2: Initialize the Diffie-Hellman key exchange protocol parameters including
the elliptic curve group $\mathbb{G}^{\prime}$ of prime order $q^{\prime}$ and
a generator $g^{\prime}\in\mathbb{G}^{\prime}$

3: Initialize four empty maps $\mathbf{State}$, $\mathbf{TdMap}$,
$\mathbf{OriCip}$, and $\mathbf{EDB}$

4: Initialize an empty list $\mathbf{CList}$

5: Initialize timestamp $u\leftarrow 0$

6: Initialize the search key $K_{1}\overset{\$}{\leftarrow}\mathbb{Z}^{*}_{q}$
and the encryption key $K_{2}\overset{\$}{\leftarrow}\mathbb{Z}^{*}_{q}$

7: Set the simulated key $K_{\Sigma}\leftarrow(K_{1},K_{2})$

8: Send the simulated database $\mathbf{EDB}$ to the server

$\mathcal{S}^{\prime}.\textsf{DataUpdate}(\mathcal{L}^{DaUpdt}_{\text{Stl}}(op,(w,id))=\text{NULL})$

Simulated Client:

1: Accumulate timestamp $u\leftarrow u+1$ and insert
$\mathbf{OriCip}[u]\leftarrow u$

2: Randomly select three elements
$(L_{0},D_{0},C_{0})\overset{\$}{\leftarrow}\mathbb{G}\times\mathbb{G}\times\mathbb{G}$

3: Compute the simulated ciphertext $L=L^{K_{1}}_{0}$, $D=D^{K_{1}}_{0}$, and
$C=C^{K_{1}}_{0}$

4: Insert records $(0,u,L_{0},D_{0},C_{0})$ and $(1,u,L,D,C)$ into
$\mathbf{CList}$

5: Send the simulated ciphertext $(L,D,C)$ to the simulated server Simulated
Server:

6: Store the simulated ciphertext into the simulated database
$\mathbf{EDB}[L]\leftarrow(D,C)$

$\mathcal{S}^{\prime}.\textsf{Search}(\mathcal{L}^{Srch}_{\text{Stl}}(w)=\text{NULL})$

Simulated client:

1: Establish a temporary secure channel with the simulated server using the
Diffie-Hellman key exchange protocol

2: Accumulate timestamp $u\leftarrow u+1$

3: Randomly sample three elements
$(e_{1},e_{2},e_{3})\overset{\$}{\leftarrow}\mathbb{G}\times\mathbb{G}\times\mathbb{G}$

4: Send the simulated search trapdoor $(K_{1},e_{1},e_{2},e_{3})$ to the
simulated server via above secure channel Simulated Server:

5: Return $a_{\text{max}}$ random elements of $\mathbb{G}$ to the simulated
client via above secure channel

$\mathcal{S}^{\prime}.\textsf{KeyUpdate}(\mathcal{L}^{KeyUpdt}_{\text{Stl}}=\text{NULL})$

Simulated Client:

1: Establish a temporary secure channel with the server using the Diffie-
Hellman key exchange protocol

2: Accumulate timestamp $u\leftarrow u+1$

3: Randomly draw the KeyUpdate token $\Delta$ from $\mathbb{Z}^{*}_{q}$

4: Update search key $K_{1}\leftarrow K_{1}\cdot\Delta$ and encryption key
$K_{2}\leftarrow K_{2}\cdot\Delta$

5: Send $\Delta$ to the server via the above secure channel Simulated Server:

6: for all ciphertext $(L,D,C)\textbf{ such that
}(D,C)\leftarrow\mathbf{EDB}[L]$ do

7: Accumulate $u\leftarrow u+1$

8: Update ciphertexts $L^{\prime}\leftarrow L^{\Delta}$, $D^{\prime}\leftarrow
D^{\Delta}$, and $C^{\prime}\leftarrow C^{\Delta}$

9: Insert ciphertext
$\mathbf{EDB}[L^{\prime}]\leftarrow(D^{\prime},C^{\prime})$

10: Insert record $(1,u,L^{\prime},D^{\prime},C^{\prime})$ into
$\mathbf{CList}$

11: Retrieve $(1,u^{\prime},L,D,C)$ from $\mathbf{CList}$

12: Retrieve $u_{0}\leftarrow\mathbf{OriCip}[u^{\prime}]$ and insert
$\mathbf{OriCip}[u]\leftarrow u_{0}$

13: Remove $\mathbf{EDB}[L]$

14: end for

$\mathbf{S}^{\prime}.\textsf{KeyLeak}(\mathcal{L}^{KeyLeak}_{\text{Stl}}=(\textsf{CUHist}(U_{\text{now}}),\textsf{DUHist}(U_{\text{now}})))$

1: Accumulate timestamp $u\leftarrow u+1$

2: Initialize a list
$\mathbf{LeakedData}\leftarrow\textsf{CUHist}(U_{\text{now}})\cup\textsf{DUHist}(U_{\text{now}})$

3: for each keyword $w$ existing in $\mathbf{LeakedData}$ do

4: Initialize an empty list $\mathbf{DB}_{w}$

5: for each records $(u^{\prime},op,(w,id))\in\mathbf{LeakedData}$ do

6: Retrieve $u^{\prime}_{0}\leftarrow\mathbf{OriCip}[u^{\prime}]$ and insert
$(u^{\prime}_{0},op,id)$ into $\mathbf{DB}_{w}$

7: end for

8: for all triplets $(u^{\prime}_{0},op,id)\in\mathbf{DB}_{w}$ in the
ascending order of $u^{\prime}_{0}$ do

9: Retrieve search token
$tk_{w}^{\prime}\leftarrow\mathbf{TdMap}[u^{\prime}_{0}]$

10: If $tk_{w}^{\prime}\neq\text{NULL}$, continue

11: Otherwise, randomly sample the search token
$tk_{w}^{\prime}\overset{\$}{\leftarrow}\\{0,1\\}^{\lambda}$

12: Insert $\mathbf{TdMap}[u^{\prime}_{0}]\leftarrow tk_{w}^{\prime}$

13: Retrieve record $(0,u^{\prime}_{0},L_{0},D_{0},C_{0})$ from
$\mathbf{CList}$

14: Retrieve record $(tk_{w},cnt_{w})$ from $\mathbf{State}[w]$

15: if $(tk_{w},cnt_{w})=(\text{NULL},\text{NULL})$ then

16: Randomly sample search token $tk_{w}\leftarrow\\{0,1\\}^{\lambda}$

17: Initialize DataUpdate counter $cnt_{w}\leftarrow 0$

18: Program random oracles $\mathbf{H}_{1}(tk_{w})$, $\mathbf{H}_{2}(tk_{w})$,
and $\mathbf{G}(tk_{w})$ with randomly selected elements from $\mathbb{G}$

19: end if

20: Program random oracle $\mathbf{H}_{1}(tk_{w}^{\prime})\leftarrow L_{0}$

21: Program random oracle
$\mathbf{H}_{2}(tk_{w}^{\prime})\leftarrow\frac{D_{0}}{\pi(tk_{w})}$

22: Program random oracle
$\mathbf{G}(tk_{w}^{\prime})\leftarrow\frac{C_{0}}{\pi(op||id)^{\frac{K_{2}}{K_{1}}}}$

23: Update $\mathbf{State}[w]\leftarrow(tk_{w}^{\prime},cnt_{w}+1)$

24: end for

25: end for

26: Expose $(K_{1},K_{2},\mathbf{State})$ to stealer
$\mathcal{A}_{\text{Stl}}$

Protocol $\mathcal{S}.\textsf{DataUpdate}$ simulates the ciphertexts by
sampling three random elements from $\mathbb{G}$ and computing the $K_{1}$
exponentiations of those random elements. Due to the one-wayness and collision
resistance of $\mathbf{H}_{1}$, $\mathbf{H}_{2}$, and $\mathbf{G}$ and the
randomness of the selected search token, a ciphertext $(L,D,C)$ generated by
the real protocol DataUpdate is indistinguishable from random numbers before
the client issues a Search query related to this ciphertext. Thus, this
protocol is indistinguishable from the real Bamboo.DataUpdate.

Protocol $\mathcal{S}.\textsf{Search}$ simulates Bamboo.Search. $\mathcal{S}$
uses the minimum timestamp $u^{\text{srch}}_{\text{min}}$ of search pattern
$\textsf{sp}(w)$ to indicate the latest simulated search trapdoor of $w$ in
the map $\mathbf{SHist}$ (Steps 4 to 6). The map $\mathbf{SHist}$ also records
a timestamp $u_{s}$ to the newest simulated ciphertext whose corresponding
search trapdoor has been programmed into random oracles $\mathbf{H}_{1}$,
$\mathbf{H}_{2}$, and $\mathbf{G}$. Any simulated ciphertext of $w$ of which
the timestamp is older than $u_{s}$ will not be re-programmed into the oracles
in the following steps. In Steps 7 to 15, $\mathcal{S}$ randomly generates a
search token for each simulated ciphertext and programs the search token into
random oracles $\mathbf{H}_{1}$, $\mathbf{H}_{2}$, and $\mathbf{G}$. The
programming process follows the rule that for any two successive ciphertexts
of the same keyword, the latter ciphertext encrypts the former ciphertext’s
search token. Under this rule, the server can correctly disclose the whole
hidden chain of the same keyword with the search token of the latest simulated
ciphertext of $w$. The above guarantees that the server cannot distinguish the
simulated search process from the real one. Besides,
$H_{L}^{K_{1}},H_{D}^{K_{1}},\text{and }H_{C}^{K_{1}}$ in the simulated search
trapdoor are also indistinguishable from $L,Msk_{D},\text{and }Msk_{C}$ of a
real search trapdoor. Since the base numbers of $L,Msk_{D},\text{and }Msk_{C}$
are hashed by randomly selected numbers (i.e., indistinguishable from random
numbers), the simulated protocol Search is indistinguishable from the real
one.

In terms of the simulated protocol KeyUpdate, it is clear that what the server
$\mathcal{A}_{\text{Srv}}$ observes in $\mathcal{S}.\textsf{KeyUpdate}$ is
identical to what $\mathcal{A}_{\text{Srv}}$ can observe in
$\texttt{Bamboo}.\textsf{KeyUpdate}$. Thus, $\mathcal{S}$ simulates an
indistinguishable protocol KeyUpdate.

To summarize, we construct an $\mathcal{S}$ that simulates an
indistinguishable ideal game from the real game in view of the server with the
given leakage functions $\mathcal{L}_{\text{Srv}}$. Hence, Bamboo is
$\mathcal{L}_{\text{Srv}}$-adaptive where Leakage functions
$\mathcal{L}_{\text{Srv}}=(\mathcal{L}^{Stp}_{\text{Srv}},\mathcal{L}^{DaUpdt}_{\text{Srv}},$
$\mathcal{L}^{Srch}_{\text{Srv}},\mathcal{L}^{KeyUpdt}_{\text{Srv}})$ are
defined as:

$\displaystyle\mathcal{L}^{Stp}_{\text{Srv}}(\lambda,a_{\text{max}})=(\lambda,a_{\text{max}}),~{}\mathcal{L}^{DaUpdt}_{\text{Srv}}(op,(w,id))=\text{NULL},$
$\displaystyle\mathcal{L}^{KeyUpdt}_{\text{Srv}}=\mathcal{L}^{\prime}_{\text{Srv}}(\textsf{KUHist}(U_{\text{now}})),$
$\displaystyle\mathcal{L}^{Srch}_{\text{Srv}}(w)=\mathcal{L}^{\prime\prime}_{\text{Srv}}(\textsf{sp}(w),\textsf{TimeDB}(w),\qquad\qquad\qquad\quad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\textsf{DUTime}(w),\textsf{KUHist}(U_{\text{now}})),$

where $\mathcal{L}^{\prime}_{\text{Srv}}$ and
$\mathcal{L}^{\prime\prime}_{\text{Srv}}$ are two stateless functions. ∎

2) Proof of $\mathcal{L}_{\text{Stl}}$-Adaptive Security

###### Proof.

We construct a simulator $\mathcal{S}^{\prime}$ to prove that Bamboo is
$\mathcal{L}_{\text{Stl}}$-adaptively secure, where leakage functions
$\mathcal{L}_{\text{Stl}}=(\mathcal{L}^{Stp}_{\text{Stl}},\mathcal{L}^{DaUpdt}_{\text{Stl}},$
$\mathcal{L}^{Srch}_{\text{Stl}},$
$\mathcal{L}^{KeyUpdt}_{\text{Stl}},\mathcal{L}^{KeyLeak}_{\text{Stl}})$ are
defined as follows:

$\displaystyle\mathcal{L}^{Stp}_{\text{Stl}}(\lambda,a_{\text{max}})=(\lambda,a_{\text{max}}),~{}\mathcal{L}^{DaUpdt}_{\text{Stl}}(op,(w,id))=\text{NULL},$
$\displaystyle\mathcal{L}^{Srch}_{\text{Stl}}(w)=\text{NULL},~{}\mathcal{L}^{KeyUpdt}_{\text{Stl}}=\text{NULL},$
$\displaystyle\mathcal{L}^{KeyLeak}_{\text{Stl}}=\mathcal{L}^{\prime}_{\text{Stl}}(\textsf{CUHist}(U_{\text{now}}),\textsf{DUHist}(U_{\text{now}})),$

where $\mathcal{L}^{\prime}_{\text{Stl}}$ is a stateless function.

Algorithm 6 presents the construction of $\mathcal{S}^{\prime}$. In the
construction of simulator $\mathcal{S}^{\prime}$, we utilize the DDH
assumption and model hash functions $\mathbf{H}_{1}$, $\mathbf{H}_{2}$, and
$\mathbf{G}$ as random oracles.

The simulated protocol Setup initializes the map $\mathbf{State}$ and two keys
$K_{1}$ and $K_{2}$ to simulate the private state $\mathbf{State}$ and the
secret key $K_{\Sigma}$, respectively. $\mathcal{S}^{\prime}$ uses
$K_{\Sigma}$ and $\mathbf{State}$ to simulate a key-compromise event. The list
$\mathbf{CList}$ records the ciphertexts and their corresponding generation
timestamp. The map $\mathbf{TdMap}$ records the search token used to generate
ciphertexts. The map $\mathbf{OriCip}$ maps the timestamp of ciphertext to the
timestamp of the original DataUpdate query that the former is key-updated
from. In the views of stealer $\mathcal{A}_{\text{Stl}}$, the simulation is
indistinguishable from Bamboo.Setup.

In the simulated DataUpdate, $\mathcal{S}^{\prime}$ simulates the ciphertext
by sampling three random numbers from $\mathbb{G}$ and computes the $K_{1}$
exponentiations of those elements (Steps 2 and 3). In Step 4,
$\mathcal{S}^{\prime}$ inserts the three elements and the simulated ciphertext
along with the timestamps to the $\mathbf{CList}$. The simulated ciphertext is
indistinguishable from the real one via Bamboo.DataUpdate even if
$\mathcal{A}_{\text{Stl}}$ compromises the secret key and the private state.
As a newly generated ciphertext is encrypted with a fresh random search token,
the ciphertext is indistinguishable from a random number unless
$\mathcal{A}_{\text{Stl}}$ holds the corresponding search token.
$\mathcal{A}_{\text{Stl}}$ cannot access the search token unless it obtains
the secret key and private state again.

The simulated Search is indistinguishable from Bamboo.Search. With the
protection of the secure channel, $\mathcal{A}_{\text{Stl}}$ cannot learn the
search trapdoor and the search results. The simulated client and server only
need to select $a_{\text{max}}$ random data and transfer those data via the
secure channel.

Similarly, the simulated protocol KeyUpdate is protected by the secure
channel, i.e., $\mathcal{A}_{\text{Stl}}$ cannot get the KeyUpdate token.
Without the token, $\mathcal{A}_{\text{Stl}}$ cannot tell whether a post-key-
updated ciphertext is updated from a pre-key-updated ciphertext even if
$\mathcal{A}_{\text{Stl}}$ obtains _one of_ the secret keys of post-key-
updated ciphertext and the pre-key-updated ciphertext. This security property
can be reduced to the DDH assumption in the group $\mathbb{G}$. Specifically,
suppose a searchable ciphertext $(L,D,C)$ can be written in the form of
$(g^{x},g^{y},g^{z})$, where $g$ is a generator of $\mathbb{G}$ and
$x,y,z\in\mathbb{Z}^{*}_{q}$. Notice that even if $\mathcal{A}_{\text{Stl}}$
learns the underlying secret key $(K_{1},K_{2})$, the entry $(op,(w,id))$, the
search token $tk_{w}^{\prime}$, and the next search token $tk_{w}$ used to
generate $(L,D,C)$, $\mathcal{A}_{\text{Stl}}$ cannot exactly know $x$, $y$,
and $z$. Specifically, suppose $\mathbf{H}_{1}(tk_{w}^{\prime})=g^{h_{1}}$,
$\mathbf{H}_{2}(tk_{w}^{\prime})=g^{h_{2}}$,
$\mathbf{G}(tk_{w}^{\prime})=g^{h_{3}}$, $\pi(tk_{w})=g^{n_{1}}$, and
$\pi(op||id)=g^{n_{2}}$, we have $x=K_{1}\cdot h_{1}$, $y=(n_{1}+h_{2})\cdot
K_{1}$, and $z=K_{2}\cdot n_{2}+K_{1}\cdot h_{3}$, wherein $h_{1}$, $h_{2}$,
and $h_{3}$ are independent and random to $\mathcal{A}_{\text{Stl}}$. Thus,
$x$, $y$, and $z$ are independent and random to $\mathcal{A}_{\text{Stl}}$ due
to the properties of the random oracles. Suppose $\mathcal{S}^{\prime}$ holds
the KeyUpdate token $\Delta\in\mathbb{Z}^{*}_{q}$, a random bit
$b\leftarrow\\{0,1\\}$ and two different ciphertexts
$(L_{0},D_{0},C_{0})=(g^{x_{0}},g^{y_{0}},g^{z_{0}})$ and
$(L_{1},D_{1},C_{1})=(g^{x_{1}},g^{y_{1}},g^{z_{1}})$ where
$x_{0},y_{0},z_{0},x_{1},y_{1},z_{1}\in\mathbb{Z}^{*}_{q}$, and
$\mathcal{A}_{\text{Stl}}$ is given $(g^{x_{0}},g^{y_{0}},g^{z_{0}})$,
$(g^{x_{1}},g^{y_{1}},g^{z_{1}})$, and
$(g^{\Delta},g^{x_{b}\cdot\Delta},g^{y_{b}\cdot\Delta},g^{z_{b}\cdot\Delta})$
(note $\mathcal{A}_{\text{Stl}}$ actually never has access to $g^{\Delta}$ in
both real and ideal games). The advantage that $\mathcal{A}_{\text{Stl}}$
guesses a correct $b^{*}\in\\{0,1\\}$ such that $b^{*}=b$ is directly reduced
to $\mathcal{A}_{\text{Stl}}$’s advantage to solve three DDH assumption
instances, which is negligible. Thus, the simulated KeyUpdate is
indistinguishable from the real one in the view of $\mathcal{A}_{\text{Stl}}$.

When a key-compromise event happens, the function KeyLeak will be called with
the input of leaked information. The goal of function KeyLeak is to expose a
secret key and a private state to $\mathcal{A}_{\text{Stl}}$ that allows
$\mathcal{A}_{\text{Stl}}$ to correctly perform search over and decrypt the
encrypted database $\mathbf{EDB}$. $\mathcal{S}^{\prime}$ first extracts
entries containing the same keywords $w$ from both
$\textsf{CUHist}(U_{\text{now}})$ and $\textsf{DUHist}(U_{\text{now}})$, and
stores those entries into $DB_{w}$ (Steps 4 to 7). Secondly, for each entry in
$DB_{w}$, $\mathcal{S}^{\prime}$ prepares the search token $tk^{\prime}$ that
can be used to generate that entry. If an entry has already been assigned a
search token, $\mathcal{S}^{\prime}$ skips this entry and processes the
successive one (Steps 8 to 10). Then, $\mathcal{S}^{\prime}$ uses the
information in $\mathbf{State}[w]$ to construct the hidden chain of $w$’s
ciphertexts (Steps 14 to 23). It programs the random oracles to construct the
hidden chain of $w$’s ciphertext (Steps 20 to 22). Finally,
$\mathcal{S}^{\prime}$ exposes the secret key $K_{\Sigma}=(K_{1},K_{2})$ and
private state $\mathbf{State}$ to $\mathcal{A}_{\text{Stl}}$. After KeyLeak,
the encrypted database contains real entries, and $\mathcal{A}_{\text{Stl}}$
can perform correct searches and decryption over the encrypted database with
the exposed search key $K_{1}$ and encryption key $K_{2}$ and private state
$\mathbf{State}$. Thus, $\mathcal{A}_{\text{Stl}}$ cannot distinguish the
simulated encrypted database from the real one.

To summarize, we construct an $\mathcal{S}^{\prime}$ that is indistinguishable
from the real game in the views of $\mathcal{A}_{\text{Stl}}$. Thus, Bamboo is
$\mathcal{S}_{\text{Stl}}$-adaptively secure where
$\mathcal{L}_{\text{Stl}}=(\mathcal{L}^{Stp}_{\text{Stl}},\mathcal{L}^{DaUpdt}_{\text{Stl}},\mathcal{L}^{Srch}_{\text{Stl}},$
$\mathcal{L}^{KeyUpdt}_{\text{Stl}},\mathcal{L}^{KeyLeak}_{\text{Stl}})$ is
defined as:

$\displaystyle\mathcal{L}^{Stp}_{\text{Stl}}(\lambda,a_{\text{max}})=(\lambda,a_{\text{max}}),~{}\mathcal{L}^{DaUpdt}_{\text{Stl}}(op,(w,id))=\text{NULL},$
$\displaystyle\mathcal{L}^{Srch}_{\text{Stl}}(w)=\text{NULL},~{}\mathcal{L}^{KeyUpdt}_{\text{Stl}}=\text{NULL},$
$\displaystyle\mathcal{L}^{KeyLeak}_{\text{Stl}}=\mathcal{L}^{\prime}_{\text{Stl}}(\textsf{CUHist}(U_{\text{now}}),\textsf{DUHist}(U_{\text{now}})),$

where $\mathcal{L}^{\prime}_{\text{Stl}}$ is a stateless function. ∎

From the above security analysis, we prove that Bamboo is
$\mathcal{L}_{\text{Srv}}$-adaptively and
$\mathcal{L}_{\text{Stl}}$-adaptively secure with the leakage functions
$\mathcal{L}_{\text{Srv}}$ and $\mathcal{L}_{\text{Stl}}$ defined in Theorem
1, respectively.


# Study of Signal Temporal Logic Robustness Metrics for Robotic Tasks
Optimization

Akshay Dhonthi12, Philipp Schillinger1, Leonel Rozo1 and Daniele Nardi3  2,3
Department of Artificial Intelligence and Robotics, Sapienza University, Rome,
Italy
1 Bosch Center for Artificial Intelligence, Renningen, Germany
Email<EMAIL_ADDRESS>1{philipp.schillinger,
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Signal Temporal Logic (STL) is an efficient technique for describing temporal
constraints. It can play a significant role in robotic manipulation, for
example, to optimize the robot performance according to task-dependent
metrics. In this paper, we evaluate several STL robustness metrics of interest
in robotic manipulation tasks and discuss a case study showing the advantages
of using STL to define complex constraints. Such constraints can be understood
as cost functions in task optimization. We show how STL-based cost functions
can be optimized using a variety of off-the-shelf optimizers. We report
initial results of this research direction on a simulated planar environment.

##  I Introduction

Many robotic manipulation tasks are sensitive to small changes in execution
parameters like target positions or stiffness. Also, applications usually
benefit significantly from improved execution time or reliability. However, it
usually is a challenge to optimize such tasks due to their sequential nature.
To handle such a challenge, _Signal Temporal Logic_ (STL) which is a formalism
to describe the temporal characteristics of trajectories is well-suited. It
provides a real-valued function, called _robustness metric_ generated from a
logical specification which can be used to evaluate the performance of the
robotic tasks.

In this work, we investigate the suitability of different robustness metrics
for the purpose of optimizing robotic manipulation tasks. There have been
efforts towards applying STL in various robotic environments. [2, 1] develop
control techniques for multi-agent systems using STL specification. [3] has
used STL rewards in Learning from Demonstration (LfD) on a 2D driving
scenario. However, most works focus on directly optimizing the controller,
which can be limiting for complex systems. Instead, we propose to use
classical black-box optimization to improve existing tasks.

##  II STL Robustness Metric

Formally, an STL specification can be understood as follows. Consider a
discrete time sequence $t:=\\{t_{k}|k\in\mathbb{Z}_{\geqslant 0}\\}$. The STL
formula $\varphi$ is defined using the predicate $\mu$ that is of the form
$f(x(t))$, where $x(t)$ is the state of the signal at time $t$ and $f$ maps
each time point to the real-value $x(t_{k})\in\mathbb{R}$. The STL syntax is
defined as:
$\varphi\;:=\;\mu\;|\;\neg\varphi\;|\;\varphi_{1}\land\varphi_{2}\;|\;\varphi_{1}\lor\varphi_{2}\;|\;\mathbf{G}_{I}\varphi\;|\;\mathbf{F}_{I}\varphi\;|\;\varphi_{1}\mathbf{U}_{I}\varphi_{2}$.
Where $I=[a,b]$ is the set of all $t_{k}\in t$ such that $a,b\in t$; $b>a\geq
0$. The operators $\neg,\land,\lor$ refer to Boolean _negation_ ,
_conjunction_ and _disjunction_ operators respectively. The temporal operators
$\mathbf{G},\textbf{F},\mathbf{U}$ refer to _globally_ , _eventually_ and
_until_ operators, respectively. A complete definition of STL can be found in
[11],[10].

The _robustness metric_ denoted as $r(\varphi,x,t)\in\mathbb{R}$ is the
quantitative semantics of the STL formula $\varphi$ that measures ”how well”
the signal $x$ is fulfilled at time $t$. The classical way of defining this
semantics is _space robustness_ [8]. This measure is positive if and only if
the signal satisfies the specification (_Soundness_ property) and the closer
the robustness is to zero, the smaller are the required changes of signal
values to change the truth value. Formally, space robustness and its
corresponding operators are defined as follows.

$\displaystyle r(\mu,x,t)=$ $\displaystyle\;f(x(t)),$ (1) $\displaystyle
r(\neg\varphi,x,t)=$ $\displaystyle\;-r(\mu,x,t),$ $\displaystyle
r(\varphi_{1}\land\varphi_{2},x,t)=$
$\displaystyle\;\operatorname{min}(r(\varphi_{1},x,t),r(\varphi_{2},x,t)),$
$\displaystyle r(\varphi_{1}\mathbf{U}_{[a,b]}\varphi_{2},x,t)=$
$\displaystyle\operatorname{max}_{t_{k1}\in[t+a,t+b]}(\operatorname{min}(r(\varphi_{1},x,t_{k1}),$
$\displaystyle\operatorname{min}_{t_{k2}\in[t+a,t+t_{k1}]}r(\varphi_{2},x,t_{k2}))).$

The remaining operators can be derived using (1). Although this is an
intuitive and practical way to determine robustness, it has limitations,
particularly due to the _min_ and _max_ functions. They are non-smooth and
non-differentiable, making it more difficult for any optimizer to find a good
solution. To better address this issue, we discuss other alternatives to space
robustness and some of the properties.

###  II-A Robustness types

To address the above issues of space robustness, several alternative
definitions have been proposed in recent years.

####  II-A1 Time Robustness

Instead of looking at how well individual signal values satisfy the
specification, time robustness shifts the signal in time to quantify
satisfaction [6]. However, time robustness is discontinuous at rising and
falling edges due to the switching from positive to negative values. This kind
of robustness is useful in cases where it is important to find a signal based
on how fast/slow they should satisfy the specification.

####  II-A2 LSE Robustness

To deal with the smoothness problem of space robustness, [5] proposed a
robustness formulation based on the _Log-Sum-Exponential_ (LSE) approximation
of the _max_ and _min_ operators. LSE corresponds to the log of a summation of
exponential terms with some scalar multiplier $k\in\mathbb{R}_{\geq 0}$ as
scaling factor. This approximation is smooth due to infinitely differentiable
approximation and can reduce the influence of spikes in the signal, but it
comes with a cost of loosing soundness property.

####  II-A3 AGM Robustness

The _Arithmetic Geometric Mean_ (AGM) is an average-based robustness that
modifies all STL operators at all time instances [10]. This definition
captures how fast the signal $x\in[-1,1]$ satisfies the specification by
computing arithmetic and geometric means. This normalizing approach can be
useful when the signals are of different units. However, AGM does not
guarantee convergence with gradient-based optimization techniques [9].

####  II-A4 Smooth Robustness

It can be seen as a combination of _LSE_ and _AGM_ robustness metrics [9]
because, unlike them, it is both sound and guarantees convergence. The use
case is similar to _LSE_ with additional feature of giving positive outcome
only if the signal satisfies the specification.

####  II-A5 Avg Robustness

It is a combination of _space_ and _time_ robustness. It captures how soon or
late a signal meets the specification [11] and therefore is suitable to
optimize both accuracy and speed in achieving a task. The Avg Robustness does
not support nested temporal operators, which can be a drawback.

####  II-A6 NEW Robustness

It is introduced recently in [7] which is based on a scale-invariant
behaviour. The metric is computed by taking weighted average of the effective
measures where the weight is defined such that it becomes traditional _space
robustness_ when it approaches infinity. The test results from the authors
show better performance comparing to _AGM robustness_ in terms of finding
feasible solutions.

###  II-B Metric Properties

[7] summarizes a number of properties that help to classify robustness
metrics. Since most of these properties are especially relevant for
optimization, we summarize them here for all considered robustness definitions
in Table I for the following property definitions. A metric is _sound_ if
$r(\varphi,x,t)\geq 0$ if and only if the signal $x$ satisfies the
specification at time $t$. A metric is _weakly smooth_ if it is continuous
everywhere and its gradient is continuous for all points. A metric is said to
_converge_ if it guaranteed to reach a local maximum with gradient-based
optimization. A metric is _shadow-lifting_ if the metric increases when making
partial progress towards the task specification. A metric is _scale-invariant_
if $\operatorname{max}(\alpha a_{1}\land\alpha
a_{2})=\alpha\;\operatorname{max}(a_{1}\land a_{2})$ for any
$\alpha\in\mathbb{R}$.

TABLE I: Comparison of properties with Robustness Metrics | Robustness Metrics
---|---
Properties | Space | LSE | Smooth | AGM | Avg | NEW
Weakly Smooth | No | Yes | Yes | Yes | No | Yes
Sound | Yes | No | Yes | Yes | Yes | Yes
Converge | No | Yes | Yes | No | No | Yes
Shadow-lifting | No | No | No | Yes | Yes | Yes
Scale-invariance | Yes | Yes | Yes | No | Yes | Yes

##  III Learning from STL

In this section, we discuss on the approach of using STL for optimizing tasks
and present STL specifications to obtain a desired behavior. Specifically, we
focus on optimizing task execution duration and final position of the robot
end-effector using CMA-ES and Bayesian optimization (BO). The optimizer gets
rewards based on the STL specifications at the end of each task execution and
generates different parameters for the next run to finally produce a learned
trajectory that fulfills the STL constraints.

Let us consider the following desired behavior: “The end-effector has to
_eventually_ visit three regions $j\in\\{A,B,C\\}$ within the time intervals
$3$ to $4$, $8$ to $10$, and $13$ to $15$ seconds, respectively”. Equation (2)
represents this as an STL specification.

$\displaystyle\varphi=$
$\displaystyle\mathbf{F}_{[3,4]}\varphi_{A}\;\land\;\mathbf{F}_{[8,10]}\varphi_{B}\;\land\;\mathbf{F}_{[13,15]}\varphi_{C},$
(2) $\displaystyle\varphi_{j}=$ $\displaystyle
x_{j,lb}<x_{j}<x_{j,ub}\;\land\;y_{j,lb}<y_{j}<y_{j,ub}.$

These specifications show the potential of STL for defining temporal
constraints for task optimization. In general, we obtain a lower reward when
the behavior fails the constraints soon. The rewards obtained using such
specifications can push the optimizer to satisfy all the conditions even when
those constraints are in different time instants. This specification is simple
to change to achieve different desired outcomes.

##  IV Experiments and Results

We conduct experiments on the 7-DOF Panda Robot in a simulated environment.
The number of evaluations per experiment is $60$, each one taking $25$ minutes
approximately. Time taken for reward computation and optimization is fast. The
robot motion profile has three point to point trajectories. The objective here
is to optimize duration parameters and $(x,y)$ end-effector position
parameters at each trajectory which totals to 9 parameters. The robot resets
to its initial position at the end of each evaluation.

Figure 1:  End-effector trajectories (blue solid lines) at each BO iteration
using the _NEW_ robustness metric, where color intensity depicts the
optimization evolution. The red line represents the highest reward execution.
The lines has colored patches which are the time domain when the end-effector
has to reach the respective regions.  Figure 2: STL Robustness values as a
function of the $(x,y)$ values provided by BO during task optimization
considering the three desired regions A, B, C (depicted by colored planes).
The used metric is _NEW_ robustness. Figure 3: The cost of Gaussian process as
a function of $(x,y)$ positions of the region B at the end of the experiment
with _Smooth_ robustness. Cost here denotes negative _STL Robustness_. The
mean of the GP is shown as colored surface. The $\pm$ variance is given as
grey surface. The green dot is the location of the minimum underlying cost.
Figure 4: The figure shows at each epoch, how close the end-effector is to the
goal during respective time frames for regions C. The optimizer used is BO.

The trace of the end-effector in Fig. 1 shows trajectory positions converging
to the goal regions. Figure 2 displays the obtained rewards, which increase
when the trajectories get closer to the desired regions. Figure 3 shows the BO
surrogate model, i.e. a _Gaussian Process_ (GP), after evaluating $60$
iterations. It is clear from Fig. 4 that the optimization is heavily affected
by the robustness metric used. Table II shows the performance of all the
metrics using _BO_ and _CMA-ES_ optimizers. The _Success Rate_ (SR) is
computed as the number of times the robot satisfies all the constraints over
$60$ epochs. The _Task Satisfaction_ (TS) is the first evaluation step when
all the STL constraints were satisfied. Some experiments are run until
convergence. _Smooth Robustness_ performs better with _BO_ while _NEW
robustness_ performs better using _CMA-ES_.

Based on observations from the experiments, optimization with the _space
robustness_ metric does not always satisfy the constraints when there are more
temporal operators. _LSE_ and _Smooth robustness_ performance is influenced by
their respective scaling factors and has to be tuned for different problems.
The convergence with _AGM_ metric is not guaranteed unless signals are
normalised, but this is not straightforward in manipulators due to the
complexity of obtaining workspace boundaries. The _NEW_ robustness metric is
suitable for defining maniputator tasks as they are robust in finding a
solution all the time while _Smooth robustness_ can converge faster given the
scaling factors are tuned.

TABLE II: Overview Table for Robustness Metric properties |  | Robustness Metrics
---|---|---
| Type | Space | LSE | Smooth | Avg | AGM | NEW
BO | SR | 12.71% | 10.0% | 29.9% | 9.92% | 20.0% | 27.41%
TS | 75 | 68 | 33 | 85 | 39 | 51
CMA | SR | 1.67% | 5.0% | 3.51% | 0.0% | 3.7% |  6.67%
TS | 58 | 48 | 32 | Fail | 50 | 29

##  V Conclusion

In this paper, we exploited STL-based constraints as cost functions to
optimize simple robotic manipulation tasks. We analyzed several STL-based cost
functions and showed their influence on optimizing simple robot trajectories
in multiple-target reaching tasks. With several experiments, it is possible to
see _Smooth_ and _NEW robustness_ are suitable with classical black-box
optimizers such as _BO_ and _CMA-ES_. Further, this work can be extended by
considering orientation parameters and nested temporal STL specifications on
other optimizers.

## References

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# Toward a global phase diagram of the fractional quantum anomalous Hall
effect

Aidan P. Reddy  Liang Fu  Department of Physics, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA

###### Abstract

Recent experiments on the twisted semiconductor bilayer system $t$MoTe2 have
observed integer and fractional quantum anomalous Hall effects, which occur in
topological moiré bands at zero magnetic field. Here, we present a global
phase diagram of $t$MoTe2 throughout the filling range $0<n\leq 1$
substantiated by exact diagonalization calculations. At a magic angle, we find
that the system resembles the lowest Landau level (LLL) to a remarkable
degree, exhibiting an abundance of incompressible fractional quantum anomalous
Hall states and compressible anomalous composite Fermi liquid states. Away
from the magic angle, particle-hole symmetry is strongly broken. Some LLL-like
features remain robust near half-filling, while others are replaced,
predominantly by charge density waves near $n=0$ and anomalous Hall Fermi
liquids near $n=1$. Among LLL-like phases, we find the anomalous composite
Fermi liquid at $n=\frac{1}{2}$ to be most robust against deviations from the
magic angle. Within the band-projected model, we show that strong particle-
hole asymmetry above the magic angle results from interaction-enhanced
quasiparticle dispersion near $n=1$. Our work sets the stage for future
exploration of LLL-like and beyond-LLL phases in fractional quantum anomalous
Hall systems.

##  I Introduction

Twisted transition metal dichalcogenide homobilayers ($t$TMDs) host
topological moiré bands that exhibit spin-valley locking and spin/valley
contrasting Chern numbers Wu et al. (2019). Owing to band topology and narrow
bandwidth, small-twist-angle bilayer MoTe2 and WSe2 are predicted to support
integer and fractional quantum anomalous Hall (QAH) states Devakul et al.
(2021); Li et al. (2021); Crépel and Fu (2023); Morales-Durán et al. (2023a).
These are chiral topological states that spontaneously break time-reversal
symmetry and exhibit integer and fractionally quantized anomalous Hall
conductance $\sigma_{\rm AH}=Ce^{2}/h$ at zero magnetic field respectively.
Recently, electronic compressibility measurements on $t$WSe2 have shown
incompressible states with $C=1$ persisting down to zero magnetic field at
filling factors $n=1$ and $3$, providing evidence for integer QAH states. In
$t$MoTe2, optically detected Landau fan diagrams reveal signatures of integer
as well as fractional QAH states, with $C=-1$, $-\frac{2}{3}$ and
$-\frac{3}{5}$ respectively Cai et al. (2023); Zeng et al. (2023).

Very recently, for the first time, the fractionally quantized anomalous Hall
effect was observed through transport measurements on $t$MoTe2 Park et al.
(2023). This observation provides convincing evidence of a topological phase
hosting fractionally charged quasiparticles at zero magnetic field, opening a
new frontier in topological physics and quantum materials research. Another
group also reported the observation of an integer QAH effect as well as a
nearly fractionally quantized anomalous Hall effect in $t$MoTe2 Xu et al.
(2023a).

Following the experimental breakthrough, recent theoretical works have studied
various FQAH states in $t$MoTe2 at specific odd-denominator filling fractions
Reddy et al. (2023); Wang et al. (2023a); Qiu et al. (2023); Goldman et al.
(2023). Using numerical exact diagonalization, our recent work Reddy et al.
(2023) showed that the FQAH state at $n=\frac{2}{3}$ appears robustly over a
broad range of twist angles. In contrast, the FQAH state at $n=\frac{1}{3}$
competes with an insulating charge density wave state that is favored at
experimentally studied twist angles $\theta>\sim 2.3^{\circ}$. These
conclusions are strongly supported by recent experiments (see below) Zeng et
al. (2023).

The formation of FQAH states in $t$TMDs results from (1) exchange-induced
spontaneous spin/valley polarization and (2) strong correlation in spin/valley
polarized Chern bands at partial filling. Broadly speaking, when a Chern band
is sufficiently flat Tang et al. (2011); Neupert et al. (2011); Sheng et al.
(2011); Sun et al. (2011); Regnault and Bernevig (2011); Song et al. (2015);
Zhang et al. (2019), its wavefunctions sufficiently resemble those of the
lowest Landau level (LLL)Kol and Read (1993); Qi (2011); Roy (2014); Claassen
et al. (2015); Tarnopolsky et al. (2019); Ledwith et al. (2020); Wang et al.
(2021); Ledwith et al. (2022); Paul et al. (2023); Dong et al. (2022); Wang et
al. (2023b); Morales-Durán et al. (2023b), and the topological gap to higher
bands is sufficiently large, the system can be approximately mapped to a
partially filled Landau level through the band-projected Hamiltonian. If this
mapping holds faithfully, the existence of FQAH states follows
straightforwardly from the well-known fractional quantum Hall states in Landau
levels. The key question is: to what extent is the phase diagram of $t$MoTe2
as a function of band filling similar to that of the LLL Kivelson et al.
(1992)? And in what aspect is it fundamentally different?

Figure 1: Schematic phase diagrams of $t$MoTe2 with respect to $n$, the number
of holes per moiré unit cell, at angles near (a) and greater than (b) the
magic angle $\theta_{m}$. (c,d) Charge gap at several Jain sequence filling
fractions at corresponding representative twist angles and two interaction
strengths. In (c,d), blue denotes FQAH, red denotes CDW, and black denotes
undetermined. Data for fractions with denominators $3$, $5$, and $7$ are
obtained from clusters with 27, 25, and 28 unit cells respectively (see
Supplemental Material). (F/IQAH: fractional/integer quantum anomalous Hall,
ACFL: anomalous composite Fermi liquid, CDW: charge density wave, AHFL:
anomalous Hall Fermi liquid.)

In this work, we map out the global phase diagram for $t$TMDs throughout the
filling range $n\leq 1$. We find that, in the vicinity of a “magic angle”, an
abundance of FQAH states appear and the phase diagram is nearly symmetric
about $n=\frac{1}{2}$, which hosts an anomalous composite Fermi liquid state
(ACFL) Goldman et al. (2023); Dong et al. (2023), see Fig. 1(a). ACFL states
may also appear at $n=\frac{1}{4}$, $\frac{3}{4}$. In these regards, the
system closely resembles the LLL. At larger twist angles, particle-hole
symmetry is strongly broken, leading to a coexistence of phases that are
familiar with others that are foreign to the LLL. The $n=\frac{1}{2}$ ACFL
states and some FQAH states survive beyond the magic angle. In contrast,
charge density waves appear at low fillings $n=\frac{1}{3},\frac{1}{4}$ and a
time-reversal-breaking Fermi liquid phase appears in the filling factor range
$\frac{2}{3}<n<1$, see Fig. 1(b).

Our findings demonstrate the remarkable robustness of ACFL state at
$n=\frac{1}{2}$ with respect to twist angle and establish its central role as
the parent of adjacent FQAH states in the phase diagram. Our work also reveals
an “anomalous Hall Fermi liquid” phase that exhibits an unquantized, filling-
dependent anomalous Hall conductivity and has no counterpart in the LLL. We
discuss the origin of similarities and differences between the many-body phase
diagram of topological bands in $t$TMDs and that of the LLL. The observable
consequences of our phase diagram for fractional quantum anomalous Hall
systems are also described and compared to recent experiments on $t$MoTe2.

##  II Charge gap phase diagram

The continuum model for AA-stacked, K-valley TMD homobilayers describes holes
in a moiré-periodic scalar potential and a layer-pseudospin “Zeeman” field
that carries a skyrmion texture Wu et al. (2019). In the vicinity of a magic
angle, several LLL-like features appear in the lowest moiré band at the
single-particle level: the bandwidth nearly vanishes and the Berry curvature
becomes nearly uniform Devakul et al. (2021); the deviation from the so-called
trace condition is minimized Morales-Durán et al. (2023a); Reddy et al.
(2023); Dong et al. (2023); and the general bound on the topological band gap
is closest to saturation Onishi and Fu (2023). Based on our large-scale
density functional theory (DFT) calculations, it is found that
$\theta_{m}\approx 2^{\circ}$ for twisted bilayer MoTe2 Reddy et al. (2023)
and $\theta_{m}\approx 1.5^{\circ}$ for twisted bilayer WSe2 Devakul et al.
(2021). We note that the use of different DFT parameters results in some
variation in the magic angle Wu et al. (2019); Tang et al. (2021); Wang et al.
(2023a); Zhang et al. (2023); Mao et al. (2023); Yu et al. (2023).

At the many-body level, our previous exact diagonalization (ED) calculations
indeed find that the neutral energy gaps of the $n=\frac{1}{3}$ and
$\frac{2}{3}$ FQAH states are both maximized near a magic angle $\theta_{m}$
Reddy et al. (2023). Here, we present a comprehensive ED study of $t$MoTe2
both near and above the magic angle and throughout the filling range $n\leq
1$, finding a plethora of incompressible and compressible states. Our results
are obtained on a finite-sized torus using the continuum model for $t$MoTe2
with the parameters of Ref. Reddy et al. (2023) projected to the lowest moiré
band and a Coulomb interaction $V(r)=\frac{e^{2}}{\epsilon r}$. Given that
previous studies have shown the ground state over a wide range of fillings
$n\leq 1$ to be fully spin/valley polarized Crépel and Fu (2023); Reddy et al.
(2023), all ED calculations in this work are performed in the fully
spin/valley polarized sector. Our model and methods are described in the
Supplemental Material of Ref. Reddy et al. (2023).

The defining property of an incompressible state is a finite charge gap in the
thermodynamic limit. Here we study the charge gap
$\Delta_{c}(N)=\mu^{+}_{N}-\mu^{-}_{N}$ with $\mu^{\pm}_{N}=\pm(E_{GS}(N\pm
1)-E_{GS}(N))$ where $N$ is the number of holes and $E_{GS}(N)$ is the ground
state energy at a given $N$ and a fixed finite system geometry. The
thermodynamic limit of $\Delta_{c}$ can be measured via compressibility and,
in a state with quasiparticles of fractional charge $e^{*}=\frac{p}{q}$, is
related to the transport activation gap $\Delta_{t}$ as
$p\Delta_{c}=q\Delta_{t}$ Girvin (2002). Note that the charge gap is distinct
from the neutral energy gap, the difference between the ground and first
excited state energies at fixed particle number.

In Fig. 1, we show the charge gaps at the Jain sequence filling fractions
$n=\frac{p}{2p+1}$ where $p=1,2,3$ and their particle-hole conjugates at two
twist angles and two interaction strengths. At $\theta=2^{\circ}$,
representative of the system near the magic angle $\theta_{m}$, the charge gap
is positive at all $p$, decreases with increasing $p$, and is nearly particle-
hole symmetric under $n\rightarrow 1-n$. By inspecting the many-body spectra,
we confirm that these incompressible states are all FQAH states (see
Supplemental Material). Remarkably, the decreasing charge gaps of FQAH states
as the filling approaches $\frac{1}{2}$ closely resembles the Jain sequence of
fractional quantum Hall states in the LLL, despite being at zero magnetic
field.

In contrast, at $\theta=2.7^{\circ}$, representative of a broad range of
angles $\theta>\theta_{m}$, the charge gap exhibits strong particle-hole
asymmetry, see Fig. 1(d). $n=\frac{1}{3}$ exhibits a large charge gap
associated with an insulating charge density wave rather than an FQAH state
Reddy et al. (2023); Morales-Durán et al. (2023a). On the other hand, the
$n=\frac{2}{3}$ state is an FQAH state with the topological order of the
particle-hole conjugate of the $\frac{1}{3}$ Laughlin state in the LLL.
Compared to that at $n=\frac{2}{3}$, other FQAH states are more fragile when
$\theta$ exceeds $\theta_{m}$. For $\epsilon^{-1}=0.1$, the charge gap at
$n=\frac{4}{7}$ become negative and that at $n=\frac{3}{5}$ is positive but
very small. When interaction strength increases to $\epsilon^{-1}=0.2$, the
charge gaps at both filling fractions increase and are positive. Compared to
the FQAH state at $n=\frac{2}{3}$, stronger interactions are necessary to
stabilize FQAH states at these filling fractions away from the magic angle.

In the Supplemental Material, we compare charge gaps at various filling
fractions between magic-angle $t$MoTe2 and the LLL on identical finite system
geometries, which show a striking similarity at _all_ carrier densities. This
indicates that the resemblance between the two systems is not limited to
specific filling fractions. We also study the evolution with respect to the
twist angle $\theta$ of the many-body spectra at several rational filling
fractions. In addition to the fractions shown here, we find that many others,
including $n=\frac{1}{5}$, $\frac{4}{5}$, $\frac{4}{9}$, $\frac{2}{7}$,
$\frac{5}{7}$, and $\frac{5}{9}$ may host FQAH states at the magic angle.

Our work reveals the physical consequences of the magic angle in $t$MoTe2.
While the $n=\frac{2}{3}$ FQAH state exists robustly over a wide range of
twist angles, only at $\theta\sim 2^{\circ}$ does the many-body phase diagram
resemble the LLL at all filling factors.

Figure 2: (a) Charge density at $n=\frac{1}{3}$. MM stacking regions appear
light blue and MX/XM stacking regions are red. Pair correlation functions
$g(\bm{r}_{0},\bm{r})$ and projected static structure factors $s(\bm{q})$ of
the $n=\frac{1}{3}$ (b,d) and $\frac{2}{3}$ (c,e) FQAH fluids at
$\theta=2^{\circ}\sim\theta_{m}$. Line cuts are taken along
$\bm{r}-\bm{r}_{0}$ going from the center to the bottom right corner of (b,c).
Line cuts with $\bm{r}_{0}$ at two high symmetry points in the moiré unit
cell, MX and MM, are both shown. $\epsilon^{-1}=0.2$ and the same system with
27 moiré unit cells is used in all panels.

##  III Fractional quantum anomalous Hall fluids

To further characterize the FQAH states, we now study their two-body
correlations. In Fig. 2, we show the pair correlation functions

$\displaystyle g(\bm{r},\bm{r}^{\prime})=\frac{\langle
n(\bm{r})n(\bm{r}^{\prime})\rangle-\delta(\bm{r}-\bm{r}^{\prime})\langle
n(\bm{r})\rangle}{\langle n(\bm{r})\rangle\langle n(\bm{r}^{\prime})\rangle}$
(1)

and projected density structure factors

$\displaystyle\begin{split}s(\bm{q})=\frac{\langle\bar{\rho}(-\bm{q})\bar{\rho}(\bm{q})\rangle}{N}-\delta_{\bm{q},\bm{0}}\end{split}$
(2)

of the $n=\frac{1}{3}$, $\frac{2}{3}$ FQAH states near the magic angle. Here
$\bar{\rho}(\bm{q})=\mathcal{P}\sum_{i}e^{-i\bm{q}\cdot\bm{r}_{i}}\mathcal{P}$
is the projected density fluctuation operator where $\mathcal{P}$ is the
projector onto the lowest moiré band and $i$ indexes holes. $\langle\rangle$
denotes an expectation value with respect to the many-body ground state(s).
The local density operator is
$n(\bm{r})=\sum_{l}\psi^{{\dagger}}_{l}(\bm{r})\psi_{l}(\bm{r})$ where
$\psi^{{\dagger}}_{l}(\bm{r})$ creates a hole at position $\bm{r}$ in layer
$l$.

The pair correlation functions of the $n=\frac{1}{3}$ and $\frac{2}{3}$ states
show clear exchange-correlation holes at small separation and remain near $1$
at large separation. At $n=\frac{1}{3}$, the pair correlation function
oscillates at intermediate range, indicating incipient crystalline order. The
projected structure factors do exhibit some non-uniformity but lack the strong
peaks expected of a crystal. Both of these correlation functions exhibit
characteristics of a fluid with strong short-range correlations. We remark
that the presence of the layer degree of freedom means that, in general,
$g(\bm{r},\bm{r})\neq 0$ even when spin/valley is fully polarized. Additional
analysis of FQAH correlation functions, including at $n=\frac{2}{5}$,
$\frac{3}{5}$ is presented in the Supplemental Material.

The charge gap phase diagram (Fig. 1) shows that, throughout a wide range of
twist angles, the system, hosts a sequence of FQAH states analogous to the
Jain sequence in the LLL. Having studied these incompressible topological
fluids, we now turn to the filling fraction $n=\frac{1}{2}$ about which they
are centered.

##  IV Anomalous composite Fermi liquid at Half Filling

Previously, we showed that metallic states analogous to the composite Fermi
liquids of the LLL but at zero magnetic field exist at $n=\frac{1}{2}$ and
$n=\frac{3}{4}$, which we dubbed “anomalous composite Fermi liquids” (ACFL)
Goldman et al. (2023). Here, we extend our earlier study to a larger system
(with 28 moiré unit cells) and a broader range of angles beyond
$\theta=2^{\circ}$. Fig. 3 shows the many-body energy spectra at
$n=\frac{1}{2}$ for three twist angles $\theta=2^{\circ}$, $3^{\circ}$, and
$3.5^{\circ}$. Near the magic angle, the ground states come in quasi-
degenerate pairs as they do in the half-filled LLL (where they are related by
center-of-mass magnetic translations Haldane and Rezayi (1985); Girvin and
Yang (2019)), and their many-body momenta also match those of the half-filled
LLL on a torus of identical geometry, showing that the $n=\frac{1}{2}$ state
is a composite Fermi liquid. Unlike incompressible candidate competing states,
the composite Fermi liquid’s finite-size ground state degeneracy is sensitive
to the system geometry and can be interpreted from trial wave functions as
arising from compact configurations of the composite Fermi sea Rezayi and Read
(1994); Jain and Kamilla (1997); Rezayi and Haldane (2000); Geraedts et al.
(2018); Wang et al. (2019, 2021).

Figure 3: Low-lying many-body energy spectra (top) and ground state momentum
distribution function $n(\bm{k})$ (bottom) at $n=\frac{1}{2}$ and several
twist angles. In each case $\epsilon^{-1}=0.1$ and the lowest 5 energy levels
in each momentum sector are shown. The system size is 28 moiré unit cells.

As the twist angle increases from $2^{\circ}$ to $3.5^{\circ}$, no ground
state level crossing occurs, suggesting that the system at $n=\frac{1}{2}$
remains in the same composite Fermi liquid phase throughout. In contrast, with
the same interaction strength $\epsilon^{-1}=0.1$, the FQAH state at
$n=\frac{2}{3}$ undergoes a phase transition at $\theta\approx 3^{\circ}$
where a level crossing between the FQAH ground state manifold and excited
states occurs Reddy et al. (2023). This observation suggests that the ACFL
state at $n=\frac{1}{2}$ is more resilient against departure from the magic
angle than the FQAH state at $n=\frac{2}{3}$. In the Supplemental Material, we
provide similar evidence for the $n=\frac{1}{4},\frac{3}{4}$ ACFL states near
the magic angle, and show that, as twist angle increases, both these ACFL
states undergo level-crossing transitions before the ACFL state at
$n=\frac{1}{2}$.

We also calculate the momentum distribution function
$n(\bm{k})=\frac{1}{N_{GS}}\sum_{i}\bra{\Psi_{i}}c^{{\dagger}}_{\bm{k}}c_{\bm{k}}\ket{\Psi_{i}}$
(where $c^{{\dagger}}_{\bm{k}}$ creates a hole in a moiré Bloch state and $i$
runs over a set of $N_{GS}$ degenerate many-body ground states), shown in Fig.
3. At $\theta=2^{\circ}$, $n(\bm{k})\approx 0.5$ is nearly constant and, in
particular, smooth. This shows that the state is not a Landau Fermi liquid in
which $n(\bm{k})$ is necessarily discontinuous at the Fermi surface in the
thermodynamic limit. At larger angle, $n(\bm{k})$ varies throughout the
Brillouin zone but still does not exhibit a sharp jump. We remark that the
momentum distribution function is experimentally accessible through angle-
resolved photoemission spectroscopy.

Since $n(\bm{k})\rightarrow 1-n(-\bm{k})$ under a particle-hole
transformation, the ground state wavefunction of ACFL state beyond the magic
angle is clearly not invariant under the particle-hole transformation.
(Whether there exists an “emergent” particle-hole symmetry at low energy is an
open question.) At $\theta=3.5^{\circ}$, the spectrum is on the verge of a
level crossing between the higher-energy partner of the quasi-degenerate ACFL
ground states, indicating a phase transition out of the ACFL. The nature of
the many-body ground state following this transition is an interesting open
question for future work.

So far we have seen that (1) near a magic angle, the system resembles the LLL
to a remarkable extent; and (2) above this magic angle, some LLL-like features
remain in the neighborhood of half filling, while other new phases appear at
fillings away from half filling. The similarity of our many-body phase diagram
at the magic angle with the LLL supports the recent argument that the
wavefunctions of the magic-angle moiré band can be approximately mapped to
those of the LLL Morales-Durán et al. (2023b). Equally important is the
departure from the LLL analog at larger twist angles, to which we now turn our
attention.

##  V Microscopic origin of particle-hole asymmetry

A generic band-projected Hamiltonian takes the form

$\displaystyle\begin{split}H&=\sum_{\bm{k}}\varepsilon(\bm{k})c^{{\dagger}}_{\bm{k}}c_{\bm{k}}+\frac{1}{2}\sum_{\bm{k}\bm{p}\bm{q}}V_{(\bm{k}+\bm{q})(\bm{p}-\bm{q})\bm{k}\bm{p}}c^{{\dagger}}_{\bm{k}+\bm{q}}c^{{\dagger}}_{\bm{p}-\bm{q}}c_{\bm{p}}c_{\bm{k}}\end{split}$
(3)

where spin is neglected. Therefore, it is determined entirely by the
dispersion $\varepsilon(\bm{k})$ and the interaction matrix elements
$V_{\bm{k}^{\prime}\bm{p}^{\prime}\bm{k}\bm{p}}$. The latter are determined in
turn by the band’s single-particle wavefunctions and the two-body interaction
potential. Any deviation between the phase diagrams of given band-projected
model and the LLL originates from a deviation in these features.

Motivated by our observation of strong asymmetry between $n$ and $1-n$ states
above the magic angle as shown in Fig. 1, we now consider the band-projected
Hamiltonian under the particle-hole transformation. Under the particle-hole
transformation $d^{{\dagger}}(\bm{r})=c(\bm{r})$ or, equivalently,
$d^{{\dagger}}_{\bm{k}}=c_{-\bm{k}}$, where $c_{\bm{k}}$ annihilates the
particle in Bloch state $\ket{\bm{k}}$, Eq. 3 can be rewritten in terms of
hole operators as

$\displaystyle\begin{split}H&=\sum_{{\bm{k}}}\tilde{\varepsilon}(\bm{k})d^{{\dagger}}_{{\bm{k}}}d_{{\bm{k}}}+\frac{1}{2}\sum_{\bm{k}\bm{p}\bm{q}}\tilde{V}_{(\bm{k}+\bm{q})(\bm{p}-\bm{q})\bm{k}\bm{p}}d^{{\dagger}}_{\bm{k}+\bm{q}}d^{{\dagger}}_{\bm{p}-\bm{q}}d_{\bm{p}}d_{\bm{k}}\end{split}$
(4)

where we neglect a constant energy shift. The full-band Slater determinant
state $\ket{\Psi_{f}}=(\prod_{\bm{k}}c^{{\dagger}}_{\bm{k}})\ket{0}$ is the
vacuum for holes, $d_{\bm{k}}\ket{\Psi_{f}}=0$. The interaction matrix
elements of holes are related to those of electrons as

$\displaystyle\begin{split}\tilde{V}_{\bm{k}^{\prime}\bm{p}^{\prime}\bm{k}\bm{p}}=V^{*}_{-\bm{k}^{\prime}-\bm{p}^{\prime}-\bm{k}-\bm{p}}.\end{split}$
(5)

$\tilde{\varepsilon}(\bm{k})$ describes the energy-momentum dispersion of a
single particle removed from an otherwise full band,

$\displaystyle\begin{split}\tilde{\varepsilon}(\bm{k})&=-\varepsilon(-\bm{k})-\Sigma(-\bm{k}).\end{split}$
(6)

Here, $\Sigma(\bm{k})$ is a self-energy term coming from the interaction
between the electron at $\bm{k}$ and all others in the full band (see
Supplemental Material for definition). In the lowest Landau level, both
$\varepsilon(\bm{k})$ and $\Sigma(\bm{k})$ are $\bm{k}$-independent, producing
particle-hole symmetry. More generally, applying a time reversal
transformation $H\rightarrow\mathcal{T}H\mathcal{T}^{-1}$ (where
$\mathcal{T}d_{\bm{k}}\mathcal{T}^{-1}=d_{-\bm{k}}$ and
$\mathcal{T}i\mathcal{T}^{-1}=-i$) to Eq. 4 and comparing to Eq. 3 shows that
particle-hole symmetry arises when the condition
$\tilde{\varepsilon}(\bm{k})=\varepsilon(-\bm{k})$ is satisfied. Any particle-
hole asymmetry between states at filling fractions $n$ and $1-n$ in the phase
diagram of a band-projected model has its origin in the inequivalence between
the energy dispersion of a particle added to an empty band and that of a
particle removed from a filled band.

Figure 4:  (a) Comparison of hole dispersion $\varepsilon(\bm{k})$ at $n=0$
and electron quasiparticle dispersion $\tilde{\varepsilon}(\bm{k})$ at $n=1$
of the lowest moiré band assuming full valley polarization at
$\theta=2^{\circ}$ and $\theta=2.7^{\circ}$. Note that
$\tilde{\varepsilon}(\bm{k})$ depends on the two-body interaction potential
for which we use a Coulomb interaction with $\varepsilon^{-1}=0.1$. (b) Color
plots of the electron quasiparticle dispersion. All dispersions are shifted by
a constant to be centered about zero energy.

To shed light on the approximate particle-hole symmetry near the magic angle
and lack thereof at larger angles, we now calculate
$\tilde{\varepsilon}(\bm{k})$ in $t$MoTe2 and compare it with the bare energy
dispersion $\varepsilon(\bm{k})$. A note on terminology before proceeding.
Since the moiré band of interest is a valence band, we define the filling
factor $n$ as the number of holes (i.e. carrying positive charge) per moiré
unit cell relative to charge neutrality. These holes are the elementary charge
carriers in the moiré superlattice. The $n=1$ IQAH state is formed when holes
completely fill the topmost moiré valence band of one spin/valley. Removing a
hole from the $n=1$ state creates a quasiparticle carrying negative charge,
which we refer to as an electron quasiparticle. The energy dispersion of a
hole at charge neutrality is simply $\varepsilon(\bm{k})$ as determined by the
continuum model, whereas the energy dispersion of an electron quasiparticle in
the QAH state, $\tilde{\varepsilon}(\bm{k})$, is affected by interaction-
induced self-energy as stated above.

In Fig. 4, we compare the hole $\varepsilon(\bm{k})$ and electron
quasiparticle $\tilde{\varepsilon}(\bm{k})$ dispersions in the lowest $t$MoTe2
moiré band at two representative twist angles, $\theta=2.0^{\circ}$ and
$2.7^{\circ}$. At $\theta=2^{\circ}$, the self-energy has an insignificant
influence on the electron quasiparticle dispersion so
$\tilde{\varepsilon}(\bm{k})\approx-\varepsilon(-\bm{k})$. On the other hand,
at $\theta=2.7^{\circ}$, the self-energy approximately doubles the electron
quasiparticle bandwidth. The overall energy scale of interactions is set by
$\frac{e^{2}}{\epsilon a_{M}}\propto\theta$. At $\theta=2^{\circ}$,
$\Delta\Sigma(\bm{k})\equiv\max(\Sigma(\bm{k}))-\min(\Sigma(\bm{k}))=0.58$ meV
and at $\theta=2.7^{\circ}$, $\Delta\Sigma(\bm{k})=7.0$ meV (given
$\epsilon^{-1}=0.1$). In going from $\theta=2^{\circ}$ to $2.7^{\circ}$, the
overall interaction energy scale grows by a small factor of
$\frac{2.7}{2}=1.35$ while $\Delta\Sigma(\bm{k})$ grows by a much larger
factor, $\frac{7.0}{0.58}\approx 12$. Therefore, the $\bm{k}$-dependence of
$\Sigma(\bm{k})$ is much stronger above than it is near the magic angle.

The physical origin of the enhanced dispersion for electron quasiparticles
above the magic angle is as follows. The self-energy can be decomposed into
Hartree and Fock contributions,
$\Sigma(\bm{k})=\Sigma^{H}(\bm{k})+\Sigma^{F}(\bm{k})$. We have directly
confirmed that the $\bm{k}$-dependent part of the self-energy above the magic
angle is dominated by the Fock term, which is an interaction-potential-
weighted average of quantum distances between intra-unit-cell wavefunctions
$\ket{u_{\bm{k}}}$ and $\ket{u_{\bm{k}+\bm{q}}}$:

$\displaystyle\begin{split}\Sigma^{F}({\bm{k}})&=-\int\frac{d^{2}q}{(2\pi)^{2}}V(\bm{q})|\bra{u_{\bm{k}+\bm{q}}}\ket{u_{\bm{k}}}|^{2}.\end{split}$
(7)

As we showed in Ref. Reddy et al. (2023), the Bloch states in the lowest moiré
band are strongly layer polarized except in the vicinity of $\gamma$, where
they are strongly layer hybridized. The change in the wavefunctions’ layer
character at $\gamma$ causes a peak in the Fock self-energy that adds
constructively with the bare energy dispersion $\varepsilon(\bm{k})$ and
thereby enhances the quasiparticle bandwidth at $n=1$.

Our finding that, near the magic angle, the quasiparticle dispersions near
$n=0$ $\varepsilon(\bm{k})$ and $n=1$ $\tilde{\varepsilon}(\bm{k})$ are both
narrow–and consequently that the condition for particle-hole symmetry
$\tilde{\varepsilon}(\bm{k})=\varepsilon(-\bm{k})$ is only weakly violated–is
consistent with our finding of approximate particle-hole symmetry in the
magic-angle many-body phase diagram (see Fig. 1). Above the magic angle, we
find in contrast that 1) $\varepsilon(\bm{k})$ is broad to begin with, 2)
$\tilde{\varepsilon}(\bm{k})$ is approximately twice as broad due to strong
$\bm{k}$-dependence of the Fock energy, and 3) $\varepsilon(\bm{k})$ has two
degenerate minima at the corners of the moiré Brillouin zone whereas
$\tilde{\varepsilon(\bm{k})}$ has a unique minimum at its center (see Fig. 4).
These properties are consistent with the enhanced-particle hole asymmetry in
the above-magic-angle many-body phase diagram shown in Fig. 1.

Figure 5: Momentum distribution function (see main text) of the many-body
ground states at several carrier densities $\frac{2}{3}<n<1$ at.
$\theta=2.70^{\circ}$, $\epsilon^{-1}=0.1$, and the system includes $36$ moiré
unit cells. Figure 6: (a) Intrinsic anomalous Hall conductance of a fully
spin/valley polarized Fermi gas in the lowest moiré band as a function of hole
density at several twist angles. The white background denotes the region
hosting the anomalous Hall Fermi liquid phase studied in this work at angles
larger than the magic angle $\theta_{m}\sim 2^{\circ}$ to which the data shown
are applicable. (b) Berry curvature distribution of the lowest moiré band.

##  VI Fermi liquid and unquantized anomalous Hall effect

With this understanding of particle-hole asymmetry in the band-projected model
for $t$MoTe2 above the magic angle, and in particular of the broadened
quasiparticle dispersion $\tilde{\varepsilon}(\bm{k})$ in the $n=1$ QAH state,
we now study its consequences at finite doping $n=1-\delta$ with $\delta>0$.
For small doping $\delta$, the low-energy physics of our system maps to a
uniform electron gas. Provided that the density of electron quasiparticles
$\delta$ is not too low, it is natural to expect that its ground state is a
Fermi liquid. The reduction of electron quasiparticle mass by interaction-
induced self energy near $n=1$ is also beneficial to the formation of a Fermi
liquid Läuchli et al. (2013). From band-projected ED calculations, we indeed
find a fully spin/valley-polarized, metallic state with a filling-dependent
anomalous Hall conductivity in the carrier density range $\frac{2}{3}<n<1$
that we refer to as an _anomalous Hall Fermi liquid_.

Fig. 5 shows the momentum distribution function $n(\bm{k})$ at three filling
fractions $n=0.89,0.83,0.78$. Unlike the ACFL state at $n=\frac{1}{2}$, at
these fillings $n(\bm{k})$ drops sharply across a circle centered at $\gamma$,
indicating the presence of a Fermi surface expected from the quasiparticle
dispersion $\tilde{\varepsilon}(\bm{k})$. Moreover, the degenerate ground
states’ many-body momenta match those expected from simply filling electrons
according to the quasiparticle dispersion $\tilde{\varepsilon}(\bm{k})$. In
Fig. 5 (d), for instance, the sixfold ground state degeneracy at
$n=\frac{7}{9}$ ($\delta=\frac{2}{9}$) comes from adding
$36\times\frac{2}{9}=8$ electrons to the 7 Bloch states closest to $\gamma$
and one of the $6$ momenta in the next available shell. Similar data obtained
from other finite system geometries are also shown in the Supplemental
Material.

In contrast to the Fermi liquid phase close to $n=1$, correlation effects are
much stronger at low fillings close to $n=0$ because the effective mass of
holes is much larger than that of electron quasiparticles, as seen in Fig. 4.
As a consequence, we find that above the magic angle, the $n=\frac{1}{4}$
state is a $2\times 2$ charge density wave (see below), whereas the
$n=\frac{3}{4}$ state is a Fermi liquid. Our finding of Fermi liquids
stabilized by interaction-enhanced dispersion near unity filling echoes
previous studies of other Chern band systems Läuchli et al. (2013);
Abouelkomsan et al. (2020, 2023).

Having established its existence, we now study the properties of the
ferromagnetic Fermi liquid state in $t$MoTe2. In Fig. 6, we show the zero-
temperature, intrinsic anomalous Hall conductance

$\displaystyle\sigma_{AH}=\frac{1}{2\pi}\frac{e^{2}}{h}\int
d^{2}\bm{k}\,\theta(\varepsilon_{F}-\varepsilon(\bm{k}))F(\bm{k})$ (8)

where $F(\bm{k})$ is the Berry curvature of the lowest moiré band in the
presence of full spin/valley polarization. We note that this this formula is
applicable only when the system is in a Fermi liquid phase. Near the magic
angle, a relatively uniform Berry curvature distribution leads to
$\sigma_{AH}\approx n\frac{e^{2}}{h}$. In contrast, at larger twist angles,
the Berry curvature has a hotspot near the band maximum as shown in Fig. 6(b).
This leads to a rapid reduction in anomalous Hall conductivity
$\sigma_{AH}<n\frac{e^{2}}{h}$ as the filling is reduced from $n=1$ and the
system enters the anomalous Hall Fermi liquid phase, before rising to the
quantized value $\sigma_{AH}=\frac{2e^{2}}{3h}$ in the $n=\frac{2}{3}$ FQAH
state.

In addition to the QAH states at $n=1,\frac{2}{3},\frac{3}{5}$, the recent
transport experiment on $t$MoTe2 reveals intriguing behavior as a function of
carrier density Park et al. (2023). Notably, it is found that (1) $R_{xy}\gg
R_{xx}$ throughout the filling range $\frac{1}{2}\leq n\leq\frac{2}{3}$; (2)
as the band filling increases from $\frac{2}{3}$ to $1$, $R_{xy}$ drops
rapidly, is comparable to $R_{xx}$ over an extended filling range, and
eventually rises to the quantized value $\frac{h}{e^{2}}$ at $n=1$. The
distinct transport behaviors at $\frac{1}{2}\leq n\leq\frac{2}{3}$ and
$\frac{2}{3}\leq n<1$ agree, respectively, with the LLL-like FQAH regime and
the anomalous Hall Fermi liquid regime of $t$MoTe2 above the magic angle. Our
theory then further predicts the existence of CDW states at $n<\frac{1}{2}$,
to which we turn our attention now.

Figure 7: Hole density and pair correlation functions of the charge density
wave states appearing above the magic angle at filling factors $n=\frac{1}{3}$
(a) and $\frac{1}{4}$ (b), exhibiting clear crystal structure. (c,d) Line cuts
of the pair correlation functions along $\bm{r}-\bm{r}_{0}$ going from the
center to the bottom right corner of (a,b). Insets to (c,d) show the projected
structure factors. MM stacking regions are red and MX/XM stacking regions are
yellow in the charge density plots of (a,b). Data are from clusters with 27
(a,c) and 28 (b,d) unit cells. $\theta=2.7^{\circ}$ and $\epsilon^{-1}=0.2$.

##  VII Charge density waves at $n<\frac{1}{2}$

By explicit calculation of the interaction-renormalized dispersion near $n=1$,
we have shown that the band-projected continuum model is strongly particle-
hole asymmetric above the magic angle. The model’s intrinsic particle-hole
asymmetry produces a particle-hole asymmetric diagram with charge density
waves at low filling factors in stark contrast to the AHFL phase near $n=1$.
In Fig. 7, we show the pair correlation functions and projected structure
factors at $n=\frac{1}{3}$, $\frac{1}{4}$ at $\theta=2.7^{\circ}>\theta_{m}$.
Unlike at smaller angles, the charge density here is peaked at MM stacking
sites that form a triangular lattice Reddy et al. (2023). The pair correlation
functions exhibit clear crystalline order with $N-1$ distinct peaks where $N$
is the total number of holes. Similarly, the projected structure factors
exhibit peaks at the moiré Brillouin zone corners and $m$ points, indicating
crystalline order with $\sqrt{3}\times\sqrt{3}$ and $2\times 2$ enlarged unit
cells at $n=\frac{1}{3}$, $\frac{1}{4}$ respectively.

In the LLL, the formation of charge density wave states is inevitable when the
low filling factor is sufficiently low or, equivalently, the ratio between the
average inter-particle spacing and the magnetic length is sufficiently large.
While the precise filling factor below which such states are favored is not
firmly agreed upon, it is widely believed to be around $\nu\approx\frac{1}{7}$
Lam and Girvin (1984); Yang et al. (2001). Moreover, particle-hole symmetry in
the LLL ensures that if a CDW of particles is the ground state at filling
factor $\nu$, a CDW of holes is the ground state at $1-\nu$. In topological
bands of $t$TMDs, the presence of CDWs at relatively high filling factors, and
the lack thereof at their particle-hole conjugates, illustrates a clear
departure of the system at $\theta>\theta_{m}$ from the LLL.

The nature of the $n=\frac{1}{3}$ FQAH-CDW and $n=\frac{1}{4}$ ACFL-CDW phase
transitions are interesting questions for future theoretical investigation.
Note that the CDW calculations presented here assume full spin/valley
polarization, which can be achieved by applying a small magnetic field. We
leave a systematic study of the magnetism of CDW states at zero field to
future work.

##  VIII Discussion and Outlook

In this work, we have begun to map out a global phase diagram of the
fractional quantum anomalous Hall effect that occurs in partially filled
topological bands of $t$TMDs. With an exact diagonalization study, we have
shown that near a magic angle $\theta_{m}$ the phase diagram bears remarkable
resemblance to that of the LLL, hosting anomalous composite Fermi liquid
phases at $n=\frac{1}{2}$, $\frac{1}{4}$, $\frac{3}{4}$ and FQAH phases at
Jain sequence filling fractions $n=\frac{1}{3}$, $\frac{2}{5}$, $\frac{3}{7}$,
$\frac{4}{7}$, $\frac{3}{5}$, and $\frac{2}{3}$.

Above the magic angle, we find the phase diagram to be strongly particle-hole
asymmetric. Charge density waves appear at $n=\frac{1}{3}$, $\frac{1}{4}$. In
contrast, anomalous Hall Fermi liquids appear at $\frac{2}{3}<n<1$. We find
that the ACFL state at half-filling is particularly robust against deviations
from the magic angle, surviving to even larger angles than $n=\frac{2}{3}$
FQAH state. Our phase diagram explains the recent observation of (1) an
incompressible trivial state at $n=\frac{1}{3}$ and (2) filling-dependent
anomalous Hall effect in the metallic region $\frac{2}{3}<n<1$ in $t$MoTe2
devices that show an $n=\frac{2}{3}$ FQAH state.

Recent many-body calculations using different continuum model parameters Wang
et al. (2023a); Dong et al. (2023) find FQAH states at both $n=\frac{1}{3}$
and $\frac{2}{3}$ in the twist angle range $\theta_{m}\sim 3-4^{\circ}$ where
the majority of experiments to date have been performed. In contrast, our work
finds that the simultaneous appearance of these FQAH states is limited to the
neighborhood of a magic angle, $\theta_{m}\sim 2^{\circ}$. While there is
uncertainty about the continuum model parameters and, therefore, the precise
value of $\theta_{m}$, the appearance of a trivial insulating state at
$n=\frac{1}{3}$ Zeng et al. (2023), consistent with a CDW, indicates that
devices studied to date are above the magic angle.

While the ED calculations presented in this work are performed specifically
for $t$MoTe2, we expect our main conclusions as stated above to hold
qualitatively for a broader class of Chern band systems with band-projected
Hamiltonians that can be approximately mapped to that of the LLL. These
include Chern bands from periodically modulated Landau levels Dong et al.
(2022), skrymion lattices in semiconductor-magnet heterostructures Paul et al.
(2023), and graphene moiré superlattices Zhang et al. (2019); Repellin and
Senthil (2020); Abouelkomsan et al. (2020); Ledwith et al. (2020); Wilhelm et
al. (2021); Parker et al. (2021) which host fractional Chern insulator states
at strong magnetic field Spanton et al. (2018); Xie et al. (2021).

The discovery of integer and fractional quantum Hall effects in the two-
dimensional electron gas at high magnetic field ushered in a revolution of
topological quantum physics that remains fruitful over forty years later Tsui
et al. (1982). The possibility of realizing analogous topological quantum
fluids in Chern bands, broadly referred to as Chern insulators and fractional
Chern insulators, was envisioned and demonstrated by proof-of-principle
lattice model studies Haldane (1988); Sun et al. (2011); Tang et al. (2011);
Neupert et al. (2011); Sheng et al. (2011); Regnault and Bernevig (2011).

Of particular interest and fundamental importance is the quantized anomalous
Hall effect at zero magnetic field Chang et al. (2023), which may occur in
time-reversal-invariant topological bands Wu et al. (2019); Song et al.
(2015); Kane and Mele (2005) and requires magnetic ordering that spontaneously
breaks time reversal symmetry Sharpe et al. (2019); Serlin et al. (2020);
Devakul et al. (2021); Crépel and Fu (2023); Reddy et al. (2023); Morales-
Durán et al. (2023a). Thanks to innovation in moiré quantum materials and
advanced theoretical guidance, the fractionally quantized anomalous Hall
effect has finally been observed Cai et al. (2023); Zeng et al. (2023); Park
et al. (2023). In the same spirit as a partially-filled Landau level, a
partially-filled topological band can exhibit a symphony of distinct phases as
a function of filling factor, each bringing its own novelty as an impetus to
extend the frontier of condensed matter physics.

Recent experiments on $t$MoTe2 have demonstrated the ability to induce phase
transitions out of QAH states by applying displacement field Cai et al.
(2023); Zeng et al. (2023); Park et al. (2023). These phase transitions have
received limited theoretical and numerical attention thus far and provide an
interesting direction for future work. Pressure, an additional _in situ_
tuning knob that has been demonstrated numerically to enhance the stability of
FQAH states, also deserves attention Morales-Durán et al. (2023a).

While our global phase diagram contains many prominent features supported both
by our numerical study and/or experiment, it is by no means complete. Further
study of the various phases we identify by other numerical methods with access
to larger system sizes such as density matrix renormalization group can
provide valuable insight and have indeed already done so in the case of the
ACFL Dong et al. (2023). Theoretical studies indicate that the IQAH state at
$n=1$ may be energetically outcompeted at larger angles Qiu et al. (2023);
Wang et al. (2023c), and the possibility of competing phases at fractional
filling not captured in the single band-projected model also deserves further
investigation Xu et al. (2023b).

_Acknowledgements–_ We thank Hart Goldman, Nisarga Paul, Ahmed Abouelkomsan,
and Emil Bergholtz for related collaborations. This work is supported by the
Air Force Office of Scientific Research (AFOSR) under Award No.
FA9550-22-1-0432 and the Simons Investigator award from the Simons Foundation.
The authors acknowledge the MIT SuperCloud and Lincoln Laboratory
Supercomputing Center for providing HPC resources that have contributed to the
research results reported within this paper.

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