# Shaping the topology of twisted bilayer graphene
via time-reversal symmetry breaking

Cunyuan Jiang1,2,3 Matteo Baggioli1,2,3<EMAIL_ADDRESS>Qing-Dong
Jiang1,4,5<EMAIL_ADDRESS>1 School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China  2 Wilczek Quantum
Center, School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China  3 Shanghai Research Center for Quantum Sciences,
Shanghai 201315,China  4 Tsung-Dao Lee Institute, Shanghai Jiao Tong
University, Shanghai 200240, China  5 Shanghai Branch, Hefei National
Laboratory, Shanghai 201315, China

###### Abstract

Symmetry breaking is an effective tool for tuning the transport and
topological properties of 2D layered materials. Among these materials, twisted
bilayer graphene (TBG) has emerged as a promising platform for new physics,
characterized by a rich interplay between topological features and strongly
correlated electronic behavior. In this study, we utilize time-reversal
symmetry breaking (TRSB) to manipulate the topological properties of TBG. By
varying the strength of TRSB, we discover a topological phase transition
between a topological insulating phase, which exhibits a pair of flat bands
with opposite Chern numbers, and a novel insulating state where the Chern
number, but not the Berry curvature, of the flat bands vanishes. We
demonstrate that this topological transition is mediated by a gap closing at
the $\Gamma$ point, and we construct a three-dimensional phase diagram as a
function of the twisting angle, the symmetry-breaking parameter, and the
mismatch coupling between AA and AB stacking regions. Finally, we show that
this novel electronic phase can be identified in the lab by measuring, as a
function of the Fermi energy, its non-quantized anomalous Hall conductivity
that is induced by the Berry dipole density of the lowest flat bands.

Twisted bilayer graphene (TBG) is a 2D quantum material composed of two
stacked layers of graphene twisted with a relative angle $\theta$ that create
a so-called moiré pattern Andrei and MacDonald (2020). TBG has attracted a lot
of attention in the recent years because of strongly-correlated electronic
phases and anomalous properties emerging around the so-called “magic-angle”,
$\theta\approx 1.09^{\circ}$ Gibney (2019). Interestingly, TBG hosts a pair of
topological electronic flat bands and provides a perfect platform to explore
the interplay between topology, strongly correlated physics and exotic
superconductivity.

On the other hand, symmetry breaking represents a valuable tool for
controlling and engineering the physical properties of 2D materials, including
their topological structure Du  _et al._ (2021). TBG is no exception. Time-
reversal symmetry breaking (TRSB) is particularly important because it enables
the emergence of several physical phenomena that are otherwise forbidden. In
particular, the Berry curvature $\Omega(\boldsymbol{q})$ (with
$\boldsymbol{q}$ being the wave vector) flips its sign under time-reversal
operation, i.e., $\Omega(\boldsymbol{q})=-\Omega(-\boldsymbol{q})$.
Consequently, in a time-reversal symmetric system, the overall Berry curvature
must vanish, leading to a zero anomalous Hall conductivity.

TRSB in TBG can be introduced in various ways such as imposing an external
magnetic field Sánchez and Stauber (2024), exploiting coherent light-matter
interactions Vogl  _et al._ (2020); Topp  _et al._ (2019), and using chiral
optical cavities Jiang  _et al._ (2024) or hBN encapsulation Long  _et al._
(2022). As for the simpler case of monolayer graphene Wang  _et al._ (2019),
TRSB can induce topological phase transitions in TBG Topp  _et al._ (2019) and
can have strong effects on the electronic spectrum flattening for example the
topological isolated bands away from the magic angle Jiang  _et al._ (2024).

In this study, leveraging on an effective model of TBG, we study in detail the
effects of TRSB in twisted bilayer graphene as a function of the twisting
angle and the different interlayer coupling strength of AA and AB stacked
regions Koshino  _et al._ (2018), that is known to appear because of the
surface relaxation effects and the consequent atomic corrugation Uchida  _et
al._ (2014); Lucignano  _et al._ (2019). In doing so, we reveal a previously
overlooked topological phase transition (TPT) that appears at a critical value
of TRSB, and that leads to a new electronic state exhibiting a pair of flat
bands with zero Chern number but non-trivial Berry curvature.

Figure 1: The electronic band structure of TBG for $u=0.817$, $u^{\prime}=1$,
$\theta=1.5^{\circ}$ by varying the strength of time-reversal symmetry
breaking $\tau$ from $0$ (a) to $0.2$ (d). The color map indicates the value
of the normalized Berry curvature for the lowest bands that is shown more
clearly in panel b of Fig.2. The integers show the Chern numbers of the lowest
electronic bands.

We model TBG using an effective Hamiltonian Catarina  _et al._ (2019)
$H_{\text{TBG}}(\mathbf{q})$,

$\left(\begin{matrix}H_{1}(\mathbf{q})&T_{\mathbf{q}_{b}}&T_{\mathbf{q}_{tr}}&T_{\mathbf{q}_{tl}}&\cdots\\\
T_{\mathbf{q}_{b}}^{\dagger}&H_{2}(\mathbf{q}-\mathbf{q}_{b})&0&0&\cdots\\\
T_{\mathbf{q}_{tr}}^{\dagger}&0&H_{2}(\mathbf{q}-\mathbf{q}_{tr})&0&\cdots\\\
T_{\mathbf{q}_{tl}}^{\dagger}&0&0&H_{2}(\mathbf{q}-\mathbf{q}_{tl})\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{matrix}\right),$ (1)

where $\mathbf{q}$ is the wave-vector, and

$H_{n}(\mathbf{q})=v_{F}\left(\begin{matrix}0&q_{x}-iq_{y}\\\
q_{x}+iq_{y}&0\end{matrix}\right)$ (2)

is the Hamiltonian of each decoupled layer. $n=1,2$ is the index of the layer
and the Fermi energy is fixed to $v_{F}\approx 5.944eV\cdot\textup{\AA}$. We
refer to Ref. Catarina  _et al._ (2019) for more details about the precise
definitions of the wave-vectors
$\mathbf{q}_{b},\mathbf{q}_{tr},\mathbf{q}_{tl}$.

Finally, the hopping matrix elements are given by

$\displaystyle T_{\mathbf{q}_{b}}=t\left(\begin{matrix}u&u^{\prime}\\\
u^{\prime}&u\end{matrix}\right),\quad
T_{\mathbf{q}_{tr}}=t\left(\begin{matrix}ue^{i\phi}&u^{\prime}\\\
u^{\prime}e^{-i\phi}&ue^{i\phi}\end{matrix}\right),$ $\displaystyle\hskip
42.67912ptT_{\mathbf{q}_{tl}}=t\left(\begin{matrix}ue^{-i\phi}&u^{\prime}\\\
u^{\prime}e^{i\phi}&ue^{-i\phi}\end{matrix}\right),$ (3)

with $\phi=2\pi/3$ and $t=0.11$. In order to take into account the different
interlayer coupling strength of AA and AB stacked regions Koshino  _et al._
(2018), we fix $u^{\prime}=1$ and keep $u\neq 1$ as a free parameter.

We then deform our initial system by adding a time-reversal breaking
deformation

$H_{\text{TBG}+\tau}(\mathbf{q})=H_{\text{TBG}}(\mathbf{q})+\tau\sigma_{z}\mathbb{I},$
(4)

where the parameter $\tau$ serves to quantify the strength of TRSB,
$\sigma_{z}$ is the Pauli spin matrix and $\mathbb{I}$ the identity matrix in
the multi-dimensional wave-vector space on which $H_{\text{TBG}}$ is defined.

We obtain the electronic band structure by direct diagonalization of
$H_{\text{TBG}+\tau}$ keeping as free parameters the twisting angle $\theta$,
the time-reversal breaking strength $\tau$ and the coupling strength $u$.
Moreover, we compute the Berry curvature of the $n$th band at wave-vector
$\boldsymbol{q}$ using the following expression

$\Omega_{n}(\boldsymbol{q})=i[\langle\partial_{q_{x}}n(\boldsymbol{q})|\partial_{q_{y}}n(\boldsymbol{q})\rangle-\langle\partial_{q_{y}},n(\boldsymbol{q})|\partial_{q_{x}}n(\boldsymbol{q})\rangle],$
(5)

where $|n(\boldsymbol{q})\rangle$ labels the eigenstate of $n$th band at wave
vector $\boldsymbol{q}$. Finally, we define the topological Chern number
$\mathcal{C}$,

$\mathcal{C}_{n}=\frac{1}{2\pi}\int_{\text{BZ}}\Omega_{n}(\boldsymbol{q})d\boldsymbol{q},$
(6)

where the integration is taken over the whole Brillouin zone (BZ).

We start by exploring the effects of TRSB at a constant twisting angle
$\theta=1.5^{\circ}$ and coupling $u=0.817$. In Fig.1, we show the electronic
band structure of TBG by dialing the symmetry breaking parameter $\tau$. Panel
(a) corresponds to the time-reversal symmetric case ($\tau=0$), where the band
structure displays a pair of weakly dispersing electronic bands crossing at
$K$ points. In this case, both the Berry curvature and the Chern number of all
the bands identically vanish. Therefore, $\tau=0$ corresponds to a
topologically trivial phase.

By breaking time-reversal symmetry, $\tau\neq 0$ (panel (b) in Fig.1), a gap
opens at the $K$ points and two isolated bands emerge in the spectrum with
non-trivial and opposite topological Chern number $\mathcal{C}=\pm 1$. The
system is now in a topological insulating phase (phase 1). This first
topological phase transition (TPT1 in Fig.1) is not new Topp  _et al._ (2019);
Jiang  _et al._ (2024), and it is indeed analogous to the case of monolayer
graphene under time-reversal symmetry breaking Wang  _et al._ (2019).

By increasing further $\tau$ (panel (c) in Fig.1), the gap at the $\Gamma$
point closes and a clear band-crossing appears in the spectrum. This gap
closing and re-opening signals the onset of a second topological phase
transition (TPT2) that leads to a final state (phase 2, panel (d) in Fig.1)
with two isolated and rather flat electronic bands with zero Chern number.
This final state is topologically trivial. Nevertheless, because of the
absence of time-reversal symmetry, the Berry curvature of the lowest bands
with vanishing Chern number is non-zero and, on the contrary, shows a strong
peak around the $\Gamma$ point (see panel (b) in Fig.2). This second
topological phase transition (TPT2) is similar in spirit to what recently
observed by dialing the external magnetic field in TBG Sánchez and Stauber
(2024).

In panel (a) of Fig.2, we show a three-dimensional phase diagram at fixed
$u^{\prime}=0.817$ as a function of $u,\tau,\theta$. For clarity, we only show
four different 2D cuts corresponding respectively to $u=0.75,0.817,0.9,1$. The
shaded colored regions indicate phase 2, while the white one represents phase
1. We observe that a larger $u$ disfavors the topological phase 1 that
survives only in the region of small $\tau$ and large $\theta$. The TRSB
parameter $\tau$ has an analogous effect as it pushes the system towards phase
2. On the contrary, a larger twisting angle makes phase 1 more robust and its
extent in the phase diagram wider.

Figure 2: (a) 3D Phase diagram as a function of twisting angle $\theta$, time
reversal symmetry breaking parameter $\tau$ and coupling $u$ with
$u^{\prime}=0.817$. The four 2D cuts correspond to $u=0.75,0.817,0.9,1$. The
blue region is the topologically trivial phase 2 with $\mathcal{C}=0$ but
$\Omega\neq 0$. The white region corresponds to the topological insulating
phase 1 with $\mathcal{C}\neq 0$. (b) The Berry curvature along the path
connecting the high-symmetry points in phase 1 (panel (b) in Fig.1) is sign
definite and opposite for the lowest valence and conduction flat bands. In
phase 2 (panel (d) in Fig.1), $\Omega$ is still peaked at the $\Gamma$ point
but not anymore sign definite, leading to a vanishing total Chern number. The
horizontal black line indicates $\Omega=0$ and the color map is the same as in
Fig.1. The vector field in the BZ is the local Berry curvature dipole
$\boldsymbol{d}_{n}(\boldsymbol{q})$ defined in (7) and computed for the
lowest conduction band. The dashed black circle highlights the domain wall
structure in the Berry dipole moment in phase 2. (c) The Hall conductivity
$\sigma_{xy}$ as a function of the Fermi energy $\epsilon_{F}$ for phase 1
(gray color) and phase 2 (blue color). The solid gray and blue lines
corresponds to panels (b) and (d) in Fig.1, while the dashed ones correspond
to a smaller angle, $\theta=1.1^{\circ}$.

In panel (b) of Fig.2, we plot the Berry curvature of the lowest valence and
conduction bands along a q-path connecting the high-symmetry points
$K_{2}$-$\Gamma$-$K_{1}$. In both cases, the Berry curvature is peaked at
$\Gamma$ point. In phase 1, $\Omega$ is sign definite and has opposite sign
for the lowest flat bands. On the contrary, in phase 2, $\Omega$ of the lowest
bands is no longer sign definite, but it crosses the horizontal axes (black
line), leading to a vanishing integral over the whole BZ, i.e.,
$\mathcal{C}=0$.

In order to characterize further the different topological features of phases
1 and 2, we use the local Berry curvature dipole density,

$\boldsymbol{d}_{n}(\boldsymbol{q})\equiv\Omega_{n}(\boldsymbol{q})\dfrac{\partial\epsilon_{n}(\boldsymbol{q})}{\partial\boldsymbol{q}}\dfrac{\partial
f(\epsilon)}{\partial\epsilon},$ (7)

where $\epsilon_{n}(\boldsymbol{q})$ is the energy dispersion of the n-th band
and $f$ is the Fermi-Dirac distribution function. The integral of
$\boldsymbol{d}$ on the whole BZ is known to generate a nonlinear anomalous
Hall coefficient even in systems with time-reversal symmetry (but broken
inversion symmetry) Sodemann and Fu (2015). In panel (b) of Fig.2 we show
$\boldsymbol{d}$ computed on the lowest conduction band on the whole BZ for
the two different phases. In phase 1, the structure of the vector field
$\boldsymbol{d}$ suggests the presence of a local source of Berry dipole at
the $\Gamma$ point. The situation is different in phase 2, where
$\boldsymbol{d}$ shows a domain wall structure around the $\Gamma$ point
(highlighted by the dashed circle) that corresponds to the location at which
the Berry curvature $\Omega$ changes sign. This suggests that the topology of
phase 1 can be understood from the presence of a Berry monopole at the
$\Gamma$ point, while that of phase 2 presents a strong dipolar structure
between opposite charges located at $\Gamma$ and $K$ points.

To connect these differences with a macroscopic measurable quantity, we
consider the anomalous Hall conductivity,

$\sigma_{xy}=\dfrac{e^{2}}{\hbar}\int_{BZ}\dfrac{d^{2}q}{(2\pi)^{2}}f(\epsilon(\boldsymbol{q}))\Omega_{n}(\boldsymbol{q}).$
(8)

In panel (c) of Fig.2, we show $\sigma_{xy}$ as a function of the Fermi energy
for the two different phases. In phase 1, $\sigma_{xy}$ displays a plateau
around $\epsilon_{F}=0$ where it attains the quantized value dictated by the
integer Chern number $+1$, $\sigma_{xy}=e^{2}\mathcal{C}/\hbar$. The width of
the plateau is controlled by the flatness of the lowest topological bands and
it decreases moving towards the magic angle (dashed gray line). On the other
hand, $\sigma_{xy}$ is not quantized in phase 2 but, despite $\mathcal{C}=0$,
it is nonzero and negative when the Fermi energy crosses the lowest flat
bands. Again, if these bands have a more flat dispersion, as expected for
twisting angles closer to the magic one, $\sigma_{xy}$ acquires a larger value
and a sharper form. We emphasize that, for the phase at $\tau=0$,
$\sigma_{xy}$ vanishes for any $\epsilon_{F}$ since the Berry curvature is
identically zero as a consequence of time reversal symmetry. The different
behavior of $\sigma_{xy}$ in the three phases suggests that they can be
distinguished in the lab and the corresponding properties can be connected to
the topological features of their electronic spectrum.

In summary, we discovered a time-reversal symmetry breaking induced
topological phase transition in TBG separating a topological insulating phase
and a topological trivial phase in which the spectrum exhibits a pair of
isolated bands with zero Chern number but nonzero local Berry curvature. We
have shown how to identify this new phase in the lab by measuring a non-zero,
and not quantized, anomalous Hall conductivity upon tuning the Fermi energy
across the lowest isolated bands and related this property to the topological
Berry curvature dipole.

We provided a clear demonstration of how TRSB can help engineering and
controlling the topological properties of TBG Adak  _et al._ (2024) leading to
novel electronic phases. We expect that our findings might be generally
applicable to several 2D moiré systems and that this novel topological phase
transition could be experimentally detected in the near future.

## Acknowledgments

The authors thank Jinhua Gao and Giandomenico Palumbo for useful discussions.
CJ and MB acknowledge the support of the Shanghai Municipal Science and
Technology Major Project (Grant No.2019SHZDZX01). MB acknowledges the support
of the sponsorship from the Yangyang Development Fund. QDJ sponsored by
National Natural Science Foundation of China (NSFC) under Grant No.12374332,
Jiaoda2030 Program Grant No.WH510363001, TDLI starting up grant, and
Innovation Program for Quantum Science and Technology Grant No.2021ZD0301900.

## References

  * Andrei and MacDonald (2020) E. Y. Andrei and A. H. MacDonald, Nature Materials 19, 1265 (2020).
  * Gibney (2019) E. Gibney, Nature 565, 15 (2019).
  * Du  _et al._ (2021) L. Du, T. Hasan, A. Castellanos-Gomez, G.-B. Liu, Y. Yao, C. N. Lau, and Z. Sun, Nature Reviews Physics 3, 193 (2021).
  * Sánchez and Stauber (2024) M. S. Sánchez and T. Stauber, arXiv preprint arXiv:2402.00884 (2024).
  * Vogl  _et al._ (2020) M. Vogl, M. Rodriguez-Vega, and G. A. Fiete, Phys. Rev. B 101, 241408 (2020).
  * Topp  _et al._ (2019) G. E. Topp, G. Jotzu, J. W. McIver, L. Xian, A. Rubio, and M. A. Sentef, Phys. Rev. Res. 1, 023031 (2019).
  * Jiang  _et al._ (2024) C. Jiang, M. Baggioli, and Q.-D. Jiang, Phys. Rev. Lett. 132, 166901 (2024).
  * Long  _et al._ (2022) M. Long, P. A. Pantaleón, Z. Zhan, F. Guinea, J. Á. Silva-Guillén, and S. Yuan, npj Computational Materials 8, 73 (2022).
  * Wang  _et al._ (2019) X. Wang, E. Ronca, and M. A. Sentef, Phys. Rev. B 99, 235156 (2019).
  * Koshino  _et al._ (2018) M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Phys. Rev. X 8, 031087 (2018).
  * Uchida  _et al._ (2014) K. Uchida, S. Furuya, J.-I. Iwata, and A. Oshiyama, Phys. Rev. B 90, 155451 (2014).
  * Lucignano  _et al._ (2019) P. Lucignano, D. Alfè, V. Cataudella, D. Ninno, and G. Cantele, Phys. Rev. B 99, 195419 (2019).
  * Catarina  _et al._ (2019) G. Catarina, B. Amorim, E. V. Castro, J. M. V. P. Lopes, J. M. V. P. Lopes, and N. Peres, “Twisted bilayer graphene: Low-energy physics, electronic and optical properties,” in  _Handbook of Graphene Set_ (John Wiley & Sons, Ltd, 2019) Chap. 6, pp. 177–231.
  * Sodemann and Fu (2015) I. Sodemann and L. Fu, Phys. Rev. Lett. 115, 216806 (2015).
  * Adak  _et al._ (2024) P. C. Adak, S. Sinha, A. Agarwal, and M. M. Deshmukh, Nature Reviews Materials (2024), 10.1038/s41578-024-00671-4.