configuration.
Theoretical interest can be paid to discuss the deep-lying physics of the
spontaneous phase broken, and the utilization of the anti-$\mathcal{PT}$
symmetry in loss-free NIM to achieve higher-order EPs and mode switching
purpose.

Before ending this article, we would like to emphasize again that the whole
structure studied here is loss- and gain-free. Consequently, in principle it
is a closed Hermitian system, and the energy should be conserved. The non-
Hermitian effects discussed above, especially the existence of complex $k$
below $a_{c}$, should not violate the principle of energy conservation. These
complex-$k$ eigenmodes might be associated with some effects similar to the
photonic bandgap effect and waveguiding effect below cutoff, which forbid the
propagation of optical field without absorbing any energy R21 ; R28 . Detailed
discussion deserves our further efforts.

##  V Conclusion

In summary, in this article we check the optical waves in coupled WGs made of
lossless NIM and PIM. We show that the guided modes in this kind of gain/loss-
free optical system can also support non-Hermitian features. A simple non-
Hermitian coupled-mode theory is utilized to explain our results. This theory
proves that the critical degeneracy point at $a_{c}$ is an EP, and the field
dynamics induced by NIM belongs to anti-$\mathcal{PT}$ symmetry. Features at
EPs including the slow-light effect are discussed. This study highlights the
non-negligible novelties of NIM, which can be utilized in studying non-
Hermitian optics to bypass many restrictions of $\mathcal{PT}$ symmetry.

## Funding

Natural National Science Foundation of China (NSFC) (12104227, 12274241);
Scientific Research Foundation of Nanjing Institute of Technology (YKJ202021).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

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# Tailoring the Deformation Behaviour of a Medium Mn Steel through Isothermal
Intercritical Annealing

X. Xu  T. W. J. Kwok  P. Gong  D. Dye<EMAIL_ADDRESS>Department of Materials,
Royal School of Mines, Imperial College London, Prince Consort Road, London,
SW7 2BP, United Kingdom  Department of Materials Science and Engineering, The
University of Sheffield, Western Bank, Sheffield, S10 2TN, United Kingdom
School of Materials, Sun Yat-Sen University (Shenzhen), and Southern Marine
Science and Engineering Guangdong Laboratory (Zhuhai), 519000, China

###### Abstract

A novel concept of varying the strain hardening rate of a medium Mn steel with
8 wt% Mn by varying the duration of the intercritical anneal after hot rolling
was explored. It was found that the stability of the austenite phase showed an
inverse square root relationship with intercritical annealing duration and
that the maximum strain hardening rate showed a linear relationship with
austenite stability. The change in austenite stability was attributed to
continuous Mn enrichment with increasing intercritical annealing duration.
Twinned martensite was also found to be the most likely product of the
martensitic transformation during deformation.

††journal: arXiv

##  1 Introduction

The latest generation of AHSS, also known as 3rd Generation AHSS (3Gen AHSS),
are multiphase steels characterised by simultaneous improvement to strength
and ductility through various plasticity enhancing mechanisms such as
Transformation Induced Plasticity (TRIP) and Twinning Induced Plasticity
(TWIP) [1]. More recently, medium Mn steels have shown great promise as
candidates for 3Gen AHSS because of their high strength, ductility and strain
hardening rate [2, 3]. Medium Mn steels are duplex ($\gamma+\alpha$), contain
between 4$-$12 wt% Mn and have been shown to exhibit a combined TWIP$+$TRIP
plasticity enhancing mechanism [4, 5, 3].

The key step in the thermomechanical processing of medium Mn steels is the
Intercritical Annealing (IA) heat treatment, typically conducted either after
hot rolling or cold rolling [6, 7]. The IA heat treatment is conducted between
the A1 and A3 temperatures, i.e. in the $\gamma+\alpha$ temperature regime and
the chosen IA temperature determines the equilibrium volume fractions of
austenite and ferrite. During IA, solute elements are able to partition
between the two phases [8]. Austenite stabilising elements such as Mn and C
diffuse to austenite, and ferrite stabilising elements such as Al and Si
diffuse to ferrite [9, 2]. Therefore, the IA temperature also influences the
austenite composition and therefore its Stacking Fault Energy (SFE) and
stability against martensitic transformation, ultimately governing the
resulting alloy’s plasticity enhancing mechanism via TWIP and/or TRIP
mechanisms [9, 8].

One of the challenges facing medium Mn steel is the slow diffusion of Mn to
the austenite phase during the IA heat treatment [10, 11, 12]. Therefore, the
duration of the IA heat treatment is also important so as to achieve the
desired Mn content at a given temperature. In the medium Mn steel literature,
most researchers approach this challenge in two different ways. The first
method is to anneal quickly at a relatively high temperature in a Continious
Annealing Line (CAL) [13, 14, Kwok2021] where IA is conducted typically above
700 °C and is complete in no longer than 10 min [15]. However, this method
typically requires a large prior defect density, i.e. cold rolled steel, in
order to enhance Mn diffusion [16, 17]. In a study by Lee et al., a cold
rolling reduction of 66% (3 $\rightarrow$ 1 mm) was necessary to enable IA to
be complete (i.e. reach thermodynamic equilibrium) within 180 s. The second
method is to anneal slowly at a relatively low temperature in a Batch
Annealing Furnace (BAF) for several hours where temperatures are typically
below 700 °C [18, 19, 20].

The usage of either a CAL or BAF ultimately aims to drive element partitioning
to completion, i.e. reaching equilibrium, at a given IA temperature. However,
we explore a novel concept of varying the Mn content in the austenite phase
through incomplete IA at a constant temperature. If the Mn content of
austenite phase can be altered by varying the IA temperature, assuming full
partitioning, the same effect can theoretically be achieved by varying the IA
duration while the Mn content tends to equilibrium. The range of Mn contents
that could be achieved by this method would vary between the bulk Mn and the
thermodynamic equilibrium content at the chosen IA temperature. If a
sufficiently low IA temperature is chosen, the equilibrium Mn content in the
austenite will be relatively high and provide a sufficiently large processing
window. Moreover, IA duration, like IA temperature, can significantly
influence the recrystallisation and recovery processes during the
thermomechanical processing of medium Mn steel and therefore alter the
microstructure. This study therefore aims to investigate the effect of IA
duration on the tensile properties and microstructural evolution in a medium
Mn steel at a constant IA temperature.

##  2 Experimental Procedures

A medium Mn steel with a nominal composition of
Fe-8Mn-2.5Al-1.5Si-0.4C-0.02Nb-0.03V in mass percent was arc melted using pure
elements to produce an ingot measuring 23 mm $\times$ 23 mm $\times$ 60 mm.
The bar was sectioned into four smaller bars measuring approximately 10 mm
$\times$ 10 mm $\times$ 60 mm. Each small bar was then quartz encapsulated in
low pressure Ar, homogenised at 1200 °C for 24 h and then quenched in water.
Before hot rolling, each bar was heated to 1000 °C for 30 min. The bars were
hot rolled in one pass at 1000 °C (50% reduction) and 4 passes at 900 °C (25%
reduction per pass) before water quenching immediately after the final pass.
The final strip thickness was approximately 1.5 mm. Coupons from the rolled
strip were intercritically annealed at 660 °C for various durations ranging
from 10 min to 24 h before air cooling. Tensile samples were then cut out from
each coupon using Electric Discharge Machining (EDM) with gauge dimensions of
approximately 1.5 mm $\times$ 1.5 mm $\times$ 19 mm. The tensile samples were
cut with the tensile direction parallel to the rolling direction. Tensile
testing was conducted at a nominal strain rate of $10^{-3}$ s-1. An
extensometer was used to measure the strain between 0 and 6-8%, strain was
measured using the crosshead displacement thereafter. One tensile specimen was
tested per annealing condition.

Correlative Electron Backscatter Diffraction (EBSD) and Energy Dispersive
Spectroscopy (EDS) were conducted on a Zeiss Sigma FE-SEM equipped with a
Bruker EBSD detector and XFlash 6160 EDS detector. Samples for EBSD were first
mechanically ground and subsequently polished with an OP-U suspension.
Transmission Electron Microscopy (TEM) and Energy Dispersive Spectroscopy
(TEM-EDS) were conducted on a JEOL JEM-F200, operated at an accelerating
voltage of 200kV. TEM samples were cut from intercritically annealed coupons
or post-mortem tensile samples via EDM, mechanically ground to a thickness
below 60 $\mu$m and electropolished in a Struers Tenupol twin-jet
electropolishing unit with a solution containing 5% perchloric acid, 35%
butyl-alcohol and 60% methanol at a temperature of $-40$ °C.

Table 1: Nominal bulk composition and measured composition of the investigated steel in mass percent. †measured using IGF. | Fe | Mn | Al | Si | C† | Nb | V
---|---|---|---|---|---|---|---
Nominal | Bal | 8 | 2.5 | 1.5 | 0.4 | 0.02 | 0.03
ICP | Bal | 8.24 | 2.3 | 1.32 | 0.361 | 0.02 | 0.03

##  3 Results

The composition of the investigated steel was measured using Inductively
Coupled Plasma (ICP) and Inert Gas Fusion (IGF) and is shown in Table 1. The
measured composition was then used to generate a property diagram using the
thermodynamic software Thermo-Calc and TCFE7 database in Figure 1. The
property diagram shows a wide fully austenitic region (790$-$1375 °C) for hot
rolling. An intercritical annealing temperature of 660 °C was chosen to obtain
a microstructure with equal volume fractions of austenite and ferrite and also
a small fraction of cementite for carbide precipitation strengthening. It
should be noted that in medium Mn steels, the phase fractions and compositions
predicted by Thermo-Calc at the IA temperature tend to remain the same at room
temperature [21, 6, Kwok2021].

Figure 1: Thermo-Calc property diagram of the measured composition. Some minor
phases have been omitted for clarity. Figure 2: (a) Engineering and (b) true
stress strain curves of the investigated steel with varying IA durations.
Figure 3: Strain hardening rate of the investigated steel with varying IA
duration. A portion of each curve where the extensometer was removed was
omitted for the sake of clarity.

###  3.1 Tensile properties

The rolled strips were intercritically annealed at 660 °C for 0 min, 10 min,
30 min, 2 h, 4 h and 24 h. Henceforth, each sample will be named 660-$X$,
where $X$ is the IA duration at 660 °C. The tensile curves from each annealing
condition are shown in Figure 2 and the strain hardening rates of each sample
are shown in Figure 3. The tensile properties are summarised in Table 2.

The 660-0 min or the as-quenched sample showed no ductility, fracturing at 724
MPa with only 2% strain. However, with only 10 mins of intercritical annealing
at 660 °C, the tensile curve changed drastically. In the 660-10 min sample,
the steel exhibited a large improvement in tensile properties with a yield
strength ($\sigma_{0.2}$) of 772 MPa, Ultimate Tensile Strength (UTS) of 1216
MPa and ductility of 27%. When IA duration was increased further up to 24 h,
there was a monotonic decrease in yield strength and UTS while there was a
monotonic increase in elongation.

From Figure 3, the strain hardening rate, obtained from the first derivative
of the true stress-strain curve, showed a similar shape. After the initial
rapid decrease, the strain hardening rate increases to a maximum at
approximately 0.1$-$0.15 true strain then decreases up to fracture. However,
the maximum strain hardening rate ($\Theta_{max}$) was observed to decrease
with increasing IA duration.

###  3.2 Evolution of Microstructure

Figure 4: EBSD IQ+PM of the undeformed (a) as-quenched or 660-0 min, (b) 660-10 min, (c) 660-30 min, inset: correlative Mn-EDS map showing that the marked triangular region which indexed as BCC was more enriched in Mn (brighter) compared to the other surrounding BCC grains (darker), (d) 660-2 h, (e) 660-4 h and (f) 660-24 h samples. Red $-$ austenite, green $-$ ferrite/$\alpha^{\prime}$ martensite. Black lines indicate High Angle Grain Boundaries (HAGBs) and white lines indicate austenite $\Sigma 3$ boundaries.  Table 2: Tensile properties of the investigated steel in different annealing conditions. Sample | $\sigma_{0.2}$ | UTS | Elongation | $\Theta_{max}$
---|---|---|---|---
| (MPa) | (MPa) | (%) | (GPa)
660-0 min | 724 | 724 | 2 | NA
660-10 min | 772 | 1216 | 27 | 4.7
660-30 min | 726 | 1167 | 30 | 4
660-2 h | 707 | 1165 | 34 | 3.5
660-4 h | 676 | 1121 | 35 | 3.2
660-24 h | 616 | 986 | 37 | 2.5

Figure 4 shows the change in microstructure with increasing IA duration at 660
°C and a summary of the phase fractions obtained through EBSD is shown in
Table 3. In the 660-0 min sample, the microstructure indexed as predominantly
BCC. However, it is likely that the BCC phase was athermal $\alpha^{\prime}$
martensite rather than $\alpha$ ferrite because of the lack of ductility in
the 660-0 min tensile sample. The microstructure was also of the laminate or
lamellar-type morphology, which was to be expected in medium Mn steels when
quenched directly after hot rolling [22].

During IA, the austenite fraction increased rapidly within the first 30 min
but slowed down after 2 h before reaching the equilibrium phase fraction as
predicted by Thermo-Calc within 24 h. However, when cross referencing the EBSD
phase maps with the correlative Mn-EDS maps, it was found that some of the
grains which indexed as BCC contained a variation in Mn content. An example is
shown in the inset of Figure 4c, where the triangular marked region was
enriched in Mn but indexed as BCC, while the surrounding dark (Mn depleted)
lamellae also indexed as BCC. It is therefore reasonable to conclude that the
triangular region was actually austenite during the IA at 660 °C but
transformed back to $\alpha^{\prime}$ martensite during cooling, while the
darker laths were intercritical ferrite grains which formed during the IA heat
treatment. The intercritical ferrite would be depleted in Mn since Mn
partitions out of ferrite into austenite during IA. This suggests that the
austenite fraction at 660 °C in samples intercritically annealed for less than
24 h might have been larger than that observed at room temperature. In a
similar manner, Lee et al. [9] showed that when the IA temperature was raised
too high, i.e. forming a high volume fraction of austenite at the IA
temperature, some of the austenite would transform back into martensite when
cooled to room temperature as the Mn content in the austenite phase became too
dilute in order to stabilise it to room temperature. This suggests that the
increasing austenite fraction observed in Table 3 might be due to an
increasing Mn content, stabilising more austenite down to room temperature.

Table 3: Phase fractions in % of intercritically annealed samples as predicted by Thermo-Calc and measured with EBSD. N.I. $-$ Non-indexed fraction. | $\gamma$ |  $\alpha$/$\alpha^{\prime}$ | $\theta$ | NbC | VC | N.I.
---|---|---|---|---|---|---
660-0 min | 6.7 | 77 | - | - | - | 16.3
660-10 min | 30 | 65.1 | - | - | - | 4.9
660-30 min | 33.6 | 62.7 | - | - | - | 3.7
660-2 h | 41.7 | 35.7 | - | - | - | 22.6
660-4 h | 42.4 | 56.7 | - | - | - | 0.9
660-24 h | 43.8 | 52.6 | - | - | - | 3.6
Thermo-Calc | 43.7 | 54.2 | 2 | 0.022 | 0.045 | N.A.

When the tensile samples were deformed to failure, EBSD phase maps were
obtained from the gauge of the postmortem samples midway between the fracture
edge and shoulder and are shown in Figure 5. The measured phase fractions are
shown in Table 4. In all the tensile samples, it was observed that the
microstructures were almost entirely BCC, i.e. ferrite or $\alpha^{\prime}$
martensite. This shows that a large fraction of the austenite phase had
transformed to $\alpha^{\prime}$ martensite. However, with increasing IA
duration, it can be seen that a small but increasing fraction of austenite
remained untransformed even after deformation. This suggests that the
austenite phase was increasingly stable against transformation with increasing
IA duration.

Figure 5: EBSD IQ+PM of the postmortem (a) as-quenched or 660-0 min, (b)
660-10 min, (c) 660-30 min, (d) 660-2 h, (e) 660-4 h and (f) 660-24 h. Red $-$
austenite, green $-$ ferrite/$\alpha^{\prime}$ martensite. Black lines
indicate HAGBs and white lines indicate austenite $\Sigma 3$ boundaries.

In order to quantify the austenite stability in medium Mn steels, Steineder et
al. [23] proposed using the stability constant $k\textsubscript{p}$, after
Ludwigson and Berger [24]. $k\textsubscript{p}$ may be calculated according to
Equation 1:

$V_{\gamma}^{-1}-V_{\gamma 0}^{-1}=k_{p}\>p^{-1}\>\epsilon^{p}$ (1)

where $V_{\gamma 0}$ is the austenite volume fraction in the unstrained
condition and $V_{\gamma}$ is the austenite volume fraction at a true strain
of $\epsilon$, here taken to be the true strain at fracture. $p$ is a strain
exponent corresponding to the autocatalytic effect and can be taken to be
equal to 1 as proposed by Matsumura et al. [25]. Equation 1 can therefore be
rewritten as Equation 2 where a high $k\textsubscript{p}$ value indicates
lower austenite stability and vice versa.

$k_{p}=\frac{V_{\gamma}^{-1}-V_{\gamma 0}^{-1}}{\epsilon}$ (2)

Based on the austenite phase fractions obtained in Figure 4 and 5, the change
in $k_{p}$ with annealing time was calculated and shown in Table 4 and Figure
6a. It can be seen that the $k_{p}$ value decreased rapidly with time up to
approximately 4 h and slowed significantly up to 24 h. However, when $k_{p}$
was plotted as a function of the inverse square root of IA duration, a good
linear fit was obtained as shown in Equation 3, where $t$ is in hours.

$k_{p}=\frac{12.3}{\sqrt{t}}+24.5$ (3)

In TRIP steels, it is known that the austenite stability strongly affects the
kinetics of martensite formation during tensile deformation [26, 27] and in
Figure 6b, a very good linear relationship was found between the $k_{p}$ value
and $\Theta_{max}$ according to Equation 4:

$\Theta_{max}\,(GPa)=0.07k_{p}+0.88$ (4)

This strongly suggests that the maximum strain hardening rate in medium Mn
steels is highly dependent on the austenite stability even though the initial
austenite volume fractions were not the same.

Figure 6: (a) Relationship between $k_{p}$ and annealing time. Inset: good
linear relationship between $k_{p}$ and the inverse square root of time. (b) A
linear relationship is obtained between $k_{p}$ and $\Theta_{max}$, i.e.
maximum strain hardening rate.

In Figure 7, the post-mortem samples of 660-10 min, 660-2 h and 660-24 h, were
further examined in a TEM. In the 660-10 min sample, very thin twin-like
structures were observed inside a lamellar grain. In the 660-2 h and 660-24 h,
similar twin-like structures were also found inside lamellar grains. Twinning
streaks were observed in the diffraction patterns in Figure 7f and i.
Therefore, it is likely that the twin-like structures were twinned martensite.
Similar twinned martensite were also observed in metastable austenitic
stainless steels [28, 29].

Table 4: Phase fractions in % of postmortem samples as measured by EBSD and calculated $k_{p}$ value. | $\gamma$ | $\alpha$ | N.I. | $k_{p}$
---|---|---|---|---
660-0 min | 4.9 | 84.7 | 10.4 | 276.9
660-10 min | 6.3 | 74.8 | 18.9 | 52.5
660-30 min | 6.79 | 62.5 | 30.7 | 44.8
660-2 h | 7.74 | 75.8 | 16.5 | 36.0
660-4 h | 8.69 | 76.2 | 15.1 | 30.5
660-24 h | 10.3 | 53.9 | 35.8 | 23.6

###  3.3 Composition

The types of plasticity enhancing mechanism in medium Mn steels are usually
attributed to both the SFE and stability of the austenite phase, both of which
are composition and grain size dependent [8, 30, 31]. In Table 5, the
compositions of four samples were measured using TEM-EDS and compared to the
equilibrium composition obtained by Thermo-Calc. Based on previous work by
Kwok et al. [32], the Mn distribution in the vacuum arc melted ingot was
assumed to be homogeneous after the homogenisation heat treatment of 1250 °C
for 24 h and therefore not susceptible to Mn banding [33]. A minimum of six
point scans were obtained for the austenite and ferrite phases each. The
composition of the austenite in the 660-0 min sample was assumed to be the
bulk composition, since no partitioning could have occured. C content was
determined using the lever rule based on the assumptions that ferrite has
negligible C solubility [8, 34] and C in the form of cementite (0.153 wt% C)
would have fully precipitated within 10 min with the volume fraction as
predicted by Thermo-Calc in Table 3. Cementite is known to precipitate rapidly
even after 1 min of IA [35] and while Si is known to retard cementite
formation, the large C supersaturation is expected to overcome any kinetic
effects imposed by Si [36]. Grain size was calculated using the Equivalent
Circle Diameter (ECD) obtained from the EBSD software ESPRIT. The SFE was
calculated according to the method proposed by Sun et al. [34] and Md30, a
measure of austenite stability against strain-induced transformation, was
calculated according to Equation 5 [31, 37, 34]:

$\begin{split}\small
Md_{30}(\mathrm{\SIUnitSymbolDegree}C)=\,&551-462C-8.1Mn-9.2Si\\\
&-1.42(-3.29-6.64\log_{10}d_{\gamma}-8)\end{split}$ (5)

where the compositions are given in mass percent and $d_{\gamma}$ is the
austenite grain diameter. Md30 is defined as the temperature at which half of
the austenite phase transforms to martensite at a true strain of 0.3 [31].
This parameter was primarily developed to measure the stability against strain
induced martensitic transformation in stainless steels [31, 37] but has been
increasingly used in medium Mn steels too [34, 38, 39, 40]. A low Md30
indicates a high austenite stability and vice versa.

From Table 5, it could be seen that the Mn content of the austenite phase in
the 660-10 min sample had largely the same Mn content as the bulk composition
but rose continuously from the 660-30 min sample to the 660-24 h sample. The
rise in Mn only after 10-30 min of intercritical annealing and the rate of
growth of austenite fraction in Table 3 are in agreement with the findings by
Kamoutsi et al. [11] and Farahani et al. [41], which showed that austenite
initially forms rapidly from martensite under Negligible Partitioning Local
Equilibrium (NPLE) mode where growth is controlled by C diffusion. But the
rate of austenite growth slows down as it switches to Partitioning with Local
Equilibrium (PLE) mode where austenite growth is controlled by Mn diffusion in
both austenite and ferrite. Thermo-Calc also predicted that the equilibrium
austenite Mn content at 660 °C would be 12.5 wt%, as compared to 11.9 wt% in
the 660-24 h sample. This suggests that the Mn content may not have reached
the equilibrium composition, even after 24 h of intercritical annealing. Al,
which was expected to partition to ferrite, varied significantly between the
measured samples, likely due to some degree of interference with the TEM
sample holder. Reasonable conclusions therefore could not be drawn from the Al
content which would affect SFE calculation. Nb was undetected in the 660-30
min, 2 h and 24 h samples, suggesting that Nb had completely precipitated in
the form of carbides. However, V largely remained in solution.

Regarding austenite stability, Samek et al. [42] showed that in multiphase
TRIP steels, a decreasing $k_{p}$ should also correlate with a decreasing
Md30. However, this was not the case in Table 5 which showed an increasing
Md30 (i.e. increasingly unstable austenite) with increasing IA duration as
opposed to Figure 6a which showed a decreasing $k_{p}$ with increasing IA
duration (i.e. increasingly stable austenite). The most likely reason is the
inaccurate determination of C. From Figure 4 it was established that the
austenite fraction at 660 °C for short annealing durations was likely much
larger than the observed fraction at room temperature. It would therefore be
incorrect to use the lever rule with the room temperature phase fractions.
While the C content can no longer be determined using the lever rule, the
increasing Mn content in the austenite phase with IA duration as seen in Table
5 gives confidence that the Md30 temperature should also be decreasing
according to Equation 5.

Figure 7: Brightfield TEM micrographs of postmortem samples. (a) Overview of 660-10min, (b) lath structure containing twin-like structures, (c) magnified view of the red square in (b) showing the twin-like structures in greater detail (red dotted lines). (d) Overview of 660-2 h, (e) higher magnification showing very fine twin-like structures along the red dotted line and (f) diffraction pattern obtained from (e) showing twinning streaks. (g) Overview of 660-24 h sample, (h) magnified view of the red square in (g) also showing twin-like structures, (f) diffraction pattern obtained from (e) also showing twinning streaks. Beam direction was parallel to $[110]_{\alpha}$. Table 5: Compositions of austenite phase from various samples measured by TEM-EDS. Standard errors given in parantheses. ND - Not Detected. $\dagger$ C content calculated based on the lever rule and on phase fractions obtained by EBSD. $\ddagger$ Composition of 660-0min sample assumed to be equal to the bulk composition. *Assuming an austenite grain size of 2.9 $\mu$m.  | Fe | Mn | Al | Si | C$\dagger$ | Nb | V | dγ | SFE | Md30 |
---|---|---|---|---|---|---|---|---|---|---|---
| (wt%) | ($\mu$m) | (mJ m-2) | ( °C) |
660-0 min$\ddagger$ | Bal | 8.2 | 2.3 | 1.3 | 0.36 | 0.02 | 0.03 | 0.4 | 12.9 | 318 |
660-10 min | Bal | 8.2 (0.1) | 5.3 (0.2) | 2.4 (0.1) | 0.69 | 0.03 (0.02) | 0.06 (0.03) | 2.1 | 46.5 | 164 |
660-30 min | Bal | 10.1 (0.1) | 2.6 (0.1) | 2.0 (0.1) | 0.62 | ND | 0.04 (0.03) | 2.7 | 26.1 | 185 |
660-2 h | Bal | 11.1 (0.1) | 1.5 (0.1) | 2.2 (0.1) | 0.50 | ND | 0.03 (0.02) | 2.8 | 13.7 | 231 |
660-24 h | Bal | 11.9 (0.1) | 2.1 (0.1) | 1.8 (0.1) | 0.48 | ND | 0.04 (0.02) | 2.9 | 19.1 | 239 |
Thermo-Calc | Bal | 12.5 | 1.6 | 1.6 | 0.48 | trace | trace | - | 16.1 | 234* |

##  4 Discussion

The results in this study have largely confirmed the initial hypothesis to be
true, that varying the IA duration was able to vary the strain hardening
behaviour of a medium Mn steel. Figure 6 showed that it was possible to
reliably predict the $k_{p}$ value and therefore the $\Theta_{max}$ of the
investigated steel with $\Theta_{max}$ showing a linear relationship with the
inverse square root of IA duration. Therefore, depending on the application,
the investigated steel could be annealed in a CAL for a short duration to
produce a stronger steel with a high $\Theta_{max}$ or annealed in a BAF for a
longer duration to produce a steel with a lower $\Theta_{max}$ but with more
ductility.

The reason behind the change in $\Theta_{max}$ with IA duration can be
attributed to the change in austenite stability. Herrera et al. [43] described
that in order to enable a ductile TRIP steel, the austenite should have a
sufficiently low stability to enable TRIP but also high enough such that TRIP
occurs over a wider strain regime. In this study, the austenite stability was
quantified using two different parameters: $k_{p}$ and Md30. The $k_{p}$
parameter was a more phenomenological approach, comparing the amount of
transformed austenite at a given strain. On the other hand, Md30 attempts to
determine the austenite stability from the composition and grain size. While
Md30 is more useful for alloy design, it is still based on empirial data and
regression analysis [31, 37]. Ultimately, austenite stability depends on many
different factors such as composition, grain size, neighbouring constituents
and texture [44, 45], making it difficult to determine from first principles.
In this study, it is acknowledged that there was some difficulty in measuring
the Md30 temperature accurately and it was the phenomenological parameter
$k_{p}$ which showed a good correlation with the tensile properties of the
investigated steel.

As mentioned, austenite stability depends on a variety of microstructural and
compositional factors. Considering the microstructural factors in the
investigated steel, it could be seen from Figure 4 and Table 5 that the
austenite morphology and grain size did not vary significantly. From Table 3,
the phase fractions of the 660-2 h, 4 h and 24 h samples were also very
similar. As such, these microstructural features were unlikely to have a
significant impact on the austenite stability. However, from Table 5, it can
be seen that the Mn content in austenite continued to increase from 10 min to
24 h. Even after 24 h of intercritical annealing, the Mn content still did not
reach the thermodynamic equilibrium as predicted by Thermo-Calc. This shows
that while the steel may be at phase equilibrium, the constituent phases may
not be at compositional equilibrium. Mn, which is an austenite stabiliser,
also strongly stabilises the austenite phase against strain induced
martensitic transformation as shown in Equation 5. Therefore, in the
investigated steel, it is highly likely that the increasing austenite
stability with IA duration was largely brought about through continuous Mn
partitioning to the austenite phase, i.e. chemical stabilisation.

In addition to dislocation glide, the austenite phase in medium Mn steels are
known to deform via TWIP$+$TRIP or just TRIP, largely depending on the SFE [8,
20, 40]. From Table 5, the SFE calculated from the 660-24 h sample was 19.1 mJ
m-2, however this is expected to be lower (between 15.3-19.1 mJ m-2) due to
the likely over-measurement of the Al content. Nevertheless, the SFE value of
the 660-24 h sample should place the steel in the proposed regime for
TWIP$+$TRIP [8, 40]. However, no austenite twins were observed in the
postmortem samples under TEM (Figure 7). The investigated steel was therefore
purely a TRIP-type medium Mn steel, which may explain the good correlation
between $k_{p}$ and $\Theta_{max}$ since there were no other plasticity
enhancing mechanisms involved. One reason why twinning did not occur in the
investigated steel might be the residual high dislocation density associated
with lath-type microstructures [46] which is known to suppress twinning [47].

From Figure 7, twin-like features closely resembling twinned martensite was
found in the postmortem 660-10 min sample. However, twinned martensite could
also be associated with athermal martensite, i.e. formed during cooling [29].
Since it was determined that samples intercritically annealed for less than 24
h might contain athermal martensite, it could not be established if the
twinned martensite formed during deformation or cooling after IA. However,
twinned martensite was still found in the 660-24 h postmortem sample as shown
in Figure 7g-i. Since athermal martensite was not expected in this sample,
this strongly suggests that the observed twinned martensite was the product of
deformation induced martensitic transformation. Medium Mn steels have shown to
be able to form a variety of different types of martensites such as strain
induced martensite [8] and stress-assisted martensite [48]. However, the
significance and role of twinned martensite in medium Mn steels are not yet
clearly understood, although the type of martensite gives certain clues as to
how it is formed. The formation of twinned martensite during deformation is
not commonly reported in medium Mn steels [49, 40] but is more often reported
in TRIP steels [42], TRIP-assisted steels [50] and Fe-Ni-C steels [28, 51].
Based on the work by Zhang and Kelly [51], and Kelly and Nutting [52] in Fe-
Ni-C steels, twinned martensite is formed during a deformation induced
martensitic transformation when the C content of the austenite is above 0.4
wt%, which is consistent with the estimated C content of the 660-24 h sample
in Table 5.

##  5 Conclusions

A novel concept of varying the IA duration at a single temperature instead of
varying the IA temperature was explored. An 8 wt% Mn medium Mn steel was hot
rolled, quenched and intercritially annealed at 660 °C for durations between
10 min and 24 h. This produced a range of austenite stabilities because of
continuous Mn enrichment of the austenite phase with increasing IA duration.
When tested in uniaxial tension, the $\Theta_{max}$ was found to decrease with
increasing IA duration. A good linear fit was found between $\Theta_{max}$ and
the austenite stability constant $k_{p}$ and thus between $\Theta_{max}$ and
the IA duration.

The deformation structures were observed with TEM and found that TRIP, rather
than TWIP$+$TRIP was the dominant plasticity enhancing mechanism despite the
calculated SFE being within the correct range for TWIP$+$TRIP. It was
postulated that the residual high dislocation density in the microstructure
might have suppressed twinning in the steel. The likely formation of twinned
martensite was observed and determined to be the primary deformation induced
martensitic product in the steel due to the high C content of the prior
austenite.

##  6 Acknowledgements

TWJK would like to thank A*STAR, Singapore for provision of a studentship. DD
acknowledges funding from the EPSRC grant Designing Alloys for Resource
Efficiency (DARE), EP/L025213/1.

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