# Kekulé spirals and charge transfer cascades in twisted symmetric trilayer
graphene

Ziwei Wang  Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford,
OX1 3PU, UK  Yves H. Kwan  Princeton Center for Theoretical Science, Princeton
University, Princeton NJ 08544, USA  Glenn Wagner  Department of Physics,
University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland  Nick
Bultinck  Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford,
OX1 3PU, UK  Department of Physics, Ghent University, Krijgslaan 281, 9000
Gent, Belgium  Steven H. Simon  Rudolf Peierls Centre for Theoretical Physics,
Parks Road, Oxford, OX1 3PU, UK  S.A. Parameswaran  Rudolf Peierls Centre for
Theoretical Physics, Parks Road, Oxford, OX1 3PU, UK

###### Abstract

We study the phase diagram of magic-angle twisted symmetric trilayer graphene
in the presence of uniaxial heterostrain and interlayer displacement field.
For experimentally reasonable strain, our mean-field analysis finds robust
Kekulé spiral order whose doping-dependent ordering vector is incommensurate
with the moiré superlattice, consistent with recent scanning tunneling
microscopy experiments, and paralleling the behaviour of closely-related
twisted bilayer graphene (TBG) systems. Strikingly, we identify a new
possibility absent in TBG: the existence of commensurate Kekulé spiral order
even at zero strain for experimentally realistic values of the interlayer
potential in a trilayer. Our studies also reveal a complex pattern of charge
transfer between weakly- and strongly-dispersive bands in strained trilayer
samples as the density is tuned by electrostatic gating, that can be
understood intuitively in terms of the `cascades' in the compressibility of
magic-angle TBG.

Introduction.— The discovery of superconductivity proximate to correlated
insulating behaviour [1, 2, 3, 4, 5, 6, 7, 8, 9] in twisted bilayer graphene
(TBG) in the `magic angle' regime [10, 11] has stimulated intense
investigation of a host of other multilayer graphene systems. Among these,
alternating-twist multilayers exhibit identical moiré patterns between each
pair of adjacent layers and have well-defined magic angles in the chiral limit
[12]. The simplest member in the group, twisted symmetric trilayer graphene
(TSTG), which is composed of three layers of graphene stacked together with
the inner layer rotated, has been experimentally studied [13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25], especially in the context of its
superconducting properties, and has received extensive theoretical
investigation [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. In the absence of external
fields, the single-particle Hilbert space of TSTG can be decomposed into a
direct sum of monolayer graphene and TBG parts [12], allowing many phenomena
in TBG to be reproduced in TSTG. If one applies a perpendicular electric
field, the TBG and graphene sectors are hybridized, allowing for further
tunability of the system.

Among the various correlated states in TBG, a novel form of translation-
symmetry breaking order dubbed the incommensurate Kekulé spiral (IKS) has been
proposed by some of us to characterize the many-body ground state at all
integer fillings except charge neutrality [53], as well as a range of non-
integer fillings [54]. The signature of the proposed state, which is
stabilized by experimentally realistic uniaxial strains, consists of a
graphene scale Kekulé-like pattern with complex spatial dependence on the
moiré scale, and has recently been observed with high-resolution scanning
tunnelling microscopy (STM) [55]. A similar pattern has also been observed in
TSTG [25], but Kekulé spiral order in TSTG is theoretically unexplored,
motivating the present work.

Figure 1: Twisted symmetric trilayer graphene. (a) The large hexagons denote
graphene Brillouin zones of individual layers and the small hexagon denotes
the moiré BZ for valley $K$. (b) Without interlayer potential, TSTG decomposes
into a TBG sector (red) and a graphene sector (blue) with opposite mirror
eigenvalues. (c) A finite interlayer potential (100 meV) breaks mirror
symmetry and hybridizes the two sectors. The color represents the relative
weight of the wavefunction on the zero-field TBG and graphene sectors. (d,e)
TBG sector conduction band at zero interlayer potential without and with
strain ($\epsilon=0.1\%$) respectively. Strain breaks $C_{3}$-symmetry,
unpinning the Dirac points, and increases the bandwidth. The corresponding
band structures in valley $K^{\prime}$ can be found using time-reversal
symmetry.

Here, we conduct extensive Hartree-Fock (HF) simulations of TSTG that
incorporate the effect of strain and allow for translation symmetry breaking.
As in the case of TBG, we find IKS to be ubiquitous for non-zero integer and
non-integer fillings of TSTG at finite strain. Furthermore, we find that
commensurate Kekulé spiral (CKS) order in unstrained TSTG can be accessed even
at zero strain by applying a large interlayer potential — a possibility absent
in TBG. These results should be contrasted with previous studies of TSTG [36,
40, 44, 50, 52] at zero strain. We also explore the process of `charge
transfer' between the TBG and graphene sectors of TSTG at zero interlayer
potential, and elucidate subtleties in defining commensurate fillings in light
of this mechanism.

Model and Symmetries.—TSTG is made by stacking three sheets of graphene with
the middle layer twisted by an angle $\theta$ relative to the top and bottom
layers [Fig. 1(a)]. The moiré patterns generated within the top and bottom
pairs of layers are identical, preserving the overall moiré periodicity of the
system. The single-particle Hamiltonian of TSTG can be decomposed [12] into a
mirror-symmetry even sector, which is effectively a renormalized version of
TBG, and a mirror-symmetry odd sector, which resembles monolayer graphene
[Fig. 1(b)]. The magic angles of TSTG and TBG are related via
$\theta_{\text{TSTG}}=\sqrt{2}\theta_{\text{TBG}}\approx 1.56^{\circ}$. In
contrast to TBG, an electric displacement field normal to the layers has a
significant effect: it produces an interlayer potential that breaks the mirror
symmetry and mixes the odd and even sectors [Fig. 1(c)]. This modifies the
dispersion, expanding the parameter space for new phases.

We also incorporate the effect of strain in the system. Homostrain (where all
layers experience the same strain) has only a small effect on TBG [56], and we
expect it to play a similarly minor role in TSTG. Therefore, we restrict our
attention to heterostrain, produced by unequal strain in the different layers.
In principle, strain could break the mirror symmetry of the system and
introduce incommensurate moiré patterns for the top and bottom pairs of
layers, potentially leading to a supermoiré pattern. However, we focus here on
the case where top and bottom layers experience identical strains, thus
preserving the moiré periodicity of the system. This choice can be justified
as STM experiments observe well defined moiré patterns in large regions of the
TSTG samples [16, 17, 25]. We also ignore the small-angle Pauli-matrix
rotations in the Dirac kinetic terms in each layer; then, absent strain or
interlayer potential, the system has exact particle-hole symmetry, which is
lost when both perturbations are included.

Phase Diagram.—We perform band-projected self-consistent HF calculations on
the model discussed above at the magic angle $\theta=1.56^{\circ}$. We use
realistic hopping parameters $w_{AA}=75$ meV and $w_{AB}=110$ meV, and include
strain $\epsilon$ and interlayer potential $\Delta V$ as free parameters of
the model. We model interactions via the dual-gate screened Coulomb potential
$V(q)=(e^{2}/2\epsilon_{0}\epsilon_{r}q)\tanh qd$, with a screening length
$d=25$ nm and relative permittivity $\epsilon_{r}=10$. To avoid double-
counting interactions, we use the ``average'' subtraction scheme [57]. Despite
the lack of exact particle-hole symmetry for $\Delta V,\epsilon\neq 0$, we
find that the phase diagrams at $\nu$ and $-\nu$ are qualitatively similar.
Hence we only present results for $|\nu|\geq 0$ in the main text, and refer
readers to Ref. [57] for $\nu<0$ results.

Unless otherwise noted, we assume collinear spin configurations, but we have
also checked that, for the parameter range in Fig. 2, relaxing this condition
does not lead to different ground states up to
$\text{SU}(2)_{K}\times\text{SU}(2)_{K^{\prime}}$ valley-dependent spin
rotations. In order to capture the incommensurate Kekulé spirals (IKS)
proposed in Ref. [53] or its commensurate counterparts, we allow for coherence
between states of momentum $\bm{k}-\tau\bm{q}/2$ in valley $\tau$ with states
of momentum $\bm{k}-\tau^{\prime}\bm{q}/2$ in valley $\tau^{\prime}$ (here
$\tau$ and $\tau^{\prime}$ take on values of $\pm 1$) for some Kekulé spiral
vector $\bm{q}$ in our HF calculations, and then optimize over all $\bm{q}$.
One could further consider the slightly more general situation where the two
spin sectors have different $\bm{q}$. As such, the one-particle density matrix
takes the form

$\braket{\hat{c}^{\dagger}_{\bm{k}-\tau\bm{q}_{s}/2,\tau
sa}\hat{c}_{\bm{k}-\tau^{\prime}\bm{q}_{s}/2,\tau^{\prime}sb}}=P_{\tau
a;\tau^{\prime}b}(\bm{k},s)$ (1)

where $s$ is spin index, $a$ and $b$ are single particle band indices, and
$\bm{q}_{s}$ is the Kekulé spiral vector with possible spin-dependence. In the
majority of cases, this spin-dependence is not relevant and we simply scan
over $\bm{q}_{\uparrow}=\bm{q}_{\downarrow}\equiv\bm{q}$. Spin-dependence of
Kekulé spiral vectors becomes relevant when both spin sectors have intervalley
coherence and the two sectors are nonequivalent, as is the case for $\nu=1$.
As such, we have allowed $\bm{q}_{\uparrow}\neq\bm{q}_{\downarrow}$ when
computing the phase diagram at $\nu=1$. For the range of interlayer potentials
considered in the main text, we found that allowing for more generic
translation symmetry breaking beyond Eq. 1 does not lead to different results.
At higher interlayer potentials, we observe a complex pattern of translational
symmetry breaking with nontrivial spin structure [57].

Figure 2: Phase diagrams as a function of strain and interlayer potential at
integer fillings $\nu$. We have performed $10\times 10$ self-consistent
Hartree-Fock calculations, scanning over all possible Kekulé spiral vectors
$\bm{q}$. $\Delta E$ is the energy difference between the $\bm{q}=0$ state
with the overall energy minimum. The acronyms are: K-IVC for Kramers-
intervalley coherence, VP for valley polarization, (I)KS for (incommensurate)
Kekulé spiral, SM for (compensated) semi-metal. Asterisk (*) denotes states
that break $C_{2}T$-symmetry. The KS*(VP) region in $\nu=3$ consists of Kekulé
spiral states with possible finite valley polarization. The red lines on the
zero-strain axis indicate commensurate Kekulé spirals. The shaded regions have
a finite charge gap. For $\nu=1$, we have shown the phases and charge gaps of
the two spin sectors separately, where $\nu_{s}$ denotes the filling of each
spin sector. As $\Delta E$ cannot be separately defined for each spin sector,
the data are simply duplicated in the two plots of $\nu=1$.

Fig. 2 shows the phase diagrams for $\nu=0,1,2,3$. Kekulé states involve non-
zero Kekulé vector $\bm{q}$ and finite intervalley coherence. To diagnose the
presence of these states and demonstrate their robustness, we show energy
differences per unit cell between the lowest-energy $\bm{q}=0$ solution and
the energy achieved by minimizing over all $\bm{q}$: non-zero values indicate
Kekulé spiral ground states.

At zero interlayer potential, the single-particle Hamiltonian decomposes into
mirror-symmetry even (TBG) and odd (graphene) sectors. For all fillings and
strains examined here, we observe no spontaneous breaking of this symmetry,
and the two sectors are only coupled by Hartree terms. Compared to pure TBG,
the inclusion of the graphene sector can lead to `charge-transfer cascades'
from the TBG sector, discussed below, but otherwise the phase diagram is
consistent with the corresponding one for TBG [53]. We find that IKS order
appears at $\nu=2,3$ at finite strain (Fig. 2). While for the range of strains
shown in Fig. 2 we find Kramers intervalley-coherent (K-IVC) state at
$\nu=0,1$, for larger strains the system transitions into IKS at $\nu=1$ and a
symmetric state at $\nu=0$, again consistent with the situation in TBG.

A non-zero interlayer potential mixes the two symmetry sectors, and the simple
picture of Hartree-coupled TBG and graphene sectors is no longer applicable.
The mixing of relatively flat TBG bands with graphene bands leads to more
dispersive central bands. This favors the formation of IKS by reducing the
minimal strain required for its stabilization, except at charge neutrality,
where no Kekulé spiral is observed for either TBG or TSTG.

At high interlayer potential of about $200$ meV, a common feature except at
$\nu=3$, is the breaking of $C_{2}T$ symmetry and the opening of a charge gap.
While strong $C_{2}T$-breaking has not been observed in monolayer graphene
under ordinary conditions, as graphene orbitals are hybridized with TBG
orbitals, the Dirac point becomes less dispersive and interaction-induced
spontaneous symmetry breaking becomes possible. For $\nu=3$, $C_{2}T$ symmetry
only spontaneously breaks in one of the two spin sectors, leaving the system
gapless. We note that Ref. [21] finds a charge gap at $\nu=2$, in qualitative
agreement with our result.

The phase diagram is most complex at $\nu=1$, where the two spin sectors have
densities of approximately 0 and 1 electron per unit cell relative to their
respective charge neutrality points (finite but small charge transfers between
the two sectors could occur). The two sectors are only coupled with Hartree
interaction, and otherwise exhibit their own physics. To capture the physics
correctly, we allow for different Kekulé vectors for the two sectors 111For
numerical efficiency we fix one spin sector to have $\bm{q}=0$ as we observe
no Kekulé spirals at $\nu=0$. We have also performed HF allowing for most
general symmetry breaking to confirm the adequacy of the method.. In
particular, we find ground states with K-IVC ($\bm{q}_{\uparrow}=0$) in the
spin sector at charge neutrality and IKS ($\bm{q}_{\downarrow}\neq 0$) in the
other spin sector.

Commensurate Kekulé Spirals.—A new phase that we identify in TSTG is the
commensurate Kekulé spiral (CKS*) 222The asterisk emphasises $C_{2}T$-symmetry
breaking, though in TSTG we do not find $C_{2}T$-conserving commensurate
Kekulé spiral states. at high interlayer potential and zero strain. In CKS*,
the spiral wavevector relates the $K_{M}$-point of valley $K$ (Fig. 1(c))
where TBG and graphene hybridization is most prominent, to the $\Gamma$-point
of valley $K^{\prime}$. As such, the resulting spiral order is commensurate
with moiré periodicity. We note that CKS* further differs from IKS in that it
breaks $C_{2}T$ symmetry. This is also true of other `starred' states,
including the KS* at $\nu=1,2,3$. At zero strain all KS* states have
commensurate $\bm{q}$ (except $\nu=3$ and large $\Delta V$), but this
commensurability is lost once any finite strain is introduced (though the
graphene scale lock-in mechanism introduced in Ref. [25] may stabilize
commensuration at finite strain). While CKS* is possible at all of
$\nu=1,2,3$, it is only robust (i.e. significant energy advantage over
competing states) at $\nu=2$, where it is the ground state up to about $\Delta
V=240$ meV.

Charge-Transfer Cascades.—At zero interlayer potential, the TBG and graphene
sectors are coupled only by Hartree terms as the mirror symmetry is unbroken.
While the total electron density is fixed, electronic charge could transfer
non-trivially between the two sectors, as first noted in Ref. [36] at $\nu=3$
and large chiral ratio. We now track this charge-transfer mechanism across
general fillings, and with finite strain. We use a HF interpolation scheme
[36] to mitigate finite-size effects on the dispersion and hence more
accurately compute the charge transfer.

Fig. 3 shows the filling of the TBG sector $\nu_{\text{TBG}}$ as a function of
total filling $\nu$. We observe plateaus at integer values of
$\nu_{\text{TBG}}$ near $\nu=2$ and $\nu=3$. This can be rationalized in terms
of the formation of a correlated insulator with a finite charge gap in the TBG
sector, such that injection of a small amount of additional charge only
changes the filling in the graphene sector. The precise width of the plateau
depends on the size of the TBG charge gap and the density of states of
graphene at the chemical potential. We also find that the sum of unsigned
Fermi volumes of all bands — a rough proxy for how `metallic' the system is —
can reach a local minimum at non-integer total fillings, meaning that the most
insulator-like filling does not necessarily occur at integer $\nu$ due to the
charge transfer.

In generating Fig. 3, we have determined the charge-transfer from self-
consistent HF calculations of the full system. However, if we treat the
Hartree interaction between the two sectors as a function of only the
densities of the respective sectors (which can be justified by assuming the
graphene sector charge density is approximately uniform), we can write

$\begin{split}E(n)&=E_{\text{TBG}}(n_{\text{TBG}})+E_{\text{graphene}}(n-n_{\text{TBG}})\\\
&\,\,\,\,+U_{0}n_{\text{TBG}}(n-n_{\text{TBG}})\end{split}$ (2)

where $n$ denotes the total electron density and $U_{0}=V(\bm{q}=0)$. We can
re-write this as

$E(n)=\tilde{E}_{\text{TBG}}(n_{\text{TBG}})+\tilde{E}_{\text{graphene}}(n-n_{\text{TBG}})$
(3)

where
$\tilde{E}_{\text{TBG}}(n_{\text{TBG}})=E_{\text{TBG}}(n_{\text{TBG}})-U_{0}n_{\text{TBG}}^{2}/2$
is the energy of TBG sector with the electrostatic energy of uniform electron
gas subtracted away, and similarly for $\tilde{E}_{\text{graphene}}$. We have
also dropped terms that depend only on total electron density. Minimization of
the total energy corresponds to $dE/dn_{\text{TBG}}=0$, giving

$\tilde{\mu}_{\text{TBG}}=\tilde{\mu}_{\text{graphene}}.$ (4)

The graphene sector can be modelled as a non-interacting Dirac cone, so that
we can estimate the charge transfer from the interacting physics of TBG sector
alone. The result is consistent with full TSTG HF calculations [57].

Figure 3: `Charge transfer plateaus' in TSTG. We plot the fillings of TBG and
graphene sectors from $10\times 10$ self-consistent HF, interpolated to
$60\times 60$, as a function of total fillings. We notice plateaus at integer
TBG fillings, which are due to a correlated charge gap in the TBG sector. In
the inset, we additionally show the total unsigned Fermi volumes of all bands
(`Fermi Volume') as a measure of the metallicity of the system. We notice
that, due to non-trivial charge transfer between the two sectors, minima in
`Fermi Volume' need not occur exactly at total integer fillings.

Comparison with Experiments.—Ref. [25] observes Kekulé patterns using STM in a
TSTG device with heterostrain of $-(0.24\pm 0.08)\%$ 333There is a factor of 2
difference between our convention for strain and that in Ref. [25]., and
$\bm{q}_{\text{Kekul\'{e}}}$ vectors, which are equivalent to our $\bm{q}$
vectors up to choice of reference point, at fillings from $-2$ to $-2.5$. From
HF numerics, we find the ground states of the TSTG under experimental
parameters and fillings are IKS states, and we show the corresponding
$\bm{q}_{\text{Kekul\'{e}}}$ in Figure. 4. We are able to reproduce many
features of the experimentally measured values, such as
$\bm{q}_{\text{Kekul\'{e}}}$ closer to $\Gamma_{M}$ at $\nu=-2$ compared to
non-integer fillings, and a jump in $\bm{q}_{\text{Kekul\'{e}}}$ when doping
away from $\nu=-2$. That $\bm{q}_{\text{Kekul\'{e}}}$ vectors lie close to the
moiré BZ boundary for non-integer fillings is also consistent with experiment.
However, overall we observe an angular offset compared to experiment.

Figure 4: (a) Filling-dependent $\bm{q}_{\text{Kekul\'{e}}}$ vectors, found
from $10\times 10$ HF, of a system with $\epsilon=-0.28\%$ and
$\varphi=97^{\circ}$, consistent with the experimental parameters in Ref.
[25]. We set $\Delta V=0$ as displacement field in the single-gate geometry is
weak ($\Delta V$ up to 40 meV does not alter the results appreciably), and
take the middle-layer graphene to be rotated counter-clockwise, as in
experiment but opposite to our usual convention. Error bars are estimated by
comparing the experimental temperature of 2K to the
$\bm{q}_{\text{Kekul\'{e}}}$-dependence of the HF variational energy. A small
random offset has been added to visually resolve points that overlap due to
the finite discretization. (b,c) $\bm{q}_{\text{Kekul\'{e}}}$-dependence of
the energies of IKS states for $\nu=-2$ and $\nu=-2.5$ respectively, where
$\Delta E=E(\bm{q}_{\text{Kekul\'{e}}})-E_{\text{min}}$.

Concluding Remarks.—Motivated by the experimental discovery of Kekulé spiral
order in Ref. [25] in TSTG, we have systematically studied its ground state
phase diagram under strain and interlayer potential in search for potential
Kekulé spiral order. At zero interlayer potential, since the system decomposes
into TBG and graphene sectors, the physics is similar to that of TBG,
including the emergence of IKS order at finite strain. At finite interlayer
potential, we find new effects such as the breaking of $C_{2}T$ symmetry and
the formation of a commensurate Kekulé spiral state. Our prediction that the
Kekulé spiral state at $\nu=2$ breaks $C_{2}T$ symmetry and opens up a charge
gap is in agreement with experimental results [21]. The commensurate spiral
state at zero strain is an addition to the family of Kekulé spiral states
introduced by the discovery of the incommensurate Kekulé spiral, and hints at
a broader relevance of KS order beyond strained samples. While the high
interlayer potential required to stabilize the phase in TSTG necessitates a
dual-gate device, precluding the direct detection of its Kekulé pattern using
STM, the mechanism behind the formation of CKS order is more general, and
could emerge in other moiré systems, especially in cases where $C_{2}$
symmetry is broken explicitly [61]. We have further investigated the normal
(non-superconducting) state of TSTG at non-integer fillings. We establish a
charge-transfer cascade mechanism between the TBG and graphene sectors in the
absence of interlayer potential, and show that this is consistent with a
picture where the two sectors are coupled by Hartree interactions and the TBG
sector density of states tracks that of standalone TBG. Notably, we find that
integer fillings of the TBG sector and `maximally insulating' densities need
not coincide with overall integer filling, which has implications for
experimentally mapping the phase diagram of TSTG.

###### Acknowledgements.

Acknowledgements.— We thank Étienne Lantagne-Hurtubise for useful discussions
about Ref. [25], and akcnowledge support from EPSRC Grant EP/S020527/1, the
European Research Council (ERC) under the European Union Horizon 2020 Research
and Innovation Programme (Grant Agreement Nos. 804213-TMCS), and Leverhulme
Trust International Professorship grant (number LIP-202-014). GW acknowledges
funding from the University of Zurich postdoc grant FK-23-134. For the purpose
of Open Access, the author has applied a CC-BY public copyright licence to any
Author Accepted Manuscript version arising from this submission.

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  * Note [1] For numerical efficiency we fix one spin sector to have $\bm{q}=0$ as we observe no Kekulé spirals at $\nu=0$. We have also performed HF allowing for most general symmetry breaking to confirm the adequacy of the method.
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  * Note [4] In the case where no interlayer potential is applied, we can work around this by defining a particle-hole symmetry operator that is the direct sum of the correct operators on each sector, i.e. $P^{\prime}=(P)_{\text{TBG}}\oplus(C_{2x}P)_{D}$.

— Supplementary Material —

##  Appendix A Strained BM Model for TSTG

The unstrained BM model in one spin and valley for twisted trilayer graphene
is given by [12]

$H=\begin{pmatrix}-iv_{0}\bm{\sigma}_{\theta/2}\bm{\nabla_{\bm{r}}}+\Delta
V/2&T(\bm{r})&0\\\
T^{\dagger}(\bm{r})&-iv_{0}\bm{\sigma}_{-\theta/2}\bm{\nabla_{\bm{r}}}&T^{\dagger}(\bm{r})\\\
0&T(\bm{r})&-iv_{0}\bm{\sigma}_{\theta/2}\bm{\nabla_{\bm{r}}}-\Delta
V/2\end{pmatrix}$ (5)

where
$\bm{\sigma}_{\theta/2}=e^{-(i\theta/4)\sigma_{z}}(\sigma_{x},\sigma_{y})e^{(i\theta/4)\sigma_{z}}$,
$T(\bm{r})=\sum_{n=1}^{3}T_{n}e^{-i\bm{q}_{n}\bm{r}}$, and $\Delta V$ is the
interlayer potential. The interlayer hopping matrices are given by
$T_{n+1}=w_{AA}\sigma_{0}+w_{AB}(\sigma_{x}\cos(n\phi)+\sigma_{y}\sin(n\phi))$
where $\phi=2\pi/3$ and the hopping vectors are given by
$\bm{q}_{1}=k_{\theta}(0,-1)$, $\bm{q}_{2,3}=k_{\theta}(\pm\sqrt{3}/2,1/2)$,
where $k_{\theta}=2k_{D}\sin(\theta/2)$ and $k_{D}=4\pi/3a_{0}$ is the Dirac
momentum. Further, in all the calculations, we will ignore the Pauli twist in
the kinetic energy term, i.e. replace $\bm{\sigma}_{\pm\theta/2}$ with simply
$\bm{\sigma}$.

The effect of strain is incorporated in two ways. Firstly, moiré lattice
vectors are modified, the details of which are developed and discussed in Ref.
[62, 63]. Secondly, strain shifts the position of Dirac points through a
vector potential [64], which is given by

$\bm{A}=\frac{\beta}{2a}(\epsilon_{xx}-\epsilon_{yy},-2\epsilon_{xy})$ (6)

Here, the $2\times 2$ strain tensor is given by

$S(\epsilon,\varphi)=R^{-1}(\varphi)\begin{pmatrix}-\epsilon&0\\\
0&\nu\epsilon\end{pmatrix}R(\varphi)$ (7)

where $\epsilon$ is the magnitude of strain and $\varphi$ the strain angle,
measured from the $x$-axis. $R(\varphi)$ is the usual rotation matrix. We will
consider heterostrain that is equal in the top and bottom layers and opposite
in the middle layer, with $\epsilon_{\text{top}}=\epsilon/2$,
$\epsilon_{\text{middle}}=-\epsilon/2$ and
$\epsilon_{\text{bottom}}=\epsilon/2$, and we will take $\varphi=0$ for all
layers unless otherwise noted. The strain vector potential is incorporated
such that $-i\bm{\nabla_{\bm{r}}}\rightarrow-i\bm{\nabla_{\bm{r}}}-\bm{A}$,
where $\bm{A}$ is the strain vector potential for the corresponding layer. In
the main text, we have considered $\epsilon>0$, corresponding to compression
of top and bottom layers and extension of the middle layer.

From Ref. [12], we know that TSTG can be decomposed into a TBG sector and a
graphene sector. The TBG sector has effective tunnelling strengths (i.e.
$w_{AA}$ and $w_{BB}$) scaled by a factor of $\sqrt{2}$, and the corresponding
magic angle is also scaled by a factor of $\sqrt{2}$. To reproduce the same
single-particle band-structure, strain should also be scaled by a factor of
$\sqrt{2}$. This is to say, there is a correspondence between a TBG system of
angle $\theta$ and strain $\epsilon$ with the TBG sector of a TSTG system of
angle $\sqrt{2}\theta$ and strain $\sqrt{2}\epsilon$. An intuitive
understanding of the scaling of strain is the following: the size of the moiré
BZ is proportional to $\theta$, while the shift of Dirac points due to strain
vector potential $\bm{A}$ is proportional to $\epsilon$. As such, the ratio
$\epsilon/\theta$ is a good measure of the effect of strain. This explains why
we found phase transitions tending to happen at a larger strain in TSTG
compared to TBG.

##  Appendix B Effect of Homostrain on Single-Particle Band Structure

In the main text, we have exclusively focused on heterostrain, where adjacent
layers of graphene experience opposite strains. Homostrain, i.e. strain acting
equally on all layers, has a much smaller effect on the band structure. In
Figure S1, we show, within the continuum model, the central bands of the TBG
sector of a TSTG system at $0.1\%$ homostrain. The band structure very closely
resembles that of unstrained system, shown in Figure 1(d) in the main text.
This is to be contrasted with heterostrained band structure in Figure 1(e).

Figure S1: Single-particle central bands of a TSTG system under homostrain of
$0.1\%$ ($\epsilon_{1}=\epsilon_{2}=\epsilon_{3}=0.1\%$). The band structure
closely resembles that of unstrained system.

##  Appendix C Hartree-Fock and Subtraction Scheme

We perform Hartree-Fock (HF) iterations self-consistently. To keep the
computation time manageable, we only retain a small number of active single-
particle bands obtained from the BM model in previous section. Unless
otherwise stated, we keep 4 bands per spin/valley in our HF calculations.

Since the BM model band structures have already taken into account of some of
the interaction, it is necessary to subtract away such effect. In practice,
this amounts to replacing the density matrix $P$ with $P-P_{0}$ in computing
the interaction term, where $P_{0}$ is the reference matrix. A few different
methods exist, but in this work we have employed the average scheme: two
central bands per spin/valley have occupation $1/2$, while bands above (below)
the central bands are empty (filled). This is similar to the scheme employed
in Ref. [36].

##  Appendix D Symmetry Properties of States Found in Phase Diagram

In Table S1, we have summarized the symmetry properties of various states
found in the main phase diagram.

Phase | $\nu_{s}$ | Valley Pol. | $C_{2}T$ | $U_{V}(1)$ | $T$ | $T^{\prime}$
---|---|---|---|---|---|---
(I)KS(*) | 1 | $0^{\dagger}$ | * | ✗ | ✓† | ✗
VP(*) | 1 | 1 | * | ✓ | ✗ | ✗
K-IVC(*) | 0 | 0 | * | ✗ | ✗ | ✓
SM | 0 | 0 | ✓ | ✓ | ✓ | ✓
Table S1: The symmetries of various states that appear in the phase diagram.
Since all states considered here can be reduced to colinear spin
configurations with appropriate
$\text{SU}(2)_{K}\times\text{SU}(2)_{K^{\prime}}$ rotations, we denote the
filling of the spin sector where the state could occur as $\nu_{s}$. For
states that break $U_{V}(1)$ symmetry, an appropriate $U_{V}(1)$ rotation is
allowed before $C_{2}T$ symmetry is considered. Asterisk(*) labels variants of
states that break $C_{2}T$ symmetry (e.g. VP preserves $C_{2}T$-symmetry but
VP* breaks $C_{2}T$-symmetry). $\dagger$: At some regions, KS* states could
have finite valley polarization and break time-reversal symmetry along the
way. Refer to Figure 2 of the main text and the next section on phase
boundaries for more discussion.

##  Appendix E Phase Boundaries at $\nu=2$

At $\nu=2$, in the main phase diagram, we show the transition from K-IVC to
KS* as we increase the interlayer potential at zero or small strain. In fact,
we also observe intermediate phases of K-IVC* (which is the $C_{2}T$-symmetry
breaking variant of K-IVC) and KS*-VP (which is a KS* state with finite valley
polarization) in our HF numerics, as shown in Figure S2. We believe that the
order of transitions depends on the details of modelling, and in an actual
experiment the intermediate phases may not be present.

Figure S2: Properties and ground states at $\nu=2$ without strain from
$10\times 10$ HF results. The phases in each parameter range is labelled above
the graph.

##  Appendix F Momentum Space Properties of Commensurate Kekulé Spiral

In Figure. S3, we show the strength of intervalley coherence (IVC) in the
moiré BZ for the CKS* state. We note that the regions where IVC strength is
significantly reduced correspond to the $K_{M}$-point of $K$-valley or
$K^{\prime}_{M}$-point of $K^{\prime}$-valley.

Figure S3: Momentum resolved intervalley coherence strength of CKS* from
$24\times 24$ self-consistent HF. The blue dots mark $K_{M}/\Gamma_{M}$
position (for $K$ and $K^{\prime}$ valleys), and the red dot marks
$\Gamma_{M}/K^{\prime}_{M}$ position.

##  Appendix G Behaviours with Large Strains

Here we consider the behaviour with strains larger than those considered in
the main text, up to $0.6\%$, at zero interlayer potential. As shown in Figure
S4, at $\nu=0$, there is a transition from K-IVC to symmetric semi-metal when
going from the unstrained system to a system with large strain of about
$0.5\%$. The Kekulé vector $\bm{q}$ remains zero wherever IVC is non-
vanishing. At $\nu=1$, there is onset of IKS from $0.4\%$ strain onwards.

Figure S4: Systems with fillings $\nu=0$ and $\nu=1$ with strain up to $0.6\%$
and zero interlayer potential. The results are obtained from $12\times 12$
self-consistent HF. On the left, ``IVC" is defined as
$||P_{+-}||^{2}/(N_{1}N_{2})$, the Frobenius norm squared of the intervalley
part of the density matrix divided by the system size. On the right, $\Delta
E$ is the energy difference between the $\bm{q}=0$ state with the overall
energy minimum. For the plot of $\nu=0$, we choose to retain, instead of all
bands closest to charge neutrality, 2 TBG bands and 2 graphene bands
respectively closest to charge neutrality per spin and valley.

##  Appendix H Comparison of Two Methods to Compute Charge Transfer

As explained in the main text, the charge transfer between the TBG sector and
graphene sector of a TSTG system can be understood in terms of matching the
chemical potential of the two sectors. This allows the determination of the
charge transfer with only Hartree-Fock calculations of TBG, where in addition
the graphene sector can be modelled as simply non-interacting. The results of
this method, labelled as `TBG HF', are compared with full TSTG HF calculation,
shown in Figure S5, and we obtain good agreement. The remaining discrepancy is
likely the result of nonequivalent band cut-off implemented for TBG and TSTG
calculations.

Figure S5: Comparison of fillings of TBG and graphene sectors obtained from
two different methods. The results are from $10\times 10$ HF calculations with
strain of $0.3\%$ and zero interlayer potential.

##  Appendix I Particle-Hole Symmetry Breaking and Negative Fillings

As was shown in Ref. [31], TSTG with interlayer potential but without strain
has exact particle hole symmetry, which is given by a combined operator
$m_{z}C_{2x}P$, where $P$ is the usual particle-hole symmetry, $m_{z}$ is
mirror reflection in the $z$-plane, and $C_{2x}$ is $\pi$-rotation with
respect to the $x$-axis. Now, applying strain would break $C_{2x}$ symmetry
for Hamiltonian of the TBG sector ($\hat{H}_{\text{TBG}}$), and break the
combined symmetry for $\hat{H}_{\text{TBG}}$ (strained graphene sector
$\hat{H}_{D}$ also breaks $m_{z}C_{2x}P$ symmetry, though the reasoning is
less trivial as it starts without $C_{2x}$ symmetry in the unstrained case).
As such, there is no particle-hole symmetry for TSTG with both strain and
interlayer potential 444In the case where no interlayer potential is applied,
we can work around this by defining a particle-hole symmetry operator that is
the direct sum of the correct operators on each sector, i.e.
$P^{\prime}=(P)_{\text{TBG}}\oplus(C_{2x}P)_{D}$..

It is easily seen that under $C_{2x}$ symmetry, for the TBG sector, strain is
mapped from $(\epsilon,\varphi)$ to $(-\epsilon,-\varphi)$, where $\epsilon$
is the magnitude of strain and $\varphi$ is the direction of strain measured
from the $x$-axis. It can be shown that under the combined $m_{z}C_{2x}P$ the
strain of the graphene sector transforms an the identical way, establishing a
correspondence between system at $(\nu,\epsilon,\varphi)$ and system at
$(-\nu,-\epsilon,-\varphi)$, where $\nu$ is the filling factor as usual.

Lacking exact particle-hole symmetry, we computed the phase diagrams for the
negative fillings as well, as shown in Figure S6. We see that there is no
qualitative difference to that of the positive fillings. From the last
paragraph, we understand that the results could also be applied to systems
with positive fillings but negative strains.

Figure S6: The phase diagram at negative fillings. The acronyms are the same
as that used in the main text, with non-IVC* denotes a $C_{2}T$-breaking state
that preserves other symmetries. Despite lack of exact particle-hole symmetry,
there is little qualitative difference to the phase diagrams of positive
fillings. Due to the rather coarse grid used in parameter space, the phase
boundaries should be interpreted as a rough guide.

##  Appendix J Translation Symmetry Breaking at High Interlayer Potential

We perform self-consistent Hartree-Fock calculations allowing for all
translation and spin symmetry breaking, and we observe the ground state to
have complex spin textures at $\nu=1$ with large interlayer potential ($>200$
meV) and finite strain. The system has significant spatial fluctuations of
spin densities, with much smaller total charge density modulation, which may
be an artefact of our numerical methods. The spin patterns partially resemble
spin spirals, but the spin rotations are not co-planar, and the results show
strong finite size dependence. As this is not the main purpose of our work,
the result remains inconclusive, but this could be the subject of future
studies with careful control of finite size effects.

Figure S7: Spin texture of TSTG with $\epsilon=0.16\%$ and $\Delta V=240$ meV,
obtained from $9\times 9$ self-consistent HF, allowing for most general
symmetry breakings. Since the system has
$\text{SU}(2)_{K}\times\text{SU}(2)_{K^{\prime}}$ symmetry, allowing for
independent spin rotations in each valley, we have only shown the spin texture
of $K$-valley here.

##  Appendix K Additional Results for Non-integer Fillings

Figure S8: Total unsigned Fermi volume of TSTG system at zero interlayer
potential, obtained from $10\times 10$ HF.