text:
\section{Introduction}
Control over the propagation of light using artificial photonic crystals\cite{PhC_John,PhC_Yablonovitch}
and electromagnetic metamaterials\cite{MetaSmith} has received enormous attention in recent decades mainly
because of its importance for many applications in the vibrant field of photonics. For example, metamaterials
such as left-handed media\cite{Veselago68,Chan18} have shown promising potential for novel
technologies\cite{Pendry00,Klimov12}. In recent decades, great progress in this field
has been often made by taking advantages of analogies with electronic systems in solid state physics.
For example, the concept of a photonic band gap material\cite{PhC_John,PhC_Yablonovitch}, a man-made system
with a periodic dielectric function, was inspired by the electronic Bloch states in a crystalline semiconductor.

More recently, there have been growing interests in using topological photonic materials\cite{Lu2014}
to manipulate the flow of light, again inspired by the recent developments of electronic topological
materials \cite{Hasan2010,Qi2011,hmwengreview}. In particular, the electronic quantum anomalous Hall (QAH) phase
is a two-dimensional (2D) bulk ferromagnetic insulator with a nonzero Chern number in the presence of spin-orbit
coupling (SOC) but in the absence of applied magnetic fields\cite{Haldane1988,QAHexp}. Its associated metallic
chiral edge states in this Chern insulator carry dissipationless unidirectional electric current. Haldane and Raghu
recently proposed\cite{photon_QAH_Haldane} to construct analogs of this intriguing QAH in photonic crystals made
of time-reversal symmetry (TRS) breaking materials to realize unidirectional optical waveguides. Subsequently,
these topological electromagnetic states in a number of gyromagnetic photonic crystals with broken TRS were
further proposed \cite{QAH_Wangzhen_theory,ochiai2009photonic,QAH_poo2011,zeromode,QAH_largeChern,Dirac_mirrorsym,Chan2018}
and observed\cite{QAH_Wangzhen_exp,QAH_poo2013,QAH_largeChern_exp,He2016}. Interestingly, photonic analog of
electronic quantum spin Hall effect in 2D topological insulators with TRS\cite{Hasan2010,Qi2011} were also
observed in bi-anisotropic photonic crystals\cite{Khanikaev2013,z2new,z2}.

Nevertheless, investigations of photonic unidirectional edge modes have mostly been limited to topological
photonic crystals periodic on the scale of the operational wavelength, and this considerably restricts the
applications of topological photonic materials. Very recently, Gao et. al.\cite{chiralhypermeta} theoretically
demonstrated topological photonic phase in chiral hyperbolic metamaterials (CHM) made of continuous TRS media
with photonic edge states robust against disorder on all length scales\cite{chiralhypermeta}.
In hyperbolic metamaterials\cite{HMM13},
which are plasmonic metamaterials, equi-frequency surfaces (EFSs) of transverse electric-field (TE) and
transverse magnetic-field (TM) modes are degenerate on the high-symmetry points in the momentum space [see Fig. 1(b)].
When the bi-anisotropic property (chirality) is introduced in hyperbolic metamaterials with TRS,
which then become CHMs, the degeneracies are broken due to the coupling between TE and TM modes\cite{Khanikaev2013}
and consequently, a nontrivial band gap is opened\cite{chiralhypermeta}. The nontrivial topology of the CHM
results from the nonzero Berry curvature due to the chirality (equivalent to the SOC in electronic topological
insulators) and broken spatial inversion symmetry in continuous medium\cite{continuousmediachern}. Furthermore,
the design principle was based on the concept of  a Floquet topological insulator (FTI)\cite{floquetrechtsman2013}
where by treating the distance $z$ propagated by a waveguide mode as a timelike coordinate, e.g.,
a honeycomb lattice of helical waveguides behaves like a FTI.

In this work, as a novel mechanism for controlling light flow, we introduce the photonic Chern insulators
made of continuous gyromagnetic hyperbolic metamaterials (GHM) with the band gap opened by the TRS-breaking
gyromagnetic effect [Figs. 1(b) and (c)]. The non-trivial topology is demonstrated by the calculated Berry
curvature and nonzero Chern number due to the broken TRS. The unidirectional backscattering-immune non-radiative
edge modes at the interface between the GHM and vacuum are uncovered by the finite-element electromagnetic simulations.
As in the photonic quantum spin-Hall insulators (QSHI) made of the CHMs\cite{chiralhypermeta}, the photonic
edge states in our Chern insulators are robust against disorder on all length scales. In contrast to the
QSHIs\cite{chiralhypermeta}, however, the light flow direction on the surface of our Chern insulators
can be easily flipped by switching the direction of applied magnetic fields. Furthermore, our Chern insulators
made of the GHM can be easily fabricated.

\section{Gyromagnetic hyperbolic metamaterial}

We consider a GHM as a hyperbolic metamaterial\cite{HMM13} with the gyromagnetic response
which is described by the constitutive relation,
\begin{equation}
\left( \begin{matrix}
\mathbf{D}  \\
\mathbf{B}  \\
\end{matrix} \right)=\left( \begin{matrix}
{{\varepsilon_0}}\hat{\varepsilon} & 0  \\
0 & {{\mu_0}}\hat{\mu}  \\
\end{matrix} \right)\left( \begin{matrix}
\mathbf{E} \\
\mathbf{H}
\end{matrix} \right),\\
\end{equation}
where
\begin{equation}
\hat{\varepsilon}=\left(
\begin{array}{ccc}
	\epsilon_{xx}  & 0 & 0 \\
	0 & \epsilon_{yy} & 0 \\
	0 & 0 & \epsilon_{zz} \\
\end{array}
\right),
\hat{\mu }=
\begin{pmatrix}
	\mu_{xx} & -i\gamma & 0 \\
	i\gamma & \mu_{yy} & 0 \\
	0 & 0 & \mu_{zz}
\end{pmatrix}
\end{equation}
are the relative permittivity and permeability tensors, respectively, and $\gamma$ is the gyromagnetic parameter,
representing the degree of TRS breaking upon application of a magnetic field in the $z$-direction (gyromagnetic effect).
For simplicity, let us set $\mu_{xx}=\mu_{yy}=\mu_{zz}=\mu=1$, $\epsilon_{xx}=\epsilon_{yy}=\alpha$ and
$\epsilon_{zz}=\beta$ such that EFS dispersion are isotropic in $X-Y$ plane.
The propagation behavior in the gyromagnetic hyperbolic metamaterials can be described by the wave equation
of the electric field $\mathbf{E}=(E_x,E_y,E_z)^T$ as
\begin{equation}
\label{eq_wave}
\mathbf{k}\times \hat{\mu}_r^{-1} \mathbf{k}\times\mathbf{E}+k_0^2 \hat{\epsilon_r} \mathbf{E}=0,
\end{equation}
where $\mathbf{k}=(k_x,k_x,k_z)$ is the wave vector, and $k_0=\omega/c$ is the wave number in vacuum. The left
side of Eq. \eqref{eq_wave} can be rewritten in the matrix form as
\begin{equation}
\label{eq_waveMatrix}
\left(
\begin{array}{ccc}
\alpha k_0^2 - \frac{k_y^2}{\mu}-Ak_z^2  & \frac{k_xk_y}{\mu}+Bk_z^2 & Ak_xk_z-Bk_yk_z \\
\frac{k_xk_y}{\mu}-Bk_z^2 & \alpha k_0^2 - \frac{k_x^2}{\mu} - Ak_z^2 & Ak_yk_z+Bk_xk_z\\
Ak_xk_z+Bk_yk_z & Ak_yk_z-Bk_xk_z & \beta k_0^2-Ak_t^2\\
\end{array}
\right)
\end{equation}
where $k_x^2+k_y^2=k_t^2$, $A=\frac{\mu}{\mu^2-\gamma^2}$, $B=\frac{\gamma}{\mu^2-\gamma^2}$. The nontrivial
solutions of $\mathbf{E}=(E_x,E_y,E_z)^T$ exist when the determinant of Eq.~\ref{eq_waveMatrix} equals to zero,
resulting in the eigen equation of
\begin{equation}
\label{eq_Det0}
\alpha ^2 \beta  k_0^2 \mu-(\alpha A\mu+\beta-A)\alpha k_t^2
+(A-2\alpha \mu A)\beta k_z^2=0.
\end{equation}

Starting with an isotropic optical medium with $\epsilon_{xx}=\epsilon_{zz}=2$ and $\gamma=0$, one has two routes
to arrive at a GHM, as illustrated in Fig. 1. The EFS dispersion of the isotropic medium is a perfect sphere,
as shown in Fig. 1(a). Furthermore, the eigenstates consisting of transverse electric mode TE ($E_x=E_y=0$, $H_z\neq0$)
and transverse magnetic modes TM ($H_x=H_y=0$, $E_z\neq0$), are degenerate and their dispersion are identical
and on the same sphere. When  the isotropic medium is transformed to a hyperbolic metamaterial
($\epsilon_{xx}=2$, $\epsilon_{zz}=-1$, and $\gamma=0$) by varying $\epsilon_{zz}$ from 2 to -1, the EFS sphere
of the TM mode splits and becomes two parabola along the $k_z$-axis while that of the TE mode remains spherical,
as depicted in Fig. 1(b) [see Fig. 2(a, b) for polarization analysis]. This is because TM "sees" $\epsilon_{xx}=\epsilon_{yy}=2$
only while, in contrast, TM "sees" $\epsilon_{x-y}=2$ and =$\epsilon_{zz}=-1$. Interestingly, the two TM parabola
touch the TE sphere at $k_z =\pm 1.0k_0$, respectively. When the gyromagnetic effect is further introduced into
the system by making $\gamma$ nonzero (e.g., $\gamma=0.8$), the system becomes a GHM. As a result, a band gap
is opened at both degenerate points $k_z =\pm1.0k_0$ [see Fig. 1(d)]. Interestingly, since the gyromagnetic effect breaks TRS,
the eigenstates near $\pm 1.0k_0$ become circularly (elliptically) polarized, while eigenstates on the upper and
lower hyperbolic bands far away from the singular points remain TM-polarized [see Fig. 2(c,d)].

\begin{figure}[htbp]
\includegraphics[width=3.2in]{figure1.pdf}
\caption{EFS evolution from (a) an isotropic medium ($\epsilon_{xx}=\epsilon_{zz}=2$ and $\gamma=0$) to (d)
a GHM ($\epsilon_{xx}=2$,$\epsilon_{zz}=-1$, and $\gamma=0.8$) via either (b) a hyperbolic metamaterial
($\epsilon_{xx}=2$,$\epsilon_{zz}=-1$, and $\gamma=0$) or (c) a gyromagnetic medium ($\epsilon_{xx}=\epsilon_{zz}=2$
and $\gamma=0.8$). TE (TM) and RCP (LCP) denote TE (TM) polarization and right-handed (left-handed) circular
polarization, respectively.}
\end{figure}

Alternatively, one can go from an isotropic medium first to a gyromagnetic medium (e.g., $\epsilon_{xx}=\epsilon_{zz}=2$
and $\gamma=0.8$) [see Fig. 1(c)] and then to the GHM. In a gyromagnetic medium, because of broken TRS, TE and TM
modes are coupled, resulting in the separation of the degenerate EFS spheres into two ellipsoids [Fig. 1(c)].
Also eigenstates becomes elliptically polarized [see Fig. 2(e,f)]. In particular, the eigenstates on the top
and bottom surfaces of the ellipsoids become nearly fully circularly polarized, while those in the vicinity
of the $k_z=0$ plane remain almost purely either (inner ellipsoid) TE or (outer ellipsoid) TM polarization.
Finally, when $\epsilon_{zz}$ is tunned to a negative value (e.g. $\epsilon_{zz}=-1$), the system becomes a GHM
and the outer ellisoid splits to form two open parabolas, as shown in Fig. 1(d).

\begin{figure}[htbp]
\includegraphics[width=3.2in]{figure2.pdf}
\caption{Polarization analysis of eigenstates. Calculated polarization of EFS dispersion for (a, b) the hyperbolic
metamaterial, (c, d) the GHM (c, d) and (e, f) the gyromagnetic medium. RCP (LCP) and TE (TM) denote
right-handed (left-handed) circular polarization and TE (TM) polarization, respectively.}
\end{figure}

\section{Topological phase transition}

To investigate the topological property of the EFS dispersions of all four kinds of optical media and metamaterials
and also to examine the topological nature of the band gaps near $k_z=\pm k_0$,
we calculate the Berry phase $\Phi =\iint{\mathbf{F}(\mathbf{k})\cdot ds}$ and hence the Chern number $\mathcal{C}=\tfrac{1}{2\pi}\Phi$
of all EFS surfaces.\cite{fukui2005chern,Chan2018} Following our previous work\cite{Chan2018},
we adopt the efficient numerical algorithm reported in Ref. [\onlinecite{fukui2005chern}] to
evaluate the Berry curvature $\mathbf{F}(\mathbf{k})$. Note that in principle, the Berry phase and the Chern number are
well defined only for a closed surface. Fortunately, our test calculations show that the Berry curvature $\mathbf{F}(\mathbf{k})$
is negligibly small when the radial wave vector $k_\rho=\sqrt{k_x^2+k_y^2}>15$.
Therefore, although the hyperbolic shaped EFS surfaces are open (Figs. 1 and 2), we find that the calculated Chern numbers would converge
well to integers as long as the surface integration is carried out from the $k_0$ up to the radial wave vector $k_\rho=\sqrt{k_x^2+k_y^2}$
being larger than $15$. The Chern numbers calculated in this way are shown in Fig. 1, and the Berry curvature distributions
are displayed in Fig.~3. We notice that the Berry curvature has the following symmetry properties:
(a) $\mathbf{F}(\mathbf{k})=\mathbf{F}(-\mathbf{k})$ if the system has the spatial inversion symmetry (IS) and
(b) $\mathbf{F}(\mathbf{k})=\mathbf{-F}(-\mathbf{k})$ if the system has the TRS symmetry. Therefore, the Berry curvature
is identically zero and hence the Chern number is zero for both isotropic medium and hyperbolic metamaterial because they have
both IS and TRS [See Figs.~1(a) and 1(b)].

\begin{figure}[htbp]
\includegraphics[width=3.2in]{figure3.pdf}
\caption{Berry curvature distribution on EFSs for the gyromagnetic hyperbolic metamaterials with
(a) $\alpha=\beta=2$, and $\gamma=0.8$, (b) $\alpha=2$, $\beta=-1$, and $\gamma=1.2$.}
\end{figure}

In contrast, the Berry curvatures become nonzero in both gyromagnetic medium and GHM since their TRS is broken,
as shown in Fig.~3. Nevertheless, in the gyromagnetic medium, the Berry curvatures of the eigenstates on each EFS ellipsoid
form an odd function of $k_z$ with respect to the out-surface element [Fig. 3(a)]. Consequently, the sum of
the Berry curvatures on the EFS surface of each ellipsoid (i.e., the Berry phase and the Chern number) remains zero [Fig. 1(c)].
This is also true for the inner ellipsoid in the GHM [Fig. 3(b)] and hence the Chern number of the inner EFS is zero [Fig. 1(d)].
However, the outer ellipsoid in the GHM now splits and hence transforms into two open parabolic surfaces with the Berry curvatures
pointing to the positive $z$-direction. Although the Berry curvatures on the upper and lower surfaces still form
an odd function of $k_z$, they are on the different surfaces. Resultantly, the upper and lower open surfaces each
acquires a Chern number of -1. Therefore, the phase transformation from the hyperbolic to gyromagnetic hyperbolic
due to the gyromagnetic effect is a topological one, and the band gaps centered at $\pm k_0$ are topological nontrivial.

\section{Nonradiative one-way photonic edge mode}
Next, we follow the method suggested in Ref. [\onlinecite{Dya88}] to calculate analytically the surface band between the dielectric and
anisotropic material. Let the interface between air ($y>0$) and GHM ($y<0$) be on the $xz$ plane ($y=0$). The wave vector in
GHM normal to the interface can then be obtained by solving Eq. \eqref{eq_Det0}

\begin{widetext}
\begin{equation}
\label{eq_ky}
k_y^{GHM} = \pm \left( \frac{\alpha  k_0^2 \left(\alpha  \mu ^2+\beta  \left(\mu
	^2-\kappa ^2\right)\right)-2 \alpha  \mu  k_x^2-\alpha
	\mu  k_z^2-\beta  \mu  k_z^2 \pm Y}{2 \alpha  \mu }\right) ^{0.5}.
\end{equation}
where
\begin{equation}
Y \equiv \sqrt{\alpha ^2 k_0^4 \left(\alpha  \mu ^2+\beta
	\left(\kappa ^2-\mu ^2\right)\right)^2+2 \alpha  k_0^2
	\mu  k_z^2 \left(-\alpha ^2 \mu ^2+\alpha  \beta
	\left(\kappa ^2+2 \mu ^2\right)+\beta ^2 \left(\kappa
	^2-\mu ^2\right)\right)+\mu ^2 (\alpha -\beta )^2 k_z^4}
\end{equation}

Then two linear independent eigen fields in GHM $\vec{E}_i(k_x,k_y,k_z)$ ($i=3,4$) can be obtained by solving the null space
of Eq. \eqref{eq_waveMatrix} with $k_y=k_y(k_x,k_z)$ from Eq. \eqref{eq_ky}. Once $\vec{E}_i$ are obtained, $\vec{H}_i$
can also be obtained via Faraday's law $\vec{H}_i=\frac{1}{Z_0}\vec{k} \times \vec{E}_i$. Likewise, two orthogonal eigen fields
in the air can be expressed as
\begin{equation}
\vec{E}_1=\left(
\begin{array}{c}
-k_z \\
0 \\
k_x \\
\end{array}
\right), \quad
\vec{E}_2=\left(
\begin{array}{c}
-k_y \\
k_x \\
0 \\
\end{array}
\right), \quad
\vec{H}_1=\frac{1}{Z_0}\left(
\begin{array}{c}
k_x k_y \\
-k_x^2-k_z^2 \\
k_y k_z \\
\end{array}
\right), \quad
\vec{H}_2=\frac{1}{Z_0}\left(
\begin{array}{c}
-k_x k_z \\
-k_y k_z \\
k_x^2+k_y^2 \\
\end{array}
\right),
\end{equation}
where $Z_0=\sqrt{\mu_0/\epsilon_0}$ is the vacuum impedance, and $k_y=i\sqrt {k_x^2 + k_z^2 - k_0^2}$. Since the tangential
components of electromagnetic fields should be continuous across the interface, we arrive at $Det(D)=0$ with
\begin{equation}
D(k_x,k_z)=
\left(
\begin{array}{cccc}
-k_z & -k_y & E_{3x} & E_{4x} \\
k_x & 0 & E_{3z} & E_{4z} \\
k_x k_y & -k_x k_z & H_{3x} & H_{4x} \\
k_y k_z & k_x^2+k_y^2 & H_{3z} & H_{4z} \\
\end{array}
\right).
\end{equation}
In order to obtain a surface mode, it is important to choose the range of the imaginary part of $k_y$ ($k_y^{GHM}$)
in the vacuum (GHM) to be positive (negative). In other words, the fields in the air and GHM must decay exponentially
along the vertical direction of the interface.
\end{widetext}

According to the bulk-edge correspondence, the gap Chern numbers of $\Delta C=-1$ implies that there is
one propagating edge state in each band gap for each air-GHM interface. Furthermore, these edge states
are chiral and topology-protected, i.e., they are reflection-free one-way edge states. To verify these
amazing predictions, we solve the Maxwell's equations for a slab made of the GHM, as illustrated in Fig.~4(a).
The calculated EFS dispersions for this slab, displayed in Fig. 4(b), show that in addition to the bulk bands
(black curves), there is indeed one edge state for both top (red curve) and bottom (blue curve) interface
between air and the GHM slab in each band gap.

We further simulate light propagation in the air-GHM interface by putting a line source along the $z$-axis
at $x = 0$. The results are displayed in Figs. 4(c)-4(f). Our simulations show clearly that light on
each interface runs without radiation in either $+x$ or $–x$ direction but not both directions, i.e.,
light propagation on the edges is unidirectional. We notice that in Fig.~4(c), both edges at $k_z=1.6k_0$
carry light towards $–x$ direction. This is because the dispersions of both bulk and edge states are
symmetric with respect to the mirror located at $k_z=0$. And the propagation along the $–x$ direction is
caused by the negative gap Chern numbers ($\Delta C=-1$) of both band gaps. Moreover, Fig.~4(g) shows
that light can overcome the obstacles on the surface, indicating light propagation is reflection-free
due to the topological protection. Finally, the direction of light propagation depends on the sign
of gyromagnetic parameter $\gamma$ [see Figs. 4(c) and 4(d)], demonstrating that one can control light
transmission direction by switching the magnetization direction of the GHM slab.

\begin{figure
\includegraphics[width=3.2in]{figure4.pdf}
\caption{One-way propagating edge states. (a) A GHM slab in air. (b) Calculated EFS dispersions for
the GHM slab in air (a). Black, red, and blue lines denote dispersions of bulk state, top and bottom edge states,
respectively. The dashed green circle denotes light cone. (c) and (d) Simulated propagations of light emitted
by a line source at $x=0$ on the top surface at $k_z=1.6k_0$ with $\gamma=-0.8$ and $\gamma= +0.8$, respectively.
(e) The same as in (c) except that the surface is now uneven, which does not stop light propagate along the negative $x$-direction.
(f) The same as in (d) but on the bottom surface.
}
\end{figure}

\begin{figure
\includegraphics[width=3.2in]{figure5.pdf}
\caption{(a) EFS dispersions and (b) decay rate of the electromagnetic waves for the GHM slab in the air.
In (a), the solid black and dashed blue curves denote the dispersions of the bulk states
and the air, respectively. In (a) and (b), the yellow regions denote the overlap regions
of the air dispersion and bulk gap regions, while the grey regions denote the complete gap regions.
(c) and (d) Electric field distributions of the edge states with $k_z=1.06$ and $k_z=0.88$, respectively.
Note that in (c), the electric field is completely confined to the edge of the GHM slab and
hence there is no radiative energy loss, because the $k_z$ is outside the light EPS sphere.
In contrast, in (d), because the $k_z$ is within the light EPS sphere, the electromagnetic wave
of the edge state propagates strongly into the air, resulting in a large radiative energy loss.
}
\end{figure}

A GHM is based on a hyperbolic metamaterial, which is generally a
bilayer superlattice consisting of a metal layer and a dielectric layer stacked
along the normal direction (see next Section for details). Thus, the GHMs
possess the property of surface plasmon polaritons (SPPs). Compared with
the electromagnetic waves in the air, the SPPs are slower waves and hence
have larger wave vectors than that of light in the air. As a result, the edge
states on a GHM may inherit this larger wave vector propagation, as shown in
Fig. 5. Therefore, there may be no intersection between the air light line and
edge states, i.e., the edge states are out of the light line. Consequently, the
edge state would propagate on the interface without radiation [Fig. 5(c)] because the
edge states cannot couple to the electromagnetic waves in the air. In strong
contrast, most of photonic topological insulators are made of photonic
crystals and consequently, the one way edge states on the interfaces of these insulators
suffer from severe radiation loss and a metal wall serving as a cladding perfect
electric conductor or a perfect magnetic conductor has to be added to the
surrounding edges to stop radiation loss into the air \cite{QAH_Wangzhen_exp}.
In practice, there is no lossless perfect conductor and even good metals such as copper
used as the cladding walls absorb the radiation significantly. This certainly hinders
their applications. In contrast, in a GHM, the air serves as the good insulator for
stopping the radiation of the edge states into the air [Fig. 5(c)].

\section{Negative refraction of edge mode}
Another interesting property of the topological surface states of the GHM
is negative refraction, similar to that of the CHM reported recently in Ref. [\onlinecite{Klimov18}].
As shown in Fig. 6(b), the $x$ component of the group velocity (energy flow) of all the surface states
of the GHM with $\gamma=-0.8$ (the blue line in the bottom panel) is positive ($v_{v,x}>0$),
while the phase velocity ($v_{ph,x} = \omega/k_x$) changes sign across $k_z=0$. As a result,
in the $k_x<0$ region the $x$ components of the group velocity and phase velocity have opposite signs
($v_{g,x}>0$ and $v_{ph,x}<0$) [see Fig. 4(b)]. This is the signature
of negative refraction \cite{HMM13,Klimov18}. On the contrary, the surface states on the GHM with $\gamma=0.8$
in the $k_x<0$ region have $v_{g,x}<0$ and $v_{ph,x}<0$ [see upper panel of Fig. 6(b)].
Following the method proposed in Ref. [\onlinecite{He18}],
we show in Fig. \ref{figNR} that by drawing a vertical line in the EFS diagram (i.e., conservation of the momentum
parallel to the interface), the group velocities of the three points intersecting the surface modes will
give the propagation directions of the surface waves [Fig. \ref{figNR}(b)]. Consequently, as shown in Fig. \ref{figNR}(c),
the incident (wave 1) and refracted (wave 3) surface waves lie on the same side of the normal line, indicating a negative refraction.
The existence of negative refraction of the topological surface states would lead to many interesting
effects\cite{Veselago68,Pendry00,Chan18,He18,Fang05,Liu07,Smolyaninov07,Lin08}
such as superlens effect\cite{Pendry00,Fang05,Liu07,Smolyaninov07} and suppression of reflected waves \cite{He18}.

\begin{figure
\includegraphics[width=3.2in]{figure6.pdf}
\caption{\label{figNR}
Schematic illustration of the negative refraction of the TPSS of the GHM.
(a) Side view and (b) the EFS of the system. (c) The incident,
reflected, and transmitted waves at the interface between the two different GHM insulators.
whose group velocities are denoted as  1, 2, and 3 in (b), respectively.
Here $v_{vg,x}$ and $v_{ph,x}$ denote the $x$ components of the group and phase velocities, respectively.}
\end{figure}

\section{Realization of the proposed gyromagnetic hyperbolic metamaterials}

The hyperbolic metamaterials are a highly anisotropic material with real parts of the principal components
of its permittivity tensor having opposite signs \cite{HMM13} and have been intensively investigated
because of this unique property. Here we consider a bilayer superlattice composed of a metal layer with
relative permittivity $\epsilon_m$ and thickness $d_m$ and a dielectric layer with relative
permittivity$\epsilon_d$ and thickness $d_d$ stacked along the $z$-direction. The effective  dielectric constant
of this metamaterial is given by
\begin{equation}
\epsilon_{eff}=\left[
\begin{array}{ccc}
\epsilon_{x-y} & 0 & 0 \\
0 & \epsilon_{x-y} & 0 \\
0 & 0 & \epsilon_{z} \\
\end{array}
\right],
\end{equation}
where
\begin{subequations}
\begin{equation} \epsilon_{x-y}=\frac{\epsilon_md_m+\epsilon_dd_d}{d_m+d_d}, \end{equation}
\begin{equation} \frac{1}{\epsilon_{z}}=\frac{d_m/\epsilon_m+d_d/\epsilon_d}{d_m+d_d}. \end{equation}
\end{subequations}

If the dielectric layer is replaced by a gyromagnetic medium layer, the superlattice becomes a gyromagnetic
hyperbolic metamaterial with its relative permeability tensors
$\begin{bmatrix}
\mu_{g} & -i\gamma & 0 \\
i\gamma & \mu_{g} & 0 \\
0 & 0 & \mu_{g} \\
\end{bmatrix}$
and
$\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{bmatrix}$
for the gyromagnetic and metal layers, respectively. The bilayer superlattice with relative
permittivity $\epsilon_g= \epsilon_d$ and thickness $d_g= d_d$ for the gyromagnetic layer, is schematically shown in Fig. 7.

\begin{figure
\includegraphics[width=3.2in]{figure7.pdf}
\caption{Proposed gyromagnetic hyperbolic metamaterial as a bilayer
superlattice composed of a gyromagnetic slab and a metal slab.}
\end{figure}

For simplicity, let us write the fields in the metal as $(\mathbf{E}^g, \mathbf{H}^g)=(E_x^g, E_y^g, E_z^g, H_x^g, H_y^g, H_z^g)$
and in the gyromagnetic medium as $(\mathbf{E}^m, \mathbf{H}^m)=(E_x^m, E_y^m, E_z^m, H_x^m, H_y^m, H_z^m)$ .
We assume that the wavelength is much larger than the thickness. Using the boundary conditions
that $D_z$, $E_x$, $E_y$, $B_z$, $H_x$ and $H_y$ must be continuous across an interface, we obtain
$\epsilon_mE_z^m=\epsilon_gE_z^g$, $E_x^m=E_x^g=E_x$, $E_y^m=E_y^g=E_y$, $H_z^m=\mu_gH_z^g$,
$H_x^m=H_x^g=H_x$, $H_y^m=H_y^g=H_y$.
Consequently, the average $D$-field and $B$-field are
\begin{subequations}
\begin{equation} D_x^{eff}=\frac{d_g\epsilon_g+d_m\epsilon_m}{d_g+d_m}E_x   \end{equation}
\begin{equation} D_y^{eff}=\frac{d_g\epsilon_g+d_m\epsilon_m}{d_g+d_m}E_y   \end{equation}
\begin{equation} D_z^{eff}=\epsilon_mE_z^m=\epsilon_gE_z^m                  \end{equation}
\begin{equation} B_x^{eff}=\frac{d_g \mu_g+d_m}{d_g+d_m}H_x -\frac{id_g\gamma}{d_g+d_m}H_y \end{equation}
\begin{equation} B_y^{eff}=\frac{d_g \mu_g+d_m}{d_g+d_m}H_y +\frac{id_g\gamma}{d_g+d_m}H_x  \end{equation}
\begin{equation} B_z^{eff}=\mu_0H_z^m=\mu_gH_z^m. \end{equation}
\end{subequations}
Therefore, the effective relative permittivity and permeability tensors are given by
\begin{subequations}
\begin{equation}
\epsilon_{eff}=\left[
\begin{array}{ccc}
\frac{d_g\epsilon_g+d_m\epsilon_m}{d_g+d_m} & 0 & 0 \\
0 & \frac{d_g\epsilon_g+d_m\epsilon_m}{d_g+d_m} & 0 \\
0 & 0 & \frac{(d_g+d_m)\epsilon_g\epsilon_m}{d_g\epsilon_m+d_m\epsilon_g} \\
\end{array}
\right],
\end{equation}

\begin{equation}
\mu_{eff}=\left[
\begin{array}{ccc}
\frac{d_g\mu_g+d_m}{d_g+d_m} & \frac{-id_g\gamma}{d_g+d_m} & 0 \\
\frac{id_g\gamma}{d_g+d_m} & \frac{d_g\mu_g+d_m}{d_g+d_m} & 0 \\
0 & 0 & \frac{(d_g+d_m)\mu_g}{d_g\epsilon_m+d_m\mu_g} \\
\end{array}
\right].
\end{equation}

\end{subequations}

We can easily find suitable materials as the gyromagnetic medium and the metal slab to
construct a GHM described above. For example, yttrium iron garnet (YIG) under an applied magnetic
field of 1600 Gauss has the effective relative permittivity $\epsilon_g=15$ and permeability $\mu_g=1.12$
and $\gamma=0.124$ at 1.94 THz~\cite{QAH_Wangzhen_theory}, and thus can be used as the gyromagnetic medium. InSb has the
effective relative permittivity $\epsilon_m$ = -10.78 at 1.94 THz~\cite{Wu2014} and thus can serve as the metal slab.
In the bilayer multilayer, the thicknesses of the gyromagnetic medium and metal layers are taken to be
the same and much less than 1.546$\times$ $10^{-4}$ m (wavelength of 1.94 THz). In this case,
the effective relative permittivity and permeability tensors can be written as

\begin{subequations}
\begin{equation}
\epsilon_{eff}=\left[
\begin{array}{ccc}
2.11 & 0 & 0 \\
0 & 2.11 & 0 \\
0 & 0 & -76.73 \\
\end{array}
\right],
\end{equation}

\begin{equation}
\mu_{eff}=\left[
\begin{array}{ccc}
1.06 & 0.062i & 0 \\
-0.062i & 1.06 & 0 \\
0 & 0 & 1.06 \\
\end{array}
\right].
\end{equation}
\end{subequations}

The EFS dispersions obtained using the effective parameters are also displayed in Fig. 8. Clearly,
the shape of the EFS dispersions is similar to that of the EFS presented in the last section,
and furthermore, the topological properties such as the Chern numbers of the bands and the topological nature
of the band gaps are identical to those shown in the last section.

\begin{figure}[htbp]
\includegraphics[width=3.2in]{figure8.pdf}
\caption{Calculated EFS of the bilayer superlattice shown in Fig. 7
made of a YIG slab as the gyromagnetic medium and an InSb layer as the metal slab.}
\end{figure}

\begin{figure}[htbp]
\includegraphics[width=3.2in]{figure9.pdf}
\caption{Calculated band gap map on the $\gamma-(\epsilon_{zz}/\epsilon_{xx})$ plane of the bilayer superlattice
shown in Fig. 7, which is made of a YIG slab as the gyromagnetic medium and an InSb layer as the metal slab.}
\end{figure}

To see the robustness of the topological band gap against the variations of the material parameters,
we show the calculated band gap as a function of both $\gamma$ and $\epsilon_{zz}/\epsilon_{xx}$ in Fig. 9.
Clearly, the band gap is significant $(>0.5)$ in a wide range of the parameters. For example,
when $\epsilon_{zz}/\epsilon_{xx}$ is in between -0.1 and -1.0, the band gap is significant for $\gamma$
being from 0.4 to 2.0. Interestingly, Fig. 9 shows that for $\epsilon_{zz}/\epsilon_{xx}$ is in
between -1.0 and -2.0, one tunes the system from the gapless to the Chern insulating state
and then back to gapless state.
Finally, it should be pointed out that the present photonic Chern insulator would be much simpler
to design and fabricate than many other photonic insulators. For example, the photonic Floquet
topological insulators reported in Ref. [\onlinecite{floquetrechtsman2013}] are composed of an array of
micrometer-scale helical waveguides. The photonic chiral hyperbolic topological insulators proposed
in Ref. [\onlinecite{chiralhypermeta}] would be based on an array of hyperbolic coils which serve as chiral resonators.
In contrast, as discussed above, the photonic Chern insulators would consist of a simple YIG/InSb bilayer
superlattice, and thus would be much easier to fabricate.

\section{Discussion and conclusions}

In the QAH effect, vacuum is an insulator for electrons and thus the chiral edge states are localized
at the surface of the Chern insulator. In contrast, vacuum to photons is often like a free electron
metal and thus light propagation on the unidirectional edge modes would generally
suffer from radiation loss. Consequently, a metal film such as copper is usually inserted between air
and the topological photonic insulator to suppress the radiation leakage\cite{QAH_Wangzhen_exp,QAH_poo2013,QAH_largeChern_exp}.
However, metals like copper could incur significant Ohmic loss. In strong contrast, the chiral edge modes
in our GMHs are not only reflection-free but also non-radiative, as demonstrated
by our finite element electromagnetic simulations (see Figs. 4 and 5). This is simply
because the chiral edge dispersions are located outside light cone
[see Fig. 4(b)]. This is an important advantage of the GMHs
over the gyromagnetic photonic crystals\cite{QAH_Wangzhen_exp,QAH_poo2013}.

In conclusion, by both theoretical analysis and electromagnetic simulations,
we have demonstrated that gyromagnetic hyperbolic metamaterials (GHM)
are photonic Chern insulators with fascinating properties.
We further show that the large topogical band gaps with gap Chern number of one in these metamaterials,
result from the simultaneous presence of the hyperbolicity and also the gyromagnetic effect,
which breaks the time-reversal symmetry and thus gives rise to nonzero Berry curvatures on the EFSs.
Remarkably, unlike many other photonic Chern insulators, the GHM Chern insulators possess
non-radiative chiral edge modes on their surfaces, and thus allow to fabricate unidirectional waveguides
without cladding metals which generally incurr considerable Ohmic loss.
Furthermore, the photonic edge states in the proposed Chern insulators are robust against disorder
on a wide range of length scales, in strong contrast to crystalline topological insulators,
and the light flow direction on the surface of the Chern insulators can be easily
flipped by switching the direction of an applied magnetic field.
We also uncover negative refraction of the topological surface wave at the boundary between
the GHMs with the opposite signs of gyromagnetic parameters. Finally, we show that
compared with other photonic topological materials such as chiral hyperbolic materials\cite{chiralhypermeta}, the present
GHM Chern insulators can be much easier to fabricate.

\begin{acknowledgments}
This work is supported by the Ministry of Science and Technology, the National Center for Theoretical Sciences
and the Thematic Research Program (AS-TP-106-M07) of the Academia Sinica of the R. O. C. (Taiwan).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
\section*{\!\!Introduction}
In this paper, we discuss the problem of integration of the
following Hamiltonian system with two degrees of freedom:
\begin{equation}\label{Ham_syst}
\dot p=-H_x,\quad \dot x=H_p,\quad
H=|p|C(x),
\end{equation}
where $C(x)$ is a positive smooth function.

We show that such systems are integrable in elliptic functions for
functions $D(x)$ of a certain form; we shall write them out further
below. The problem of integration is interesting not only as an
independent problem, but also from the standpoint of applications.
In particular, such Hamiltonian systems arise in the study of
rapidly varying solutions of the two-dimensional wave equation. For
example, in the case of long linear gravity surface waves in an
infinite basin with varying bottom given by $D(x) > 0$, the function
$C(x)=\sqrt{g D(x)}$ ($g$ is the free-fall acceleration), and the
corresponding equation has the form \cite{Stoker,Peli}
\begin{equation}\label{Weq1}
\frac{\partial^2 u}{\partial t^2}-\langle \nabla,
C^2(x)\nabla\rangle u=0,\quad C^2(x)=gD(x),\quad x\in \mathbb{R}^2.
\end{equation}
The initial (or some other additional) conditions for system
\eqref{Ham_syst} are generated by the conditions for equation
\eqref{Weq1}. We consider the initial conditions for system
\eqref{Ham_syst} corresponding to the Cauchy problem for
\eqref{Weq1} with localized initial conditions
\begin{equation}\label{Weq2}
u|_{t={t_\circ}}=V\left(\frac{x-x^{\circ}}{l}\right),\quad
\left.\frac{\partial u}{\partial t}\right|_{t_{\circ}}=0,
\end{equation}
where $V (y)$ is a smooth function decreasing rapidly enough as
$|y|\to \infty$, the parameter $l$ characterizes the source size,
and $x^{\circ}$ is a point in whose neighborhood the initial
perturbation is localized. Such conditions for \eqref{Weq1} give the
following Cauchy problem \cite{DobSekTirVol,DobShafTir,DobrNaz} for
system \eqref{Ham_syst}:
\begin{equation}\label{cond}
p|_{t_\circ}=\mathbf{n}(\psi)=\begin{pmatrix} \cos\psi\\
\sin\psi\end{pmatrix}, \quad \psi\in[0,2\pi],\quad
x|_{t_\circ}=x^{\circ}\in \mathbb{R}^2.
\end{equation}
The Hamiltonian system we consider makes it possible to describe the
water waves.

By $X(\psi,t),P(\psi,t)$ we denote the solutions of problem
\eqref{Ham_syst}, \eqref{cond}. At each moment of time $t$, the ends
of these trajectories determine the following smooth closed curves
\begin{equation*}
\Gamma_t=\{x=X(\psi,t),\quad p=P(\psi,t),\quad \psi \in S^1\}
\end{equation*}
in the 4-D phase space $\mathbb R^4_{xp}$, which are called
\emph{wave fronts in the phase space}. According to
\cite{DobSekTirVol,DobShafTir,DobrNaz}, at each $t$, the asymptotic
solution of problem \eqref{Weq1}, \eqref{Weq2} is determined by
$\Gamma_t$ and turns out to be localized in a neighborhood of the
curves $\gamma_t=\{x=X(\psi,t),\;\psi \in S^{1}\}$---the projections
of $\Gamma_t$ on $\mathbb{R}_{xp}^4$---which are called \emph{fronts
in the configuration (physical) space}. In contrast to $\Gamma_t$,
the curves $\gamma_t$ may be non-smooth and have the
self-intersection points. Under $t=0$, the set $\gamma_t$ is the
point~$x^{\circ}$.

As mentioned above, we here we will show that, in the cases where
the bottom is shaped as an underwater bank, i.e., the depth of the
water layer is determined by the function
\begin{equation}\label{bank}
D(\varrho,\varphi)=D(\varrho)\equiv
\frac{b+\varrho^2}{a+\varrho^2},\,\qquad a>b>0\;\;\text{are constants},\quad
\varrho\;\text{\;is the polar radius},
\end{equation}
and in the case where the underwater ridge-shape is determined by
the function
\begin{equation}\label{ridge}
D(x_1,x_2) = \frac{x_1^2+b}{x_1^2+a},\qquad a>b>0\;\;\text{are constants}\,.
\end{equation}
The above-written Hamiltonian system is integrable in elliptic
functions. The exact analytic solutions obtained in these cases are
given in sect.~\ref{result}. These formulas can be used to construct
an asymptotic solution of problem \eqref{Weq1}, \eqref{Weq2} and
asymptotic solutions of stationary equations with localized
right-hand sides, for example, the Helmholtz equation \cite{Anikin}
$$
-\omega^2 v-\varepsilon^2\langle \nabla, C^2(x)\nabla\rangle
v=\frac{1}{\varepsilon}V\left(\frac{x-x^{0}}{\varepsilon}\right), \qquad
\omega=\mathrm{const},\quad x\in \mathbb{R}^2.
$$
The corresponding formulas will be given in the expanded version of
the work. In sect.~\ref{visual}, we describe an algorithm for
constructing fronts, which makes it possible to visualize them and
facilitates dealing with the obtained expressions. In
sect.~\ref{proof}, we outline the proofs.

\section{Exact analytic solutions}\label{result}

Since the function determining the bottom shape is symmetric in both
cases under study, we can without loss of generality assume that
$x^{\circ}= \left(\begin{smallmatrix}
-\xi\\0\end{smallmatrix}\right)$, where $\xi>0$.

Let us consider the situation where the function describing the
depth of the water layer has the form \eqref{bank}, i.e., determines
the underwater bank. We assume that the free-fall acceleration
$g=1$. Then, in the polar $(\varrho,\varphi)$-coordinates
$\{x_1=\varrho \cos\varphi,\,x_2=\varrho \sin\varphi\}$, the
Hamiltonian becomes
\begin{equation}\label{Ham_new} \mathscr
H(\varrho,\varphi;u,v)=\sqrt{u^2+\frac{v^2}{\varrho^2}}\cdot
\sqrt{\frac{\varrho^2+b}{\varrho^2+a}},
\end{equation}
where $u:=p_{\varrho}$, $v:=p_{\varphi}$ are the momenta
corresponding to the system $(\varrho,\varphi).$ The initial
conditions \eqref{cond} in these coordinates become
\begin{equation}\label{InD2}
u|_{t_\circ}=-\cos \psi,\qquad v|_{t_\circ}=-\xi\sin
\psi,\qquad \varrho|_{t_\circ}=\xi,\qquad \varphi|_{t_\circ}=\pi.
\end{equation}

\begin{theorem}\label{theorem1}
The solution of the Hamiltonian system with Hamiltonian
\eqref{Ham_new} and initial conditions \eqref{InD2} is given by the
expressions
\begin{equation}\label{SOL}
\left\{
\begin{aligned}
\varrho&=\sqrt{\wp(\bop)+\delta-a},\\
 \varphi&=\pi \pm \bigg\{1+2\,b\,
\frac{\zeta(\varkappa)}{\wp'(\varkappa)}\bigg\}h\cdot(\bop-\po) \pm
\frac{b\,h}{\wp'(\varkappa)}
\ln\frac{\sigma(\bop-\varkappa)\,\sigma(\po+\varkappa)}
{\sigma(\bop+\varkappa)\,\sigma(\po-\varkappa)},\\
u&=-\frac12
\sqrt{\frac{\xi^2+b}{\xi^2+a}}\,
\frac{\wp'(\bop)}{\wp(\bop)+\delta-a+b}\,
\frac{1}{\sqrt{\wp(\bop)+\delta-a}},\qquad
v=-\xi\sin\psi\,,
\end{aligned}
\right.
\end{equation}
where the variables $\bop, \po$ and $\varkappa$ are solutions of the
following transcendental equations
\begin{equation}\label{p0pkappa}
\wp(\po)=
\xi^2-\delta+a, \quad \wp(\varkappa)=\frac13(a+b-\frak h),\qquad
t-t_\circ=\delta\cdot
(\bop-\po)-\zeta(\bop)+\zeta(\po).
\end{equation}
The expressions for the constants $\delta, \alpha$, and $\beta$ in
terms of the parameters of the problem have the form
\begin{equation}\label{alpha}
\begin{aligned}
\delta&=\frac{1}{3}\,({\frak h}+2\,a-b), \qquad
\alpha=\frac{4}{3}\,\Bigl({\frak h}^2-
2\,(a-2\,b)\,{\frak h}+a^2-a\,b+b^2\Bigr),\\[1ex]
\beta&=\frac {4}{27}\,\Bigl({\frak h}-a+2\,b\Bigr)
\Bigl(2\,{\frak h}^2-4\,(a-2\,b){\frak h}+(2\,a-b)(a+b)\Bigr),
\end{aligned}
\end{equation}
where
\begin{equation}\label{h}
h^2={\frak h}=\xi^2\,\frac{\xi^2+a}{\xi^2+b}\sin^2\psi\,.
\end{equation}
Here, as always in the sequel, the notation
$\wp(z)=\wp(z;\alpha,\beta)$, $\zeta(z)=\zeta(z;\alpha,\beta)$,
$\sigma(z)=\sigma(z;\alpha,\beta)$ is used for the Weierstrass
elliptic functions \cite{Ahiezer}.
\end{theorem}
\begin{remark}
Note that $\po, \bop$, and $\varkappa$ need not be real. They lie on
the edges of the parallelogram $(0,\omega,\omega+\omega',\omega')$,
where $\omega$ is the pure real period of the Weierstrass
$\wp$-function and $\omega$ is the pure imaginary one. In this case,
$\bop$ lies on the same edge of the parallelogram as $\po$. We also
note that equations \eqref{p0pkappa} have infinitely many roots. For
definiteness, we assume that the roots are taken from the first
positive branch. In sect.~\ref{visual}, we describe an algorithm for
constructing fronts, which illustrates these formulas.
\end{remark}

In the case where the function determining the depth of the water
layer has the form \eqref{ridge} and describes the underwater ridge,
the solutions are given by the following theorem.
\begin{theorem}
The solution of the Hamiltonian system \eqref{Ham_syst} with
Hamiltonian
$$
\mathscr H(x,y;u,v)=
\sqrt{p_{x_1}^2+p_{x_2}^2}\sqrt{(x_1^2+b)/(x_1^2+a)}
$$
and initial conditions \eqref{cond} has the form
\begin{equation}\label{res_case2}
\left\{
\begin{aligned}
x_1&=-\sqrt{\wp(\bop;\alpha,\beta)+\delta-a},\qquad
x_2=\sqrt{1-\frak h^{-1}}t+ \sqrt{\frak
h-1}\,(b-a)\cdot(\bop-\po),\\
p_{x_1}&=\frac{1}{2}\,\frac{\gamma}{\sqrt{\frak h-1}}
\frac{\wp'\big({\frak
p}(t)\big)} {\wp\big({\frak p}(t)\big)+\delta-a+b}
\sqrt[-2]{\wp\big({\frak p}(t)\big)+\delta-a},\qquad
p_{x_2}=\gamma.
\end{aligned}
\right.
\end{equation}
The variables $\bop$ and $\po$ are solutions of the transcendental
equations
\begin{equation}\label{new2'}
\frac{1}{h}\,t=\delta\cdot
(\bop-\po)-\zeta(\bop;\alpha,\beta)+
\zeta(\po;\alpha,\beta),\qquad
\po=\wp^{-1}(\xi^2-\delta+a;\alpha,\beta).
\end{equation}
The expressions for the constants $\delta, \alpha$, and $\beta$ in
terms of the parameters of the problem have the form
\begin{equation}\label{alpha'}
\begin{aligned}
\delta&=\frac{1}{3}\,\Bigl((b-a)\,
\frak h+3\,a-2\,b\Bigr), \qquad
\alpha=\frac{4}{3}\,\Bigl((b-a)^2\frak h^2-
(b-a)\,b\,\frak h+b^2\Bigr),\\[1ex]
\beta&=
\frac {4}{27}\,\Bigl((b-a)\,\frak h+b\Bigr)
\Bigl((b-a)\,\frak h-2\,b\Bigr)\Bigl(2\,(b-a)\,\frak h-b\Bigr),
\end{aligned}
\end{equation}
where
\begin{equation}
h^2=\frak{h}=\frac{\xi^2+b}{(\xi^2+b)-(\xi^2+a)\sin^2\psi}, \qquad
\gamma=\sin \psi.
\end{equation}
\end{theorem}

\section{Visualizing the fronts. An algorithm}\label{visual}

Here we describe an algorithm for constructing, in particular, the
fronts $\gamma_t$ by using the software package
\textsc{Wolfram~Mathematica}. This algorithm makes it easy to visualize
the formulas obtained in the preceding section and illustrate the
application of these formulas. On the other hand, the problem of
visualizing fronts is also interesting from the standpoint of
applications, because, as mentioned above, the asymptotic solution
of problem \eqref{Weq1}, \eqref{Weq2} is localized in a neighborhood
of the front \cite{DobSekTirVol,DobShafTir,DobrNaz}.
\begin{figure}[h!] \centering
\includegraphics[width=0.7\linewidth]{1_1}
\caption{The graph of $y=\wp(x)-\xi^2+\delta-a$ under
$\psi=\frac{9}{10}\pi$}\label{figp0}
\end{figure}
\begin{figure}[h!]
\centering
\begin{minipage}[h]{0.7\linewidth}
\begin{minipage}[h]{0.6\linewidth}
{\includegraphics[width=0.85\linewidth]{t100}}
\end{minipage}
\hfill
\begin{minipage}[h]{0.35\linewidth}
{\includegraphics[width=0.6\linewidth]{t100plus}}
\end{minipage}
\caption{The graph of the front at $t=100$} \label{figt100}
\end{minipage}
\end{figure}

\section*{\!\!Algorithm}

\emph{Step}-1: We fix the following parameters of the problem: the
constants $a=100$, $b=1$ determining the bottom shape and the
constant $\xi=0.5$ characterizing the source position.

\emph{Step}-2: For each fixed $\psi \in [0,\pi]$, there exist
infinitely many roots of the equation
\begin{equation}\label{alg_p2}
\wp(p_{\circ})=\xi^2-\delta+a.
\end{equation}
As $p_{\circ1}(\psi)$ and $p_{\circ2}(\psi)$ we take, respectively,
the first and second positive roots (see Fig.~\ref{figp0}). In the
figure, the red point corresponds to $p_{\circ1}$ and the blue
point, to $p_{\circ2}$. We determine $\varkappa(\psi)$ by the
formula \eqref{p0pkappa}, where we again take the first positive
root.
\begin{remark}
As already mentioned, the roots of \eqref{alg_p2} need not be real.
We must seek the complex roots in one of the forms $ip_{\circ}$,
$p_{\circ}+\omega'$, and $ip_{\circ}+\omega$, where $\omega$ is the
pure real period of the Weierstrass $\wp$ and $\omega'$ is its pure
imaginary period, $p_{\circ}$ is real; as  $p_{\circ1}$ and
$p_{\circ2}$ we must again take the first two positive roots.
\end{remark}

\emph{Step}-3: We fix a moment of time $t$. Let
$p_{i}(\psi,t)\,(i=1,2)$ be the first positive root of the equation
$t=\delta\cdot [p-p_{\circ i}(\psi))-\zeta(p)+\zeta(p_{\circ
i}(\psi)]$. If $p_{\circ i}$ is not real, then we seek $p_{i}$ in
the same manner as $p_{\circ i}$.

\emph{Step}-4: For each fixed $t$, we plot the curves in polar
coordinates $\varrho,\varphi$, using formulas \eqref{SOL}. In the
case $\psi \in \left[0,\frac{\pi}{2}\right]$, we take $p_{\circ1}$
and $p_1$ as parameters, and in the case $\psi \in
\left[\frac{\pi}{2},\pi\right]$, we take $p_{\circ2}$ and $p_2$.

\emph{Step}-5: By symmetry, we reflect the obtained graph about the
horizontal axis. Figure~\ref{figt100} shows the front at $t = 100$;
the dashed curve in this figure corresponds to $\psi \in
\left[0,\frac{\pi}{2}\right]$ and the solid curve, to $\psi \in
\left[\frac{\pi}{2},\pi\right]$.

\section{Proof of theorem \ref{theorem1} in a nutshell}\label{proof}

Solving the Hamiltonian system \eqref{Ham_syst} with Hamiltonian
\eqref{Ham_new} and initial conditions \eqref{InD2}, we get
\begin{equation*}
v=\gamma \quad (=\mathrm{const})\,,\quad
\left(u^2+\frac{\gamma^2}{\varrho^2}\right)
\frac{\varrho^2+b}{\varrho^2+a}=\frac{\gamma^2}{h^2}
\quad( = \mathrm{const}).
\end{equation*}
Making use of the integral $\mathscr
H(\varrho,\varphi;u,v)=\gamma/h$, we obtain the following autonomous
dynamics for the variable $\varrho$
\begin{equation*}
\varrho\,\dot\varrho=
\frac{\sqrt{(\varrho^2+b)\left[\varrho^2(\varrho^2+a)-
(\varrho^2+b)h^2\right]}}{(\varrho^2+a)}\,.
\end{equation*}
Making the change $z=\varrho^2+a$ and applying the shift
$s=z-\delta$ to reduce the integral to the canonical Weierstrass
form
\begin{equation*}
\int\limits^{\varrho^2+a-\delta}_\infty\!\!\!
\frac{(s+\delta)\,ds}{\sqrt{4\,s^3-\alpha\,s-\beta}}=t,
\end{equation*}
we derive an expression for $\varrho$ in system \eqref{SOL}. Using
the obtained result and the integral $\mathscr{H}=\gamma/h$, we
obtain the formula for $u$. Substituting the obtained expression
into the Hamiltonian system, one obtains the equation
\begin{equation*}
\dot \varphi=
h\left(1+ \frac{b}{\wp(\bop)+\delta-a}
\right)\frac{1}{\wp(\bop)+\delta}\,.
\end{equation*}
Its integration leads to the logarithmic elliptic integral that can
be easily evaluated \cite{Ahiezer}. Simplifying the obtained
expression, we arrive at the solutions \eqref{SOL}.

\section*{\!\!Acknowledgments}

The authors wish to express their gratitude to S.~Yu.~Dobrokhotov
for the statement of the problem and to M.~Babich and M.~Pavlov for
useful discussions.


text:
\section{Introduction}

Working within the context of semi-classical quantum gravity,
Krasnikov~\cite{Krasnikov} and Sushkov~\cite{Sushkov} have recently
provided examples of two classes of spacetimes containing time
machines for which the gravitational vacuum polarization is
arbitrarily small all the way to the chronology horizon. In related
developments Kay, Radzikowski, and Wald~\cite{KRW}, and Cramer and
Kay~\cite{Kay-Cramer} have shown that these geometries suffer from
diseases on the chronology horizon itself. More recently, I have
argued~\cite{Reliability} that we should not physically trust
semi-classical quantum gravity once we reach the chronology horizon.

In this brief report I wish to present yet another class of spacetimes
for which the gravitational vacuum polarization can be made
arbitrarily small. Implications for chronology
protection~\cite{HawkingI,HawkingII} are briefly discussed.

\section{The Roman ring}

Given {\em one} wormhole, it appears (classically) to be absurdly
easy to turn it into a time
machine~\cite{Morris-Thorne,MTY,Kim-Thorne,Outrageous}, though
quantum effects vitiate this particular
approach~\cite{HawkingI,HawkingII,Kim-Thorne,Outrageous,Visser}.

Given {\em two} wormholes, it appears (even including quantum
effects) to be relatively easy to turn the compound system into a
time machine without each individual wormhole itself being a time
machine~\cite{Visser94,Lyutikov}.

Given {\em many} wormholes, I shall now show that it appears to be
even easier to turn the conglomeration into a time machine (with
all sub-collections of wormholes not themselves being time machines).

The key technical result is that, for any spacetime of non-trivial
topology, the gravitational vacuum polarization may be estimated
by adiabatic techniques to be

\begin{equation}
\langle T^{\mu\nu}(x) \rangle
\approx \Delta_\gamma {\hbar\over s_\gamma(x,x)^4} t^{\mu\nu}.
\end{equation}

\noindent
Here $s_\gamma(x,x)$ is the length of the shortest spacelike geodesic
connecting the point $x$ to itself, while $t^{\mu\nu}$ is a
dimensionless tensor built out of the metric and tangent vectors
to this geodesic. $\Delta_\gamma$ is the van Vleck determinant
associated with this geodesic.

For the one-wormhole system this van Vleck determinant can be
estimated (insofar as the throat of the wormhole is reasonably
thin) to be close to $1$.  For the two-wormhole system the van
Vleck determinant is a complicated function of relative positions
and velocities. I will now provide a simple class of  multiple
wormhole configurations in which the van Vleck determinant is
calculable---and thereby show that there exists a class of geometries
for which the van Vleck determinant can be made arbitrarily small
all the way down to the chronology horizon.

\setlength{\unitlength}{1.0cm}
\begin{figure}[tb]
\begin{center}
\begin{picture}(5,5)(-1,0)
{
\thicklines
\put(0,0){\vector(0,1){5}}
\put(3,0){\vector(0,1){5}}
}
\put(0,1){\circle*{0.2}}
\put(3,3){\circle*{0.2}}
{
\thinlines
\put(0,1){\line(1,0){3}}
\put(0,1){\line(3,2){3}}
}
\put(1.5,0.50){L}
\put(3.1,1.9){T}
\end{picture}
\end{center}
\caption{Schematic representation of a single (chronology
respecting) wormhole. We identify two timelike lines that are
separated by a spatial jump $L$ and time shift $T$, with $T\ll L$.}
\end{figure}

Start by taking $N$ identical wormholes in otherwise flat Minkowski
spacetime. (These may be taken to be simple cut-and-paste wormholes
of the type discussed in~\cite{Visser,Visser89a,Visser89b}.) For
simplicity assume that all wormhole mouths are at rest with respect
to each other; so each wormhole is characterized by a spatial jump
$L$ and time-shift $T$ with $T \ll L$.

If we look at a geodesic that wraps once through a single wormhole,
the invariant interval is simply $s_1^2 = L^2 - T^2 \gg 0$.

Now arrange the $N$ wormholes in a big symmetric polygon, so that
the exit mouth of one wormhole is a normal-space distance of $\ell$
from the entrance mouth of the next wormhole.  (We will want $\ell
\ll L$, so that the normal space distance travelled to get from one
wormhole to the next is less than the distance then ``jumped'' by
going through the wormhole.)

\setlength{\unitlength}{1.0cm}
\begin{figure}[tb]
\begin{center}
\bigskip
\begin{picture}(5,5)
\put(0,1){\circle{0.5}}
\put(1,0){\circle{0.5}}
\put(0,4){\circle{0.5}}
\put(1,5){\circle{0.5}}
\put(4,0){\circle{0.5}}
\put(5,1){\circle{0.5}}
\put(4,5){\circle{0.5}}
\put(5,4){\circle{0.5}}
\put(0,1){\vector(+1,-1){1}}
\put(4,0){\vector(+1,+1){1}}
\put(5,4){\vector(-1,+1){1}}
\put(1,5){\vector(-1,-1){1}}
\put(0,0){$\ell$}
\put(0,5){$\ell$}
\put(5,0){$\ell$}
\put(5,5){$\ell$}
\put(+2.5,-0.0){L}
\put(+2.5,+5.0){L}
\put(-0.0,+2.5){L}
\put(+5.0,+2.5){L}
\end{picture}
\bigskip
\end{center}
\caption{Schematic representation of a Roman ring. This
example contains four wormholes. In each wormhole the two mouths
are separated by a spatial jump $L$. The normal space distance from
the exit mouth of one wormhole to the entrance mouth of the next
is $\ell$.}
\end{figure}

This ``Roman ring'' is a generalization of the two-wormhole ``Roman
configuration''~\cite{Morris-Thorne,Visser,Visser94,Lyutikov}.
(The realization that wormholes generically seem to imply time
travel can be traced back to an observation by Tom Roman~\cite{Outrageous}.
The ``Roman configuration'' was the first two-wormhole time machine
constructed by Morris and Thorne~\cite{Morris-Thorne}.) The
$N$-wormhole ``Roman ring'' can holistically be close to forming
a time machine even if its individual components are perfectly well
behaved.

Consider a closed geodesic that wraps once around the entire
compound system: this consists of $N$ segments of length $\ell$
in normal space, plus $N$ time-shifts of magnitude $T$ from going
through the $N$ wormholes. The invariant interval of this
once-through-the-system geodesic is

\begin{equation}
s_\gamma^2 = (N\ell)^2 - (N T)^2 = N^2 (\ell^2-T^2).
\end{equation}

\noindent
Because it is the normal-space distance between the wormhole mouths
that enters here ($\ell$; not the spatial jump $L$) we can easily
make this geodesic timelike.

For a conformally coupled massless field the symmetries enforce

\begin{equation}
t^{\mu\nu} = \eta^{\mu\nu} - 4 t^\mu t^\nu.
\end{equation}

\noindent
Here $\eta^{\mu\nu}$ is the spacetime metric, which is flat except
at the wormholes themselves; $t^\mu$ is the tangent vector to the
geodesic and is given (up to rotations) by

\begin{equation}
t^\mu
= {(T,0,0,\ell)\over\sqrt{\ell^2-T^2}}
= {N(T,0,0,\ell)\over s_\gamma}.
\end{equation}

\noindent
So anywhere along the geodesic $\gamma$ we can estimate

\begin{equation}
\langle T^{\mu\nu} \rangle
\approx
\Delta_\gamma {\hbar\over s_\gamma^4}
\left[
\eta^{\mu\nu} - 4 {N^2 (T,0,0,\ell)^\mu (T,0,0,\ell)^\nu\over s_\gamma^2}
\right].
\end{equation}

Of course, the only reason we are bothering with this Roman ring
geometry is because the extreme symmetry makes it relatively
easy to calculate the van Vleck determinant. Using the thin-throat
approximation, plus the tidal reformulation of the van Vleck
evolution equation presented in~\cite{Visser93}, we may calculate
the van Vleck determinant at the surface of any of the wormhole
throats to be~\cite{Visser94}

\begin{equation}
\Delta_\gamma = \left[ {N\over U_{N-1}(1+{\ell\over R})}\right]^2.
\end{equation}

\noindent
Here $U_N(x)$ is a Chebyshev polynomial of the second kind, and
$R$ is the radius of each wormhole mouth (assumed spherical). The
time shift $T$, and spatial jump $L$  quietly cancel out of the
calculation for the van Vleck determinant.

Proving the above result is a combinatoric agony that is presented
in excruciating detail in~\cite{Visser94}. That calculation was
carried out for a slightly different configuration: a geodesic that
wraps $N$ times through a single wormhole, but that calculation
can just as easily be adapted to the present case; a geodesic that
wraps once through $N$ wormholes---note that the high degree of
symmetry in the Roman ring configuration is essential for this
purpose.

If we are satisfied with the situation $\ell\gg R$ (a
perfectly sensible constraint---manipulating wormholes is likely to
be quite difficult enough without having them bump into each
other) then we can approximate

\begin{equation}
\Delta_\gamma \approx N^2 \left({R\over 2\ell}\right)^{2(N-1)}
\end{equation}

\noindent
As a consistency check, compare this with the results quoted by
Kim and Thorne~\cite{Kim-Thorne}, Lyutikov~\cite{Lyutikov}, and
Visser~\cite{Visser94}.

The physical interpretation for this result is simple: the van
Vleck determinant measures geometrically induced deviations from
the inverse-square law~\cite{Visser}. By assumption, we are sitting
right on top of one wormhole mouth, and the above result can be
thought of as due to $N-1$ defocussing events which occur as we
move through the $N-1$ other wormholes in the system to get back
to our starting point.

The van Vleck determinant may now be made as small as desired simply
by adding more wormholes to the system.

At the throat of any one of the wormholes, for any $\ell > T$, we
have

\begin{eqnarray}
\langle T^{\mu\nu} \rangle &\approx&
\hbar\; {N^2\over s_\gamma^4} \;
\left({R\over2\ell}\right)^{2(N-1)}
\nonumber\\
&&\times \left[ \eta^{\mu\nu} - 4 N^2
{(T,0,0,\ell)^\mu \; (T,0,0,\ell)^\nu \over s_\gamma^2} \right].
\end{eqnarray}

\noindent
So at fixed $T$ and $\ell$ we have

\begin{eqnarray}
\langle T^{\mu\nu} \rangle &\approx&
\hbar\; {1\over N^2(\ell^2-T^2)^2}
\; \left({R\over2\ell}\right)^{2(N-1)}
\nonumber\\
&&\times \left[ \eta^{\mu\nu} -
4 {(T,0,0,\ell)^\mu \; (T,0,0,\ell)^\nu \over \ell^2 - T^2} \right].
\end{eqnarray}

\noindent
In particular $\langle T^{\mu\nu} \rangle \to 0$ as $N \to \infty$.

Now suppose the whole system is adiabatically shrunk, keeping $T$
fixed but letting $\ell \to T^+$. The ``reliability
horizon''~\cite{Reliability}, the location at which we should
cease to believe the applicability of semi-classical quantum gravity,
will be located at $\sqrt{\ell^2 - T^2} = \ell_{Planck}$.

(General arguments supporting this designation are provided
in~\cite{Reliability}. In the present more specific context it
suffices to realise that once $\sqrt{\ell^2 - T^2} < \ell_{Planck}$,
any quantum field [including gravitons] propagating on this background
will be subject to Planck scale physics.)

At the throat of any one wormhole, when the system is at the
reliability horizon, we have

\begin{eqnarray}
\langle T^{\mu\nu} \rangle &\approx&
\hbar\; {1\over N^2 \ell_{Planck}^4}
\; \left({R\over2\ell}\right)^{2(N-1)}
\nonumber\\
&&\times \left[ \eta^{\mu\nu} -
4 {(\ell,0,0,\ell)^\mu \; (\ell,0,0,\ell)^\nu \over \ell_{Planck}^2} \right].
\end{eqnarray}

\noindent
Again $\langle T^{\mu\nu} \rangle \to 0$ as $N \to \infty$.

Thus with enough wormholes, we can arrange the gravitational vacuum
polarization,  and therefore the back-reaction, to be arbitrarily
small all the way down to the reliability horizon.

\section{Implications}

This counter-example is enough to show that it is impossible to
come up with a chronology protection theorem that makes reference
only to the ``reliable region''---and so it is impossible to come
up with chronology protection theorem that is {\em physically
reliable} within the context of semi-classical quantum gravity.

In this regard I am completely in agreement with Krasnikov~\cite{Krasnikov}
and Sushkov~\cite{Sushkov}, though the current class of models is
obtained in a radically different (and perhaps more physically
transparent) manner.

My interpretation is perhaps a little different: I view this not
as a vindication for time travel enthusiasts but rather as an
indication that resolving issues of chronology protection requires
a fully developed theory of quantum gravity~\cite{Reliability}.

\acknowledgements

This research was supported by the U.S. Department of Energy.  I
wish to thank Tom Roman for reading the manuscript and providing
helpful comments.


text:
\section{Introduction}
\seclabel{Intro}

In 1943, \citet{Hadwiger43} made the following conjecture, which is widely considered to be one of the most important open problems in graph theory; see \citep{Toft-HadwigerSurvey96} for a survey\footnote{All graphs in this paper are undirected, simple and finite. Let $G$ be a graph. The vertex set and edge set of $G$ are denoted by $V(G)$ and $E(G)$. For $v\in V(G)$, let $N_G(v):=\{w\in V(G):vw\in E(G)\}$. If $X\subseteq V(G)$ then $G[X]$ denotes the subgraph induced by $X$.
If $vw$ is an edge of $G$ then $G/vw$ is the graph obtained from $G$ by contracting $vw$; that is, the edge $vw$ is deleted and the vertices $v$ and $w$ are identified. A \emph{minor} of $G$ is a graph that can be obtained from a subgraph of $G$ by contracting edges.
A \emph{$k$-colouring} of $G$ is a function that assigns one of at most $k$ colours to each vertex of $G$, such that  adjacent vertices receive distinct colours. $G$ is \emph{$k$-colourable} if $G$ admits a $k$-colouring. }.

\medskip\noindent\textbf{Hadwiger Conjecture.}
Every graph with no $K_t$-minor is $(t-1)$-colourable.
\smallskip

The Hadwiger Conjecture is true\footnote{If $G$ has no $K_1$-minor then $V(G)=\emptyset$ and $G$ is $0$-colourable. If $G$ has no $K_2$-minor then $E(G)=\emptyset$ and $G$ is $1$-colourable. If $G$ has no $K_3$-minor then $G$ is a forest, which is $2$-colourable.  \citet{Hadwiger43} and \citet{Dirac52} independently proved that if $G$ has no $K_4$-minor (so-called \emph{series-parallel} graphs) then $G$ is $3$-colourable. The Hadwiger Conjecture with $t=5$ implies the Four-Colour Theorem, since planar graphs contain  no $K_5$-minor. In fact, \citet{Wagner37} proved that the Hadwiger Conjecture with $t=5$ is equivalent to the Four-Colour Theorem, and is therefore true \citep{Gonthier-NAMS08,RSST97}.  \citet{RST-Comb93} proved that the Hadwiger Conjecture with $t=6$ also is a corollary of the Four-Colour Theorem.} for $t\leq 6$. \citet{Kostochka82,Kostochka84} and \citet{Thomason84,Thomason01} independently proved that for some constant $c$, every graph $G$ with no $K_t$-minor has a vertex of degree at most $ct\sqrt{\log t}$ (and this bound is best possible). It follows that $G$ is $ct\sqrt{\log t}$-colourable. This is the best known such upper bound. In particular, the following conjecture is unsolved:

\smallskip\noindent\textbf{Weak Hadwiger Conjecture.}
For some constant $c$,
every graph with no $K_t$-minor is $ct$-colourable.
\smallskip

This conjecture motivated \citet{KawaMohar-JCTB07} to prove the following relaxation; see \citep{Kawarabayashi-CPC08} for a recent extension to graphs with no odd $K_t$-minor.

\begin{theorem}[\citet{KawaMohar-JCTB07}]
\thmlabel{RelaxedHadwiger}
For each $t\in\mathbb{Z}^+$ there exists $N_t\in\mathbb{Z}^+$ such that every graph with no $K_t$-minor admits a vertex partition into $\CEIL{\frac{31}{2}t}$ parts, and each connected component of the subgraph induced by each part has at most $N_t$ vertices.
\end{theorem}

With $N_t=1$ the vertex partition in \thmref{RelaxedHadwiger} is a colouring. So
\thmref{RelaxedHadwiger} is a relaxation of the Weak Hadwiger Conjecture. It would be interesting to improve the bound of  $\frac{31}{2}t$  in \thmref{RelaxedHadwiger}. Indeed,  \citet{KawaMohar-JCTB07} write,
\begin{quote}
``The $\frac{31}{2}t$ bound can be improved slightly by fine-tuning parts of the proof in \citep{BKMM}.
However, new methods would be needed to go below $10t$.''
\end{quote}
The main contribution of this paper is to improve  $\frac{31}{2}$  in \thmref{RelaxedHadwiger} to $\frac{7}{2}$.

\begin{theorem}
\thmlabel{Main}
For each $t\in\mathbb{Z}^+$ there exists $N_t\in\mathbb{Z}^+$ such that every graph with no $K_t$-minor admits a vertex partition into $\CEIL{\frac{7t-3}{2}}$ parts, and each connected component of the subgraph induced by each part has at most $N_t$ vertices.
\end{theorem}

There are three main ingredients to the proof of \thmref{Main}. The first ingredient is a list colouring argument due to  \citet{KawaMohar-JCTB07}, which is described in  \secref{ListColouring}. The second ingredient is a sufficient condition for a graph to have a set of edges whose contraction increases the connectivity. This condition generalises previous results by \citet{Mader-DM88}, and is presented in \secref{Contractibility}. The third ingredient, the ``new methods'' alluded to above, is the following recent result by \citet{NorineThomas}.

\begin{theorem}[\citet{NorineThomas}]
\thmlabel{NT}
For each $t\in\mathbb{Z}^+$ there exists $N_t\in\mathbb{Z}^+$ such that every $(t+1)$-connected graph with at least $N_t$ vertices has a $K_t$-minor.
\end{theorem}

\section{List Colouring}
\seclabel{ListColouring}

A key tool in the proofs of \twothmref{RelaxedHadwiger}{Main} is the notion of list colouring, independently introduced by \citet{Vizing76} and \citet{ERT80}. A \emph{list-assignment} of a graph $G$ is a function $L$ that assigns to each vertex $v$ of $G$ a set $L(v)$ of colours. $G$ is \emph{$L$-colourable} if there is a colouring of $G$ such that the colour assigned to each vertex $v$ is in $L(v)$. $G$ is \emph{$k$-choosable} if $G$ is $L$-colourable for every list-assignment $L$ with $|L(v)|\geq k$ for each vertex $v$ of $G$. If $G$ is $k$-choosable then $G$ is also $k$-colourable---just use the same set of $k$ colours for each vertex. See \citep{Woodall-ListColouringSurvey} for a survey on list colouring.

As well as being of independent interest, list colourings enable inductive proofs about ordinary colourings that might be troublesome without using lists. Most notable, is the proof by \citet{Thomassen-JCTB94} that every planar graph is $5$-choosable. This proof, unlike most proofs of the 5-colourability of planar graphs, does not use the fact that every planar graph has a vertex of degree at most $5$. Given that there are graphs with no $K_t$-minor and minimum degree $\Omega(t\sqrt{\log t})$, this suggests that list colourings might provide an approach for attacking the Hadwiger Conjecture. List colourings also provide a way to handle small separators---first colour one side of the separator, and then colour the second side with the vertices of the separator precoloured. This idea is central in the proofs of \twothmref{RelaxedHadwiger}{Main}.

We need the following definitions. Let $G$ be a graph. For $A,B\subseteq V(G)$, the pair $\{A,B\}$ is a \emph{separation} of $G$ if $G=G[A]\cup G[B]$ and $A-B\neq\emptyset$ and $B-A\neq\emptyset$. In particular, there is no edge between $A-B$ and $B-A$. The set $A\cap B$ is called a \emph{separator}, and each of $A-B$ and $B-A$ are called \emph{fragments}. If $|A\cap B|\leq t$ then $\{A,B\}$ is called a \emph{$t$-separation} and $A\cap B$ is called a \emph{$t$-separator}. By Menger's Theorem, $G$ is $t$-connected if and only if $G$ has no $(t-1)$-separation and $|V(G)|\geq t+1$. For  $Z\subseteq V(G)$, a separation $\{A,B\}$ of $G$ is \emph{$Z$-good} if $\{A-Z,B-Z\}$ is also a separation of $G-Z$; otherwise it is \emph{$Z$-bad}. Observe that $\{A,B\}$ is \emph{$Z$-bad} if and only if $A-B\subseteq Z$ or $B-A\subseteq Z$.

\thmref{Main} follows from the next lemma (with $Z=\emptyset$ and $L(v)=\{1,\dots,\CEIL{\frac{7t-3}{2}}\}$ for each $v\in V(G)$).

\begin{lemma}
\lemlabel{ListColouring}
Let $G$ be a graph containing no $K_t$-minor.
Let $Z\subseteq V(G)$ with $|Z|\leq2t-1$.
Let $L$ be a list assignment of $G$ such that:
\begin{itemize}
\item $|L(v)|=1$ for each vertex $v\in Z$ (said to be ``precoloured''),
\item $|L(w)|\geq\frac{7t-3}{2}$ for each vertex $w\in V(G)-Z$.
\end{itemize}
Then there is a function $f$ such that:
\begin{enumerate}
\item[(C1)] $f(v)\in L(v)$ for each vertex $v\in V(G)$,
\item[(C2)] for each colour $i$, if $V_i:=\{v\in V(G):f(v)=i\}$ then each component of $G[V_i]$ has at most $N_t+2t-1$ vertices (where $N_t$ comes from \thmref{NT}), and
\item[(C3)] $f(v)\neq f(w)$ for all $v\in Z$ and $w\in N_G(v)-Z$.
\end{enumerate}
\end{lemma}

\begin{proof}
We proceed by induction on $|V(G)|$.

\textbf{Case I:} First suppose that $|V(G)|\leq N_t+2t-1$.
For each vertex $v\in Z$, let $f(v)$ be the element of $L(v)$.
For each vertex $w\in V(G)-Z$, choose $f(w)\in L(w)$ such that $f(w)\neq f(v)$ for every vertex $v\in Z$. This is possible since $|L(w)|\geq\frac{7t-3}{2}>2t-1\geq|Z|$.
Thus (C1) and (C3) are satisfied. (C2) is satisfied since $|V(G)|\leq N_t+2t-1$.
Now assume that $V(G)\geq N_t+2t-1$.

\textbf{Case II:} Suppose that some vertex $x\in V(G)-Z$ has degree less than $\frac{7t-3}{2}$ in $G$.
Let $f$ be the function obtained by induction applied to $G-x$ with $Z$ precoloured.
Choose $f(x)\in L(x)$ such that $f(x)\neq f(y)$ for each $y\in N_G(x)$.
This is possible since $|L(x)|\geq \frac{7t-3}{2}>\deg(x)$.
Thus $x$ is in its own monochromatic component.
Hence (C1), (C2) and (C3) are maintained.
Now assume that every vertex in $V(G)-Z$ has degree at least $\frac{7t-3}{2}$.

\textbf{Case III:} Suppose that $G$ has a $Z$-good $t$-separation $\{A,B\}$.
Let $P:=Z-B$ and $Q:=Z\cap A\cap B$ and $R:=Z-A$ and $X:=(A\cap B)-Z$.
Thus $P,Q,R,Z$ are pairwise disjoint.
Since $Z=P\cup Q\cup R$, we have $|P|+|Q|+|R|\leq 2t-1$.
Since $A\cap B=Q\cup X$, we have $|Q|+|X|\leq t$ and $|Q|+2|X|\leq 2t$.
Thus  $|P|+|R|+2|Q|+2|X|=(|P|+|Q|+|R|)+(|Q|+2|X|)\leq 4t-1$.
Without loss of generality, $|P|\leq|R|$.
Thus  $2|P|+2|Q|+2|X|\leq 4t-1$, implying $|P|+|Q|+|X|\leq 2t-1$.
That is, $|A\cap(B\cup Z)|\leq 2t-1$.

Now $B\cup Z\neq V(G)$, as otherwise $A-B\subseteq Z$ and $\{A,B\}$ would be $Z$-bad. Thus the induction hypothesis is applicable to $G[B\cup Z]$ with $Z$ precoloured. (This is why we need to consider $Z$-good and $Z$-bad separations.)\ Hence there is a function $f$ such that:
\begin{enumerate}
\item[(C$1'$)] $f(v)\in L(v)$ for each vertex $v\in B\cup Z$,
\item[(C$2'$)] for each colour $i$, if $V'_i:=\{v\in B\cup Z:f(v)=i\}$ then each component of $G[V'_i]$ has at most $N_t+2t-1$ vertices, and
\item[(C$3'$)] $f(v)\neq f(w)$ for all $v\in Z$ and $w\in (B\cap N_G(v))-Z$.
\end{enumerate}

Let $L'(w):=\{f(w)\}$ for each vertex $w\in A\cap(B\cup Z)$.
Let $L'(v):=L(v)$ for each vertex $v\in A-(B\cup Z)$.
Now apply induction to $G[A]$ with list assignment $L'$, and $A\cap(B\cup Z)$ precoloured.
This is possible since $|A\cap(B\cup Z)|\leq 2t-1$. Hence there is a function $f$ such that:
\begin{itemize}
\item[(C$1''$)] $f(v)\in L'(v)$ for each vertex $v\in A$,
\item[(C$2''$)] for each colour $i$, if $V''_i:=\{v\in A:f(v)=i\}$ then each component of $G[V''_i]$ has at most $N_t+2t-1$ vertices, and
\item[(C$3''$)] $f(v)\neq f(w)$ for all neighbours $v\in A-(B-Z)$ and $w\in A\cap(B\cup Z)$.
\end{itemize}

Since $L'(v)\subseteq L(v)$, conditions (C$1'$) and (C$1''$) imply (C1).
Since there is no edge between $A-B$ and $B-A$ in $G$,
(C$3'$) and (C$3''$) imply that
every component of $G[V_i]$ is a component of $G[V_i']$ or $G[V_i'']$ or $G[Z]$.
Since $N_t+2t-1\geq|Z|$, conditions (C$2'$) and (C$2''$) imply (C2).
Hence (C1), (C2) and (C3) are satisfied.
Now assume that every $t$-separation of $G$ is $Z$-bad.

\textbf{Case IV:}
Every vertex in $V(G)-Z$ has degree at least $\frac{7t-3}{2}\geq\frac{3}{2}k+|Z|-2$, where $k:=t+1$.
Thus \thmref{Hope} below implies that $G$ has a $(t+1)$-connected minor $H$ with at least $|V(G)|-|Z|\geq N_t$ vertices.
By \thmref{NT}, $H$ and thus $G$, has a $K_t$-minor. This contradiction completes the proof.
\end{proof}

\section{Contractibility}
\seclabel{Contractibility}

The main result in this section is \thmref{Hope}, which was used in the proof of \lemref{ListColouring}. The proof reduces to questions about contractibility that are of independent interest. \citet{Mader-DM88} proved the following sufficient condition for a given vertex to be incident to an edge whose contraction maintains connectivity\footnote{\thmref{Mader} is a special case of Theorem 1 in \citep{Mader-DM88} with $\mathfrak{S}=\{\{v,w\}:w\in N_G(v)\}$. Reference \citep{Mader-DM88} cites reference \citep{Mader-AM71} for the proof of Theorem 1 in \citep{Mader-DM88}. The proof of our \thmref{Contractibility} was obtained by following a treatment of Mader's work by \citet{Kriesell-CPC05}.}. See references \citep{Kriesell-GC02, Mader-DM05} for surveys of results in this direction.

\begin{theorem}[\citet{Mader-DM88}]
\thmlabel{Mader}
Let $v$ be a vertex in a $k$-connected graph $G$, such that every neighbour of $v$ has degree at least $\frac{3}{2}k-1$. Then $G/vw$ is $k$-connected for some edge $vw$ incident to $v$.
\end{theorem}

The following strengthening of \thmref{Mader} describes a scenario when there is an edge whose contraction \emph{increases} connectivity.

\begin{theorem}
\thmlabel{Contractibility}
Let $v$ be a vertex in graph $G$, such that $N_G(v)$ is the only minimal $(k-1)$-separator,
and every neighbour of $v$ has degree at least $\frac{3}{2}k-1$. Then $G/vw$ is $k$-connected for some edge $vw$ incident to $v$.
\end{theorem}

The condition in \thmref{Contractibility} is equivalent to saying that every $(k-1)$-separation of $G$ is $\{v\}$-bad.
Thus \thmref{Contractibility} is a special case of the following theorem  (with $Z=\{v\}$).

\begin{theorem}
\thmlabel{Hope}
Suppose that $G$ is a graph and for some $Z\subset V(G)$,
\begin{itemize}
\item every $(k-1)$-separation of $G$ is $Z$-bad, and
\item every vertex in $\cup\{N_G(v)-Z:v\in Z\}$ has degree at least $\frac{3}{2}k+|Z|-2$ in $G$.
\end{itemize}
Then $G$ has a set of at most $|Z|$ edges, each with one endpoint in $Z$, whose contraction gives a $k$-connected graph.
\end{theorem}

\begin{proof}
We proceed by induction on $|Z|$.
If $Z=\emptyset$, or $N_G(v)\subseteq Z$ for each $v\in Z$, then $G-Z$ is $k$-connected.
Now assume that $N_G(v)\not\subseteq Z$ for some $v\in Z$.
By assumption, every vertex in $N_G(v)-Z$ has degree at least $\frac{3}{2}k+|Z|-2$ in $G$.
By \lemref{Induction} below there is an edge $vw$ with $w\in N_G(v)-Z$,
such that  every $(k-1)$-separation of $G/vw$ is $(Z-\{v\})$-bad.
For every vertex $x\in V(G/vw)$,
If contracting $vw$ decreases the degree of some vertex $x$, then $x$ is a common neighbour of $v$ and $w$, and $\deg_{G/vw}(x)=\deg_G(x)-1$. Thus $\deg_{G/vw}(x)\geq\frac{3}{2}k+|Z-\{v\}|-2$.
By induction, $G/vw$ has a set $S$ of at most $|Z-\{v\}|$ edges whose contraction gives a $k$-connected graph.
Thus $S\cup\{vw\}$ is a set of at most $|Z|$ edges in $G$ whose contraction gives a $k$-connected graph.
\end{proof}

\begin{lemma}
\lemlabel{Induction}
Suppose that $G$ is a graph and for some $Z\subset V(G)$ and for some vertex $v\in Z$ with $N_G(v)-Z\neq\emptyset$,
\begin{itemize}
\item every $(k-1)$-separation of $G$ is $Z$-bad, and
\item every vertex in $N_G(v)-Z$ has degree at least $\frac{3}{2}k+|Z|-2$ in $G$.
\end{itemize}
Then there is an edge $vw$ with $w\in N_G(v)-Z$, such that
\begin{itemize}
\item every $(k-1)$-separation of $G/vw$ is $(Z-\{v\})$-bad.
\end{itemize}
\end{lemma}

\begin{proof}

Suppose on the contrary that for each $w\in N_G(v)-Z$, the contracted graph $G/vw$ has a $(Z-\{v\})$-good $(k-1)$-separator. This separator must contain the vertex obtained by contracting $vw$. Thus $G$ has a $Z$-good $k$-separator containing $v$ and $w$.
Let \ensuremath{\mathbb{S}}\ be the set of $Z$-good $k$-separations $\{A,B\}$ of $G$ such that $v\in A\cap B$ and $A\cap B\cap (N_G(v)-Z)\neq\emptyset$. We say $\{A,B\}\in\ensuremath{\mathbb{S}}$ \emph{belongs} to $x$ for each $x\in A\cap B\cap (N_G(v)-Z)$.
As proved above, for each $w\in N_G(v)-Z$, some separation in \ensuremath{\mathbb{S}}\ belongs to $w$.

For each separation $\{A,B\}\in\ensuremath{\mathbb{S}}$,
\begin{equation}
\eqnlabel{FragmentContainsNeighbour}
(A-B)\cap(N_G(v)-Z)\neq\emptyset\quad\text{and}\quad
(B-A)\cap (N_G(v)-Z)\neq\emptyset\enspace;
\end{equation}
otherwise $\{A-\{v\},B\}$ or $\{A,B-\{v\}\}$ would be a $Z$-good $(k-1)$-separation of $G$.

Say $\{A,B\}\in\ensuremath{\mathbb{S}}$ belongs to $x\in N_G(v)-Z$,
and $\{C,D\}\in\ensuremath{\mathbb{S}}$ belongs to $y\in (N_G(v)-Z)-\{x\}$.
Let $S:=A\cap B$ and $T:=C\cap D$ be the corresponding separators in $G$.
Let $A':=A-B$ and $B':=B-A$ and $C':=C-D$ and $D':=D-C$ be the corresponding fragments in $G$.
Let $U:=(S\cap{C'})\cup(S\cap T)\cup (T\cap{A'})$.
Thus $U$ separates ${A'}\cap{C'}$ and $B'\cup D'$, as illustrated in \figref{Separators}.

\begin{figure}[h]
\begin{center}
\includegraphics{Separators}
\caption{\figlabel{Separators} Separator $S$ and its fragments $A'$ and $B'$.
Separator $T$ and its fragments $C'$ and $D'$.
The induced separator $U$ is shaded.}
\end{center}
\end{figure}

Suppose that ${A'}\cap{C'}\not\subseteq Z$.
Since $\{A,B\}$ is $Z$-good, $B'\not\subseteq Z$.
Since $B'\cup D'\not\subseteq Z$,
$$\mathcal{U}:=\big\{({A'}\cap{C'})\cup U,{B'}\cup{D'}\cup U\big\}$$
is a $Z$-good separation of $G$, whose separator is $U$.
Thus $|U|\geq k$.
That is, $|S\cap{C'}|+|S\cap T|+|T\cap{A'}|\geq k$.
Now $|S\cap{C'}|+|S\cap T|=|S|-|S\cap{D'}|\leq k-|S\cap{D'}|$.
Hence $k-|S\cap{D'}|+|T\cap{A'}|\geq k$, implying $|T\cap{A'}|\geq|S\cap{D'}|$.
Similarly, $|S\cap{C'}|\geq|T\cap{B'}|$.
By symmetry:
\begin{align}
A'\cap {C'}\not\subseteq Z\quad\Longrightarrow\quad&
|T\cap {A'}|\geq|S\cap{D'}|
\quad\text{and}\quad
|S\cap {C'}|\geq|T\cap{B'}|\eqnlabel{BlahOne}\\
A'\cap D'\not\subseteq Z\quad\Longrightarrow\quad&
|T\cap {A'}|\geq|S\cap{C'}|
\quad\text{and}\quad
|S\cap{D'}|\geq|T\cap{B'}|\eqnlabel{BlahTwo}\\
B'\cap C'\not\subseteq Z\quad\Longrightarrow\quad&
|T\cap {B'}|\geq|S\cap{D'}|
\quad\text{and}\quad
|S\cap {C'}|\geq|T\cap{{A'}}|\eqnlabel{BlahThree}\\
B'\cap D'\not\subseteq Z\quad\Longrightarrow\quad&
|T\cap{B'}|\geq|S\cap{{C'}}|
\quad\text{and}\quad
|S\cap{D'}|\geq|T\cap{{A'}}|\eqnlabel{BlahFour}
\end{align}

Choose a separation $\{A,B\}\in\ensuremath{\mathbb{S}}$ that minimises $\min\{|A-B|,|B-A|\}$.
Let $x$ be a vertex in $N_G(v)-Z$ such that $\{A,B\}$ belongs to $x$.
Define the separator $S$, and the fragments $A'$ and $B'$ as above.
Without loss of generality, $|A'|\leq|B'|$.
By \eqnref{FragmentContainsNeighbour}, there is a vertex $y\in (N_G(v)-Z)\cap{A-B}$.
Let $\{C,D\}$ be a separator in \ensuremath{\mathbb{S}}\ that belongs to $y$.
Define the separator $T$, and the fragments $C'$ and $D'$ as above.

Suppose that $A'\cap C'\not\subseteq Z$ and $B'\cap D'\not\subseteq Z$.
By \twoeqnref{BlahOne}{BlahFour}, $|T\cap A'|=|S\cap D'|$.
Define $U$ and $\mathcal{U}$ as above.
Thus $\mathcal{U}$ is a $Z$-good separation of $G$, whose separator is $U$.
Now $|U|=|S\cap{C'}|+|S\cap T|+|S\cap{D'}|=|S|\leq k$.
Thus $\mathcal{U}$ is a $Z$-good $k$-separation.
Observe that $v\in S\cap T\subseteq U$ and $y\in{A'}\cap T\subseteq U$.
Thus $\mathcal{U}\in\ensuremath{\mathbb{S}}$ and $\mathcal{U}$ belongs to $y$.
One fragment of $\mathcal{U}$ is ${A'}\cap{C'}\subseteq A_s-\{y\}$ since $y\in T$.
Thus $|{A'}\cap{C'}|<|A_s|$, which contradicts the choice of $\{A,B\}$.

Thus $A'\cap C'\subseteq Z$ or $B'\cap D'\subseteq Z$.
By symmetry,  $A'\cap D'\subseteq Z$ or $B'\cap C'\subseteq Z$.
It follows that
${A'}-Z\subseteq T$ or ${B'}-Z\subseteq T$ or
${C'}-Z\subseteq S$ or ${D'}-Z\subseteq S$.
The choice of $S$ will not be used in the remainder.
So without loss of generality, $A'-Z\subseteq T$.

We claim that $A'-Z$ or $B'-Z$ or $C'-Z$ or $D'-Z$ has at most $\ensuremath{\protect\tfrac{1}{2}}\max\{|S-T|,|T-S|\}$ vertices.
If $B'-Z\subseteq T$, then $(A'-Z)\cup(B'-Z)\subseteq T-S$, implying
$A_S-Z$ or $B'-Z$ has at most $\ensuremath{\protect\tfrac{1}{2}}|T-S|$ vertices, as claimed.
Now assume that $B'-Z\not\subseteq T$.
Without loss of generality, $B'\cap C'\not\subseteq Z$.
By \eqnref{BlahThree}, $|S\cap C'|\geq|T\cap{A'}|=|A'-Z|$.
If $|A'-Z|\leq\ensuremath{\protect\tfrac{1}{2}}|S-T|$ then the claim is proved.
Otherwise, $|S\cap C'|\geq|A'-Z|>\ensuremath{\protect\tfrac{1}{2}}|S-T|$.
Thus $|S\cap{D'}|<\ensuremath{\protect\tfrac{1}{2}}|S-T|$ (since $S-T$ is the disjoint union of $S\cap{C'}$ and $S\cap{D'}$).
If $D'-Z\subseteq S$ then $|D'-Z|\leq |D'\cap S|<\ensuremath{\protect\tfrac{1}{2}}|S-T|$.
So assume that $D'-Z\not\subseteq S$. Thus ${D'}\cap{B'}\not\subseteq Z$.
By \eqnref{BlahFour},
$|S\cap{D'}|\geq|T\cap{A'}|=|A'-Z|>\ensuremath{\protect\tfrac{1}{2}}|S-T|$, which is a contradiction.

Hence $|Q-Z|\leq\ensuremath{\protect\tfrac{1}{2}}\max\{|S-T|,|T-S|\}$ for some fragment $Q\in\{A',B',C', D'\}$.
Now $\max\{|S-T|,|T-S|\}=\max\{|S|,|T|\}-|S\cap T|\leq k-1$ since $v\in S\cap T$.
Thus $|Q|\leq\ensuremath{\protect\tfrac{1}{2}}(k-1)+|Z|$.
By \eqnref{FragmentContainsNeighbour}, there is a vertex $w\in (N_G(v)-Z)\cap Q$.
Then $N_G(w)\subseteq Q\cup S$ or $N_G(w)\subseteq Q\cup T$.
Since $v\in S\cap T\cap Z$ and $|S-\{v\}|\leq k-1$ and $|T-\{v\}|\leq k-1$ and  $w\in Q$, we have
$\deg(w)\leq \ensuremath{\protect\tfrac{1}{2}}(k-1)+|Z|+(k-1)-1=\frac{3k-5}{2}+|Z|$.
This contradicts the assumption that each vertex in $N_G(v)-Z$ has degree at least $\frac{3}{2}k+|Z|-2$.
\end{proof}

We now show that the degree bound in \thmref{Contractibility} is best possible. The proof is an adaptation of a construction by \citet{Watkins-JCT70} that shows that the degree bound in \thmref{Mader} is best possible. For odd $k\geq5$ and $n\in[4,k-1]$, let $p:=\ensuremath{\protect\tfrac{1}{2}}(k-1)$. Start with the lexicographic product $C_n\cdot K_p$, which consists of $n$ disjoint copies $H_1,\dots,H_n$ of $K_p$, where every vertex in $H_i$ is adjacent to every vertex in $H_{i+1}$, and $H_j$ means $H_{j\bmod{n}}$. Let $G$ be the graph obtained by adding a new vertex $v$ adjacent to one vertex $w_i$ in each $H_i$, as illustrated in \figref{Construction}. It is straightforward to verify  that there are $k$ internally disjoint paths in $G$ between each pair of distinct vertices in $V(G)-\{v\}$. Thus $N_G(v)$ is the only minimal $(k-1)$-separator in $G$ (since $\deg(v)=n\leq k-1$). For each  neighbour $w_i$ of $v$, observe that  $\deg(w_i)=(p-1)+2p+1=\frac{3}{2}(k-1)$, but in $G/vw$ the set $V(H_i)\cup V(H_{i+2})$ is a $2p$-separator, implying $G/vw_i$ is not $k$-connected. Thus the degree bound of $\frac{3}{2}k-1$ in \thmref{Contractibility} is best possible.

\begin{figure}
\begin{center}
\includegraphics{Construction}
\end{center}
\caption{\figlabel{Construction} Contracting $vw_i$ produces a $(k-1)$-separation.}
\end{figure}

\section{Final Remarks}

Seymour and Thomas conjectured the following strengthening of \thmref{NT}.

\begin{conjecture}[Seymour and Thomas]
\conjlabel{Apex}
For each $t\in\mathbb{Z}^+$ there exists $N_t\in\mathbb{Z}^+$ such that every $t$-connected graph $G$ with at least $N_t$ vertices and no $K_t$ minor contains a set $S$ of $t-5$ vertices such that $G-S$ is planar.
\end{conjecture}

\citet{KNRWb,KNRWa} proved this conjecture for $t\leq 6$. Recently, \citet{NorineThomas} proved it for $t\leq8$. If true,
\conjref{Apex} could be used instead of \thmref{NT} to make small improvements to \thmref{Main}.

\smallskip
Given that list colourings are a useful tool in attacking the Hadwiger Conjecture, it is interesting to ask what is the least function $f$ such that every graph with no $K_t$-minor is $f(t)$-choosable. Since every graph with no $K_t$-minor has a vertex of degree at most $ct\sqrt{\log t}$, it follows  that $f(t)\leq ct\sqrt{\log t}$, and this is the best known bound. In particular, the following conjecture is unsolved.

\medskip\noindent\textbf{Weak List Hadwiger Conjecture.}
For some constant $c$,
every graph with no $K_t$-minor is $ct$-choosable.
\smallskip

\citet{KawaMohar-JCTB07} discuss this conjecture and suggest that it might be true with $c=\frac{3}{2}$. We dare to suggest the following.

\medskip\noindent\textbf{List Hadwiger Conjecture.}
Every graph with no $K_t$-minor is $t$-choosable.
\smallskip

This conjecture is true for $t\leq 5$ \citep{Skrekovski-DM98,HMS-DM08}. The $t=6$ case is open.

\section*{Acknowledgements}

Thanks to Ga\v{s}per Fijav\v{z}, Vida Dujmovi{\'c}, Matthias Kriesell, and Attila P{\'o}r for helpful discussions.

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