\section{INTRODUCTION}

Geometrically frustrated magnets are fertile ground for exotic spin-lattice coupling phenomena.\cite{S.Ji_PRL_ZnCr2O4,NPhys_Matsuda2007,SpinDrivenJT_PRL,OrderByDistortion_PRL} %
Because of the topology of the lattices and the intricate magnetic interactions, %
the frustrated magnets tend not to exhibit magnetically ordered states even at low temperatures.
To relieve the geometrical spin frustration, they often exhibit `spin-driven' crystal lattice distortions, which lower the symmetry of the lattices and lift the vast ground-state degeneracy. %

From 2000s, the spin-lattice coupling phenomena have been intensively investigated using spinel compounds, for instance $A$Cr$_2$O$_4$ or $A$V$_2$O$_4$ ($A=$ Zn, Cd, Hg and Mg), which are regarded as Heisenberg antiferromagnets with pyrochlore lattices.\cite{SpinDrivenJT_PRL,OrderByDistortion_PRL,HalfMagPlateauCdCr2O4_PRL,NPhys_Matsuda2007,S.Ji_PRL_ZnCr2O4} %
The strong spin frustration on the pyrochlore lattices are relieved by cubic-to-tetragonal (or orthorhombic) structural transitions, and consequently, antiferromagnetic orderings appear at low temperatures. %
Previous theoretical studies have pointed out that the symmetry-lowering structural transitions result in bond-order states in which exchange interactions for nearest neighbor bonds in a tetrahedron are enhanced or reduced reflecting the magnetic orderings, specifically the spin correlation $\langle\mbox{\boldmath $S$}_i\cdot\mbox{\boldmath $S$}_j\rangle$ on each bond.\cite{SpinDrivenJT_PRL,OrderByDistortion_PRL} %
This phenomenon may be called `spin-driven' bond order. %

It is also known that the spin-lattice coupling effect also plays a crucial role in magnetic-field-induced phase transitions in Cr-spinel oxides, which exhibit 1/2-magnetization plateau states under applied magnetic field.\cite{HalfMagPlateauCdCr2O4_PRL,PRL2004_Penc,NPhys_Matsuda2007} %
In the case of HgCr$_2$O$_4$, previous x-ray and neutron diffraction measurements have detected changes in nearest neighbor bond length reflecting the bond order in the 1/2-magnetization plateau phase.\cite{NPhys_Matsuda2007} %
Although these extensive studies have established the importance of the spin-lattice coupling in the frustrated magnets, only a few experimental studies have directly observed the changes in exchange interactions in the bond-order states thus far.\cite{SKimura_HgCr2O4} %

In this paper, we report inelastic neutron scattering (INS) measurements on a triangular lattice antiferromagnet, which is a typical example of geometrically frustrated spin systems, CuFeO$_2$ (CFO) under applied magnetic fields of up to 13.4 T. %
We have observed spin-wave spectra in a magnetic field induced 1/5-magnetization plateau phase, which appears above $\sim$ 12.5 T below $\sim$ 10 K. %
We have determined the exchange interactions in the 1/5-magnetization plateau phase by calculating the spin-wave spectra using a Hamiltonian including the spin-lattice coupling effects for nearest neighbor (NN) exchange interactions. %
As a result, we have revealed that CFO exhibits a spin-driven bond order in the 1/5-magnetization plateau phase. %

\begin{figure*}[t]
\begin{center}
	\includegraphics[clip,keepaspectratio,width=16cm]{Fig1.eps}
	\caption{(Color Online) (a) The $H$-$T$ magnetic phase diagram of CFO (redrawn from Ref. \onlinecite{Mitsuda_2000}). %
	Open and filled symbols denote the transition temperatures (or fields) observed with increasing and decreasing temperature (or field), respectively. %
	(b) A schematic of the experimental setup for the present measurements (top view). %
	The $Q$-$E$ space measured with $E_i=8.6$ meV is shown by a pink shaded area. %
	 The observed INS spectra in (c) zero field, (e) under applied field of 10 T, and (f) 13.4 T. %
	 The calculated INS spectra for (d) the 4SL phase $(H=0$ T) and (g) the 5SL phase ($H=13.4$ T). %
	 The gray solid lines show the calculated spin-wave dispersion relation on the $(-h,1-h,1/2)$ line. %
	 [(h)-(j)] Schematics showing the oxygen displacements and the splitting of the NN exchange interactions with the magnetic structures in the (h) 4SL, (i) FE-ICM, and (j) 5SL phases.
	$a_{\rm m}$ and $b_{\rm m}$ denote the monoclinic basis. }
	\label{8.6meV}
\end{center}
\end{figure*}

CFO has been extensively investigated as a geometrically frustrated magnet from 1990s.\cite{Mitsuda_1991,Mekata_1993} %
The crystal structure of CFO is shown in the inset of Fig. \ref{8.6meV}(a); the magnetic Fe$^{3+}$ ions are arranged in equilateral triangular lattice layers, which are separated by O$^{2-}$-Cu$^+$-O$^{2-}$ dumbbells. %
The Curie-Weiss temperature of this system has been estimated to be $\Theta_{CW}\sim -100$ K.\cite{Petrenko_2000,Mekata_1993}  %
On the other hand, CFO undergoes a magnetic phase transition from the paramagnetic (PM) phase to an incommensurate collinear magnetic phase, which is referred to as the partially disordered (PD) phase\cite{Mitsuda_1994_PD},  at $T_{\rm N1} = 14$ K in zero magnetic field. %
The large difference between $T_{\rm N1}$ and $|\Theta_{CW}|$ indicates the existence of the strong spin frustration in this system.
The magnetic phase transition at $T_{\rm N1}$ is accompanied by a structural transition from the original trigonal structure to a monoclinic structure,\cite{Terada_CuFeO2_Xray,Ye_CuFeO2} so that the geometrical frustration is partly relieved. %
With further decreasing temperature from $T_{\rm N1}$, the system enters a collinear four-sublattice (4SL) antiferromagnetic ground state at $T_{\rm N2}\sim 11.2$ K. %
The  spin arrangement in the 4SL phase is shown in Fig. \ref{8.6meV}(h). %
When a magnetic field is applied along the $c$ axis at low temperatures, CFO exhibits successive magnetic phase transitions, as shown in the $H$-$T$ magnetic phase diagram in Fig. \ref{8.6meV}(a). %
The first-field-induced phase is referred to as the ferroelectric incommensurate-magnetic (FE-ICM) phase. %
The magnetic structure in this phase is a distorted screw-type structure,\cite{CompHelicity,Haraldsen_SW_FEphase} which breaks the inversion symmetry of the system and accounts for the ferroelectricity in this phase.\cite{Kimura_CuFeO2} %
The second-field-induced phase is the 1/5-magnetization plateau phase. %
Figure \ref{8.6meV}(j) shows the spin arrangement on a triangular lattice layer in this phase.\cite{Mitsuda_2000} %
Because the magnetic unit cell has five spins on its basal plane, this phase has been referred to as the five sublattice (5SL) phase.

Interestingly, these magnetic-field-induced phase transitions are accompanied by distinct changes in crystal structure.\cite{Terada_CFO_Pulse,Terada_14.5T} %
This indicates that the spin-lattice coupling effect plays an important role for the field-induced transitions in CFO.\cite{Vishwanath_PRL_spin-phonon} %
Terada \textit{et al.} have explained the field-induced lattice deformations in terms of changes in Fe-O-Fe bonding angle.\cite{Terada_CFO_Pulse,Terada_LatticeModulation} %
They have pointed out that the antiferromagnetic (AF) NN interaction is enhanced by increasing the bonding angle, and vice versa.\cite{Terada_LatticeModulation} %
As a result, positions of the Fe$^{3+}$ and O$^{2-}$ ions are shifted so as to lower the  exchange energy in each of the magnetically ordered phases. %
Moreover, in CFO, an O$^{2-}$ ion is surrounded by three Fe$^{3+}$ ions, and thus a displacement of an O$^{2-}$ ion can affect three Fe-O-Fe bonds. %
This situation can lead to a variety of bond-order states associated with the magnetic orderings. %

Recently, we have identified the bond order in the 4SL phase.\cite{SingleDomainSW2} %
By means of INS measurements using a single crystal of CFO, we have revealed that the NN exchange interaction, $J_1$, splits into two interactions of $J_1^{(1)}$ and $J_1^{(2)}$; $J_1^{(1)}$ is a strong AF interaction connecting antiferromagnetically coupled NN spins and $J_1^{(2)}$ is a weak AF interaction connecting ferromagnetically coupled NN spins, as shown in Fig. \ref{8.6meV}(h). %
The atomic displacements associated with this bond order result in doubling of the unit cell along the [110] direction, %
which is consistent with the previous x-ray diffraction results.\cite{Terada_14.5T,Terada_CuFeO2_Xray,Ye_CuFeO2}
A similar bond order and resulting atomic displacements were also found in the FE-ICM phase (see Fig. \ref{8.6meV}(i) and Refs.\onlinecite{SingleDomain_SW_CFGO,Haraldsen_SW_FEphase,Terada_14.5T,CFAO_xray}). %
On the other hand, in the 5SL phase, the changes in exchange interactions were not directly observed because of the difficulty of measuring the spin-wave spectra under high magnetic fields. %
In order to understand the role of the spin-lattice coupling effect in the field-induced transitions in this system, however, it is indispensable to determine the exchange interactions in the 5SL phase. %
In the present study, we have thus performed INS measurements on CFO under applied magnetic fields of up to 13.4 T. %

\section{EXPERIMENT}

The neutron scattering experiment was carried out using the chopper spectrometer LET at the ISIS spallation neutron source.\cite{LET} %
The detector coverage used in this experiment was from $-30^{\circ}$ to $50^{\circ}$. %
We used a vertical field superconducting cryomagnet whose maximum field is 13.5 T. %
The vertical open-angle of the magnet is from $-10^{\circ}$ to $15^{\circ}$. %
A number of incident energies ($E_i$) were selected by the multi-$E_i$ method. %
In the present analysis, we mainly used data measured with $E_i =$ 3.6 and 8.6 meV, for which the energy resolutions are estimated to be 0.049 and 0.17 meV, respectively. %
A single crystal of CFO was grown by the floating zone method,\cite{Zhao_FZ} and was cut into a plate shape with dimensions of 22, 5.0 and 3.7 mm for [001], [110] and $[1\bar{1}0]$ direction, respectively. %
During the experiment, we applied uniaxial pressure of $\sim 5$ MPa on the $[1\bar{1}0]$ surfaces of the sample using a duralumin clamp. %
This is because CFO has three magnetic domains in the magnetically ordered phases owing to the threefold rotational symmetry about the $c$ axis, and volume fractions of the three domains can be controlled by a small uniaxial pressure applied in the triangular lattice plane.\cite{SingleDomainSW} %
The sample with the uniaxial-pressure clamp was mounted in the cryomagnet so that the $c$ axis is parallel to the magnetic field. %
By measuring elastic magnetic Bragg reflections in the 5SL phase, we have found that the magnetic domain having the magnetic modulation wave vector parallel to the [110] direction has the volume fraction of 64 \%, and dominates over the other two domains (14\% and 22\%). %
The huge combined data set were handled by the HORACE software of ISIS.\cite{HORACE}

\begin{figure}[t]
\begin{center}
	\includegraphics[clip,keepaspectratio,width=8.5cm]{Fig2.eps}
	\caption{(Color Online) The (a) observed and (b) calculated spin wave excitation spectra along the $(h,h,0)$ line in the 5SL phase. %
	The calculated spin-wave dispersion relations are shown by solid gray lines. %
	(c) The magnetic structure in the 5SL phase and the definitions of $J_2$, $J_3$ and $J_z$. %
	Because of the minus sign between the spin angular momentum and the magnetic moment, the direction of the magnetic field is taken to be $-z$ direction. %
	The gray dashed lines show the magnetic unit cell for the 5SL phase. %
	 }
	\label{3.6meV}
\end{center}
\end{figure}

\section{RESULTS AND DISCUSSIONS}

Figures \ref{8.6meV}(c), \ref{8.6meV}(e), and \ref{8.6meV}(f) show the INS spectra measured with $E_i = 8.6$ meV, in the 4SL ($H=0$ T), FE-ICM (10 T), and 5SL (13.4 T) phases, respectively. %
These spectra were measured by setting the angle between the (110) direction of the crystal and the direction of the incident neutrons to $68^{\circ}\sim 73^{\circ}$, as shown in Fig. \ref{8.6meV}(b). %
The observed data were projected on to the $(-h,1-h,1/2)$ line, on which magnetic Bragg reflections appear in all the three phases.  %
In zero magnetic field, we found a distinct spin-wave branch lying in the energy range of $E = 2\sim 3.5$ meV, as shown in Fig. \ref{8.6meV}(c). %
This corresponds to the higher energy branch observed in the previous INS measurements in the 4SL phase.\cite{SingleDomainSW,SingleDomainSW2} %
This spin-wave spectrum also indicates that the inelastic scattering signals from the two minority domains were negligible in the present experiment, %
because the observed spectrum is almost the same as that in the nearly `single-domain' 4SL phase in Ref. \onlinecite{SingleDomainSW}. %
In Fig. \ref{8.6meV}(d), we show a calculated INS spectrum along the $(-h,1-h,\frac{1}{2})$ line in the single-domain 4SL phase, which also qualitatively agrees with the present result. %
In the FE-ICM phase, the excitation spectrum became rather diffusive, as shown in Fig. \ref{8.6meV}(e). %
This is characteristic of the incommensurate and noncollinear magnetic ordering in the FE-ICM phase, and is similar to the magnetic excitation spectra observed in the FE-ICM phase of CuFe$_{1-x}$Ga$_x$O$_2$ $(x=0.035)$.\cite{SingleDomain_SW_CFGO,Haraldsen_SW_FEphase} %
In the 5SL phase, we found that distinct spin-wave branches were retrieved, as shown in Fig. \ref{8.6meV}(f). %

\begin{table*}[t]
\begin{center}
\begin{tabular}{ccccccccccc}
\hline
\hline
magnetic phase &sample         & $J_1^{(1)}$ & $J_1^{(2)}$ & $J_1^{(3)}$ & $J_1^{(4)}$ & $J_2$      & $J_3$     & $J_{z}$    & $D$ & Ref. \\
\hline
4SL  ($H=0$ T)                 &CuFeO$_2$     & $ -0.176$    & $-0.060$     & -                   & -                  & $-0.041$ & $-0.142$ & $-0.071$ & $0.064$ & \onlinecite{SingleDomainSW2}\\
FE-ICM  ($H=0$ T)   &CuFe$_{1-x}$Ga$_x$O$_2$ ($x=0.035)$ & $ -0.169$    & $-0.066$     & -                   & -                  & $-0.070$ & $-0.098$ & $-0.070$ & $0.014$ & \onlinecite{SingleDomain_SW_CFGO}\\
5SL  ($H=13.4$ T)                    &CuFeO$_2$ &  $ -0.18$    & $-0.06$     & $-0.10$      & $-0.14$     & $-0.06$ & $-0.15$ & $-0.06$  & $0.064$ & This work\\
\hline
\hline
\end{tabular}
\caption{The Hamiltonian parameters in the 4SL phase (from Ref. \onlinecite{SingleDomainSW2}),\cite{comment1} the FE-ICM phase (from Ref. \onlinecite{SingleDomain_SW_CFGO}) and the 5SL phase (in meV). %
 }
\label{Table_J}
\end{center}
\end{table*}

In Fig. \ref{3.6meV}(a), we show the spin-wave excitation spectrum along the $(h,h,0)$ line measured with $E_i=3.6$ meV at $H=13.4$ T and $T=1.7$ K. %
The observed spectrum seems to be consistent with a theoretical prediction by Haraldsen \textit{et al}, in which the spin-wave excitations under applied magnetic fields were calculated using Monte Carlo simulations and variational method for two dimensional triangular lattice.\cite{Haraldsen_CFO_Prediction} %
However, in CFO, exchange interactions between adjacent triangular lattice layers are quite important as was pointed out in the previous theoretical study by Fishman \textit{et al}.\cite{Fishman_Stacking_PRB} %
In the present study, we have thus employed the three dimensional magnetic structure of the 5SL phase to calculate the spin-wave spectra. %

Figure \ref{3.6meV}(c) shows the magnetic structure in the 5SL phase.\cite{Mitsuda_2000} %
The magnetic unit cell in the 5SL phase contains two triangular lattice layers, each of which has five spins, and therefore, the 5SL phase actually has ten sublattices.\cite{Fishman_Stacking_PRB} %
To calculate the spin-wave spectrum in the 5SL phase, we have employed a conventional linear spin-wave theory with a Hamiltonian, %
\begin{eqnarray}
\mathcal{H}=-\frac{1}{2}\sum_{i\ne j}J_{ij}\mbox{\boldmath $S$}_i\cdot \mbox{\boldmath $S$}_j-D\sum_i(\mbox{\boldmath $S$}_i^z)^2+ g\mu_B\sum_i\mbox{\boldmath $S$}_i^zH,
\label{Hamiltonian}
\end{eqnarray}
where  $S=5/2$, and $D$ is a uniaxial anisotropy. The gyromagnetic ratio, $g$, is assumed to be 2. $H$ and $\mu_B$ are the applied magnetic field and the Bohr magneton, respectively. %
Similarly to our previous work on the spin-wave excitations in the 4SL phase,\cite{SingleDomainSW2} nearest, second, and third neighbor exchange interactions within the triangular lattice layers ($J_1, J_2$ and $J_3,$) and an exchange interaction between the adjacent layers ($J_z$) are employed in Eq. (\ref{Hamiltonian}). %
As for $J_1$, we have introduced the spin-lattice coupling effect in the same spirit as the previous works on the 4SL and FE-ICM phases.\cite{SingleDomainSW2,SingleDomain_SW_CFGO,Haraldsen_SW_FEphase} %
The displacements of each O$^{2-}$ ion are assumed from the spin arrangements of the three neighboring Fe$^{3+}$ ions. %
Hereby, $J_1$ splits into four different NN interactions, $J_1^{(1)}, J_1^{(2)}, J_1^{(3)}$ and $J_1^{(4)}$ as shown in Fig. \ref{8.6meV}(j). %

The procedure of the spin-wave calculation for the 5SL phase is essentially the same as those in the previous works on the 4SL phase.\cite{SingleDomainSW2,Fishman_SW} %
We have applied a Holstein-Primakoff $1/S$ expansion about the classical limit to the Hamiltonian of Eq.(\ref{Hamiltonian}). %
We express the spins, $\mbox{\boldmath $S$}_i$, on each sublattice using Fourier transformed boson operators. %
By solving the Heisenberg equation of motion for the boson operators, the spin-wave dispersion relations and the INS cross section were calculated. %
To obtain the resolution-convoluted neutron scattering spectra, the experimental resolutions for $E_i=3.6$ and 8.6 meV were taken into account. %
The Hamiltonian parameters were adjusted so that the calculations reproduce the observed data, and finally, the best fit was obtained for the parameters shown in Table \ref{Table_J}.\cite{comment3} %
The calculated spectra for the $(h,h,0)$ and $(-h,1-h,\frac{1}{2})$ lines with $E_i=3.6$ and 8.6 meV are shown in Fig. \ref{3.6meV}(b) and \ref{8.6meV}(g), respectively. %

Comparing the parameters for the 5SL phase with those in the 4SL and FE-ICM phases, we have found that $J_1^{(1)}$ and $J_1^{(2)}$ are nearly common in all the three phases. %
This is consistent with the fact that the spin arrangements and the oxygen displacements associated with $J_1^{(1)}$ and $J_1^{(2)}$ in the 5SL phase are the same as those in the 4SL phase, as shown in Figs. \ref{8.6meV}(h) and \ref{8.6meV}(j). %
As for $J_1^{(3)}$ and $J_1^{(4)}$, their magnitudes are found to be smaller than that of $J_1^{(1)}$.
This is also reasonable because they connect two ferromagnetically coupled spins. %
In addition, the exchange paths of $J_1^{(3)}$ and $J_1^{(4)}$ include the O$^{2-}$ ions surrounded by three up spins, and the Fe-O-Fe bonding angles between the three up spins are expected to remain the same as each other. %
Therefore, the `reductions' in magnitudes of $J_1^{(3)}$ and $J_1^{(4)}$ are expected to be smaller than that of $J_1^{(2)}$. %
We have found that the experimentally determined parameters are in good agreement with this scenario. %
This is the direct evidence for the spin-driven bond order in the 5SL phase. %
Moreover, the present results have also demonstrated that the field-induced phase transition from the FE-ICM phase to the 5SL phase is accompanied by the bond-order transition. %

In contrast to the drastic changes in $J_1$, the spin-lattice coupling effects on the distant interactions ($J_2, J_3, J_z$) are relatively small,\cite{SingleDomainSW2} and these interactions in the 5SL phase are comparable to those in the 4SL and FE-ICM phases. %
This might be because a direct exchange interaction between the NN Fe$^{3+}$ ions, which is assumed to be ferromagnetic,\cite{Mekata_1993} competes with the Fe-O-Fe superexchange interaction, and therefore $J_1$ is highly sensitive to the small lattice distortions as compared to $J_2$, $J_3$ and $J_z$. %

In summary, we have investigated the spin-wave excitations and the spin-lattice coupling in the 5SL phase of CFO, by means of the INS measurements under applied field of 13.4 T. %
Comparing the observed spin-wave spectra with the calculations including the spin-lattice coupling effects for the NN exchange interactions, we have revealed that CFO exhibits the spin-driven bond order in the 5SL phase. %
It should be emphasized that we have constructed the model of the bond order by taking into account the fact that an O$^{2-}$ ion belongs to three Fe-O-Fe bonds in CFO. %
The present results suggest the importance of topology of exchange-interactions paths for understanding the exotic spin-lattice coupling phenomena, specifically spin-driven bond order, in geometrically frustrated magnets. %

The inelastic neutron scattering experiment at ISIS was carried out along the proposal No. BR1220149. %
We are grateful to Dr. T. Guidi for technical support in the experiment. %
This work was supported by Grants-in-Aid for Young Scientist (B) (Grant Nos. 23740277 and 25800203). %
N.T. is supported by the JSPS Postdoctoral Fellowships for Research Abroad. %
The images of the crystal and magnetic structures in this paper were depicted using the software VESTA\cite{VESTA} developed by K. Monma.


\section{Introduction\label{Introduction}}
During the last decade intensive debates have centered around the
question what happens when a system evolves from a band to a Mott
insulator~\cite{Mott1949,Gebhard} in the context of different models
in different
dimensions~\cite{Nakamura2000,Zhang2004,Tam2006,Ejima2007,Fabrizio1999,
Zhang2003,Batista2004,Go2011,Garg2006,Kancharla2007PRL,Paris2007,Craco2008,Chen2010}.
like the extended
 Hubbard model~\cite{Nakamura2000,Zhang2004,Tam2006,Ejima2007}
and the ionic
Hubbard model~\cite{Fabrizio1999,Zhang2003,Batista2004,Go2011} in one
dimension, or
the ionic Hubbard model~\cite{Garg2006,Kancharla2007PRL,Paris2007,Chen2010}
in two dimensions.  A
model that has gained a lot of attention in recent years is the
bilayer Hubbard model on a square lattice.  The discovery of bilayer
band splitting in angle-resolved photoemission spectroscopy
experiments~\cite{Damascelli2003} for YBa$_2$Cu$_3$O$_{6+x}$
(YBCO)~\cite{Fournier2010} suggested the consideration of such a model
as a minimal model for describing double-layered YBCO compounds.

In fact, the bilayer Hubbard model (see Fig.~\ref{Fig1:model}) was investigated by several groups
within the dynamical mean field theory approximation
(DMFT)~\cite{Fuhrmann2006,Hafermann2009} and cellular
DMFT~\cite{Kancharla2007}. Transitions from metal to band insulator at
small $U/t$ and from Mott to band insulator at large $U/t$ were
reported with increasing $t_{\perp}/t$.  However, by definition
spatial fluctuations are completely ignored in DMFT~\cite{Georges1996}
and such features like an intermediate non-Fermi-liquid phase in the
single-layered Hubbard model at half
filling~\cite{Zhang2007,Park2008,Gull2008} are not captured by this
approach.  Moreover, only a small cluster size of $N_c=2\times2$ (two
sites in one layer and two sites in another layer) was used in
previous cellular DMFT calculations for the bilayer Hubbard
model~\cite{Kancharla2007}.  For such sizes, the $C_4$ rotational
symmetry of the square lattice is broken, resulting in an artificially
enhanced local pair within each plane as the interaction $U/t$ is
switched on.  As reported in Ref.~\onlinecite{Lee2010}, such a choice
is unable to describe an intermediate orbital-selective phase in a
two-orbital Hubbard model. On the other hand, the dynamical cluster
approximation (DCA) with clusters sizes of $N_c=2 \times 16$
(Ref.~\onlinecite{Maier2011}), the determinant quantum Monte Carlo
method~\cite{Bouadim2008} and the Gutzwiller
approximation~\cite{Lanata2009} -which do not suffer from the previous
cluster limitations- were recently employed mainly to understand the
nature of the superconducting state in the bilayer Hubbard model away
from half filling.  In view of the above results, and due to the
limitations of the various approaches used, there is still not a full
understanding of the transition from band to Mott insulating behavior
as a function of $t_{\perp}/t$ in the bilayer Hubbard model at half
filling.

\begin{figure}[htbp]
\includegraphics[width=0.4\textwidth]{bilayer_withlabel}
\caption{(Color online) Cartoon of the bilayer Hubbard Hamiltonian. The $N_c=2 \times 4$ region studied here using DCA is shown in darker colors. }
\label{Fig1:model}
\end{figure}

In the present work we concentrate on this question and study the
bilayer Hubbard model at and away from half filling as the band
splitting ratio $t_{\perp}/t$ is increased up to $4$ at a large
interaction strength of $U/t=6.0$ in the framework of the
DCA~\cite{Hettler1998,Maier2005}.  We consider one plaquette in each
layer, i.e., a cluster size of $N_c=2 \times 4$, which keeps the
rotation symmetry of the square lattice and allows for short-range
spatial fluctuations.  We focus on the lower temperature regime, which
couldn't be accessed in previous studies~\cite{Maier2011} due to the
larger cluster sizes. We use an interaction-expansion continuous-time
quantum Monte Carlo algorithm as an impurity
solver~\cite{Rubtsov2005,Assaad2007,Gull2011}. Note that the value of
the critical interaction strength for the metal-insulator transition
is sensitive to the level of approximation
considered~\cite{Fuhrmann2006,Hafermann2009,Kancharla2007,Tocchio2013}.
Within DCA and for the cluster sizes considered in this work, the
value $U/t=6.0$ is deep in the insulating phase.
In order to distinguish
between a Mott and a band insulating regime we analyze the behavior of (i) the density of states
near the Fermi level,
 (ii) the imaginary part of the self energy at the lowest Matsubara
frequency and (iii) the momentum resolved electron density.

At small interlayer hopping $t_{\perp}/t$ we observe Mott insulating
behavior in the DCA momentum sectors ($\pi,0$)/($0,\pi$) and band
insulating behavior in the DCA momentum sectors ($0,0$)/($\pi,\pi$) as
was also obtained in cellular DMFT with $N_c=4$ in the single-band
Hubbard model~\cite{Park2008}.  With increasing $t_{\perp}/t$ the Mott
behavior is rapidly suppressed in the DCA momentum sectors ($\pi,0$)
and ($0,\pi$), and at $t_{\perp}/t\simeq2.5$ it becomes dominant in
the DCA momentum sectors ($\pi,\pi$) of the bonding band and ($0,0$)
of the anti-bonding band. At $t_{\perp}/t\simeq3.0$ a tiny scattering
rate is observed in all momentum sectors at the Fermi level,
indicating a transition from a Mott to a band insulator. Such a
momentum-dependent evolution of the insulating state has not been
reported in previous studies and will be analyzed in detail in the
present work.

\begin{figure}
\includegraphics[width=0.48\textwidth]{bilayer_sectors}
\caption{(Color online) The Fermi surface for (a) $t_{\perp}/t=0.0$
  and (b) 2.0 at half-filling in the weak coupling limit of
  $U/t=0$. The colored areas indicate the patches for momentum
  clusters $\rm{K}=(0,0)$, $(\pi,0)$, $(0,\pi)$, and $(\pi,\pi)$ of
  the dynamical cluster approximation with $N_c = 2 \times
  4$.}\label{Fig1:sectors}
\end{figure}

In particular, we find a monolayer plaquette singlet Mott insulator
(m-PSMI) with strong intralayer plaquette order but weak interlayer
antiferromagnetic (AF) correlations at small $t_{\perp}/t$ and a
bilayer plaquette singlet Mott insulator (b-PSMI) with strong AF
correlations between plaquettes belonging
to different layers  at intermediate values
of $t_{\perp}/t$. At the critical value $t_{\perp}/t\simeq3.0$
we observe
a transition from a Mott to a band insulating state with tiny
intralayer spin-spin correlations. Such a momentum-dependent phase
behavior results from the competition and cooperation of short-range
spatial fluctuations and interlayer coupling $t_{\perp}$ with the help
of the Coulomb interaction $U$.

The paper is organized as follows. In Sec.~\ref{Formalism} we present
the model and dynamical cluster approximation. In Sec.~\ref{Results}
we present the density of states, the self-energy and electron density
at each DCA momentum sector as well as spin-spin correlations and we
discuss the nature of Mott insulator and band insulator at
half-filling as well as non-Fermi liquid behavior  away from half-
filling. Finally, in Sec.~\ref{Conclusions} we summarize our findings.

\section{Model and method}\label{Formalism}

The bilayer Hubbard Hamiltonian can be written as
\begin{eqnarray}
	H&=&-\sum_{\langle ij\rangle m\sigma}
	t_m(c^+_{jm\sigma}c_{im\sigma}+{\rm h.c.})-
        \mu\sum_{im\sigma}n_{im\sigma}\nonumber\\
	&-& t_\perp\sum_{i\sigma}(c^+_{i1\sigma}c_{i2\sigma}+{\rm
	h.c.})+U\sum_{im}n_{im\uparrow}n_{im\downarrow}
	\label{eq:1},
\end{eqnarray}
where $c_{im\sigma}(c^{\dag}_{im\sigma})$ annihilates (creates) an
electron with spin $\sigma$ at site $i$ and layer $m \in (1,2)$, and
$\mu$ is the chemical potential. $t_m$ is the intralayer hopping
matrix element between sites $i$ and $j$ in layer $m$ and $t_{\perp}$
is the interlayer hopping parameter which induces a band splitting
into a bonding and an anti-bonding band. For $t_m = t$ ($m$=1,2) with
$t$ as energy unit throughout this paper, the energy dispersion is
given as $\epsilon^{\text{A,B}} ({\rm k}) = \epsilon ({\rm k}) \pm
t_{\perp}$, where $\epsilon ({\rm k}) = -2t (\cos {\rm k_x} + \cos {\rm
  k_y})$ and 'A' and 'B' indices denote anti-bonding (antisymmetric)
($+$) and bonding (symmetric) ($-$) states,
respectively~\cite{Fuhrmann2006}.

The DCA is the cluster extension of single-site DMFT and the
self-consistent equation can be written in momentum space with the
assumption that the self-energy is constant in the Brillouin zone
sectors that are considered.  The cluster Green's functions are
calculated by integration of each sector:
\begin{eqnarray}\label{DCAE}
\overline{G}_{\sigma}({\bf K},i\omega_n)=\frac{1}{N}\sum_{\tilde{\bf K}}
\frac{1}{i\omega_n+\mu-\epsilon_{{\bf
K+\tilde{K}}}^{{\text{A,B}}}-\Sigma_{\sigma}({\bf K},i\omega_n)},
\end{eqnarray}
where $N$ is the number of ${\tilde{\bf K}}$ points in each Brillouin
zone sector, $\mu$ the chemical potential, ${\bf K}$ is the cluster
momentum, $\epsilon_{{\bf K+\tilde{K}}}^{{\text{A,B}}}$ is the
dispersion relation for antibonding and bonding states, $\omega_n$ are
the Fermionic Matsubara frequencies, and the summation over
$\tilde{\bf K}$ is performed in each Brillouin zone sector.  In our
calculations, we considered a DCA cluster with $N_c = 2\times 4$,
where ${\bf K} = (0,0)$, $(0,\pi)$, $(\pi,0)$ and $(\pi,\pi)$  (see Fig.~\ref{Fig1:sectors}).  The
converged self-energy $\Sigma_{\sigma}({\bf K},i\omega_n)$ is
evaluated by means of Eq.~(\ref{DCAE}) and the Dyson equation and we
employed the interaction expansion continuous-time quantum Monte Carlo
approach as an impurity
solver~\cite{Rubtsov2005,Assaad2007,Gull2011}. All calculations
presented below are for a temperature $T/t=0.1$ and more than $5
\times 10^6$ QMC samplings are employed to measure the impurity
Green's function.

In the non-interacting case ($U/t=0$), the model shows a band
insulating state at $t_{\perp}/t\geq4$ due to the complete separation
of bonding and anti-bonding bands, characterized by a formation of
fully localized dimers between layers at half-filling.

\section{Results}\label{Results}

\subsection{Half-filling}

In Fig.~\ref{Fig1:DOS} we show the density of states $\rho(\omega)$ of
the model at $T/t=0.1$ and $U/t=6.0$ for small, intermediate, and
large band splittings with ratios $t_{\perp} / t=0.5$, $2.0$, and
$4.0$, respectively.  At $t_{\perp} / t = 0.5$ the gap is nonzero and
dominated by the interaction $U$, and the bonding and antibonding
states both contribute to the lower and upper Hubbard bands. On the
other hand, at $t_{\perp}/t=4.0$, the gap, which is also significant,
is dominated by the band splitting $2t_{\perp}$. In this case, only
the bonding state contributes to the band below the Fermi level while
the antibonding one contributes to the band above the Fermi level. At
intermediate $t_{\perp} / t=2.0$ the gap amplitude is reduced to a
smaller value compared to the gap amplitudes at $t_{\perp} / t=0.5$
and $4.0$. Here, both bonding and antibonding states significantly
contribute to the bands above and below the Fermi level. However, the
peak positions remain located at the same frequency as those for
$t_{\perp} / t=0.5$ which indicates that Mott behavior is still
present, i.e.  the peak position is only dependent on $U/t$ and not on
$t_{\perp} / t$.

\begin{figure}[htb]
\includegraphics[width=0.45\textwidth]{halffilling_dos}
\caption{(Color online) The density of states $\rho(\omega)$ for
  chemical potentials $\mu / t=3.0$ with half-filling at $U/t=6.0$ and
  $T/t=0.1$. The band splitting is $t_{\perp}/t=0.5$ (solid line),
  $2.0$ (dashed line), and $4.0$ (dotted line). The Fermi level is at
  $\omega=0$.}\label{Fig1:DOS}
\end{figure}

In order to obtain a deeper insight into the nature of the phases as a
function of the band splitting, we analyze in the following the
cluster self-energy at the various DCA momentum sectors $\bf K$.
Figs.~\ref{Fig2:momentum} (a) and (b) show the imaginary part of the
DCA cluster self-energy ${\rm Im}\, \Sigma({\bf K},\omega_0)$ at the
lowest Matsubara frequency $\omega_0$ and the electron density
$\langle n({\bf K})\rangle$, respectively, as a function of
$t_{\perp}/t$ for $U/t=6.0$ at half filling where $\mu / t=3.0$. Both
quantities are shown for bonding (B) and antibonding (A) bands for the
cluster momenta ${\bf K}=(0,0)$, $ (0,\pi)$ (identical to $(\pi,0)$)
and $(\pi,\pi)$

At small band splitting $t_{\perp} / t < 0.5$,  momentum selective
Mott insulating phases with large scattering rates ${\rm
  Im}\,\Sigma({\bf K},\omega_0)$ are present in both bonding and
antibonding bands at ${\bf K}=(\pi,0)$ and $(0,\pi)$ momentum sectors
where the electron density $\langle n({\bf K})\rangle$ indicates almost
half-filling, while the scattering rates at ${\bf K}=(0,0)$ and
$(\pi,\pi)$ momentum sectors remain small suggesting a band insulating
behavior with empty or fully filled electron density $\langle n({\bf
  K})\rangle$ in these sectors.  Further analysis on spin-spin
correlations (Fig.~\ref{Sup1}) shows strong intralayer but weak
interlayer antiferromagnetic correlations (Fig.~\ref{Sup1} (a) and
(b)).  Moreover, the spin-spin correlation $\langle s_{i,m}^z
s_{i+1,m'}^z \rangle$ for different layers and nearest neighbor sites
displays very weak ferromagnetic order at $t_{\perp} / t = 0.5$ (see
Fig.~\ref{Sup1} (c)).  We denote the state at small $t_{\perp} / t$
m-PSMI (monolayer plaquette singlet Mott insulator).  This m-PSMI
state has been reported to be present in the one-band Hubbard model
investigated by a 4-site cluster-DMFT approach~\cite{Gull2008}.

\begin{figure}[htb]
\includegraphics[width=0.45\textwidth]{sigma_halffilling3}
\caption{(Color online) (a) Imaginary part of the DCA cluster
  self-energy ${\rm Im} (\Sigma({\bf K},\omega_0))$ at the lowest
  Matsubara frequency $\omega_0$ and (b) electron density $\langle
  n({\rm K})\rangle$ as a function of $t_{\perp}/t$ at each DCA
  momentum sector ${\rm K}$.  Results are shown at a temperature
  $T/t=0.1$ and an interaction strength $U/t=6.0$ for chemical
  potential $\mu/t=3.0$ (half-filling $\langle n_i\rangle =
  1.00$). ${\bf K}$  are the DCA cluster momenta and indices
  ${\rm A}$ and ${\rm B}$ stand for antibonding and bonding,
  respectively.}\label{Fig2:momentum}
\end{figure}

As $t_{\perp} / t$ increases, the scattering rates ${\rm
  Im}\,\Sigma({\bf K},\omega_0)$ at the ${\bf K}=(\pi,0)$ and
$(0,\pi)$ momentum sectors rapidly decrease towards zero while the
scattering rate at the $(\pi,\pi)$ momentum sector of the bonding band
(see $\Sigma_B({\bf{K}}=(\pi,\pi))$ in Fig.~\ref{Fig2:momentum} (a))
and that at the $(0,0)$ momentum sector of the antibonding band (see
$\Sigma_A({\bf{K}}=(0,0))$ in Fig.~\ref{Fig2:momentum} (a)) develop
dramatically.  In terms of the electron density $\langle n({\bf
  K})\rangle$ in Fig.~\ref{Fig2:momentum} (b), the $(\pi,\pi)$
momentum sector of the bonding band and the $(0,0)$ momentum sector of
the antibonding band are filled with about one electron at $t_{\perp} /
t=2.2$. The strong scattering is caused by interactions between
electrons in the $(\pi,\pi)$ momentum sector of the bonding band and
the $(0,0)$ momentum sector of the antibonding band at $t_{\perp} /
t=2.2$.  The spin-spin correlations exhibit intermediate intralayer as
well as interlayer antiferromagnetic correlations. Specifically, the
ferromagnetic correlations for different layers and nearest neighbor
sites reach a maximum in Fig.~\ref{Sup1} (c).  This means that the
plaquette singlet orderings in each layer develop AF correlations. We
denote this state bilayer plaquette singlet Mott insulator (b-PSMI).

When $t_{\perp}/t$ is further increased, tiny scattering rates
${\rm Im}\,\Sigma({\bf K},\omega_0)$ are observed in the $(\pi,\pi)$
momentum sector of the bonding band and the $(0,0)$ momentum sector of
the antibonding band between $t_{\perp}/t = 3.0$ and $4.0$, while the
electron densities $\langle n({\bf K})\rangle$ are not fully occupied
(or empty) as shown in Figs.~\ref{Fig2:momentum}~(a) and (b).  Such an
insulator has been denominated a covalent band
insulator~\cite{Sentef2009}.

In the region $t_{\perp} / t > 4.0$, all scattering rates disappear
and the system is in a band insulating state, where the electron
density $\langle n({\bf K})\rangle$ are fully filled (or empty) in all
momentum sectors and all bands. The ratio $t_{\perp} / t $ at which
the Mott to band insulator phase transition happens is smaller than
the the ratio for the metal-insulator transition in the noninteracting
case ($t_{\perp} / t = 4.0$). This is due to the fact that strong
correlation narrows the bandwidth of the bonding and antibonding bands
and consequently a smaller band splitting is required for opening a
band gap. The interlayer spin-spin correlations indicate a strong
dimer state, while the remaining spin-spin correlations are extremely
weak as shown in Fig.~\ref{Sup1}.  The state at $t_{\perp} / t = 4.0$
is a band insulator with isolated dimers between layers.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{spinspincorrelation}
\caption{(Color online) The spin-spin correlations $\langle s_{i,m}^z
  s_{i',m'}^z\rangle$ as a function of $t_{\perp}/t$ (a) between $i$ and
  nearest neighbors site $i'=i+1$ in the same layer $m$, (b) between
  layer $m$ and $m'=m+1$ in the same site $i$, and (c) between $i$ and
  nearest neighbor site $i'=i+1$ as well as layer $m$ and $m'=m+1$ for
  $U/t=6.0$ and $T/t=0.1$.}
\label{Sup1}
\end{figure}

The momentum-dependent evolution of the insulating states can be well
understood from the evolution of the Fermi surface in the weak
coupling limit as a function of $t_{\perp}$. As shown in
Fig.~\ref{Fig1:sectors} (a), at $t_{\perp}/t=0$, all the Fermi
surfaces are located in the momentum sectors $(\pi,0)$ and $(0,\pi)$,
indicating that poles determined by $\omega+\mu-\epsilon_{{\bf
    K+\tilde{K}}}^{{\text{A,B}}}-{\rm Re} \Sigma_{\sigma}({\bf
  K},\omega)|_{\omega=0}=0$ are only present in these two sectors. As
the interaction $U$ becomes larger than the critical value of the Mott
metal-to-insulator transition, the opening of a gap at the Fermi level
indicates ${\rm Im} G({\bf K},\omega=0)\rightarrow0$, which requires
large scattering rates at the positions of the poles.  Therefore, Mott
physics occurs in momentum sectors $(\pi,0)$ and $(0,\pi)$.  On the
other hand, at $t_{\perp}/t=2.0$ (see Fig.~\ref{Fig1:sectors} (b)),
almost all the Fermi surfaces from the bonding band enter the momentum
sector $(\pi,\pi)$ while those from anti-bonding band are mostly in
the momentum sector $(0,0)$.  In order to be a Mott insulator at large
interaction $U$, large scattering rates are again inevitable in the
momentum sectors $(\pi,\pi)$ of the bonding band and $(0,0)$ of the
anti-bonding band since poles are now located in these sectors. Thus,
the momentum-dependent evolution of the insulating states is a
consequence of the evolution of the Fermi surface.

\subsection{Away from half-filling}

\begin{figure}
\includegraphics[width=0.45\textwidth]{selfdos1}
\caption{(Color online) (a), (b) Imaginary part of the DCA cluster
  self-energy ${\rm Im} (\Sigma({\bf K},\omega_0))$ at the lowest
  Matsubara frequency $\omega_0$ as a function of
  $t_{\perp}/t$. Results are shown at a temperature $T/t=0.1$ and an
  interaction strength $U/t=6.0$ for different values of the chemical
  potential: (a) $\mu/t=0.0$ (heavily doped case) and (b) 2.0 (close
  to half-filled case). The indices '${\bf K}$' represent the DCA
  cluster momenta and '${\rm A}$' and '${\rm B}$' indicate antibonding
  and bonding, respectively. (c), (d) Density of states $\rho(\omega)$
  for different chemical potentials $\mu / t$ at $U/t=6.0$ and
  $T/t=0.1$. The band splitting is (c) $t_{\perp}/t=0.5$ and (d)
  $t_{\perp}/t=4.0$. The Fermi level is at
  $\omega=0$.}\label{Fig3:momentum}
\end{figure}

We concentrate in what follows on the origin of non-Fermi liquid or
pseudogap, which has been discussed extensively in the
literature~\cite{Stanescu2006,Sakai2009,Gull2010,Sordi2011,Sakai2012},
based on the bilayer Hubbard model away from half-filling.  In the
heavily doped case ($\mu / t=0.0$), we observe at $t_{\perp}/t = 0.5$
for $T/t=0.1$ a Fermi liquid like metallic behavior with ${\rm
  Im}\,\Sigma({\bf K}, \omega_0)$ approaching small finite values due
to the finite temperature effect (see Fig.~\ref{Fig3:momentum} (a)). A
quasiparticle peak is present at the Fermi level in the DOS (see
Fig.~\ref{Fig3:momentum} (c)).  Close to the half-filled case, such as
$\mu / t=2.0$ at $t_{\perp}/t = 0.5$, large but finite scattering
rates are observed in the antibonding ${\bf K}=(0,\pi)$ and the
bonding ${\bf K}=(0,\pi)$ sectors due to the enhancement of intralayer
short-range AF correlations. As a result, a pseudogap appears in the
DOS at the Fermi level (see Fig.~\ref{Fig3:momentum} (c)), reminiscent
of the non-Fermi liquid behavior observed in the single-band and
multi-band Hubbard
models~\cite{Zhang2007,Park2008,Lee2010,Gull2008,Lee2010,Lee2011}. At
$t_{\perp}/t =4.0$, though the scattering rates in all momentum
sectors and all bands vanish due to the strong interlayer AF
correlations (see Fig.~\ref{Fig3:momentum} (b)), pseudogaps exist in
both heavily doped and nearly half-filled cases, indicating that a
strong scattering rate is not a necessary condition for the appearance
of non-Fermi liquid behavior; rather the short-range AF correlations
alone can be responsible for the non-Fermi liquid behavior.

Furthermore, in order to confirm the transitions from the Fermi liquid
to Mott insulator via non-Fermi liquid as a function of electron
doping (or chemical potential), we plot the ${\rm
  Im}\,\Sigma(i\omega_n)$ at the lower temperature $T/t=0.05$ and
$t_{\perp}/t = 0.5$ for $U/t=6.0$ in Fig.~\ref{Fig6:selfenergy}.  In
the case of $\langle n\rangle=0.60$, $\rm{Im}\,\Sigma (i\omega_0)$
converges to almost zero indicating a Fermi-liquid behavior. As the
electron density $\langle n\rangle$ approaches half-filling between
$\langle n\rangle=0.87$ and $0.96$, $\rm{Im}\,\Sigma (i\omega_0)$
converges to finite values. Such a behavior is typical for non-Fermi
liquid states.  Finally, at half-filling with $\langle n\rangle=1.00$
the system is a Mott insulator with $\rm{Im} \Sigma (i\omega_0)$
exhibiting a diverging behavior.

\begin{figure}[htbp]
\includegraphics[width=0.45\textwidth]{onsiteselfenergy}
\caption{(Color online) The imaginary part of on-site self-energy
  $\rm{Im}\,\Sigma ({i\omega_n})$ as a function of Matsubara frequency
  $\omega_n$ for $T/t=0.05$, $t_{\perp}/t=0.5$, and $U/t=6.0$ in
  different electron densities $\langle n\rangle$.}
\label{Fig6:selfenergy}
\end{figure}

\section{Conclusions}\label{Conclusions}

In conclusion, we investigated a two-dimensional bilayer Hubbard model
on the square lattice as a function of band splitting and doping at a
large interaction value $U/t=6.0$ by means of the dynamical cluster
approximation with a $N_c=2 \times 4$ site cluster with short-range
spatial as well as quantum fluctuations.  The scattering rate, which
indicates the degree of Mott physics, rapidly disappears in the momentum
sectors ($\pi,0$) and ($0, \pi$) with increasing interlayer hopping
$t_{\perp}/t$.  In fact we find
a momentum-selective phase reentrant behavior from band insulating
states
at weak interlayer hopping $t_{\perp}/t\simeq0.5$ to Mott insulating
behavior
at $t_{\perp}/t\simeq2.0$ and then from Mott insulating to band
insulating behavior at strong interlayer hopping $t_{\perp}/t\simeq
3.0$.  These transitions are identified from the scattering rates in
the ($\pi,\pi$) momentum sector of the bonding band and the ($0,0$)
momentum sector of the anti-bonding band at half filling.  Interesting
phases are established with two consecutive phase transitions from a
monolayer plaquette singlet Mott insulator (m-PSMI) to a band
insulator through an intermediate phase, called bilayer plaquette
singlet Mott insulator (b-PSMI), where Mott physics is more present in
the $(\pi,\pi)$ sector of the bonding band and the $(0,0)$ of the
anti-bonding band, rather than in the $(\pi,0)$ and $(0, \pi)$ sectors
as usually observed in an antiferromagnetic Mott insulator.  We
attribute the unusual consecutive phases to competition and
cooperation between short-range spatial correlations with quantum
fluctuations and interlayer hopping $t_{\perp}/t$.  The
transition of Mott to band insulator with the absence of large
scattering rates in all momentum sectors is found at large interlayer
hopping $t_{\perp}/t\simeq3.0$.
Furthermore, since the momentum-dependent
evolution of the insulating behavior with $t_{\perp}/t$  is strongly controlled by the
evolution of the Fermi surface in the weak-coupling limit,
we expect that consideration of larger clusters and different geometries
in DCA won't change this scenario qualitatively.

Finally we also find that away from half
filling, non-Fermi liquid behavior is dominated by antiferromagnetic
correlations rather than the finite scattering rate at the Fermi
level. We suggest that this non-Fermi liquid behavior might be related
to anomalous phenomena like the Fermi arc or hole pocket. We expect
that momentum-selective phenomena may exist in many cases which calls
for further studies in various models and real materials where the
short-range spatial fluctuations are emphasized.

\section{Acknowledgement}\label{Acknowledgement}

We would like to thank J. Kune\v{s}, L. F. Tocchio, F. Becca, I.I. Mazin, M. Capone,
and C.  Gros for useful discussion. We gratefully acknowledge
financial support from the Deutsche Forschungsgemeinschaft through
grants FOR 1346 and SFB/TRR 49. Y.Z is supported by National Natural
Science Foundation of China (No. 11174219), Shanghai Pujiang Program
(No.  11PJ1409900), Research Fund for the Doctoral Program of Higher
Education of China (No.  20110072110044) and the Program for Professor
of Special Appointment (Eastern Scholar) at Shanghai Institutions of
Higher Learning. H.L is supported by ERC/FP7 through the Starting Independent Grant "SUPERBAD", Grant Agreement No. 240524.


\section{Introduction}
A submanifold of a Riemannian manifold is \emph{umbilical} if it is equally curved in all tangent directions. More precisely, an isometric immersion $f\colon\, M^m\to \tilde{M}^n$ between Riemannian manifolds is umbilical if there exists a normal vector field $H$ along $f$ such that its second fundamental form $\alpha_f\in \mbox{Hom}(TM\times TM, N_fM)$
with values in the normal bundle satisfies
$$\alpha_f(X,Y)=\<X,Y\>H\,\,\,\mbox{for all}\,\,\,X,Y\in \mathfrak{X}(M).$$

The classification of umbilical submanifolds of space forms is very well known. For a general symmetric space $N$, it was shown by Nikolayevsky (see Theorem $1$ of \cite{al}) that any umbilical submanifold of
 $N$ is an umbilical submanifold of a product of space forms totally geodesically embedded in $N$. This makes the classification of umbilical submanifolds of a product of space forms an important problem.
For submanifolds of dimension $m\geq 3$ of a product $\Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ of space forms
whose curvatures satisfy $k_1+k_2\neq 0$, the problem was reduced in \cite{mt1} to the classification of $m$-dimensional umbilical submanifolds  with codimension two of $\Sf^n\times \R$ and $\Hy^n\times \R$, where  $\Sf^n$ and $\Hy^n$ stand for the sphere and hyperbolic space, respectively. The case of  $\Sf^n\times \R$ (respectively, $\Hy^n\times \R$) was carried out in \cite{mt2} (respectively, \cite{mt3}), extending previous results in \cite{st} and \cite{vv} (respectively, \cite{ckv}) for  hypersurfaces.

In this paper we extend the results of \cite{mt1} to the surface case.  In this case, the argument in one of the steps of the proof for the higher dimensional case  (see Lemma~$8.2$ of \cite{mt1}) does not apply, and requires more elaborate work. This is carried out in Lemma \ref{le:kerS=0} below, which shows that the difficulty is due to the existence of new interesting families of examples in the surface case. Indeed, our main result (see Theorem \ref{thm:umb} below) states that, in addition to the examples that appear already in higher dimensions, there are precisely two distinct two--parameter families
 of complete embedded flat umbilical surfaces that lie substantially in $\Hy_{k}^{3}\times \R^2$ and $\Hy_{k_1}^{3}\times \Hy_{k_2}^{3}$, respectively. These are discussed in Section~$3$.

\section{Preliminaries}

 Let  $f\colon\,M\to \Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ be  an isometric immersion of a Riemannian manifold.
 We always assume that $M$ is connected.
  Denote by ${\cal R}$  and ${\cal R}^\perp$ the curvature tensors of the tangent and normal bundles
  $TM$ and $N_f M$, respectively,  by $\alpha=\alpha_f\in \Gamma(T^*M\otimes T^*M\otimes N_f M)$ the second fundamental form of $f$ and by $A_\eta=A^f_\eta$ its shape operator  in the normal direction $\eta$, given by  $$\<A_\eta X,Y\>=\<\alpha(X,Y),\eta\>$$ for all $X,Y\in \mathfrak{X}(M)$. Set
 $$L=L^{{f}}:=\pi_2\circ {f}_*\in \Gamma(T^*M\otimes T\Q_{k_2}^{n_2})\,\,\,\mbox{and}\,\,\,K=K^{{f}}:=\pi_2|_{N_fM}\in \Gamma((N_fM)^*\otimes T\Q_{k_2}^{n_2}),$$
 where $\pi_i\colon\, \Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}\to \Q_{k_i}^{n_i}$ denotes the canonical projection, $1\leq i\leq 2$, and
by abuse of notation also  its derivative, which we  regard as a  section of $T^*({\Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}})\otimes T\Q_{k_i}^{n_i}$.

\subsection{The fundamental equations}

The tensors $R\in \Gamma(T^*M\otimes TM)$, $S\in \Gamma(T^*M\otimes N_fM)$ and $T\in \Gamma((N_fM)^*\otimes N_fM)$ given by
\be\label{tensorsa} R=L^tL,\,\,\,\,S=K^tL\,\,\,\,\mbox{and}\,\,\,\,T=K^tK,
\ee
or equivalently, by
$$L=f_*R+S\,\,\,\,\mbox{and}\,\,\,\,K=f_*S^t+T,$$
  were introduced in  \cite{ltv} (see also \cite{mt1}), where
   they were shown to
  satisfy the algebraic relations
\be\label{tensors}
S^tS=R(I-R),\,\,\,\,\,TS=S(I-R)\,\,\,\,
\,\mbox{and}\,\,\,\,\,\,
SS^t=T(I-T),
\ee
as well as the
differential equations
\be\label{eq:pid1}(\n_XR)Y=A_{SY}X+S^t\a(X,Y),\ee
\be\label{eq:pid2}(\n_XS)Y=T\a(X,Y)-\a(X,RY)\ee
and
\be\label{eq:pid4}(\n_X T)\xi=-SA_\xi X-\a(X,S^t\xi)\ee
for all   $X, Y\in \mathfrak{X}(M)$ and all $\xi \in \Gamma(N_fM)$.
In particular, from the first and third equations of (\ref{tensorsa}) and (\ref{tensors}), respectively, it follows that $R$ and $T$ are  nonnegative operators whose eigenvalues lie in $[0,1]$.

 The Gauss, Codazzi and Ricci equations of $f$ are, respectively,
 \be\label{eq:gauss}
 {\cal R}(X,Y)Z=(k_1(X\wedge Y-X\wedge R Y-R X\wedge Y) +\kappa R X\wedge R Y)Z
 +A_{\alpha(Y,Z)}X -A_{\alpha(X,Z)}Y,\ee
 \be\label{eq:codazzi}
 (\nabla^\perp_X \alpha)(Y,Z)-(\nabla^\perp_Y \alpha)(X,Z)=\<k_1 X-\kappa RX,Z\>SY-\<k_1 Y-\kappa RY,Z\>SX\ee
 and
 \be\label{eq:ricci}
 {\cal R}^\perp(X,Y)\eta=  \alpha(X,A_\eta Y) -\alpha(A_\eta X,Y)+\kappa(SX\wedge SY)\eta,
 \ee
 where $\kappa=k_1+k_2$.

 \subsection{The flat underlying space}

 In order to study isometric immersions $f\colon\,M\to \Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$,  it is useful to consider their compositions   $F=h\circ f$ with the
 canonical isometric embedding
\be\label{eq:j}
h\colon\,\Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}\to\R_{\sigma(k_1)}^{N_1}\times \R_{\sigma(k_2)}^{N_2}=\R_{\mu}^{N_1+N_2}.
\ee
 Here, for $k\in \R$ we set  $\sigma(k)=1$ if $k<0$ and  $\sigma(k)=0$ otherwise, and as a subscript of an Euclidean space it means the index of the corresponding flat metric.
 Also, we denote  $\mu=\sigma(k_1)+\sigma(k_2)$, $N_i=n_i+1$ if $k_i\neq 0$ and  $N_i=n_i$   otherwise, in which case $\Q_{k_i}^{n_i}$  stands for $\R^{n_i}$.

Let $\tilde\pi_i\colon\,\R_{\mu}^{N_1+N_2}\to \R_{\sigma(k_i)}^{N_i}$, $1\leq i\leq 2$, denote the canonical projection. Then, the normal space of $h$ at each point $z\in \Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ is spanned by $k_1\tilde\pi_1(h(z))$ and $k_2\tilde\pi_2(h(z))$, and its second fundamental form  is given by
\be\label{eq:sffj} \alpha_h(X,Y)=-k_1\<\pi_1 X, Y\>\tilde\pi_1\circ h-k_2\<\pi_2X, Y\>\tilde\pi_2\circ h.\ee
Therefore,  if $k_i\neq 0$,  $1\leq i\leq 2$,
 then, setting $r_i=|k_i|^{-1/2}$, the unit  vector field  $\nu_i=\nu_i^{F}=\frac{1}{r_i}\tilde\pi_i\circ F$ is  normal  to $F$  and
   we have
$$\tilde{\nabla}_X\nu_1=\frac{1}{r_1}\tilde\pi_1F_*X=\frac{1}{r_1}(F_*X-h_*LX)=\frac{1}{r_1}(F_*(I-R)X-h_*SX)$$
and
$$\tilde{\nabla}_X\nu_2=\frac{1}{r_2}\tilde\pi_2F_*X=\frac{1}{r_2}h_*LX=\frac{1}{r_2}(F_*RX+h_*SX),$$
where $\tilde \n$ stands for the derivative in $\R_{\mu}^{N_1+N_2}$.
 Hence
 \be\label{eq:nu}^F\nabla_X^\perp \nu_1=-\frac{1}{r_1}h_*SX, \,\,\,\,\,\,\,\,\,\,\,\,\,\,A^{F}_{\nu_1} = -\frac{1}{r_1}(I-R),\ee

 \be\label{eq:nu'}^F\nabla_X^\perp \nu_2=\frac{1}{r_2}h_*SX\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,A^{F}_{\nu_2} = -\frac{1}{r_2}R.\ee

\subsection{Reduction of codimension}

 An isometric immersion $f\colon\, M^m\to \Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ is said to \emph{reduce codimension on the left by $\ell$} if there exists a totally geodesic inclusion $j_1\colon\, \Q_{k_1}^{m_1}\to \Q_{k_1}^{n_1}$, with $n_1-m_1=\ell$, and an isometric immersion $\bar f\colon\, M^m\to \Q_{k_1}^{m_1}\times \Q_{k_2}^{n_2}$ such that $f=(j_1\times id)\circ \bar f$. Similarly one defines what it means by $f$ reducing codimension \emph{on the right}.

We will need the  following result  from \cite{mt1} on reduction of codimension. In the statement, $U$ and $V$ stand for $\ker T$ and $\ker (I-T)$, respectively. Notice that the third equation in (\ref{tensors}) implies that   $S(TM)^\perp$ splits orthogonally as $S(TM)^\perp=U\oplus V$, with $U=(I-T)(S(TM)^\perp)$ and $V=T(S(TM)^\perp)$. Also,
given an isometric immersion $f\colon\, M\to \tilde M$ between Riemannian manifolds, its \emph{first normal space} at $x\in M$ is the subspace $N_1(x)$ of $N_fM(x)$ spanned by the image of its second fundamental form at $x$.

\begin{proposition}\po\label{cor:redcod} Let $f\colon\,M^m\to \Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ be an isometric immersion. If $U\cap N_1^\perp$ (respectively, $V\cap N_1^\perp$ ) is a vector subbundle of $N_fM$ with rank $\ell$  satisfying $\nabla^\perp (U\cap N_1^\perp)\subset N_1^\perp$ (respectively, $\nabla^\perp (V\cap N_1^\perp)\subset N_1^\perp$), then $f$ reduces codimension on the left (respectively, on the right) by $\ell$.
\end{proposition}

\subsection{Frenet formulae for space-like curves in $\R_{1}^4$}

We briefly recall the definition of the Frenet curvatures and the Frenet frame of a unit-speed space-like curve $\gamma\colon I\to \R_{1}^4$
in the four dimensional Lorentz space, as well as the coresponding Frenet formulae, which will be needed in the sequel.

Thus, we assume that $t(s)=\gamma'(s)$ satisfies $\<t(s), t(s)\>=1$ for all $s\in I$.
Assume first that $\<\gamma''(s), \gamma''(s)\>\neq 0$ for all $s\in I$. Define
$\hat{k}_1(s)=\|\gamma''(s)\|=|\<\gamma''(s), \gamma''(s)\>|^{1/2}$
and $n_1(s)=\gamma''(s)/\hat{k}_1(s)$ for all $s\in I$. Denote $\epsilon_1=\<n_1, n_1\>$. Suppose that $v(s)=n_1'(s)+\epsilon_1\hat{k}_1(s)t(s)$ satisfies
$\<v(s), v(s)\>\neq 0$ for all $s\in I$. Define $\hat{k}_2(s)=\|v(s)\|$ and $n_2(s)=v(s)/\hat{k}_2(s)$.  Let $n_3(s)$ be chosen so that
$\{t(s), n_1(s), n_2(s),n_3(s)\}$ is a positively-oriented orthonormal basis of $\R_{1}^4$ and set $\epsilon_3=\<n_3,n_3\>$. Then the following Frenet formulae hold, where $\hat{k}_3$ is defined by the third equation:
$$\left\{\begin{array}{l}t'=\hat{k}_1n_1,\\
n_1'=-\epsilon_1\hat{k}_1t+\hat{k}_2n_2\\
n_2'=\epsilon_3\hat{k}_2n_1+\hat{k}_3n_3\\
n_3'=\epsilon_1\hat{k}_3n_2.
\end{array}\right.$$

Lesser known are the formulae in the case in which  $\gamma''(s)$ is a nonzero light-like vector everywhere, i.e., $\tilde{n}_1(s)=\gamma''(s)$ satisfies
$\<\tilde{n}_1(s), \tilde{n}_1(s)\>=0$ for all $s\in I$. We carry them out in more detail below.

First notice that $\<t, \tilde{n}_1\>=0$. Here, and in the next computations, we drop the ``s"
for simplicity of notation and understand that all equalities hold for all $s\in I$. Thus,
$$\<\tilde{n}_1', t\>=-\<t',\tilde{n}_1\>=-\<\tilde{n}_1,\tilde{n}_1\>=0.$$
Moreover, $\<\tilde{n}_1', \tilde{n}_1\>=0$, hence $\tilde{n}_1'$ is space-like. Define $\tilde{k}_1=\|\tilde{n}_1'\|$ and $\tilde{n}_2$ by $\tilde{n}_1'=\tilde{k}_1\tilde{n}_2.$
Now let $\tilde{n}_3\in \{t, \tilde{n}_2\}^\perp$ be the unique vector such that
$$\<\tilde{n}_3, \tilde{n}_3\>=0\,\,\,\,\mbox{and}\,\,\,\,\<\tilde{n}_1, \tilde{n}_3\>=1,$$
that is, $\{\tilde{n}_1, \tilde{n}_3\}$ is a pseudo-othonormal basis of the time-like plane $\{t, \tilde{n}_2\}^\perp$.
Since
$$\<\tilde{n}_2', t\>=-\<\tilde{n}_2, t'\>=-\<\tilde{n}_2, \tilde{n}_1\>=0$$
and
$$\<\tilde{n}_2', \tilde{n}_1\>=-\<\tilde{n}_2, \tilde{n}_1'\>=-\tilde{k}_1,$$
we have
$$\tilde{n}_2'=\<\tilde{n}_2', \tilde{n}_1\>\tilde{n}_3+\<\tilde{n}_2', \tilde{n}_3\>\tilde{n}_1=-\tilde{k}_1\tilde{n}_3-\tilde{k}_2\tilde{n}_1,$$
where
$$\tilde{k}_2=\<\tilde{n}_3', \tilde{n}_2\>.$$
Finally, since
$$0=\<\tilde{n}_3',t\>=\<\tilde{n}_3', \tilde{n}_1\>=\<\tilde{n}_3', \tilde{n}_3\>,$$
we have
$$\tilde{n}_3'=\<\tilde{n}_3', \tilde{n}_2\>\tilde{n}_2=\tilde{k}_2\tilde{n}_2.$$

In summary, for a unit-speed space-like curve $\gamma\colon I\to \R_{1}^4$ with light-like curvature vector $\gamma''$,
one can define two Frenet curvatures $\tilde{k}_1$ and $\tilde{k}_2$ and a pseudo-orthonormal Frenet frame $\{t, \tilde{n}_1, \tilde{n}_2, \tilde{n}_3\}$
with respect to which the Frenet formulae are
$$
\left\{\begin{array}{l} t'=\tilde{n}_1\vspace{1ex}\\
\tilde{n}_1'=\tilde{k}_1\tilde{n}_2\vspace{1ex}\\
\tilde{n}_2'=-\tilde{k}_2\tilde{n}_1-\tilde{k}_1\tilde{n}_3\vspace{1ex}\\
\tilde{n}_3'=\tilde{k}_2\tilde{n}_2.
\end{array}\right.
$$
In both cases, a unit-speed space-like curve $\gamma\colon I\to \R_{1}^4$ is completely determined by its Frenet curvatures, up
to an isometry of $\R_1^4$.

\section{Flat umbilical surfaces in $\Hy_{k}^{3}\times \R^2$ and $\Hy_{k_1}^{3}\times \Hy_{k_2}^{3}$.}

We present below two families  of complete flat properly embedded umbilical surfaces, the first one  in $\Hy_{k}^{3}\times \R^2$ and the second in $\Hy_{k_1}^{3}\times \Hy_{k_2}^{3}$,  each of which depending on two parameters.

\begin{example}\po\label{ex:1} {\em Let $F\colon \R^2\to \R_{1}^6=\R_{1}^4\times \R^2$, where $\R_{1}^4$ has signature $(-, +, +, +)$, be given by
\be\label{eq:F}F(s,t)=\left(a_1\cosh \frac{s}{c}, a_1\sinh \frac{s}{c}, a_2\cos \frac{t}{c}, a_2\sin \frac{t}{c}, b_1\frac{s}{c}, b_2\frac{t}{c}\right),\ee
with
\be\label{eq:cond}a_1^2-a_2^2=r^2\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,a_1^2+b_1^2=c^2=a_2^2+b_2^2.\ee
Then $F(\R^2)\subset \Hy_{k}^3\times \R^2$, where  $k=-{1}/{r^2}$, by the first relation in (\ref{eq:cond}).
If $\{e_1, \ldots, e_6\}$ is the orthonormal basis of $\R_1^6$ with respect to which $F$ is given by (\ref{eq:F}),
then the subspaces $V_1$ and $V_2$ of $\Le^6$ spanned by $\{e_1, e_2, e_5\}$ and $\{e_3, e_4, e_6\}$
 can be identified
with $\R_1^3$ and $\R^3$, respectively, and
$$F=\gamma_1\times \gamma_2\colon \R\times \R=\R^2\to V_1\times V_2=\R_1^3\times \R^3=\R_1^6,$$
where $\gamma_1$ and $\gamma_2$ are the helices in $\R_1^3$ and $\R^3$, respectively, parameterized by
$$\gamma_1(s)=\left(a_1\cosh \frac{s}{c}, a_1\sinh \frac{s}{c},b_1\frac{s}{c}\right)$$
and
$$\gamma_2(t)=\left(a_2\cos \frac{t}{c}, a_2\sin \frac{t}{c},  b_2\frac{t}{c}\right).$$
By the  relations on
the right  in (\ref{eq:cond}), both $\gamma_1$ and $\gamma_2$ are unit-speed curves,
hence $F$ is an isometric immersion. Since $F(\R^2)\subset \Hy_{k}^3\times \R^2$,
there exists an isometric immersion $f\colon \R^2\to \Hy_{k}^3\times \R^2$ such that
 $F=h\circ f$, where  $h\colon \Hy_{k}^3\times \R^2\to \R_1^6$ denotes  the inclusion.
 It is easily checked that the second fundamental form of $f$ satisfies
$$\alpha_f\left(\frac{\d}{\d s},\frac{\d}{\d t}\right)=0$$
and
$$
\alpha_f\left(\frac{\d}{\d s}, \frac{\d}{\d s}\right)=H(s,t)=\alpha_f\left(\frac{\d}{\d t}, \frac{\d}{\d t}\right),
$$
where
$$h_*H(s,t)=\frac{ka_1a_2}{c^2}\left(a_2\cosh \frac{s}{c}, a_2\sinh \frac{s}{c}, a_1\cos \frac{t}{c}, a_1\sin \frac{t}{c}, 0,0\right).$$
Hence $f$ is   umbilical  with mean curvature vector field $H$.

In view of (\ref{eq:cond}), one can write
$$a_1^2=r^2\frac{(1-\lambda_1)}{\lambda_2-\lambda_1},\,\,\,\,a_2^2=r^2\frac{(1-\lambda_2)}{\lambda_2-\lambda_1},\,\,\,\,b_1^2=r^2\frac{\lambda_1}{\lambda_2-\lambda_1},\,\,\,\,b_2^2=r^2\frac{\lambda_2}{\lambda_2-\lambda_1},\,\,\,\,c^2=\frac{r^2}{\lambda_2-\lambda_1},$$
with  $0<\lambda_1< \lambda_2<1$.
Then, one can check that the  curvature vector $\gamma_i''$ of  $\gamma_i$, $1\leq i\leq 2$, satisfies
\be\label{eq:curvvector}\<\gamma_i'', \gamma_i''\>=k(\lambda_j-\lambda_i)(1-\lambda_i), \,\,\,1\leq i\neq j\leq 2,\ee
and that  the second Frenet curvature (torsion) of $\gamma_i$ satisfies
\be\label{torsion}\tau_i^2=-k\lambda_i|\lambda_j-\lambda_i|,\,\,\,\,1\leq i\neq j\leq 2.\ee  }
\end{example}

\begin{example}\po \label{ex:2}{\em Let $\R_2^8=\R_1^4\times \R_1^4$ denote Euclidean space of dimension $8$ endowed with
an inner product of signature $(-, +,+,+,-,+,+, +)$, and let $F\colon \R^2\to \R_2^8$ be given by
\be\label{eq:Fb}F(s,t)=\left(a_1\cosh \frac{s}{c}, a_1\sinh \frac{s}{c}, a_2\cos \frac{t}{c}, a_2\sin \frac{t}{c}, a_3\cosh \frac{t}{d}, a_3\sinh \frac{t}{d}, a_4\cos \frac{s}{d}, a_4\sin \frac{s}{d} \right),\ee
with
\be\label{eq:condb}a_1^2-a_2^2=r_1^2,\,\,\,\,\,\,\,\,a_3^2-a_4^2=r_2^2\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,
\frac{a_1^2}{c^2}+\frac{a_4^2}{d^2}=1=\frac{a_2^2}{c^2}+\frac{a_3^2}{d^2}.\ee
The first pair of relations in (\ref{eq:condb}) implies that $F(\R^2)\subset \Hy_{k_1}^3\times \Hy_{k_2}^3\subset \R_1^4\times \R_1^4$, with $k_i=-1/r_i^2$
 for  $1\leq i\leq 2$. If $\{e_1, \ldots, e_4, f_1, \ldots, f_4\}$ is the orthonormal basis of $\R_2^8$ with respect to which $F$ is given by (\ref{eq:Fb}),
then the subspaces $V_1$ and $V_2$ of $\R_2^8$ spanned by $\{e_1, e_2, f_3, f_4\}$ and $\{f_1, f_2, e_3, e_4\}$
 can also be identified
with $\R_1^4$, and
$$F=\gamma_1\times \gamma_2\colon \R\times \R=\R^2\to V_1\times V_2,$$
where $\gamma_1$ and $\gamma_2$ are the curves  parameterized by
$$\gamma_1(s)= \left(a_1\cosh \frac{s}{c}, a_1\sinh \frac{s}{c},  a_4\cos \frac{s}{d}, a_4\sin \frac{s}{d} \right)$$
and
$$\gamma_2(t)=\left(a_3\cosh \frac{t}{d}, a_3\sinh \frac{t}{d}, a_2\cos \frac{t}{c}, a_2\sin \frac{t}{c}\right).$$
 In view of the second pair of relations in (\ref{eq:condb}), both $\gamma_1$ and $\gamma_2$ are unit-speed curves,
hence $F$ is an isometric immersion. Since $F(\R^2)\subset \Hy_{k_1}^3\times \Hy_{k_2}^3$,
there exists an isometric immersion $f\colon \R^2\to \Hy_{k_1}^3\times \Hy_{k_2}^3$ such that
 $F=h\circ f$, where  $h\colon \Hy_{k_1}^3\times \Hy_{k_2}^3\to \R_2^8$ denotes  the inclusion.
 One can  easily check that the second fundamental form of $f$ satisfies
 $$\alpha_f\left(\frac{\d}{\d s},\frac{\d}{\d t}\right)=0$$
and
$$
\alpha_f\left(\frac{\d}{\d s}, \frac{\d}{\d s}\right)= H(s,t)
=\alpha_f\left(\frac{\d}{\d t}, \frac{\d}{\d t}\right)
$$
where
$$\begin{array}{l} h_*H(s,t)=\frac{k_1a_1a_2}{c^2}\left(a_2\cosh \frac{s}{c}, a_2\sinh \frac{s}{c}, a_1\cos \frac{t}{c}, a_1\sin \frac{t}{c},0,0,0,0\right)+ \vspace{1ex}\\
\frac{k_2a_3a_4}{d^2}\left(0,0,0,0,a_4\cosh \frac{t}{d}, a_4\sinh \frac{t}{d}, a_3\cos \frac{s}{d}, a_3\sin \frac{s}{d}\right).
\end{array}$$
It follows that $f$ is  umbilical with mean curvature vector field $H$.

By the conditions in (\ref{eq:condb}), one can write
$$a_1^2=r_1^2\frac{(1-\lambda_1)}{\lambda_2-\lambda_1},\,\,\,\,a_2^2=r_1^2\frac{(1-\lambda_2)}{\lambda_2-\lambda_1},\,\,\,\,\,
a_3^2=r_2^2\frac{\lambda_2}{\lambda_2-\lambda_1},\,\,\,\,a_4^2=r_2^2\frac{\lambda_1}{\lambda_2-\lambda_1},$$
$$c^2=\frac{r_1^2}{\lambda_2-\lambda_1}\,\,\,\mbox{and}\,\,\,\,\,d^2=\frac{r_2^2}{\lambda_2-\lambda_1},$$
with  $0<\lambda_1< \lambda_2<1$.  Then, the  curvature vector $\gamma_i''$ of the curve  $\gamma_i$, $1\leq i\leq 2$, satisfies
\be\label{eq:curvvectorb}\<\gamma_i'', \gamma_i''\>=(\lambda_i-\lambda_j)(\kappa\lambda_i-k_1), \,\,\,1\leq i\neq j\leq 2,\,\,\,\kappa=k_1+k_2.\ee
If $\kappa\lambda_i-k_1\neq 0$, one can check that $\gamma_i$, $1\leq i\leq  2$,  has constant Frenet curvatures
$\hat{k}^i_\ell$, $1\leq \ell\leq 3$, given by
\be\label{eq:hatk1}(\hat{k}^i_1)^2=|(\lambda_i-\lambda_j)(\kappa\lambda_i-k_1)|,\ee
\be\label{eq:hatk2}(\hat{k}^i_2)^2=\frac{\kappa^2|\lambda_i-\lambda_j|\lambda_i(1-\lambda_i)}{|\kappa\lambda_i-k_1|}\ee
and
\be\label{eq:hatk3}(\hat{k}^i_3)^2=\frac{k_1k_2|\lambda_i-\lambda_j|}{|\kappa\lambda_i-k_1|},\,\,\,1\leq j\neq i\leq 2.\ee

If $\kappa\lambda_i-k_1= 0$, that is, the curvature vector of $\gamma_i$ is light-like, then one can check that $\gamma_i$ has constant Frenet curvatures $\tilde{k}^i_1$ and $\tilde{k}^i_2$, $1\leq i\leq 2$ (see Subsection $2.4$), given by
\be\label{eq:tildek1}(\tilde{k}^i_1)^2=\frac{k_1k_2(\kappa\lambda_j-k_1)^2}{\kappa^2},\,\,\,1\leq j\neq i\leq 2,\ee
and
\be\label{eq:tildek2}(\tilde{k}^i_2)^2=\frac{(k_1-k_2)^2}{4k_1k_2},\,\,\,1\leq i\leq 2.\ee}
\end{example}

It is also easily checked that the isometric immersions in both of the preceding examples have the frame of
coordinate vector fields $\{\frac{\d}{\d s},\frac{\d}{\d t}\}$ as a frame of principal directions for the associated
tensor $R$, with corresponding eigenvalues $\lambda_1$ and $\lambda_2$, respectively.  Moreover, they are clearly injective
and proper, hence embeddings. Therefore, all surfaces in both families are properly embedded and isometric to the plane.

\section{The main step}

 Umbilical submanifolds of $\Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ were studied in \cite{mt1} according to the possible structures of the tensor $S$. When $\ker S=\{0\}$, it
was shown that $R$ must be a constant multiple of the identity tensor whenever the dimension of the submanifold is at least three (see \cite{mt1}, Lemma $8.2$), which corresponds to case $(i)$ in the statement of Lemma \ref{le:kerS=0}  below. We  now show that in the surface case
the only exceptions are the surfaces of the two families in the preceding section.

\begin{lemma}\po\label{le:kerS=0} Let $f\colon\,M^2\to \Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$, $k_1+k_2\neq 0$, be an umbilical isometric immersion. Assume that   $\ker S=\{0\}$ at some point $x\in M^2$. Then  one of the following  holds:
\begin{itemize}
\item[(i)] there exist umbilical isometric immersions $f_i\colon\, M^2\to \Q_{\tilde k_i}^{n_i}$, $1\leq i\leq 2$, with $\tilde{k}_1=k_1\cos^2\theta$ and $\tilde k_2=k_2\sin^2\theta$ for some $\theta\in (0, \pi/2)$,  such that $f=(\cos \theta f_1,\sin\theta f_2)$.
\item[(ii)] after interchanging the factors, if necessary, we have $k_2=0$, $n_1\geq 3$, $n_2\geq 2$ and $f=j\circ \tilde f$, where $j\colon \Q_{k_1}^{3}\times \R^2\to \Q_{k_1}^{n_1}\times \R^{n_2}$ and $\tilde f\colon M^2\to \Q_{k_1}^{3}\times \R^2$ are isometric immersions such that $j$ is totally geodesic and $\tilde f(M^2)$ is an open subset of a surface as in Example \ref{ex:1}.
\item[(iii)]  $k_i<0$ and  $n_i\geq 3$, $1\leq i\leq 2$,  and $f=j\circ \tilde f$, where
$j\colon \Q_{k_1}^{3}\times \Q_{k_2}^{3}\to \Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ and  $\tilde f\colon M^2\to \Q_{k_1}^{3}\times \Q_{k_2}^{3}$ are isometric immersions such that $j$ is totally geodesic and $\tilde f(M^2)$ is an open subset of a surface as in Example \ref{ex:2}.
\end{itemize}
\end{lemma}

\proof Let ${\cal U}\subset M$ be the maximal  connected open  neighborhood of  $x$ where   $\ker S=\{ 0\}$.
Let $\lambda_1$ and $\lambda_2$  be the eigenvalues of $R$ on ${\cal U}$. Fix an orthonormal frame $\{X_1,X_2\}$ of eigenvectors of $R$, with $X_i$ associated to $\lambda_i$,  and define $\xi_i:=SX_i$ for $i=1$, $2$. Thus, from (\ref{tensors}) we have
\begin{equation}\label{nnf}
\langle\xi_i,\xi_j\rangle = \langle S^tSX_i,X_j\rangle = \delta_{ij}\lambda_i(1-\lambda_i)
\end{equation}
and
\begin{equation}\label{evT}
T\xi_i=TSX_i=(1-\lambda_i)\xi_i
\end{equation}
for $i$, $j=1,2$. We can write equations (\ref{eq:gauss})--(\ref{eq:ricci})  in the frames $\{X_1, X_2\}$ and $\{\xi_1, \xi_2\}$, in terms of the Gaussian curvature $K$ of $M^2$ and the mean curvature vector $H$ of $f$, as
\be\label{eq:gauss3}
K=k_1(1-\lambda_1)(1-\lambda_2)+k_2\lambda_1\lambda_2+|H|^2,
\ee
\be\label{eq:codazzi3}
\nabla_{X_i}^{\perp}H=(\kappa \lambda_j-k_1)\xi_i, \,\,\,1\leq i\neq j\leq 2,
\end{equation}
and
\be\label{eq:ricci3}
\mathcal{R}^{\perp}(X_1,X_2)=\kappa (\xi_1\wedge\xi_2),
\ee
whereas equations  (\ref{eq:pid1})--(\ref{eq:pid4}) become
\begin{equation}
\label{cdt1b}
\left(\nabla_{X_i} R\right)X_j=\langle \xi_j,H\rangle X_i+\delta_{ij} S^tH,
\ee
\be
\label{cdt2b}
\left(\nabla_{X_i} S\right)X_j=\delta_{ij}(T-\lambda_jI)H
\ee
and
\be\label{cdt3b}
\left(\nabla_{X_i} T\right)\xi=-\langle \xi,H\rangle \xi_i-\langle \xi,\xi_i\rangle H
\end{equation}
for $i$, $j=1,2$ and all $\xi\in \Gamma(N_fM^2)$.
Define the Christoffel symbols $\Gamma_{11}^2$ and $\Gamma_{22}^1$ by
\begin{equation}\label{lcc}
\nabla_{X_1}X_1=\Gamma_{11}^2X_2\,\,\,\,\,\mbox{and}\,\,\,\,\,\,
\nabla_{X_2}X_2=\Gamma_{22}^1X_1.
\end{equation}
Substituting
\begin{align*}
\left(\nabla_{X_i}R\right)X_j &= \nabla_{X_i}RX_j-R\nabla_{X_i}X_j\\
                              &= X_i(\lambda_j)X_j + (\lambda_jI-R)\nabla_{X_i}X_j
\end{align*}
into  (\ref{cdt1b}) yields
\be \label{cdlamda}X_i(\lambda_j)=\delta_{ij}2\langle\xi_i,H\rangle\ee
and
\begin{equation}\label{relation}
\langle\xi_i,H\rangle=(\lambda_j-\lambda_i)\Gamma_{jj}^i,\,\,\,\,1\leq i\neq j\leq 2.
\end{equation}
On the other hand, from    (\ref{cdt2b})  we get
\begin{equation}\label{pnc-f}
\nabla^{\perp}_{X_i}\xi_j=-\Gamma_{ii}^j\xi_i, \,\,\,1\leq i\neq j\leq 2.
\end{equation}
Using  (\ref{eq:codazzi3}),    (\ref{cdlamda}) and (\ref{pnc-f}) we obtain
\begin{align*}
\mathcal{R}^{\perp}(X_1,X_2)H= &\nabla^{\perp}_{X_1}\nabla^{\perp}_{X_2}H-\nabla^{\perp}_{X_2}\nabla^{\perp}_{X_1}H-\nabla^{\perp}_{[X_1,X_2]}H\vspace{1.5ex}\\
                             =& \nabla^{\perp}_{X_1}(\kappa \lambda_1-k_1)\xi_2-\nabla^{\perp}_{X_2}(\kappa \lambda_2-k_1)\xi_1
                             +(\kappa \lambda_2-k_1)\Gamma_{11}^2\xi_1\vspace{1.5ex}\\&-(\kappa \lambda_1-k_1)\Gamma_{22}^1\xi_2\vspace{1.5ex}\\
                             =& \kappa  X_1(\lambda_1)\xi_2+(\kappa \lambda_1-k_1)\nabla^{\perp}_{X_1}\xi_2
                             -\kappa  X_2(\lambda_2)\xi_1-(\kappa \lambda_2-k_1)\nabla^{\perp}_{X_2}\xi_1\vspace{1.5ex}\\
                             &+(\kappa \lambda_2-k_1)\Gamma_{11}^2\xi_1-(\kappa \lambda_1-k_1)\Gamma_{22}^1\xi_2\vspace{1.5ex}\\
                             =& 2\kappa \langle\xi_1,H\rangle\xi_2-(\kappa \lambda_1-k_1)\Gamma_{11}^2\xi_1
                             -2\kappa \langle\xi_2,H\rangle\xi_1+(\kappa \lambda_2-k_1)\Gamma_{22}^1\xi_2\vspace{1.5ex}\\
                             &+(\kappa \lambda_2-k_1)\Gamma_{11}^2\xi_1-(\kappa \lambda_1-k_1)\Gamma_{22}^1\xi_2\vspace{1.5ex}\\
                             =&-\kappa (2\langle\xi_2,H\rangle+(\lambda_1-\lambda_2)\Gamma_{11}^2)\xi_1
                             +\kappa (2\langle\xi_1,H\rangle+(\lambda_2-\lambda_1)\Gamma_{22}^1)\xi_2.
\end{align*}
In view of (\ref{relation}), the above equation becomes
\begin{equation}\label{ricci-H-def}
\mathcal{R}^{\perp}(X_1,X_2)H=-3\kappa (\langle\xi_2,H\rangle\xi_1-\langle\xi_1,H\rangle\xi_2).
\end{equation}
Comparing the preceding equation with
\begin{equation}\label{ricci-H}
\mathcal{R}^{\perp}(X_1,X_2)H=\kappa (\langle\xi_2,H\rangle\xi_1-\langle\xi_1,H\rangle\xi_2),
\end{equation}
which follows from  (\ref{eq:ricci3}), and using that $\kappa\neq 0$, we get $\langle\xi_1,H\rangle=0=\langle\xi_2,H\rangle$,
i.e.,
\begin{equation}\label{eq:hperp} H\in \Gamma(S(TM)^{\perp}).
\end{equation}
 In particular, we obtain from   (\ref{cdlamda}) that $\lambda_1$ and $\lambda_2$ assume constant values in $(0,1)$ everywhere on ${\cal U}$.
If ${\cal U}$ were a proper subset of $M^2$, then $\lambda_1$ and $\lambda_2$ would assume  the same values on the boundary of ${\cal U}$, hence $\lambda_i(1-\lambda_i)\neq 0$ on an open connected neighborhood
of $\bar{{\cal U}}$, $1\leq i\leq 2$, contradicting the maximality of ${\cal U}$ as an open  connected neighborhood of $x$ where   $\ker S=\{ 0\}$. It follows that   ${\cal U}=M^2$.
Moreover, if $\lambda_1=\lambda_2$ then $f$ is as in $(i)$ by Proposition $5.2$ of \cite{mt1}. From now on we assume that $\lambda_1\neq \lambda_2$.

Under this assumption,  we obtain from (\ref{relation}) and (\ref{eq:hperp}) that  $\Gamma_{11}^2=0=\Gamma_{22}^1$. In particular, we have $K=0$ everywhere,
and then  (\ref{eq:gauss3}) gives
\begin{equation}\label{eq:H}
|H|^2=-k_1(1-\lambda_1)(1-\lambda_2)-k_2\lambda_1\lambda_2.
\end{equation}

Set $\xi=H$ in (\ref{cdt3b}). By using (\ref{eq:codazzi3}) and (\ref{eq:H}),  we obtain
\begin{equation}\label{ndc-TH}
\nabla^{\perp}_{X_i}TH=k_2\lambda_j\xi_i, \,\,\,\, 1\leq i\neq j\leq 2,
\end{equation}
and similarly
\begin{equation}\label{ndc-(I-T)H}
\nabla^{\perp}_{X_i}(I-T)H=-(1-\lambda_j)k_1\xi_i, \,\,\, 1\leq i\neq j\leq 2.\\
\end{equation}
In particular, bearing in mind (\ref{eq:hperp}) and the fact that $T$ leaves $S(TM)$ invariant, as follows from
the second equation in (\ref{tensors}), we obtain that  both $TH$ and $(I-T)H$ have constant length on $M^2$, hence either $TH=0$, $TH=H$ or both $TH$ and $(I-T)H$ are nonzero  everywhere.
Therefore $L_1=U\cap \{H\}^{\perp}=U\cap N_1^\perp$ and $L_2=V\cap \{H\}^{\perp}=V\cap N_1^\perp$ have constant dimensions on $M^2$, which are, accordingly,
 $(\rank\, U -1, \rank\, V)$, $(\rank\, U, \rank\, V\,-1)$ or $(\rank\, U-1, \rank\, V-1)$.
Moreover,  equations (\ref{eq:codazzi3}) and (\ref{eq:hperp})
imply that $\nabla^{\perp}_{TM}L_{i}\subset \{H\}^{\perp}$ for $i=1,2$. Hence, the assumptions of Proposition \ref{cor:redcod} are satisfied, and we conclude that
there are three corresponding possibilities for the pairs  $(n_1,n_2)$ of  \emph{substantial} values of $n_1$ and $n_2$: $(3, 2)$, $(2, 3)$ and $(3,3)$.

We first consider the case  $(n_1,n_2)=(3,2)$. This is the case in which $TH=0$, and hence $k_2=0$ by (\ref{ndc-TH}). Thus we have $k_1<0$ from (\ref{eq:H}),
and we may assume that $f$ takes values in $\Hy_{k}^3\times \R^2$, with $k=k_1<0$.

 Set $F=h\circ f$, where $h\colon \Hy_{k}^3\times \R^2\to \R_1^6$ denotes the inclusion.
The second fundamental form of $F$ is given by
$$\alpha_F(X, Y)=\<X, Y\>h_*H+\frac{1}{r}\<(I-R)X, Y\>\nu,$$
where $r=(-k)^{-1/2}$ and $\nu=\frac{1}{r}\tilde \pi_1\circ F$. Therefore
\be\label{eq:zi}\alpha_F(X_i, X_j)=\delta_{ij}(h_*H+\frac{1}{r}(1-\lambda_i)\nu):=\delta_{ij}Z_i=\tilde \nabla_{X_j}F_*X_i,\,\,\,1\leq i, j\leq 2.\ee
Notice that
$$\<Z_1, Z_2\>=|H|^2+k(1-\lambda_1)(1-\lambda_2)=0$$
 by (\ref{eq:H}),
 and that
\be\label{eq:zinorm}\<Z_i, Z_i\>=k(\lambda_j-\lambda_i)(1-\lambda_i),\,\,\,\,\,\,1\leq i\neq j\leq 2.\ee
Moreover, since
$$\tilde{\pi}_2(h_*H)=h_*\pi_2 H=h_*(f_*S^tH+TH)=0,$$
it follows that
$$\tilde{\pi}_2Z_i=0, \,\,\,1\leq i\leq 2.$$
Using (\ref{eq:codazzi3}), we have
\begin{eqnarray*} \tilde \nabla_{X_i}h_*H
&=& h_*\hat \nabla_{X_i}H+\alpha_h(f_*X_i, H)
\vspace{1ex}\\
&=&-F_*A^f_HX_i+h_*\nabla^\perp_{X_i}H+\frac{1}{r}\<\pi_1f_*X_i, H\>\nu\vspace{1ex}\\
&=& -|H|^2F_*X_i-k(1-\lambda_j)h_*\xi_i+\frac{1}{r}\<f_*(I-R)X_i-SX_i,H\>\nu\vspace{1ex}\\
&=&k(1-\lambda_j)\left((1-\lambda_i)F_*X_i-h_*\xi_i\right),\,\,\,\,\,\,1\leq i\neq j\leq 2.
\end{eqnarray*}
On the other hand,
$$ \tilde \nabla_{X_i}\nu
= \frac{1}{r}(F_*(I-R)X_i-h_*SX_i)
= \frac{1}{r}((1-\lambda_i)F_*X_i-h_*\xi_i).
$$
Therefore
\be \label{eq:xizj}\tilde\nabla_{X_i}Z_j=0,\,\,\,\mbox{if}\,\,i\neq j,\ee
and
\be\label{eq:h1}\tilde\nabla_{X_i}Z_i=k(\lambda_i-\lambda_j)((1-\lambda_i)F_*X_i-h_*\xi_i),\,\,\,1\leq i\neq j\leq 2.\ee
Also, using that
$$\nabla_{X_i}^\perp \xi_i=-\frac{1}{|H|^2}\<\nabla_{X_i}H, \xi_i\>H=-\lambda_iH,$$
as follows from (\ref{nnf}), (\ref{eq:codazzi3}) and (\ref{eq:H}), we obtain that
\begin{eqnarray} \label{eq:h2}\tilde \nabla_{X_i}h_*\xi_j&=&h_*\hat \nabla_{X_i}\xi_j+\alpha_h(f_*X_i, \xi_j)\nonumber\vspace{1ex}\\
&=&-F_*A^f_{\xi_j}X_i+h_*\nabla^\perp_{X_i}\xi_j+\frac{1}{r}\<\pi_1f_*X_i, \xi_j\>\nu\nonumber\vspace{1ex}\\
&=&-\delta_{ij}\lambda_iZ_i, \,\,\,\,1\leq i, j\leq 2.
\end{eqnarray}
It follows from (\ref{eq:zi}), (\ref{eq:xizj}), (\ref{eq:h1}) and (\ref{eq:h2}) that the subspaces $V_i=\spa \{F_*X_i, Z_i, h_*\xi_i\}$, $1\leq i\leq 2$,  are constant. Moreover, they are orthogonal to each other,
hence $\R_1^6$ splits orthogonally as $\R_1^6=V_1\oplus V_2$.

Since $\Gamma_{11}^2=\Gamma_{22}^1=0$, for each $x\in M^2$ there exists an isometry  $\psi\colon\, W=I_1\times I_2\to U_x$
of a product of open intervals $I_j\subset \R$, $1\leq j\leq 2$, onto an open neighborhood of $x$,  such that $\psi_*\frac{\d}{\d s}=X_1$
and $\psi_*\frac{\d}{\d t}=X_2$, where $s$ and $t$ are the standard coordinates on $I_1$ and $I_2$, respectively.  Write $g=F\circ \psi$. In terms of the coordinates $(s,t)$, the fact that $\alpha_F(X_1, X_2)=0$
translates into
$$\frac{\d^2 g}{\d s\d t}=0,$$
which implies that there exist smooth curves $\gamma_1\colon I_1\to V_1$ and $\gamma_2\colon I_2\to V_2$ such that $g=\gamma_1\times \gamma_2$.
By (\ref{eq:zi}),  (\ref{eq:h1}) and (\ref{eq:h2}), each $\gamma_i$ is a helix in $V_i$  with curvature vector $\gamma_i''=Z_i$ and binormal vector $h_*(\xi_i/|\xi_i|)$, $1\leq i\leq 2$.  It follows from (\ref{eq:zinorm}) that  (\ref{eq:curvvector}) holds for $\gamma_i$, whereas (\ref{eq:h2}) implies that the second Frenet curvature of $\gamma_i$ satisfies
$$\tau_i^2=\frac{\lambda_i^2|\<Z_i, Z_i\>|}{\<\xi_i, \xi_i\>}=-k\lambda_i|\lambda_j-\lambda_i|,\,\,\,\,1\leq i\neq j\leq 2.$$
Therefore, the helices $\gamma_1$ and $\gamma_2$ are precisely, up to congruence, those given in Example~\ref{ex:1}. Moreover, since the curvature vector $Z_i$ along $\gamma_i$  spans a two-dimensional subspace of $V_i$ orthogonal to the axis of $\gamma_i$  and $\tilde \pi_2 Z_i=Z_i$, $1\leq i\leq 2$, it follows that the subspace $\R^2$ in the orthogonal
decomposition $\R_1^6=\R_1^4\oplus \R^2$ adapted to $\Hy_k^3\times \R^2$ is spanned by the axes of $\gamma_1$ and $\gamma_2$. We conclude that  $g$ is (the restriction to $W$ of ) an isometric immersion as in Example \ref{ex:1}.

We have shown that, for each $x\in M^2$, there exists an open neighborhood $U_x$ of $x$ such that $f(U_x)$ is contained in a
 surface as in Example~\ref{ex:1} in a totally geodesic $\Q_{k_1}^{3}\times \R^2\subset \Q_{k_1}^{n_1}\times \R^{n_2}$. A standard
 connectedness argument now shows that $f$ is as in the statement.

The case $(n_1, n_2)=(2,3)$  is entirely similar and leads to the same conclusion with the factors interchanged. Let us consider the case $(n_1, n_2)=(3,3)$,
so we may now assume that $f$ takes values in $\mathbb{Q}^3_{k_1}\times\mathbb{Q}^3_{k_2}$. Here both $TH$ and $(I-T)H$ are nonzero everywhere,
and  we can choose unit vector fields $\xi_3\in\ker T$,   $\xi_4\in\ker(I-T)$ and write $H=\rho_3\xi_3+\rho_4\xi_4$,
where $\rho_k=\langle\xi_k,H\rangle\not=0$ for $k=3$, $4$. Applying (\ref{cdt3b}) to $\xi=\xi_k$, with $k=3$, $4$,   we get
 \begin{equation*}
T\nabla^{\perp}_{X_i}\xi_3=\rho_3\xi_i\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,
(I-T)\nabla^{\perp}_{X_i}\xi_4=-\rho_4\xi_i
\end{equation*}
for $i=1$, $2$. Therefore, for $i=1$, $2$ we obtain
\begin{equation}\label{ncd-imS-perp}
\nabla^{\perp}_{X_i}\xi_3=\frac{\rho_3}{1-\lambda_i}\xi_i\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,
\nabla^{\perp}_{X_i}\xi_4=-\frac{\rho_4}{\lambda_i}\xi_i.
\end{equation}
Using (\ref{nnf}), the preceding equations yield
\be\label{nxixii}
\nabla^{\perp}_{X_i}\xi_i=-\lambda_i\rho_3\xi_3+(1-\lambda_i)\rho_4\xi_4,\,\,\,\,1\leq i\leq 2.
\ee
On the other hand, we have
\begin{equation}\label{TH-(I-T)H}
(I-T)H=\rho_3\xi_3\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,
TH=\rho_4\xi_4.
\end{equation}
Thus, combining (\ref{ndc-TH}), (\ref{ndc-(I-T)H}) and (\ref{TH-(I-T)H}) we get
\begin{equation}\label{rho}
\rho_3^2=-k_1(1-\lambda_1)(1-\lambda_2)\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,
\rho_4^2=-k_2\lambda_1\lambda_2.
\end{equation}
In particular, we must have $k_1, k_2<0$, so  $f$ takes values in $\Hy_{k_1}^3\times \Hy_{k_2}^3$.

Set $F=h\circ f$, where $h\colon \Hy_{k_1}^3\times \Hy_{k_2}^3\to \R_1^4\times \R_1^4=\R_2^8$ denotes the inclusion.
The second fundamental form of $F$ is given by
$$\alpha_F(X, Y)=\<X, Y\>h_*H+\frac{1}{r_1}\<(I-R)X, Y\>\nu_1+\frac{1}{r_2}\<RX, Y\>\nu_2,$$
where $r_i=(-k_i)^{-1/2}$ and $\nu_i=\frac{1}{r_i}\tilde \pi_i\circ F$,  $1\leq i\leq 2$. Therefore
\be\label{eq:zib}
\alpha_F(X_i, X_j)=\delta_{ij}(h_*H+\frac{1}{r_1}(1-\lambda_i)\nu_1+\frac{1}{r_2}\lambda_i\nu_2):=\delta_{ij}Z_i=\tilde{\nabla}_{X_j}F_*X_i,\,\,\,1\leq i\leq 2.\ee
Notice that
$$\<Z_i, Z_j\>=|H|^2+k_1(1-\lambda_i)(1-\lambda_j)+k_2\lambda_i\lambda_j,\,\,\,\,\,\,1\leq i,  j\leq 2.$$
It follows from (\ref{eq:H}) that
$$\<Z_1, Z_2\>=0$$
and
\be\label{eq:zinormb}
\<Z_i, Z_i\>=(\lambda_i-\lambda_j)(\kappa\lambda_i-k_1),\,\,\,\,\,\,1\leq i\neq j\leq 2.
\ee
Using (\ref{eq:codazzi3}), we obtain
\begin{eqnarray*} \tilde \nabla_{X_i}h_*H
&=& h_*\hat \nabla_{X_i}H+\alpha_h(f_*X_i, H)
\vspace{1ex}\\
&=&-F_*A^f_HX_i+h_*\nabla^\perp_{X_i}H\vspace{1ex}\\
&=& -|H|^2F_*X_i+(\kappa \lambda_j-k_1)h_*\xi_i.
\end{eqnarray*}
On the other hand,
$$
 \tilde \nabla_{X_i}\nu_1
= \frac{1}{r_1}(F_*(I-R)X_i-h_*SX_i)= \frac{1}{r_1}((1-\lambda_i)F_*X_i-h_*\xi_i)
$$
and
$$
 \tilde \nabla_{X_i}\nu_2
= \frac{1}{r_2}(F_*RX_i+h_*SX_i)
= \frac{1}{r_2}(\lambda_iF_*X_i+h_*\xi_i).
$$
Using (\ref{eq:H}), it follows that
$$\tilde\nabla_{X_i}Z_j=0,\,\,\,\mbox{if}\,\,i\neq j,$$
whereas
\be\label{eq:h1b}\tilde\nabla_{X_i}Z_i=-\<Z_i,Z_i\>F_*X_i+\kappa (\lambda_j-\lambda_i)h_*\xi_i,\,\,\,1\leq i\neq j\leq 2.\ee
Also,
\begin{eqnarray} \label{eq:h2b}\tilde\nabla_{X_i}h_*\xi_j&=&h_*\hat \nabla_{X_i}\xi_j+\alpha_h(f_*X_i, \xi_j)\nonumber\vspace{1ex}\\
&=&-F_*A^f_{\xi_j}X_i+h_*\nabla^\perp_{X_i}\xi_j+\frac{1}{r_1}\<\pi_1f_*X_i, \xi_j\>\nu_1+\frac{1}{r_2}\<\pi_2f_*X_i, \xi_j\>\nu_2
\nonumber\vspace{1ex}\\
&=&\delta_{ij}(-\lambda_i Z_i+\rho_4 h_*\xi_4+\frac{\lambda_i}{r_2}\nu_2),
\end{eqnarray}
where we have used (\ref{nxixii}).

If $\kappa\lambda_i-k_1\neq 0$, that is,  $\<Z_i, Z_i\>\neq 0$,  define
$$
W_i = \tilde\nabla_{X_i}h_*\xi_i-\frac{\<\tilde\nabla_{X_i}h_*\xi_i,Z_i\>}{\<Z_i, Z_i\>}Z_i=\frac{-k_2\lambda_i}{\kappa \lambda_i-k_1}Z_i+\rho_4h_*\xi_4+\frac{\lambda_i}{r_2}\nu_2, \,\,\,1\leq i\leq 2.
$$
Then the vectors $F_*X_i, Z_i, h_*\xi_i$ and $W_i$ are pairwise orthogonal and the subspaces $V_i=\spa \{F_*X_i, Z_i, h_*\xi_i, W_i\}$, $1\leq i\leq 2$,  are  orthogonal to each other.
Using the second equations in  (\ref{rho}) and (\ref{ncd-imS-perp}), we obtain that
$\tilde \nabla_{X_i}W_j=0$
and
\be\label{eq:nablawi}\tilde \nabla_{X_i}W_i=\frac{k_1k_2(\lambda_i-\lambda_j)}{\kappa\lambda_i-k_1}h_*\xi_i, \,\,\,\,1\leq i\neq j\leq 2.\ee
It follows that the subspaces $V_1$ and $V_2$   are constant,
and that $\R_2^8$ also splits orthogonally as $\R_2^8=V_1\oplus V_2$.

If $\kappa\lambda_i-k_1=0$, define
$$\zeta_i=\frac{\kappa}{2k_1k_2(\kappa\lambda_j-k_1)}(-2\kappa\tilde\nabla_{X_i}h_*\xi_i +(k_2-k_1)Z_i), \,\,\,1\leq i\neq j\leq 2.$$
Then $\<\zeta_i, \zeta_i\>=0$, $\<\zeta_i, Z_i\>=1$ and $\zeta_i\in \spa\{F_*X_i, h_*\xi_i\}^\perp$. Moreover, the subspaces $V_i=\spa \{F_*X_i, Z_i, h_*\xi_i, \zeta_i\}$, $1\leq i\leq 2$,  are  orthogonal to each other. Furthermore, since
\be\label{eq:nablazetai}
\tilde \nabla_{X_i}\zeta_i=\frac{k_1^2-k_2^2}{2k_1k_2}h_*\xi_i,
\ee
it follows that $V_1$ and $V_2$   are constant
and that $\R_2^8$ also splits orthogonally as $\R_2^8=V_1\oplus V_2$.

Since $\Gamma_{11}^2=\Gamma_{22}^1=0$, for each $x\in M^2$ there exists an isometry  $\psi\colon\, W=I_1\times I_2\to U_x$
of a product of open intervals $I_j\subset \R$, $1\leq j\leq 2$, onto a neighborhood of $x$,  such that $\psi_*\frac{\d}{\d s}=X_1$
and $\psi_*\frac{\d}{\d t}=X_2$, where $s$ and $t$ are the standard coordinates on $I_1$ and $I_2$, respectively.  Write $g=F\circ \psi$. In terms of the coordinates $(s,t)$, the fact that $\alpha_F(X_1, X_2)=0$
translates into
$$\frac{\d^2 g}{\d s\d t}=0,$$
which implies that there exist smooth curves $\gamma_1\colon I_1\to V_1$ and $\gamma_2\colon I_2\to V_2$ such that $g=\gamma_1\times \gamma_2$.

If $\kappa\lambda_i-k_1\neq 0$, it follows from  (\ref{eq:zib}),  (\ref{eq:h1b}), (\ref{eq:h2b}) and (\ref{eq:nablawi}) that  $\gamma_i$ is a unit-speed space like curve  in $V_i$  with constant Frenet curvatures $\hat{k}^i_\ell$, $1\leq \ell\leq 3$, and Frenet frame $\{F_*X_i, \hat{Z}_i, h_*\hat{\xi}_i, \hat{W}_i\}$, where $\hat{Z}_i$, $\hat{\xi}_i$ and $\hat{W}_i$ denote the unit vectors in the direction of $Z_i$, $\xi_i$ and $W_i$, respectively.  Moreover, by (\ref{eq:zib}) and (\ref{eq:zinormb}) we have
$$(\hat{k}^i_1)^2=|\<Z_i,Z_i\>|=|(\lambda_i-\lambda_j)(\kappa\lambda_i-k_1)|,$$
whereas from (\ref{eq:h1}) and (\ref{eq:nablawi}) we obtain, respectively, that
$$(\hat{k}^i_2)^2=\frac{\kappa^2(\lambda_j-\lambda_i)^2\<\xi_i,\xi_i\>}{|\<Z_i, Z_i\>|}=\frac{\kappa^2|\lambda_j-\lambda_i|\lambda_i(1-\lambda_i)}{|\kappa\lambda_i-k_1|}$$
and
$$(\hat{k}^i_3)^2=\frac{k^2_1k^2_2(\lambda_i-\lambda_j)^2\<\xi_i, \xi_i\>}{(\kappa\lambda_i-k_1)^2|\<W_i, W_i\>|}=\frac{k_1k_2|\lambda_i-\lambda_j|}{|\kappa\lambda_i-k_1|},\,\,\,1\leq j\neq i\leq 2.$$

If $\kappa\lambda_i-k_1= 0$, it follows from  (\ref{eq:zib}),  (\ref{eq:h1b}), (\ref{eq:h2b}) and (\ref{eq:nablazetai}) that  $\gamma_i$ is a unit-speed space like curve  in $V_i$  with light-like curvature vector, constant Frenet curvatures $\tilde{k}^i_\ell$, $1\leq \ell\leq 2$, and Frenet frame $\{F_*X_i, {Z}_i, h_*\hat{\xi}_i, \zeta_i\}$, where  $\hat{\xi}_i$ is the unit vector in the direction of  $\xi_i$.  Moreover, from (\ref{eq:h1b}) we obtain that
$$(\tilde{k}^i_1)^2=\frac{k_1k_2(\kappa\lambda_j-k_1)^2}{\kappa^2},$$
whereas from  (\ref{eq:nablazetai}) it follows that
$$(\tilde{k}^i_2)^2=\<\xi_i, \xi_i\>\frac{(k_1^2-k_2^2)^2}{4k^2_1k^2_2}=\frac{(k_1-k_2)^2}{4k_1k_2}.
$$
Comparying with (\ref{eq:hatk1}), (\ref{eq:hatk2}) and (\ref{eq:hatk3}) in the first case, and with (\ref{eq:tildek1}) and (\ref{eq:tildek2}) in the second, we see that $\gamma_1$ and $\gamma_2$ are precisely, up to congruence, the curves given in Example~\ref{ex:2}.

Now observe that
$$\tilde{\pi}_2F_*\xi_i=h_*\pi_2 f_*\xi_i=h_*(f_*\xi_i+SX_i)=\lambda_iF_*X_i+h_*\xi_i,$$
whereas
$$\tilde{\pi}_2h_*\xi_i=h_*\pi_2\xi_i=h_*(f_*S^t\xi_i+T\xi_i)=(1-\lambda_i)(\lambda_iF_*X_i+h_*\xi_i),$$
where we have used that
$$S^t\xi_i=S^tSX_i=R(I-R)X_i=\lambda_i(1-\lambda_i)X_i$$
and
$$T\xi_i=TSX_i=S(I-R)X_i=(1-\lambda_i)SX_i=(1-\lambda_i)\xi_i.$$
On the other hand,
$$\tilde{\pi}_2 Z_i=h_*\pi_2H+\frac{\lambda_i}{r_2}\pi_2\nu_2=\rho_4h_*\xi_4+\frac{\lambda_i}{r_2}\nu_2.$$
Since
$$\tilde{\pi}_2 h_*\xi_4=h_*\pi_2\xi_4=h_*(f_*S^t\xi_4+T\xi_4)=h_*\xi_4,$$
we obtain that
$$\tilde{\pi}_2(\rho_4h_*\xi_4+\frac{\lambda_i}{r_2}\nu_2)=\rho_4h_*\xi_4+\frac{\lambda_i}{r_2}\nu_2.$$
If $\<Z_i, Z_i\>\neq 0$, it follows that  $\tilde{\pi}_2 W_i$ and $\tilde{\pi}_2 Z_i$ are colinear. Similarly,
$\tilde{\pi}_2 \zeta_i$ and $\tilde{\pi}_2 Z_i$ are colinear if $\<Z_i, Z_i\>=0$. It follows that $\tilde {\pi}_2(V_i)$ is spanned by
$$\lambda_iF_*X_i+h_*\xi_i\,\,\,\,\,\mbox{and}\,\,\,\,\,\rho_4h_*\xi_4+\frac{\lambda_i}{r_2}\nu_2.$$
Therefore, the subspaces  $\tilde {\pi}_2(V_1)$ and $\tilde {\pi}_2(V_2)$ (and hence also  $\tilde {\pi}_1(V_1)$ and $\tilde {\pi}_1(V_2)$)
are mutually orthogonal, thus the first (respectively, second) factor $\R_1^4$ in the decomposition $\R_1^4\times \R_1^4$ adapted to the
product $\Hy_{k_1}^{3}\times \Hy_{k_2}^{3}$ splits orthogonally as $\R_1^4=\tilde{\pi}_1(V_1) \oplus \tilde{\pi}_1(V_2)$  (respectively,
$\R_1^4=\tilde{\pi}_2(V_1) \oplus \tilde{\pi}_2(V_2)$). We conclude that  $g$ is (the restriction to $W$ of ) an isometric immersion as in Example \ref{ex:2}, and the conclusion follows as in the preceding case.
\vspace{2ex}\qed

\section{The main result}

We are now in a position to state and prove our main result.

\begin{theorem}\po\label{thm:umb} Let $f\colon\,M^2\to \Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$, $k_1+k_2\neq 0$,  be an umbilical non totally geodesic isometric immersion.
Then one of the following possibilities holds:
\begin{itemize}
\item[$(i)$]  $f$ is an umbilical isometric immersion into a slice of $\Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$;
\item[$(ii)$] there exist umbilical isometric immersions $f_i\colon\, M^2\to \Q_{\tilde k_i}^{n_i}$, $1\leq i\leq 2$, with $\tilde{k}_1=k_1\cos^2\theta$ and $\tilde k_2=k_2\sin^2\theta$ for some $\theta\in (0, \pi/2)$,  such that $f=(\cos \theta f_1,\sin\theta f_2)$;
\item[(iii)] after interchanging the factors, if necessary, we have $k_2=0$, $n_1\geq 3$, $n_2\geq 2$ and $f=j\circ \tilde f$, where $j\colon \Q_{k_1}^{3}\times \R^2\to \Q_{k_1}^{n_1}\times \R^{n_2}$ and $\tilde f\colon M^2\to \Q_{k_1}^{3}\times \R^2$ are isometric immersions such that $j$ is totally geodesic and $\tilde f(M^2)$ is an open subset of a surface as in Example \ref{ex:1};
\item[(iv)]  $k_i<0$ and  $n_i\geq 3$, $1\leq i\leq 2$,  and $f=j\circ \tilde f$, where
$j\colon \Q_{k_1}^{3}\times \Q_{k_2}^{3}\to \Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ and  $\tilde f\colon M^2\to \Q_{k_1}^{3}\times \Q_{k_2}^{3}$ are isometric immersions such that $j$ is totally geodesic and $\tilde f(M^2)$ is an open subset of a surface as in Example \ref{ex:2};
\item[$(v)$] after possibly reordering the factors, we have $k_1>0$ (respectively, $k_1\leq 0$) and $f\circ \tilde\Pi=j\circ  \Pi\circ \tilde f$ (respectively, $f=j\circ  \Pi\circ \tilde f$), where $\tilde \Pi\colon\, \tilde M^2\to M^2$ is the universal covering of $M^2$, $\tilde f\colon\,\tilde M^2\to \R\times \Q_{k_2}^{2+\delta}$ (respectively, $\tilde f\colon\, M^2\to \R\times \Q_{k_2}^{2+\delta}$)  is  an umbilical isometric immersion with  $\delta\in \{0,1\}$, $j\colon\, \Q_{k_1}^{1}\times \Q_{k_2}^{2+\delta}\to \Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ is  totally geodesic  and $ \Pi\colon\,\R\times \Q^{2+\delta}_{k_2}\to  \Q_{k_1}^1\times \Q^{2+\delta}_{k_2}$ is  a locally isometric covering map (respectively, isometry).
\end{itemize}
\end{theorem}
\proof
If $S$ vanishes everywhere on  $M^2$, then $f$ is as in  $(i)$  by Lemma $8.1$ in \cite{mt1}. If $\ker S=\{0\}$ at some point $x\in M^2$, then $f$ is as in $(ii)$, $(iii)$ or $(iv)$ by Lemma \ref{le:kerS=0}. Then, we are left with the case in which there is  an open subset ${\cal U} \subset M^2$ where $\ker S$ has rank one.  In this case, the argument in the proof of Theorem $1.4$ of \cite{mt1} applies and shows that $f$ is as in $(v)$. \qed