# Always-Real-Eigenvalued Non-Hermitian Topological Systems

Yang Long  Haoran Xue  Baile Zhang  Corresponding author<EMAIL_ADDRESS>Division of Physics and Applied Physics, School of Physical and Mathematical
Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371,
Singapore

###### Abstract

The effect of non-Hermiticity in band topology has sparked many discussions on
non-Hermitian topological physics. It has long been known that non-Hermitian
Hamiltonians can exhibit real energy spectra under the condition of parity-
time ($PT$) symmetry—commonly implemented with balanced loss and gain—but only
when non-Hermiticity is relatively weak. Sufficiently strong non-Hermiticity,
on the other hand, will destroy the reality of energy spectra, a situation
known as spontaneous $PT$-symmetry breaking. Here, based on non-reciprocal
coupling, we show a systematic strategy to construct non-Hermitian topological
systems exhibiting bulk and boundary energy spectra that are always real,
regardless of weak or strong non-Hermiticity. Such nonreciprocal-coupling-
based non-Hermiticity can directly drive a topological phase transition and
determine the band topology, as demonstrated in a few non-Hermitian systems
from 1D to 2D. Our work develops so far the only theory that can guarantee the
reality of energy spectra for non-Hermitian Hamiltonians, and offers a new
avenue to explore non-Hermitian topological physics.

††preprint: AIP/123-QED

Conservation of energy in quantum physics demands real eigenenergies for a
closed system that is described by a Hermitian Hamiltonian. In presence of
energy exchange with the surrounding environment, the energy conservation is
broken. In such a situation, a non-Hermitian Hamiltonian will arise, leading
to complex eigenvalues. However, the non-Hermiticity is not a sufficient
condition for the existence of complex spectra, which means that it is
possible to find a non-Hermitian system with real eigenvalues.

Parity-time($PT$) symmetry, one of the major discoveries in non-Hermitian
quantum physics, claims that a class of non-Hermitian Hamiltonians with
$PT$-symmetry can still have real spectra [1, 2]. This counterintuitive
discovery fundamentally overturned the past perception that only Hermitian
operators can have real eigenvalues. However, the $PT$ symmetry approach has a
long-lasting limitation: it takes effect only when the non-Hermiticity is
relatively weak [3, 4, 2, 5, 6]; in other words, sufficiently strong non-
Hermiticity will destroy the reality of energy spectra even when $PT$ symmetry
is still respected — a situation known as spontaneous $PT$-symmetry breaking
[7, 8, 2, 9, 10, 11]. Therefore, the $PT$ symmetry cannot guarantee the
reality of energy spectra. A $PT$-symmetric Hamiltonian belongs to a more
general class of non-Hermitian Hamiltonians known as pseudo-Hermitian
Hamiltonians [12, 13], but pseudo-Hermiticity does not guarantee the reality
of energy spectra either. For example, let us consider $H=i\sigma_{z}$, which
is pseudo-Hermitian since $H^{\dagger}=\eta H\eta^{-1}$, where
$\eta=\sigma_{x}$, but $H$ has complex eigenvalues of $\pm i$. To our
knowledge, there is no theory so far that can guarantee the reality of energy
spectra in non-Hermitian systems.

Recently, there have been a lot of efforts in constructing topological states
in non-Hermitian systems [14, 15, 16, 17]. This combination of topological
physics and non-Hermitian physics challenges the conventional understanding of
topological phases [18, 19, 20, 21, 22, 19, 23, 24, 25, 26, 27, 28, 29, 30],
as they were previously defined and classified based on their Hermitian
Hamiltonians [31, 32]. As a result, there have been many developments for non-
Hermitian topological invariants to characterize non-Hermitian topology [21,
22, 3, 33, 23, 34]. Another challenge that is still under active exploration
is whether the non-Hermitian topologial systems can exhibit real energy
spectra. Most studies along this line have adopted the $PT$ symmetry in order
to maintain the real spectra as much as possible [35, 36, 37, 38]. However,
all these systems will necessarily exhibit complex spectra in presence of
sufficiently strong non-Hermiticity. Moreover, the nontrivial topology in non-
Hermitian topological systems is not necessarily determined by Hermitian
parameters [21, 39, 40, 41, 22], but can also be induced directly by non-
Hermiticity. For example, recent studies have shown that by introducing
deliberately designed loss and gain, a topological phase transition can be
induced to generate topological states in non-Hermitian systems [11, 4, 42,
43, 44]. Non-reciprocal coupling is another form of non-Hermiticity, but
discussions on its directly induced topological phase transition [e.g., to
induce a one-dimensional (1D) nontrivial Zak phase] have been relatively few.

In this Letter, we develop an approach to construct non-Hermitian topological
systems whose energy spectra are always real, regardless of weak or strong
non-Hermiticity. Due to the always real energy spectra, the constructed non-
Hermitian systems will not exhibit winding topology in the complex energy
plane [14, 20], as in the recently discovered skin effect [18, 21, 19], but
will have nontrivial topology defined in momentum space. In particular, the
nontrivial topology can be either inherited from the Hermitian counterpart, or
induced by non-Hermiticity directly. The approach is developed by the non-
commutative matrix production between a non-uniform diagonal matrix and a
Hermitian Hamiltonian matrix. The resultant non-Hermitian systems can be
implemented with non-reciprocal coupling. We demonstrate the effectiveness of
this approach in several concrete examples. In a 1D non-Hermitian system, we
show that the band topology characterized by Zak phase can be inherited from
the Hermitian Su-Schrieffer-Heeger (SSH) model, or induced by the non-
Hermiticity. Starting with a two-dimensional (2D) $C_{3}$-symmetric Hermitian
system that is topologically trivial, we show that the non-zero topological
polarization and fractional charges can emerge after introducing non-Hermitian
terms, leading to topological corner states. All these examples exhibit real
enerey spectra for both bulk and boundary states, even in presence of strong
non-Hermiticity.

Let us consider the following equation, which is commonly studied in the
regular Sturm–Liouville theory [45]:

$H_{0}\Psi_{n}=E_{n}M\Psi_{n},$ (1)

where $H_{0}$ is a Hermitian matrix, $M$ is a real diagonal matrix with
diagonal elements $M_{ii}>0$ and $\Psi_{n}$ is the column vector. One can
obtain the eigenvalues of Eq. 1 by solving $|H_{0}-E_{n}M|=0$ and $E_{n}$ will
be real ($E_{n}\in\mathbb{R}$) according to the regular Sturm–Liouville theory
[45]. Here, we consider $M={\rm
diag}[\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}...\varepsilon_{N}]$ with
$\varepsilon_{i}>0$, and its determinant is nonzero (i.e, $|M|\neq 0$) so that
$M^{-1}$ exists. Based on these facts, we can construct a system with the
Hamiltonian $H$ as:

$H=M^{-1}H_{0}.$ (2)

Clearly, this new system can have real eigenvalues $E_{n}$. However, the
Hamiltonian $H$ in Eq. 2 is generally non-Hermitian (i.e, $H\neq
H^{\dagger}$). To understand this non-Hermiticity, after performing the
Hermitian operation on $H$, one can obtain that
$H^{\dagger}=(M^{-1}H_{0})^{\dagger}=H_{0}M^{-1}$ and usually
$H^{\dagger}=H_{0}M^{-1}\neq M^{-1}H_{0}=H$, as a result of the non-
commutativity for the matrix production, except for some special cases such as
when $H_{0}$ is diagonal or the components in $M$ are uniformly identical,
namely $M=c\mathbb{I}$, where $\mathbb{I}$ is the identity matrix and $c$ is
the arbitrary constant.

Although $H$ in Eq. 2 is non-Hermitian, it can have real eigenvalues, which
means that there is no point gap in the bandstructure but only one special
line gap that is perpendicular to the real axis in complex energy plane. Based
on this special line gap, it is possible for the non-Hermitian Hamiltonian of
Eq. 2 to possess nontrivial topology [14, 20]. Interestingly, the topology can
be either inherited from a topologically non-trivial $H_{0}$, or induced by
the $M$ matrix with a trivial $H_{0}$.

Figure 1: 1D non-Hermitian topological system with real spectra. (a) System
settings. $t_{1}=\kappa_{0}\tau$ and $t_{2}=\kappa_{0}(1-\tau)$,
$\kappa_{0}=1$, $\tau\in(0,1)$, $\varepsilon\in\mathbb{R}$. (b,c)
Bandstructures when $\tau=0.6$ and $\tau=0.4$. Here, $\varepsilon=0.9$. The
red number on each band denotes the Zak phase. (d) The topological edge states
will localize toward one end more than the other when $\varepsilon\neq 1$. The
system has 80 sites ($N=80$). The $\psi_{n}$ denotes the field on the $n$-th
site. The blue line with circle markers represents the field distribution. The
red and green dotted lines denote the field amplitudes at the left and right
ends, respectively.

1D non-Hermitian system from the SSH model. As the first example, we discuss a
1D non-Hermitian system with coupling settings shown in Fig. 1(a), similar to
the conventional SSH model [46]. The corresponding Hamiltonian is:

$\displaystyle H$
$\displaystyle=\sum_{n}t_{1}a^{\dagger}_{n}b_{n}+\frac{t_{1}}{\varepsilon}b^{\dagger}_{n}a_{n}+t_{2}a^{\dagger}_{n+1}b_{n}+\frac{t_{2}}{\varepsilon}b^{\dagger}_{n}a_{n-1},$
(3)

where $t_{1}=\kappa_{0}\tau$, $t_{2}=\kappa_{0}(1-\tau)$, $\tau\in(0,1)$, and
$a^{\dagger}$($a$) and $b^{\dagger}$($b$) are the creation (annihilation)
operators. It can be regarded as the production of the inverse matrix of
$M={\rm diag}[1,\varepsilon,1,\varepsilon,...]$ and the Hamiltonian of the
conventional SSH model
$H_{0}=\sum_{n}\left(t_{1}b^{\dagger}_{n}a_{n}+t_{2}b^{\dagger}_{n}a_{n-1}+C.C\right)$,
namely, $H=M^{-1}H_{0}$. The Hamiltonian of Eq. 3 in the momentum space can be
expressed as:

$\displaystyle H_{k}$
$\displaystyle=(t_{1}+t_{2}\cos(ka))\left(\frac{1+\varepsilon}{2\varepsilon}\sigma_{x}+i\frac{\varepsilon-1}{2\varepsilon}\sigma_{y}\right)$
(4)
$\displaystyle+t_{2}\sin(ka)\left(\frac{1+\varepsilon}{2\varepsilon}\sigma_{y}-i\frac{\varepsilon-1}{2\varepsilon}\sigma_{x}\right),$

where $k$ is the Bloch wave vector, $a$ is the lattice constant and
$\sigma_{i}$ are the Pauli matrices. This Hamiltonian satisfies the chiral
symmetry: $H_{k}=-\sigma_{z}H_{k}\sigma_{z}$. We can easily obtain its energy
spectra:
$E_{k}=\pm\frac{1}{\sqrt{\varepsilon}}\sqrt{t_{1}^{2}+t_{2}^{2}+2t_{1}t_{2}\cos(k)}$,
as plotted in Fig. 1(b,c). The spectra are always real even for strong non-
Hermiticity (i.e, $\varepsilon\gg 1$ or $\varepsilon\ll 1$). Note that some
other non-Hermitian systems based on non-reciprocal couplings, e.g., the
Hatano-Nelson model [47], can exhibit real energy spectra under the open
boundary condition, but these real spectra are a result of boundary effect,
being extremely sensitive to boundary conditions [48, 49, 50, 51, 52, 53]. In
contrast, the real spectra of our systems are insensitive to boundary
conditions (see detailed discussions in Supplementary Material [54]).

Figure 2: Non-Hermiticity-induced 1D topological phase transition. (a) The
unit cell when $\varepsilon<1$. The direction and thickness of the arrow
represent the coupling direction and strength, respectively. (b) The
bandstructure of non-Hermitian system ($\varepsilon=0.5$) with trivial band
topology. The red number on each band labels the Zak phase. The blue solid and
gray dotted lines denote the bandstructures with $\varepsilon=0.5$ and
$\varepsilon=1$, respectively. (c) The unit cell when $\varepsilon>1$. (d) The
bandstructure of non-Hermitian system ($\varepsilon=1.5$) with non-trivial
band topology. (e) The topological interface states exist in the bulk gap of
the junction system composed of two topologically distinct lattices
($\varepsilon=0.5$ and $\varepsilon=1.5$). Each lattice has 120 sites. The
interface and bulk states are represented by the red and gray points,
respectively. (f) The field distribution of the topological interface state at
$E=1$. The $\psi_{n}$ denotes the field on the $n$-th site. Here, $t_{0}=1$.

The topology of our 1D non-Hermitian system can be characterized by the Zak
phase: $c_{n}=-i\int_{-\pi/a}^{\pi/a}\langle w_{n}|\partial_{k}|u_{n}\rangle
dk$, where $H_{k}|u_{k,n}\rangle=E_{k,n}|u_{k,n}\rangle$ and
$H^{\dagger}_{k}|w_{k,n}\rangle=E_{k,n}|w_{k,n}\rangle$. The Zak phases
$(c_{1},c_{2})$ for the two bands are: $(c_{1},c_{2})=(0,0)$ for
$t_{1}>t_{2}$, and $(c_{1},c_{2})=(-\pi,\pi)$ for $t_{1}<t_{2}$ [55, 54]. The
non-zero Zak phase for $t_{1}<t_{2}$ indicate that there are two degenerate
topological edge states localized at the ends of a finite system, as a
manifestation of fractional charges at the ends [56, 57, 58, 59, 60]. From Eq.
3, we can see that the mirror symmetry is broken when $\varepsilon\neq 1$. The
topological states can be affected by this non-Hermitian parameter. For a
finite system with $N$ sites, the ratio between the amplitudes of the left and
right edge states is $\varepsilon$ [54]:
$\frac{|\psi_{1}|^{2}}{|\psi_{N}|^{2}}=\varepsilon$, where $\psi_{1/N}$ denote
the field on the left/right end site. As shown in Fig. 1(d), when
$\varepsilon\neq 1$, the topological edge states can localize toward one end
more than the other [54].

Non-Hermiticity-induced topological phase transition in 1D system. After
demonstrating the topology inherited from the Hermitian topological system
$H_{0}$ in Eq. 3, we now show that the topology can also be induced by the
non-Hermitian parameter $\varepsilon$ that is applied to an originally trivial
1D $H_{0}$. As shown in Fig. 2(a,c), the Hamiltonian is:

$\displaystyle H=t_{0}\sum_{n}$
$\displaystyle\frac{1}{\varepsilon}b_{n}^{\dagger}a_{n}+\varepsilon
a_{n}^{\dagger}b_{n}+\frac{1}{\varepsilon}b_{n}^{\dagger}c_{n}+\varepsilon
c_{n}^{\dagger}b_{n}$ (5)
$\displaystyle+\varepsilon(a_{n+1}^{\dagger}c_{n}+c_{n}^{\dagger}a_{n+1}),$

which is constructed by a trivial Hermitian system
$H_{0}=t_{0}\sum_{n}b_{n}^{\dagger}a_{n}+c_{n}^{\dagger}b_{n}+a_{n+1}^{\dagger}c_{n}+C.C.$
and $M={\rm
diag}[\varepsilon^{-1},\varepsilon,\varepsilon^{-1},\varepsilon^{-1},\varepsilon,\varepsilon^{-1}...]$,
namely, $H=M^{-1}H_{0}$. The Hamiltonian of Eq. 5 in the momentum space can be
written as:

$H_{k}=t_{0}\begin{pmatrix}0&\varepsilon&\varepsilon e^{-ika}\\\
\frac{1}{\varepsilon}&0&\frac{1}{\varepsilon}\\\ \varepsilon
e^{ika}&\varepsilon&0\end{pmatrix},$ (6)

where $k$ is the wave vector, $a$ is the lattice constant. Obviously, when
$\varepsilon=1$, the $H_{k}$ represents a gapless Hermitian system. However,
after setting $\varepsilon\neq 1$, the spectra of $H_{k}$ of Eq. 6 can open a
gap as presented in Fig. 2(b,d).

Importantly, when $\varepsilon<1$ ($\varepsilon>1$), the system becomes
trivial (topological). Compared with the system with $\varepsilon<1$ in Fig.
2(b), the system with $\varepsilon>1$, as shown in Fig. 2(d), can lead to the
non-zero Zak phases for the first and third bands. After combining the trivial
and topological systems into a chain, topological interface states emerge in
the bulk gaps, as shown in Fig. 2(e,f). This shows that the non-Hermitian
parameter $\varepsilon$ not only makes the $H$ of Eq. 2 become non-Hermitian,
but is also able to drive a topological phase transition.

Figure 3: Non-Hermiticity-induced higher-order topological states in 2D
$C_{3}$-symmetric lattice. (a) The unit cell when $\varepsilon<1$, which
contains 9 sites. (b) The bandstructure of the non-Hermitian system with
$\varepsilon=1/3$. The blue solid and gray dotted lines denote the
bandstructures with $\varepsilon=1/3$ and $\varepsilon=1$, respectively. (c)
The unit cell when $\varepsilon>1$. (d) The bandstructure of the non-Hermitian
system with $\varepsilon=3$. (e) The eigenvalues for a finite triangular
system in (f) with $\varepsilon=3$. The red, blue and gray points denote the
corner, edge and bulk states, respectively. (f) The field distribution of
topological corner states at $E=-2.58$. The size of red points indicates the
field amplitude on the sites. Here, $t_{0}=1$.

Non-Hermiticity-induced higher-order topological states. We then proceed to
demonstrate the non-Hermiticity-induced higher-order topological states in a
2D lattice with $C_{3}$ symmetry. The unit cell of the system is shown in Fig.
3(a,c). When $\varepsilon=1$, the system corresponds to a trivial
$C_{3}$-symmetric Hermitian system. The condition of $\varepsilon\neq 1$ can
not only make the system non-Hermitian, but also open a gap at the valley
point $K$, as illustrated in Fig. 3(b,d) [54]. Notably, the systems with
$\varepsilon<1$ and $\varepsilon>1$ correspond to trivial and topological
ones, respectively.

The higher-order topological properties of the $C_{3}$-symmetric non-Hermitian
systems can be described by the topological index and symmetry indicator [59,
61, 62]. The topological index is given by
$\chi=([K_{1}^{(3)}],[K_{2}^{(3)}])$, where $[K_{1}^{(3)}]=\sharp
K_{1}^{(3)}-\sharp\Gamma_{1}^{(3)}$, $[K_{2}^{(3)}]=\sharp
K_{2}^{(3)}-\sharp\Gamma_{2}^{(3)}$ are the symmetry indicators. The
$\sharp\Pi_{p}^{(n)}$ is the number of bands below the bandgap with the
eigenvalue of $C_{3}$ operator $e^{i2\pi(p-1)/n}$ ($p=1,2,...,n$) at the high-
symmetry point $\Pi=\Gamma,K$. We find that $\chi=(-1,1)$ when $\varepsilon>1$
and $\chi=(0,0)$ when $\varepsilon<1$. Remarkably, the topological bulk
polarization $\bm{P}$ and fractional corner charge $Q$ can be determined by
the topological index [59]:
$\bm{P}=\left(\frac{2}{3}[K_{1}^{(3)}]+\frac{4}{3}[K_{2}^{(3)}]\right)(\bm{a}_{1}+\bm{a}_{2})\mod
1$, where $\bm{a}_{1,2}$ are primitive unit lattice vectors,
$Q=\frac{1}{3}[K_{2}^{(3)}]\mod 1$. When $\varepsilon<1$, $\bm{P}=\bm{0}$ and
$Q=0$. When $\varepsilon>1$, $\bm{P}=\frac{2}{3}(\bm{a}_{1}+\bm{a}_{2})$ and
$Q=1/3$. Due to the non-zero $\bm{P}$ and $Q$ when $\varepsilon>1$, we can see
that there are corner states in the bulk gap for a finite system, as shown in
Fig. 3(e, f).

Non-Hermitian effects on topological edge states in a 2D honeycomb lattice.
Here, we discuss the non-Hermitian topological valley system in 2D honeycomb
lattice, which has the unit cell in Fig. 4(a). The effective non-Hermitian
Hamiltonian at the valley point $K$
($\bm{k}\rightarrow\bm{k}_{0}+\delta\bm{k}$, $\bm{k}_{0}$ is the wave vector
of $K$) is [54]:

$\displaystyle H_{k}$ $\displaystyle=v_{F}\delta
k_{x}\left(\frac{1+\varepsilon}{2\varepsilon}\sigma_{x}+i\frac{\varepsilon-1}{2\varepsilon}\sigma_{y}\right)$
(7) $\displaystyle+v_{F}\delta
k_{y}\left(\frac{1+\varepsilon}{2\varepsilon}\sigma_{y}-i\frac{\varepsilon-1}{2\varepsilon}\sigma_{x}\right)+m\sigma_{z},$

where $\varepsilon\neq 1$, $m_{a}=-m_{b}=m$ and $v_{F}=-\frac{3}{2}t_{1}a$,
$a$ is the lattice constant. Setting $m\neq 0$ can open a gap at the valley
point $K/K^{\prime}$, as shown in Fig. 4(b). The topological properties of
this non-Hermitian valley system can be described by the valley Chern number
of the lower band: $C_{v}=-\frac{{\rm sign}[m]}{2}$ [54, 63]. After combining
two lattices with different valley Chern numbers ($m>0$ and $m<0$) along the
zigzag boundary, there are topological edge states existing in the bulk gap
[64], as shown in Fig. 4(c). These edge states have the dispersion relation
[54]: $E=\pm\frac{v_{F}}{\sqrt{\varepsilon}}k$, where $k$ is the wave vector
of an edge state. This dispersion shows that the $\varepsilon$ can affect the
propagation velocity of edge states $|v_{\rm
edge}|=|\partial_{k}E|=\frac{|v_{F}|}{\sqrt{\varepsilon}}$. When
$\varepsilon<1$ ($\varepsilon>1$), the propagation velocity of edge states can
become faster (slower) than in the Hermitian case: $|v_{\rm edge}|>|v_{F}|$
($|v_{\rm edge}|<|v_{F}|$).

Figure 4: Non-Hermitian topological valley system. (a) The unit cell of the 2D
honeycomb lattice. The $t_{1}$ and $t_{2}$ are the coupling strengths,
$t_{2}=\frac{t_{1}}{\varepsilon}$, $\varepsilon\neq 1$. (b) The bandstructure
of the non-Hermitian system with $\varepsilon=2.0$. Here, $t_{1}=-1.0$,
$m_{a}=-m_{b}=m$. The bandstructures with $m=0.2$ and $m=0$ are represented by
the blue solid and gray dashed lines, respectively. (c) The dispersion of the
topological edge states on the zigzag interface between two non-Hermitian
systems with $m=0.25$ and $m=-0.25$.

Experimental proposals. The non-Hermitian systems shown above can be achieved
by electric circuits [65, 66, 67, 68, 69]. The electric diode, which makes the
electric current flow directionally, can be used to readily establish non-
reciprocal coupling [70]. As an example, we demonstrate the circuit
realization of 1D non-Hermitian topological system in Ref. 111See
supplementary materials for more details about the Hamiltonians, the non-
sensitivity of our systems to boundary conditions, the effect of the non-
diagonal $M$, topologically valley-protected edge states and experimental
proposals, which include the Ref. [81, 82, 83, 84, 85, 86, 87, 88, 89]. There
are many approaches for classical wave systems to achieve non-reciprocal
coupling. For example, the non-reciprocal coupling in optical wave systems can
be achieved by exploiting dynamically modulated media [71], metamaterials [72]
or S-bending waveguides [73]. For acoustic or elastic wave systems, one can
achieve non-reciprocal coupling by using piezophononic media [74],
metamaterials [75] or additional electric setup [17].

To summarize, we propose a systematic strategy to construct real-eigenvalued
non-Hermitian topological systems. Compared with previous non-Hermitian
systems whose real spectra exist only when non-Hermiticity is relatively weak,
our systems exhibit energy spectra that are always real, regardless of weak or
strong non-Hermiticity. Furthermore, the nonreciprocal-coupling-based non-
Hermiticity is able to affect many properties of topological states. For
example, it is able to determine the topology by inducing a topological phase
transition. Our work offers a new avenue to explore non-Hermitian topological
physics [76, 77, 78, 79, 80], and would be useful in novel photonic and
acoustic devices with non-reciprocal coupling.

###### Acknowledgements.

This research is supported by Singapore National Research Fundation
Competitive Research Program under Grant no. NRF-CRP23-2019-0007, and
Singapore Ministry of Education Academic Research Fund Tier 3 under Grant No.
MOE2016-T3-1-006 and Tier 2 under Grant No. MOE2018-T2-1-022 and Grant No.
MOE2019-T2-2-085.

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