# LMCanvas: Object-Oriented Interaction to Personalize Large Language Model-
Powered Writing Environments

Tae Soo Kim<EMAIL_ADDRESS>School of Computing, KAISTDaejeonRepublic
of Korea ,  Arghya Sarkar<EMAIL_ADDRESS>New York UniversityNew YorkUSA ,
Yoonjoo Lee<EMAIL_ADDRESS>School of Computing, KAISTDaejeonRepublic
of Korea ,  Minsuk Chang<EMAIL_ADDRESS>Naver AI
LabSeongnamRepublic of Korea and  Juho Kim<EMAIL_ADDRESS>School of
Computing, KAISTDaejeonRepublic of Korea

(2023)

###### Abstract.

Large language models (LLMs) can enhance writing by automating or supporting
specific tasks in writers’ workflows (e.g., paraphrasing, creating analogies).
Leveraging this capability, a collection of interfaces have been developed
that provide LLM-powered tools for specific writing tasks. However, these
interfaces provide limited support for writers to create personal tools for
their own unique tasks, and may not comprehensively fulfill a writer’s
needs—requiring them to continuously switch between interfaces during writing.
In this work, we envision LMCanvas, an interface that enables writers to
create their own LLM-powered writing tools and arrange their personal writing
environment by interacting with “blocks” in a canvas. In this interface, users
can create text blocks to encapsulate writing and LLM prompts, model blocks
for model parameter configurations, and connect these to create pipeline
blocks that output generations. In this workshop paper, we discuss the design
for LMCanvas and our plans to develop this concept.

Generative AI, Large Language Models, Writing Support Tools, Object-Oriented
Interaction

††copyright: acmcopyright††journalyear: 2023††copyright:
rightsretained††conference: CHI 2023 Workshop on Generative AI and HCI; Apr
28, 2023; Virtual††booktitle: CHI 2023 Workshop on Generative AI and HCI), Apr
28, 2023, Virtual††doi: 10.1145/XXXXXXX.XXXXXXX

##  1\. Introduction

The advent of large language models (LLMs)—e.g., GPT-3 (Brown et al., 2020),
GPT-NeoX (Black et al., 2022), Jurassic-1 (Lieber et al., 2021), LaMDA
(Thoppilan et al., 2022)—has transformed the writing process. Instead of
manually drafting passages of text, writers can now hand over this effort to
these models and almost instantly generate passages from an initial sentence
or phrase. Beyond their generative capabilities, LLMs demonstrate significant
few-shot and zero-shot performance (Brown et al., 2020) meaning that they are
able to perform previously unseen tasks with only an instruction and/or a
couple of examples—i.e., a prompt. By leveraging this ability of LLMs, writers
can potentially automate or augment specific tasks in their workflows by using
adequate prompts and, thus, further facilitate the writing process. For
instance, based only on prompt examples provided for GPT-3 (OpenAI, 2022),
writers can use LLMs to correct grammar, create an outline, produce analogies,
or even change the point-of-view of a scene.

To seize the opportunity presented by LLMs, an assortment of products and
interfaces have been created that leverage these models to provide writers
with specific tools that automate steps in their writing workflows. For
example, tools such as WordTune (Labs, 2022a) and NotionAI (Labs, 2022b)
provide editing buttons that the user can click after selecting text to
automatically rewrite it, change its tone, summarize it, elaborate on it, etc.
Additionally, a variety of LLM-powered copywriting tools (AI, 2022; CopyAI,
2022) have also been created that provide writers with a variety of template
forms that they can fill to generate specific types of writing (e.g., video
description or script, blog introduction, article headline). Similarly in
academia, various interfaces have been designed to leverage LLMs to support
specific tasks: generate various forms of figurative language (Chakrabarty et
al., 2022), summarize a writer’s writing (Dang et al., 2022), brainstorm and
combine ideas (Di Fede et al., 2022), or propagate writing edits across a
story (Lee et al., 2022a).

Figure 1. Illustrations for the concatenate, split, and input interactions
supported in text blocks.

While the proliferation of these LLM-driven tools means that various writing
tasks can now be supported, the individual needs and challenges of writers
might not be fulfilled by these tools. Due to their type of writing, their
fluency with a language, or other factors such as their style and workflow, a
writer may have specific needs and challenges during their writing process.
However, while existing interfaces provide a general set of tools, they
provide limited or no support for the writer to create their own tools to
support their unique tasks. Further, an interface may not provide a
comprehensive set of tools that supports all of the writer’s tasks and, thus,
the writer may need to constantly switch between multiple interfaces to
support their workflows. As a result, the writer needs to scatter and adapt
their writing workflow across a variety of interfaces.

In this work, we envision a canvas-based interface that enables writers to
create their own personalized LLM-driven tools and configure them into one
cohesive writing environment. Inspired by object-oriented interaction (Xia et
al., 2016; Ciolfi Felice et al., 2016; Xia et al., 2018, 2017; Han et al.,
2022; Beaudouin-Lafon, 2000) and block-based programming (Resnick et al.,
2009), we present the design for LMCanvas, an interface that enables users to
interact with text and model blocks to flexibly create and arrange LLM-powered
tools. Through the interface, users can create text blocks to encapsulate both
their writing and LLM prompts, keep drafts as separate blocks, and organize
them in the canvas. By connecting text blocks to model blocks (i.e., blocks
that represent a set of model configurations), users can create LM pipelines,
tools, that generate outputs as text blocks based on the input text and model
block. After creating a set of tools, the user can flexibly arrange them in
the canvas to create a writing environment customized to their needs and
preferences. In this workshop paper, we discuss our design of the envisioned
LMCanvas and future work to develop this concept.

##  2\. LMCanvas: Design Concept

In our envisioned interface, LMCanvas, writers can create four types of
objects in an infinite canvas: text blocks, model blocks, and pipeline blocks.
All of these blocks can be flexibly moved, copy-pasted, deleted, and connected
to each other. Below, we detail the specific interactions that we aim to
support for each type of block.

###  2.0.1. Text Blocks

In the canvas, the writer can create text blocks, which are objects that the
writer can type text into and edit. When writing with LLMs, text can represent
different types of content: actual writing, prompts, examples for prompts,
generated outputs, etc. By compartmentalizing text into modular blocks, our
interface allows the writer to flexibly organize and structure these different
forms of text in their writing environment. For example, the writer can use a
text block as their main text editor, maintain text blocks on the side
containing alternative versions for certain paragraphs, and keep a text block
with a prompt template to reuse when creating LLM-powered tools.

Figure 2. Model blocks represent a set of parameter configurations that the
writer can configure, copy, and reuse. By connecting model blocks to text
blocks, writers can create pipeline blocks that allow them to generate outputs
based on the nested text and parameters.

To support flexible use of text blocks, we design the following interactions
specific to these blocks. Resizing allows the writer to change the format of
text blocks for different types of usage (e.g., a larger block for a text
editor) or to decrease their size to decrease clutter in the screen. When the
writer decides that they do not have to keep two text blocks separate anymore
(e.g., decided on the final versions for the first two verses of their poem),
they can concatenate these blocks by drag-and-dropping one text block into the
other. Alternatively, if the writer needs to modularize or separate certain
parts of a text (e.g., to only draft one part of a paragraph), they can split
off text by selecting it and dragging it outwards—creating a new text block.
To allow writers to create reusable LLM-powered tools, the interface allows
the user to create text blocks to which they can input other text blocks.
Specifically, the user types the “`[[input]]`” command in a text block to
create an “input prong”. Then they can attach other blocks into this prong to
replace the “`[[input]]`” command with the content of the attached text block.
Finally, as writers may want writing support tools to act on selected text
(e.g., generate metaphor for selected phrase or edit selected text to be
shorter), the interface also allows users to create select blocks by typing
the “`[[select]]`” command in a text block. The content of these blocks are
replaced by any text that the user selects in the canvas.

Figure 3. The output container of pipeline blocks can be connected to text
blocks to add generations as continuations, or to the input prongs of text
blocks to chain pipelines.

###  2.0.2. Model Blocks

LLMs possess various parameters that control the generation process. For
example, the temperature parameter determines the probability of the model
generating more out-of-distribution or improbable text. Prior work has
demonstrated that, when writing with LLMs, different configurations of these
parameters can satisfy different user needs (Lee et al., 2022b). Thus, to
support writers to set, test, and reuse parameter configurations, LMCanvas
allows users to create multiple model blocks with different combinations of
parameters (Figure 2). These blocks represents an instance of parameter
configurations that the writer can reconfigure by clicking on a parameter and
using the displayed widgets to change its value.

###  2.0.3. Pipeline Blocks

To generate text with the LLM, the user can connect a text block to a model
block to create a pipeline block (Figure 2). When a writer clicks on
“generate” in a pipeline block, the interface uses the nested text block as
input and the model block as the parameter configurations to generate an
output, which is presented as a text block. To test multiple inputs and
parameter configurations, the writer can also expand a pipeline block by
adding additional text and model blocks. In this case, when the pipeline block
generates, it produces a generation for each pairing of text and model blocks
inside the pipeline block.

By default, each time a pipeline block generates, it adds the generation as a
text block in the output container—the box containing “1” that prongs out of
the pipeline block as seen in Figure 2. However, through drag-and-drop, the
writer can connect this output container to (1) a text block to add the
generations from the pipeline as continuations to that text block (left in
Figure 3), or (2) an input prong to chain multiple pipeline blocks together
and create more complex tools (Wu et al., 2021) (right in Figure 3).
Additionally, if the writer connects the output of a pipeline to a select
block, the interface replaces any text selected across all text blocks in the
canvas with the generation produced by the pipeline. These various forms of
connecting pipeline blocks can enable the writer to create a variety of tools
from the same basic blocks.

##  3\. Future Work

In this workshop paper, we outlined the foundational objects and interactions
that we aim to support in our envisioned LMCanvas. Our goal with this
interface is to enable writers to more effectively leverage LLMs by
personalizing their use to fit their unique workflows, needs, and challenges.
At the current stage of the project, we have developed an initial prototype
that supports the three objects presented in this paper and their basic
interactions. With this prototype, we are planning to conduct formative
studies to understand the various tools that writers can create with LMCanvas,
the benefits and drawbacks of the interfaces, and additional blocks and
interactions that writers might need. Based on the findings, we plan to
improve and expand on the concept. Beyond the directions for improvement to be
distilled from the formative study, there are additional future directions
that we plan to pursue with LMCanvas.

First, an additional benefit of representing writing as text blocks is that
this can enable the interface to maintain a separate history for each text
block. With this modularized history, writers can check on and revert changes
for only specific parts of their writing, and they can also reflect back on
their generation attempts by seeing what inputs and parameter configurations
were previously used. We are planning to implement this modularized history
for text blocks and to enable users to interact with it—e.g., dragging the
text input that generated a text block out from the history and into the
canvas.

Second, in future versions of LMCanvas, we aim to support various types of
output containers for pipeline blocks. Currently, the prototype only supports
containers that keep generated text blocks as a list. However, when dealing
with a large quantity of generated outputs, writers may need alternative
methods to look at and explore generated outputs. For example, generations
could be encoded in a scatterplot (Matejka et al., 2018) to enable the writer
to visualize the output space.

Finally, identifying effective prompts (i.e., prompt engineering) is a major
hurdle in leveraging LLMs. While tools have been designed to facilitate this
in well-defined tasks where there is a “ground-truth” (Strobelt et al., 2022),
there is limited work that investigated how to support prompt engineering in
open-ended and more creative tasks. Through our initial versions of LMCanvas,
we aim to investigate mechanisms to facilitate prompt engineering in open-
ended writing and to incorporate these into the interface. For example, the
interface could allow writers to drag-and-drop text blocks into pipeline
blocks as positive or negative examples, and leverage these in the back-end to
produce outputs more aligned to the writers’ preferences.

###### Acknowledgements.

This work was supported by KAIST-NAVER Hypercreative AI Center.

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# Quarter-Metal Phases in Multilayer Graphene: Ising-XY and Annular Lifshitz
Transitions

Mainak Das and Chunli Huang  Department of Physics and Astronomy, University
of Kentucky, Lexington, Kentucky 40506-0055, USA

(August 28, 2024)

###### Abstract

Recent experiments have uncovered a distinctive magnetic metal in lightly-
doped multilayer graphene, coined the quarter metal. This quarter metal
consolidates all the doped carriers, originally distributed evenly across the
four (or twelve) Fermi surfaces of the paramagnetic state, into one expansive
Fermi surface by breaking time-reversal and/or inversion symmetry. In this
work, we map out a comprehensive mean-field phase diagram of the quarter-metal
in rhombohedral trilayer graphene within the four dimensional parameter space
spanned by the density $n_{e}$, interlayer electric potential $U$, external
magnetic field parallel to the two-dimensional material plane $B_{\parallel}$
and Kane-Mele spin-orbit coupling $\lambda$. We found an annular Lifshitz
phase transition and a Ising-XY phase transition and locate these phase
boundaries on the experimental phase diagram. The movement of the Ising-XY
phase boundary offers insights into $\lambda$. Our analysis reveals that it
moves along the line $\partial n_{e}/\partial B_{\parallel}\sim-0.5\times
10^{11}\text{cm}^{-2}\text{T}^{-1}$ within the $n_{e}$-$B_{\parallel}$
parameter space when $\lambda=30\mu$eV. Additionally, we estimated the in-
plane spin susceptibility of the valley-Ising quarter-metal
$\chi_{{}_{\parallel}}\sim 8\leavevmode\nobreak\
\mu\text{eV}\leavevmode\nobreak\ \text{T}^{-2}$. Beyond these quantitative
findings, two general principles emerge from our study: 1) The valley-XY
quarter metal’s dominance in the $n_{e}-U$ parameter space grows with an
increasing number of layers due to the reduce valley polarization variations
within the Fermi sea. 2) Layer polarization near the band edge plays an
important role in aiding the re-entrance of the paramagnetic state at low
density. The insights derived from the quarter metal physics may shed light on
the complex behaviors observed in other regions of the phase diagram.

##  I Introduction

The recent discoveries of symmetry-broken electronic phases [1, 2, 3, 4, 5] in
the density-interlayer potential ($n_{e}-U$) parameter space of multilayer
graphene have opened new avenues to explore strongly correlated physics [6, 7,
8, 9, 10, 11, 12, 13, 14, 15] and energy-efficient electronic devices [16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. The observed phases
include spin-singlet superconductor, spin-polarized superconductors and
generalized metallic ferromagnets, where the order parameter is characterized
by the 15 generators of the $SU(4)$ space spanned by spin ($s$) and valley
($\tau$) degrees of freedom. Among the various magnetic phases, the so-called
quarter metal is particular noteworthy because of its simplicity and its
analogies to the relatively well-understood $\nu=\pm 1$ graphene quantum Hall
magnet. In the large hole-density regime of the quarter metal, the area
enclosed by the Fermi surface extracted from magnetic oscillation data
precisely corresponds to $4\pi^{2}$ times the carrier density. This implies
that, despite the 4-fold degenerate density-of-states implied by time-reversal
and inversion symmetry, all holes are enclosed by a single Fermi surface.
Progressing to lower hole density in this phase, Zhou. et. al [1] found a
Lifshitz transition where the simple Fermi sea becomes an annular Fermi sea.
Despite its extensive capability, magnetic oscillations on its own does not
provide insight into the nature of the order-parameter. Indeed, the order
parameter can rotate in 15 the dimensional space without altering the area
enclosed by the Fermi surface(s). Such variations in the order-parameter leads
to variations in groundstate energy, which we found to be much smaller than
the Coulomb energy scale. We termed this energy landscape as magnetic
anisotropy energy landscape, and it holds pivotal importance in the study of
magnetism. The intrinsic spin-orbit coupling (SOC) of graphene profoundly
influences this magnetic anisotropy energy landscape. Since Kane and Mele
first introduced it [16], numerous experiments have employed varied techniques
to measure the SOC [31, 32, 33, 34, 35, 36, 37].

Figure 1: The displacement of the valley-Ising and valley-XY phase-boundary
induced by in-plane magnetic field depends on the strength of spin-orbit
coupling $\lambda$. Experimental observation of these shifts can be used to
estimate $\lambda$.

In this article, we present a comprehensive study of quarter metal phase
diagram in the parameter space spanned by $n_{e},U$, in-plane magnetic field
$B_{\parallel}$ and SOC in rhombohedral trilayer graphene (RTG). Our findings
indicate two distinct magnetic ground states for the quarter metal, termed the
valley-XY phase and valley-Ising phase, separated by a first order phase
transition. The momentum distribution of their holes either forms a
conventional Fermi sea or adopts an annular shape, with the two types being
separated by an annular Lifshitz transition (ALT). This ALT corresponds to the
enhanced resistance (depicted in yellow) line running from
$(n_{e},D)\sim(-4.5\times 10^{11}\text{cm}^{-2},0.35\text{V/nm})$ to
$(n_{e},D)\sim(-4.2\times 10^{11}\text{cm}^{-2},0.3\text{V/nm})$ in Fig. 1.e
of Ref. [3]. The microscopic origin of magnetic anisotropic energy is
attributed to the non-identical nature of wavefunctions across opposite
valleys, SOC and $U$. Notably, the Ising-XY phase-boundary shift generated by
an in-plane field $B_{\parallel}$ is strongly influenced by the strength of
Kane-Mele spin-orbit coupling $\lambda$. This correlation enables us to relate
the slope $dn_{e}/dB_{\parallel}$ to the strength of the spin-orbit coupling,
as shown in Fig. 7. Additionally, we have calculated the in-plane magnetic
susceptibility within the valley-Ising phase. These theoretical outcomes can
be readily corroborated with experimental data to provide deeper insights into
graphene magnetism.

The remainder of this article is structured as follows: In Sec. II, we study
the quarter metal phase diagram without SOC. Sec. III discuss the role of
layer-polarization in assisting the reentrance of paramagnet state at low
densities. Sec. IV examines the impact of SOC on the magnetic states within
the quarter metal. In Sec. V, we use the displacement of the Ising-XY phase
boundary to estimate the magnitude of SOC. In Sec. VI, we compare the quarter
metal phase diagram between Bernal bilayer graphene and rhombohedral trilayer
graphene. Finally, we summarize our work in Sec. VII.

##  II Quarter-Metal Phase Diagram without Spin-Orbit Coupling

In this section, we study the evolution of the quarter-metal groundstate in
the $n_{e}-U$ parameter space when spin-orbit coupling is neglected. We first
discuss the bandstructure of the valley-Ising phase $\ket{\theta_{v}=0}$ and
the valley-XY phase $\ket{\theta_{v}=\pi/2}$ obtained from self-consistent
mean-field calculation. Then we compare their band-energy and Fock exchange-
energy evolution as a function of $n_{e}$. These analysis reveal that the
first-order phase transition between $\ket{\theta_{v}=\pi/2}$ and
$\ket{\theta_{v}=0}$ is driven by the decrease of exchange energy associated
with the winding of pseudospin enforced by the band topology of multilayer
graphene.

In the absence of spin-orbit coupling, the spin-quantization axis of a spin-
polarized groundstate can be rotated without any energy cost. In contrast,
there is a preferred direction to polarize valley degree of freedom because
the wavefunctions and energy dispersion are different in the two valleys.

We use self-consistent Fock approximation to estimate the groundstate energy
of the quarter metal state. The itinerant exchange effect, as described by
Herring [38], is arguably the most important factor in shaping the total
energy of a quarter metal. This assertion is supported by the oscillation
data, which strongly suggest that time-reversal and/or inversion symmetry of
the quarter metal is broken to maximize its pseudospin polarization. Given
these observations, it seems likely that the self-consistent mean-field
approximation would work better for the quarter metal than for other
identified magnetic states, such as the half-metal. The mean field eigenvalue
equation is
$\hat{H}_{\bm{k}}|\psi_{n\bm{k}}\rangle=\epsilon_{n\bm{k}}|\psi_{n\bm{k}}\rangle$
where

$\hat{H}_{\bm{k}}=\hat{T}_{\bm{k}}+\hat{\Sigma}^{F}_{\bm{k}}.$ (1)

The band-Hamiltonian $\hat{T}_{\bm{k}}$ is parameterized by the experimentally
informed [1] Slonczewski-Weiss-McClure parameters and the Fock self-energy is
given by
$\hat{\Sigma}^{F}_{\bm{k}}=-\sum_{n\bm{k}^{\prime}}V_{\bm{k}-\bm{k}^{\prime}}\hat{\rho}_{\bm{k}^{\prime}}$.
Here $V_{\mathbf{q}}$ is the Fourier components of the gate-screened Coulomb
potential $V_{\mathbf{q}}=\frac{2\pi
k_{e}}{\mathcal{A}}\tanh(|\mathbf{q}|d)/(\epsilon_{r}|\mathbf{q}|)$ where
$\mathcal{A}$ is area of the sample, Coulomb constant $k_{e}=1.44$ eVnm,
screening constant $\epsilon_{r}=15$ and $d=5$ nm is the distance from the
gate to the material. The density-matrix is constructed from the eigenvector
and eigenvalue of the self-consistent Hamiltonian:

$\hat{\rho}_{\bm{k}}=\sum_{n}n_{F}(\epsilon_{n\bm{k}})|\psi_{n\bm{k}}\rangle\langle\psi_{n\bm{k}}|.$
(2)

The total energy of the self-consistent mean-field approximation is a sum of
the band-energy $E_{b}^{\theta_{v}}$ and the Fock exchange energy
$E_{ex}^{\theta_{v}}$,

$\displaystyle
E_{b}^{\theta_{v}}=\int\frac{d^{2}k}{4\pi^{2}}\text{Tr}[\hat{T}_{\bm{k}}\hat{\rho}_{\bm{k}}^{\theta_{v}}],$
(3) $\displaystyle
E_{ex}^{\theta_{v}}=-\frac{1}{2}\int\frac{d^{2}k}{4\pi^{2}}\text{Tr}[\hat{\Sigma}_{\bm{k}}^{F}\hat{\rho}_{\bm{k}}^{\theta_{v}}],$
(4)

where the superscript $\theta_{v}$ stands for the valley-polarization of the
converged density matrix:

$\int\frac{d^{2}k}{4\pi^{2}}\text{Tr}[\hat{\rho}_{\bm{k}}^{\theta_{v}}(\tau_{x}\hat{\bf{x}}+\tau_{z}\hat{\bf{z}})]\sim\sin(\theta_{v})\hat{\bf{x}}+\cos(\theta_{v})\hat{\bf{z}}.$
(5)

Here the trace Tr operates on the $24$ dimensional space generated by the
6$p_{z}$ orbitals, the $2$ spin and $2$ valley degree of freedoms. More
discussion of the Hamiltonian can be found in Ref. [20] and Ref. [39]. We used
a Lagrange multipliers to generate the energy landscape associated with order
parameter rotations.

Figure 2: Mean field phase diagram shows 4 unique phases of quarter metal
distinguished by their valley-polarization (XY/Ising) and Fermi surface
topology (Simple Fermi Sea/Annular Fermi Sea). The evolution of band and Fock
energy along the white dashed line, as well as the bandstructure of the black
and red points are further expanded in Fig. 3.  Figure 3: a) Band-structure
(along $k_{y}a=0$) of valley-Ising phase at $n_{e}=-2.9\times
10^{11}\text{cm}^{-2}$ and b) valley-XY phase at $n_{e}=-2.5\times
10^{11}\text{cm}^{-2}$ respectively at $U=25\leavevmode\nobreak\ \text{meV}$.
The chemical potential $(\mu)$ is marked by gray horizontal dashed line in
both cases. c,d) Band and Fock exchange energy difference of valley-Ising and
valley-XY phase per charge as a function of density (marked by white dashed
line in Fig. 2) shows the energy difference is increasing while approaching
charge neutrality. The band energy in Ising phase is higher than valley-XY
phase, whereas exchange energy of valley-XY phase is higher than valley-Ising
phase. e) Difference in layer pseudospin average $L^{\theta_{v}}_{x}$ per
charge in valley-Ising and valley-XY state decreases while approaching charge
neutrality. This indicates why exchange energy difference between these two
states are increasing with decreasing hole density. f) The total energy
difference between valley-Ising and valley-XY phase $v.s.$ $n_{e}$ identifies
the phase-transition at $n_{e}\approx-2.78\times 10^{11}$cm-2. The positive
slope indicates the chemical potential of the Ising state is higher than the
XY state.

As shown in Fig. 2, we identified four unique phases of the quarter metal.
These phases are characterized by differences in the number of Fermi surfaces
and their valley-polarization. They are separated by first-order Ising-XY
transition boundaries and first-order annular-Lifshitz transition phase
boundaries. The simplest phase among these four appears at large $|n_{e}|$ and
$U$. It has an Ising-like valley-polarization ($|\theta_{v}=0\rangle$ or
$|\theta_{v}=\pi\rangle$) and all holes are enclosed in a simply-connected
Fermi surface (SFS). Designated as the Ising-SFS, this phase is colored blue
in Fig.2 Its unconventional relationship between orbital magnetization and
valley-polarization is discussed in Ref. [39]. As $|n_{e}|$ decreases in the
interlayer potential range $30<U<35$meV, the Ising-SFS undergoes an annular
Lifshitz transition (ALT) where an electron-like Fermi surface appears at the
centre of the Fermi sea. This leads to an expansion of the hole-like Fermi
surface without changing the valley/spin polarization. We refer to this state
as the Ising-AFS which is represented as green in Fig. 2. In contrast, when
$|n_{e}|$ decreases at smaller interlayer potential $17<U<30$meV, we found
that the Ising SFS goes through a magnetic phase transition and changes its
valley order parameter from $\theta_{v}=0$ to $\theta_{v}=\pi/2$. We dubbed
this state as XY-SFS because it enclosed all the holes in a single Fermi
surface. Note that the Fermi surface is intrinsically tied to the valley-order
parameter: it adopts a $C_{6}$ symmetry for $\theta_{v}=\pi/2$ and $C_{3}$ for
$\theta_{v}=0$. As $|n_{e}|$ reduces to smaller values, XY-SFS also encounters
an ALT. We’ve labelled this phase as XY-AFS, and it is distinguished by purple
color in Fig. 2.

Fig. 3.a) and b) shows the bandstructure $\epsilon_{n,kx,ky=0}$ $v.s.$ $k_{x}$
for the valley-Ising and valley-XY states respectively. While the low-energy
electronic states in multilayer graphene are centered at the zone-corner, we
fold the bands into the $\Gamma$ point to visualize valley-mixing [40]. The
annular Lifshitz transition appears because the bandstructure has a local
minimum at $\bm{k}=0$, as shown in Fig. 3b). The valence band-projected
valley-polarization shows that the quasiparticles in valley-XY states is a
linear superposition of states from opposite valleys. Such valley-mixing is
enabled by tripling of the area of the real-space unit cell. In contrast, the
valley Ising state merely redistribute the holes all into one valley (say
$\tau_{z}=-1$ in Fig. 3.a) without changing the valley content of the spinor.

In order to understand the Ising-XY phase transition, we break down the energy
difference between $\ket{\theta_{v}=0}$ and $\ket{\theta_{v}=\pi/2}$ into the
band contribution and exchange contribution across various $n_{e}$. As
illustrated by the purple curve in Fig. 3.c), the band-energy of the Ising
state is higher than that of the XY state. This is because the $XY$ state
requires a lesser degree of quasiparticle redistribution from the paramagnetic
state. Consequently, from a band energy perspective, the XY state is favored
over the Ising state as the quarter metal’s magnetic ground state.
Importantly, this preference is reinforced at lower $|n_{e}|$ thus the band
energy contributes to driving the density-induced first-order Ising-XY phase
transition. The chemical potential difference between Ising and XY state from
the band-energy can be inferred from the slope in Fig. 3.c),

$\frac{\partial(E_{b}^{\theta_{v}=0}-E_{b}^{\theta_{v}=\pi/2})}{\partial
n_{e}}\equiv\mu_{b}^{\theta_{v}=0}-\mu_{b}^{\theta_{v}=\pi/2}>0.$ (6)

Figure 4: a) The band energy of the Ising state with respect to paramagnetic
state decreases while approaching charge neutrality. b) The Ising Fock energy
increases linearly with respect to paramagnetic Fock energy as $|n_{e}|$
deceases. c) Layer pseudospin orientation of the holes enclosed in a $C_{3}$
Fermi surface in the valley-Ising quarter metal. As we circle the Fermi
surface anticlockwise, the pseudospin rotates by $6\pi$. Result is obtained at
$n_{e},U=-2.9\times 10^{11}\text{cm}^{-2},25\leavevmode\nobreak\ \text{meV}$.
d) Layer pseudospin orientation of the hole-like Fermi pockets shown in one
valley in paramagnetic phase at same $n_{e},U$. e) The layer pseudospin
$L_{z}$ of paramagnetic phase changes faster than valley-Ising phase with
respect to density, and consequently the exchange energy gap between valley-
Ising and paramagnetic phase shrinks towards charge neutrality.

Fig. 3.d) shows that the XY state’s exchange-energy is elevated compared to
the Ising state’s exchange-energy. This energy difference can be traced back
to the non-uniform alignment of quasiparticle valley-polarization in the XY
state, as shown by the color variations in Fig.3.a). However, as $|n_{e}|$
decreases, the difference between these exchange energies
$E_{ex}^{\theta_{v}=\pi/2}-E_{ex}^{\theta_{v}=0}$ narrows, leading to

$\frac{\partial(E_{ex}^{\theta_{v}=0}-E_{ex}^{\theta_{v}=\pi/2})}{\partial
n_{e}}\equiv\mu_{ex}^{\theta_{v}=0}-\mu_{ex}^{\theta_{v}=\pi/2}>0.$ (7)

This trend is attributed to the reduction of interlayer coherence
$L_{x}^{\theta_{v}=\pi/2}-L_{x}^{\theta_{v}=0}$ with decreasing $|n_{e}|$, as
shown in Fig. 3e). The interlayer coherence and layer-polarization (per area)
is defined as
$L_{j}^{\theta_{v}}=\int\frac{d^{2}k}{4\pi^{2}}L_{j}^{\theta_{v}}(\bm{k})$
where $j=x,y,z$:

$\displaystyle L_{x}^{\theta_{v}}(\bm{k})$
$\displaystyle=\text{Tr}[\hat{\rho}_{\bm{k}}^{\theta_{v}}(\ket{A1}\bra{B3}+\ket{B3}\bra{A1})],$
(8) $\displaystyle L_{y}^{\theta_{v}}(\bm{k})$
$\displaystyle=-i\text{Tr}[\hat{\rho}_{\bm{k}}^{\theta_{v}}(\ket{A1}\bra{B3}-\ket{B3}\bra{A1})],$
(9) $\displaystyle L_{z}^{\theta_{v}}(\bm{k})$
$\displaystyle=\text{Tr}[\hat{\rho}_{\bm{k}}^{\theta_{v}}(\ket{A1}\bra{A1}-\ket{B3}\bra{B3})].$
(10)

Here $\ket{A1}$ and $\ket{B3}$ represent the outermost layer sublattices. Note
the antisymmetric layer coherence remains zero for both type of valley-
polarization $L_{y}^{\theta_{v}=0}=L_{y}^{\theta_{v}=\pi/2}=0$. For all
$n_{e}$ and $U$, although the layer-polarization is greater in magnitude than
layer coherence, for example :
$L_{z}^{\theta_{v}=0}/|n_{e}|\sim-7.584,L_{z}^{\theta_{v}=\pi/2}/|n_{e}|\sim-7.583$
and $L_{x}^{\theta_{v}=0}/|n_{e}|\sim-1.283,\leavevmode\nobreak\
L_{x}^{\theta_{v}=\pi/2}/|n_{e}|\sim-1.228$ at $n_{e},U=-2.9\times
10^{11}\leavevmode\nobreak\ \text{cm}^{-2},25\leavevmode\nobreak\ \text{meV}$,
their difference $(L_{z}^{\theta_{v}=\pi/2}-L_{z}^{\theta_{v}=0})/|n_{e}|\sim
10^{-3}$ is an order of magnitude smaller than those shown in Fig. 3.e). This
is because the $L_{z}$ is mainly determined by the interlayer potential and
the position of the Fermi level.

As a result, while the exchange energy leans towards the Ising state as the
magnetic ground state of quarter metal, this inclination becomes less
pronounced at smaller $|n_{e}|$. In this sense, the exchange energy also
contributes to driving the Ising-XY phase transition as $|n_{e}|$ reduces.
Fig. 3.f) plots the total energy difference between Ising and XY and indicated
the phase-boundary at $n_{e}\sim-2.78\times 10^{11}$cm-2.

##  III Re-entrance of Paramagnetic Metal at Low Density and the Layer
Vorticity

In the large interlayer potential ($U$) and lowest-density ($n_{e}$) region of
the $n_{e}-U$ phase-diagram for multilayer graphene, an interesting question
arises: Why does the spin-and-valley polarized magnetic quarter-metal
transition into a paramagnetic metal as density ($|n_{e}|$) decreases, in
sharp contrast to the two-dimensional electron gas (2DEG)? In this section, we
discuss how the distinctive topological band properties of multilayer graphene
leads to a narrowing energy gap between the magnetic states and paramagnetic
state as $|n_{e}|$ decreases. This trend provides a crucial ingredient in
understanding the re-entrance of the paramagnetic phase observed at low
densities [1, 41].

Fig. 4.a) and b) plot the band-energy and exchange-energy per particle of the
Ising and XY states relative to the paramagnetic state. The first feature to
notice is that the energy difference of the Ising and XY magnetic states
relative to the paramagnetic state significantly surpasses the energy gap
between the Ising and XY states themselves, i.e., the magnetic anisotropic
energy (c.f. Fig. 3.c,d and Fig. 4.a,b). A second important feature is the
swift decline in exchange energy per particle of the magnetic state relative
to the paramagnetic state as $|n_{e}|$ decreases. Meanwhile, the band-energy
difference per particle stays relatively constant. This disparity, which is on
the order of a magnitude, contrasts significantly with behaviors seen in 2DEG.
In 2DEG, as $n_{e}$ decreases, the kinetic energy difference between the fully
spin-polarized state and the paramagnetic state,

$x(E_{b}^{FM}-E_{b}^{PM})/n_{e}=\frac{1}{r_{s}^{2}}\text{Ry}$ (11)

changes more quickly than their Fock energy difference

$(E_{ex}^{FM}-E_{ex}^{PM})/n_{e}=-\frac{8\sqrt{2}}{3\pi}\frac{(\sqrt{2}-1)}{r_{s}}\text{Ry},$
(12)

where $\pi(r_{s}a_{B})^{2}=\frac{1}{n_{e}}$ and $\text{Ry}=13.6$eV. We found
that such different density-dependence of the band and Fock energies between
2DEG and multilayer graphene arises from the topological band properties of
multilayer graphene. In a $N$-layer rhombohedral graphene electronic system,
the first valence and conduction band can be approximately described by an
effective two-orbital Hamiltonian comprised of the outermost layer $p_{z}$
orbitals, e.g. $A_{1}-B_{N}$. These two degrees of freedom $A_{1}$ and $B_{N}$
form a layer pseudospin and it winds $2\pi N$ around the Brillouin zone
corner, leading to a vortex structure in the momentum space. The exchange
energy, which favors a uniform distribution of layer pseudospin in momentum
space challenges this vortex configuration. See [42] for more details.

Fig. 4.d) and e) depict the valence-band projected layer pseudospin
orientation inside the hole-like Fermi surface for both the valley-Ising and
paramagnetic phases. Arrows indicate the in-plane layer-polarization
$L_{x}(\bm{k}),L_{y}(\bm{k})$ and (c.f Eq. (8)) the color indicates the values
of $L_{z}(\bm{k})$. These vortex cores are highlighted by the black dots. The
vortex cores are situated close to the valence band maximum and their
corresponding layer polarization $(L_{x},L_{y},L_{z})\sim(0,0,-1)$. As the
pseudospins approach the vortex cores (or the band-edge), they changes rapidly
to align their pseudospin towards the $(0,0,-1)$ directions.

In the paramagnetic state, the Fermi level is situated very close to the
valence band edge with a Fermi energy of $\epsilon_{F}\sim
2.5\leavevmode\nobreak\ \text{meV}$ at $n_{e},U=-2.9\times
10^{11}\leavevmode\nobreak\ \text{cm}^{-2},25\leavevmode\nobreak\ \text{meV}$.
Thus, $L_{z}^{PM}$ changes rapidly with decreasing $|n_{e}|$. On the other
hand, in the quarter metal (be it Ising or XY), the area of the single hole-
like Fermi surface expands by a factor of 12 compared to the area enclosed by
the small hole-like Fermi pocket in the paramagnetic state. This causes the
Fermi energy to be situated further from the valence band edge even when both
states share the same values of $n_{e}$ and $U$, with $\epsilon_{F}\sim 4.8$
meV. Therefore, $L_{z}^{\theta_{v}=0,\pi/2}$ changes slowly with decreasing
$|n_{e}|$. In Fig. 4e), we show
$(L_{z}^{\theta_{v}=0}-L_{z}^{PM})\Delta_{ex}/|n_{e}|$ $v.s.$ $n_{e}$ and note
that this slope closely resembles the slope in
$(E_{ex}^{\theta_{v}=0}-E_{ex}^{PM})/|n_{e}|$ $v.s.$ $n_{e}$ shown in Fig.
4b). Thus, this density-dependence of the layer-polarization can reduce the
exchange energy difference between the quarter metal and paramagnetic state
thereby significantly influencing the re-emergence of the paramagnetic state.

Figure 5: a) Modification of the phase diagram in Fig. 2 due to a Ising spin-
orbit coupling of $\lambda=30\leavevmode\nobreak\ \mu$eV. b) valley and spin
texture of valley-Ising state with simply connected Fermi surface (Ising-SFS)
for $n_{e},U=-2.9\times 10^{11}\leavevmode\nobreak\
\text{cm}^{-2},18\leavevmode\nobreak\ \text{meV}$.c) valley and spin texture
of valley-XY state with simply connected Fermi surface (XY-SFS) for
$n_{e},U=-2.4\times 10^{11}\leavevmode\nobreak\
\text{cm}^{-2},18\leavevmode\nobreak\ \text{meV}$.

##  IV Effect of Spin-Orbit Coupling On Quarter-Metal

It is not unreasonable to assume that SOC plays a more important role in
shaping the magnetic anisotropy landscape compare to the lattice-scale Coulomb
interaction [43]111in preparation, Guopeng Chunli. When we approximate
lattice-scale Coulomb interaction by a delta-function interaction, it is clear
that it won’t alter the distribution of particles with identical spin-valley
quantization. Notably, unlike the zeroth-Landau level wavefunctions in
graphene where sublattice and valley degrees of freedom are identical, the
quasiparticle wavefunctions in RTG exhibit rather complex orbital dependence.
Nevertheless, a direct mean-field calculation suggests that inter-valley
scattering with a magnitude of $u_{\perp}=0.5\leavevmode\nobreak\ \text{meV}$
has very small impact on the quarter-metal phase diagram [see Appendix A].
Therefore, our analysis of magnetic anisotropy energy of quarter metal will
center on the SOC.

Since spin-orbit coupling that is off-diagonal in layer degree of freedom is
suppressed by the application of large electric displacement field, we focus
on the simple momentum independent Kane-Mele-like SOC,

$\hat{H}_{KM}=\lambda\leavevmode\nobreak\
\hat{\sigma}_{z}\hat{\tau}_{z}\hat{s}_{z}.$ (13)

This Hamiltonian reduces the full $SU(2)$ spin-rotation symmetry down to
$U(1)$ symmetry as it only commutes with the generator of $s_{z}$ rotation:
$[\hat{H}_{KM},e^{i\phi_{s}\hat{s}_{z}}]=0$. Note
$\hat{\sigma}_{z}=\ket{A1}\bra{A1}-\ket{B3}\bra{B3}$ here is taken as the
layer polarization between the $A1-B3$ sublattice [45, 42]. This form of SOC
can be derived by considering the point group symmetry of rhombohedral
trilayer graphene [46, 47].

When adding Eq. 13 (with $\lambda=30\leavevmode\nobreak\ \mu$eV) to Eq. 1, and
carry out the self-consistent calculations, we obtained the phase diagram
shown in Fig.5.a). A quick comparison between Fig. 5.a) and Fig. 2 shows that
even a modest value of SOC ($\lambda=30\leavevmode\nobreak\ \mu$eV) has a
pronounced effect on the $n_{e}-U$ phase diagram for the quarter metal. Note
$\lambda$ has almost no effect on the annular Lifshitz transition (ALT) phase
boundaries. This is because the two phases on both side of the ALT phase
boundary have identical spin-valley order parameters. Thus, SOC lowers their
energy in a similar way. In contrast, the Ising-XY phase boundary is shifted
dramatically in the direction that expands the Ising phase, strongly
influencing the magnetic anisotropy energy landscape.

The principal effect of $\hat{H}_{KM}$ is to introduce correlation between the
quasiparticle valley-polarization and its spin-quantization. In order to
understand this correlation, we plot the momentum space distribution of spin
and valley-polarization for the holes in the Ising and XY state. For the Ising
ground state, the spin polarization of all the holes are aligned in the $+z$
direction while their valley-polarization are all aligned in the $-z$
direction. This state is represented by the notation
$|\theta_{v},\theta_{s}\rangle$, specifying the valley-polarization angle
$\theta_{v}$ and the spin polarization angle $\theta_{s}$. This state is
called the Ising state because the energies of
$|\theta_{v}=\pi,\theta_{s}=0\rangle$ and its time-reversed counterpart
$|\theta_{v}=0,\theta_{s}=\pi\rangle$ are the same. The effect of SOC in the
Ising phase is to introduce an uniaxial easy-axis normal to two-dimensional
material plane. Note while the spin-quantization axis contains spatial
information (in this case, it’s perpendicular to the two-dimensional material
plane), the valley-polarization, lacks any spatial interpretation. In addition
to the the Ising-state, we identified another stationary solution characterize
an order parameter defined by non-zero values of $\tau_{x}$, $s_{x}$ and
$s_{z}\tau_{z}$. We named this stationary solution the vallley-XY, denoted as
$\ket{\theta_{s}=\pi/2,\theta_{v}=\pi/2}$. Note the total energy of this state
remains unchanged under rotations generated by $e^{i\phi_{s}s_{z}}$ and
$e^{i\phi_{v}\tau_{z}}$ rotations, justifying the valley-XY nomenclature. Fig.
5.c) shows the valley-polarization and spin-polarization of
$\ket{\theta_{s}=\pi/2,\theta_{v}=\pi/2}$ for the holes enclosed inside the
$C_{6}$ Fermi surface. Navigating around the Fermi surface clockwise, the
valley-polarization exhibits an oscillatory pattern about the $\tau_{x}$ axis.
This specific valley-polarization configuration aims to minimize both the
exchange and band energy: it minimizes the exchange energy by predominantly
aligning the valley-polarization along the x-direction and introduces a
valley-population imbalance across six unique momentum regions to further
reduce band energy. The presence of a finite SOC generates a similar
oscillatory pattern in spin so that the valley-polarization
($\bra{\psi_{nk}}\hat{\bm{\tau}}\ket{\psi_{nk}}$) is almost locally anti-
parallel to spin-polarization ($\bra{\psi_{nk}}\hat{\bm{s}}\ket{\psi_{nk}}$)
in momentum space. Contrary to the Ising state, which reduces its energy
linearly with SOC, the XY state decrease its energy quadratically in
$\lambda$.

Figure 6:  The effect of Kane-Mele spin-orbit coupling (SOC) on the order
parameters of valley-Ising and valley-XY quarter metals is illustrated on the
two Bloch sphere. The $\pm z$ directions on the these spheres correspond to
the $|\uparrow,\downarrow\rangle$ and $|K,K^{\prime}\rangle$ degrees of
freedom. In the valley-Ising state, SOC pins the spin and valley polarization
in the $\pm z$ and $\mp z$ directions. Conversely, in the valley-XY phase, it
aligns them in directions orthogonal to $z$. The XY-phase features two soft-
modes, arising from independent rotations of spin and valley order parameter
within the magnetic easy-plane. Refer to the maintext (Sec. IV) for discussion
on the magnitude of the arrows.

The valley-XY and valley-Ising order parameters are visualized on two Bloch
spheres in Fig. 6. Arrows touching the Bloch sphere represent maximium
polarization, as seen in the valley-Ising state. Note, the valley-XY phase
lacks maximum spin and valley polarization because of its non-uniform momentum
distribution of spin-polarization and valley-polarization as shown in Fig.
5.c).

In the Appendix C, we use the Lagrange multiplier
$\hat{H}_{L}(\bm{n})=\lambda_{L}[\bm{n}\cdot\hat{\bm{s}}+\bm{n}\cdot(e^{i\frac{\pi}{2}\hat{\tau}_{x}}\hat{\bm{\tau}}e^{-i\frac{\pi}{2}\hat{\tau}_{x}})]$
to further expand on the energy landscape of other possible quarter metal
phases in the presence of spin-orbit coupling.

##  V Displacement of the Ising-XY phase boundary

Figure 7: a) The slope of the phase boundary for $n_{e},U=-2.9\times
10^{11}\leavevmode\nobreak\ \text{cm}^{-2},18\leavevmode\nobreak\ \text{meV}$
indicates that the rate at which $B_{\parallel}$ lowers the energy of valley
XY state is less than the rate at which $\lambda$ lowers the energy of the
valley Ising state. b) $\frac{M^{XY}_{\parallel}}{(\mu_{XY}-\mu_{I})}$ is
calculated along the phase boundary which is equal to the slope
$\frac{\partial n_{e}}{\partial B_{\parallel}}$ indicated in Fig. 1 for the
specific $n_{e}$ and $U$.

In this section, we extend the electronic phase diagram of the quarter metal
from the two-dimensional $n_{e}-U$ parameter space to the more comprehensive
four dimensional $n_{e}-U-\lambda-B_{\parallel}$ space. Here, $\lambda$
denotes the Kane-Mele SOC (introduced in the last section) and $B_{\parallel}$
represents the in-plane Zeeman field. In this expanded parameter landscape, we
first examine the movement of Ising-XY phase-boundary as a function of
$\lambda$ and $B_{\parallel}$, keeping $n_{e},U$ fixed. Then, we study the
movement of this phase-boundary as a function of $n_{e}$ and $B_{\parallel}$,
keeping $\lambda,U$ fixed. The latter analysis enables us to correlate
$\lambda$ to the experimentally observable metric $dn_{e}/dB_{\parallel}$.

We compute the $n_{e}-U-\lambda-B_{\parallel}$ phase diagram by solving the
following self-consistent mean-field equation at various density $n_{e}$ and
interlayer potential $U$:

$\mathcal{H}_{\bm{k}}=\mathcal{T}_{\bm{k}}+\Sigma_{\bm{k}}^{F}\;\;,\;\;\mathcal{T}_{\bm{k}}=\hat{T}_{\bm{k}}+\hat{H}_{KM}+\hat{H}_{B_{\parallel}},$
(14)

where $\hat{T}_{\bm{k}}$ and $\Sigma_{\bm{k}}^{F}$ are described in Eq. 1. The
Kane-Mele SOC, $\hat{H}_{KM}$ is defined in Eq. 13 and the Zeeman energy is
given by the following

$\hat{H}_{B_{\parallel}}=-\frac{1}{2}g\mu_{B}\hat{s}_{x}B_{\parallel}.$ (15)

Here $g=2$ is the gyromagnetic ratio, $\mu_{B}$ is the Bohr magneton, and
$\hat{s}_{x}$ is the 1st Pauli matrix. The valley Ising-XY phase boundary is
defined by the following equation:

$E^{\theta_{v}=0}(n_{e},U,\lambda,B_{\parallel})=E^{\theta_{v}=\pi/2}(n_{e},U,\lambda,B_{\parallel}).$
(16)

where the total energy is

$\displaystyle
E^{\theta_{v}}(n_{e},U,\lambda,B_{\parallel})\equiv\frac{1}{2}\int\frac{d^{2}k}{4\pi^{2}}\text{Tr}[(\hat{\mathcal{T}}_{\bm{k}}+\mathcal{H}_{\bm{k}})\hat{\rho}_{\bm{k}}^{\theta_{v}}].$
(17)

Fig. 7.a) shows the movement of the Ising-XY phase-boundary as a function of
$B_{\parallel}$ and $\lambda$ for fixed $n_{e}$ and $U$. To leading order in
$B_{\parallel}$, the total energy of the state
$\ket{\theta_{v}=\pi/2,\theta_{s}=\pi/2}$ is lowered while
$\ket{\theta_{v}=0,\theta_{s}=\pi}$ is unaffected. In contrast, to leading
order in $\lambda$, the total energy of the state
$\ket{\theta_{v}=0,\theta_{s}=\pi}$ is lowered while
$\ket{\theta_{v}=\pi/2,\theta_{s}=\pi/2}$ is unaffected. The slope
$d(g\mu_{B}B_{\parallel})/d\lambda>1$ means SOC is more effective in reducing
the energy of the Ising state compared to the energy reduction achieved by the
Zeeman effect on the XY-state $\ket{\theta_{v}=\pi/2,\theta_{s}=\pi/2}$. This
is simply because the out-of-plane spin-polarization of the Ising state is
greater than the net in-plane spin-polarization of the XY-state, as shown in
Fig. 5.b) and c). Note that when the Zeeman field reaches a sufficiently high
value (i.e. $g\mu_{B}B_{\parallel}\gtrapprox\lambda$), the Ising state
manifests field-induced in-plane spin-polarization, such that
$\theta_{s}=\pi-\delta$ and $\theta_{v}=\delta$, with $\delta\ll\pi$.

Next, we consider the Ising-XY phase-boundary as a function of $n_{e}$ and
$B_{\parallel}$ for fixed $\lambda$ and $U$. Since $n_{e}$ and $B_{\parallel}$
are thermodynamical conjugate to chemical potential and magnetic-moments, a
small variation of density $n_{e}$ around Ising-XY boundary at $n_{e}^{*}$ and
$B_{\parallel}$ around $0$, results in the following energy changes:

$\displaystyle E^{\theta_{v}=0}(n_{e},U,\lambda,B_{\parallel})$
$\displaystyle=E^{*}+\mu^{\theta_{v}=0}dn_{e}-M_{\parallel}^{\theta_{v}=0}dB_{\parallel}$
(18) $\displaystyle E^{\theta_{v}=0}(n_{e},U,\lambda,B_{\parallel})$
$\displaystyle=E^{*}+\mu^{\theta_{v}=\pi/2}dn_{e}-M_{\parallel}^{\theta_{v}=\pi/2}dB_{\parallel}$
(19)

where
$E^{*}=E^{\theta_{v}=0}(n_{e}^{*},U,\lambda,B_{\parallel}=0)=E^{\theta_{v}=\pi/2}(n_{e}^{*},U,\lambda,B_{\parallel}=0)$
is the equilibrium free energy energy when $B_{\parallel}=0$ is zero. As the
spins of the Ising phase are pinned out of the two-dimensional material plane
by the Kane-Mele SOC, the in-plane spin-moment vanishes at leading order in
$B_{\parallel}$, $M_{\parallel}^{\theta_{v}=0}=0$. Thus, the Ising-XY phase
boundary evolves in $n_{e}-B_{\parallel}$ space according to

$\frac{\partial n_{e}}{\partial
B_{\parallel}}=\frac{M_{\parallel}^{\theta_{v}=\pi/2}}{\mu^{\theta_{v}=\pi/2}-\mu^{\theta_{v}=0}}.$
(20)

Note $M_{\parallel}^{\theta_{v}=\pi/2}$ is the magnitude of the spin
magnetization so the sign of $\frac{\partial n_{e}}{\partial B_{\parallel}}$
is determined by the denominator:

$\mu^{\theta_{v}=\pi/2}-\mu^{\theta_{v}=0}\equiv\mu^{\theta_{v}=\pi/2}_{b}+\mu^{\theta_{v}=\pi/2}_{ex}-\mu^{\theta_{v}=0}_{b}-\mu^{\theta_{v}=0}_{ex}<0.$
(21)

We used Eq. 6 and Eq. 7 to establish the above relationship. Fig. 7.b) shows
that the ratio
$M_{\parallel}^{\theta_{v}=\pi/2}/(\mu^{\theta_{v}=\pi/2}-\mu^{\theta_{v}=0})$
has very weak dependence on the strength of SOC $\lambda$. Given that
$M_{\parallel}^{\theta_{v}=\pi/2}=g\mu_{B}S_{x}^{\theta_{v}=\pi/2}$ and
considering $S_{x}^{\theta_{v}=\pi/2}\approx 1$ per hole at the densities
where $XY$ state are the groundstate, it is clear that the weak $\lambda$
dependence in $\frac{\partial n_{e}}{\partial B_{\parallel}}$ is attributed to
the small increment of $|\mu^{\theta_{v}=\pi/2}-\mu^{\theta_{v}=0}|$ as
$\lambda$ increases.

Our analysis enables to compute the in-plane spin susceptibility
$\chi_{\parallel}$ of the Ising-state $\ket{\theta_{v}=0,\theta_{s}=\pi}$:

$\chi_{{}_{\parallel}}=\frac{\partial^{2}E^{\theta_{v}=0}}{\partial
B_{\parallel}^{2}}\sim 8\leavevmode\nobreak\ \mu\text{eV\leavevmode\nobreak\
T}^{-2}.$ (22)

See Appendix B for more about in-plane spin susceptibility. Since
$1/\chi_{{}_{\parallel}}$ represents the resistance to collectively rotating
the spin away from the uniaxial magnetic easy-axis, $1/\chi_{{}_{\parallel}}$
increases with the strength of spin-orbit coupling.

##  VI Quarter Metals in Bernal Bilayer Graphene and Beyond

| RTG | BBG
---|---|---
$n_{e}$ |  $-2.7\times 10^{11}$ cm-2 |  $-1.0\times 10^{11}$ cm-2
$E_{ex}/|n_{e}|$ |  $-4.645$ meV |  $-2.349$ meV
$\langle\tau_{x}\rangle/|n_{e}|$ | $1$ | $0.86$
$\langle\tau_{z}\rangle/|n_{e}|$ | $0$ | $0$
$\tau_{z,\bm{k}}$ |  |
Table 1: Comparison of the valley-XY quarter metal in rhombohedral trilayer
graphene (RTG) and Bernal bilayer graphene (BBG). Here $n_{e}$,
$E_{ex}/|n_{e}|$ and $\langle\tau_{x}\rangle$ represent density, the exchange
energy per particle (measured respect to the paramagnetic state), and the
order parameter, respectively.
$\tau_{z,\bm{k}}=\langle\psi_{nk}|\tau_{z,k}|\psi_{nk}\rangle$ denotes the
valley polarization of the hole-occupied Bloch states and
$\langle\tau_{z}\rangle=\sum_{k}\tau_{z,\bm{k}}=0$. Note the stronger
variation of $\tau_{z,\bm{k}}$ in BBG raises the exchange energy per hole in
comparison to RTG. For these calculations, we set $U=28$ meV.

So far, our study has primarily focused on the quarter metal phase diagram of
rhombohedral trilayer graphene (RTG). In this section, we contrasts the phase
diagram of RTG with Bernal bilayer graphene (BBG). BBG is another system where
experimental studies [3, 8, 9] have confirmed the presence of the quarter
metal. Our objective here is to elucidate the core principles of magnetism
that could be universally applicable to all multilayer graphene electron gas
systems, while also highlighting specific nuances that are harder to predict.

Figure 8: Mean-field phase diagram of quarter-metal phases in BBG shows
valley-XY phase is the groundstate close to the half-metal. As the hole
density decreases, the valley-XY state transitions into the valley-Ising
state.

The mean-field phase diagram of BBG is shown in Fig. 8, which agrees
qualitatively with Ref. [36]. Both RTG and BBG’s quarter metal phase diagrams
feature the valley-XY and valley-Ising states, yet they differ significantly
in certain aspects. First, quarter metal in BBG occurs at density $|n_{e}|$
that is 3 to 5 times smaller than that in RTG. This is consistent with the
experimental observations in Ref. [41, 1]. Second, the valley-XY phase in BBG
occupies a very small region in the $n_{e}-U$ phase diagram. Finally, there is
a notable distinction in the magnetic anisotropic energy landscape: in RTG,
the valley-Ising transitions into the valley-XY as $|n_{e}|$ decreases,
whereas in BBG, the transition occurs from the valley-XY to the valley-Ising
with decreasing $|n_{e}|$. We’ll first address the initial two points using
general principles inherent to multilayer graphene systems. Subsequently,
we’ll discuss the difference in magnetic anisotropic energy landscape, which
exemplifies the nuanced specifics that challenge predictability.

Given that BBG’s quarter metal is observed at a density roughly five times
less than RTG, the area of the Fermi sea in BBG contracts by the same factor
when compared to RTG. While this reduced Fermi sea area doesn’t alter the spin
and valley order parameters of the Ising phase, it does enhance the variation
of valley-polarization within the Fermi sea for the valley-XY phase. This
variation raises valley-XY’s exchange energy rendering it less stable as the
ground state. This underlines the reason why the valley-XY phase occupies a
smaller $n_{e}-U$ phase space in BBG compared to RTG. To further elucidate
this point, we plot the valley-polarization $\tau_{z}$ of the valley-XY phase
in BBG in Table 1, obtained from self-consistent mean-field calculation.
Similar to RTG, as the Bloch states traverse the Fermi surface, the vector
$\bm{\tau}_{\bm{k}}=(\tau_{x,\bm{k}},\tau_{z,\bm{k}})$ exhibits oscillatory
behavior reminiscent of a domain wall. Yet, a closer look at the color bar in
Table 1 shows that the variations of $\tau_{z,\bm{k}}$ in BBG are more
pronounced than in RTG.

We now turn our attention to the magnetic anisotropic energy landscapes of
quarter metal in RTG and BBG. In both systems, the exchange energy invariably
promotes the Ising phase as the ground state, while the band energy gives
always preference to the XY phase. However, their mean-field energy
differences are minimal and are swayed by detailed factors such as layer
polarization. For BBG, our observations indicate that, at large $|n_{e}|$, the
band-energy difference is greater than the exchange-energy difference. This
means the valley-XY phase is the dominant ground state. In contrast, RTG
behaves oppositely.

##  VII Summary

In conclusion, we have addressed four pertinent questions concerning the low-
density portion of the magnetic phase diagram in rhombohedral trilayer
graphene:

1) Magnetic anisotropy of the quarter-metal groundstate: We identified two
competing groundstate for the quarter metal: the valley-Ising state and the
valley-XY state, detailed extensively in Sec. II. At large values of $|n_{e}|$
and $U$, the valley-Ising state is the quarter metal groundstate. As we
decrease $|n_{e}|$ and $U$, it undergoes a first order phase transition to
become the valley-XY state. This transition is driven by both the band and
exchange energy. Although the exchange effect unambiguously favors the valley-
Ising state as the ground state, the exchange energy of the valley-XY state
becomes similar to the valley-Ising state with reducing $|n_{e}|$. This
narrowing of exchange energy gap between them is due to the subtle differences
in their layer polarization.

2) Sublattice pseudospin polarization and the re-entrance of paramagnetic
state:  We found that the exchange energy difference between the quarter metal
and the paramagnetic state, denoted as $E_{ex}^{\theta_{v}}-E_{ex}^{PM}$, can
be strongly influenced by the layer-pseudospin. Given that the area of the
Fermi sea in the quarter metal is 12 times larger than the Fermi pockets in
the paramagnetic state, the Fermi level of the quarter metal is situated
further from neutrality compared to that of the paramagnetic state. As a
result, as $|n_{e}|$ reduces, the layer polarization in the paramagnetic phase
increases more rapidly than in the quarter metal phase, resulting in
$\partial(E_{ex}^{\theta_{v}}-E_{ex}^{PM})/\partial n_{e}>0$.

3) Magnetic anisotropy and Kane-Mele spin-orbit coupling: We examined the
influence of Kane-Mele spin-orbit coupling (SOC) on the quarter-metal phase
diagram. This SOC term lowers the energy of a many-body state where the
z-component of the spin-polarization ($s_{z}$) is anti-parallel to the
z-component of the valley-polarization ($\tau_{z}$). Here $\hat{z}$ is the
unit vector normal to the two-dimensional material plane. In the case of
valley Ising quarter metal, this SOC establishes a magnetic easy-axis. Here,
the spin-quantization axis of all Bloch states are uniformly aligned along
$\pm\hat{z}$ and that the valley-polarization is everywhere antiparallel to
it, i.e $\mp\hat{z}$. For the valley-XY quarter metal, this SOC introduces a
magnetic easy-plane. In this scenario, both the spin-polarization and valley-
polarization of the Bloch states predominantly orient themselves perpendicular
to $\hat{z}$. Minor oscillations in $\tau_{z}$ and $s_{z}$ are observed within
the Fermi sea. This anisotropy indicates that while the Kane-Mele SOC gives
preference to the valley-Ising as the ground state, an in-plane magnetic field
favors the valley-XY as the ground state.

4) Displacement of Ising-XY boundary: In our study of the Ising-XY boundary in
the four dimensional parameter space — defined by density $n_{e}$, interlayer
potential $U$, in-plane magnetic field $B_{\parallel}$ and SOC strength
$\lambda$ – we found that the Ising-XY phase boundary follows the trajectory
$\partial n_{e}/\partial B_{\parallel}\sim-0.5\times
10^{11}\text{cm}^{-2}\text{T}^{-1}$ within the $n_{e}$ and $B_{\parallel}$
parameter space. Furthermore, we have computed the in-plane spin
susceptibility of the valley-Ising quarter-metal to be
$\chi_{{}_{\parallel}}\sim 8\leavevmode\nobreak\
\mu\text{eV}\leavevmode\nobreak\ \text{T}^{-2}$.

We centered our attention on RTG due to its experimental significance and its
representation as the most rudimentary form of rhombohedral-stacked multilayer
graphene. It is noteworthy that the density-of-states (DOS) in RTG
significantly exceeds that of its ABA Bernal-stacked trilayer graphene
counterpart. Levering on this pronounced DOS, Ref. [6, 5] has recently
identified correlated metals and insulators in rhombohedral-stacked pentalayer
graphene even without the application of an external electric displacement
field.

Acknowledgement: We acknowledgement insightful conversation with Nemin Wei,
Andrea Young and Haoxin Zhou. We are grateful to the University of Kentucky
Center for Computational Sciences and Information Technology Services Research
Computing for their support and use of the Morgan Compute Cluster and
associated research computing resources.

##  Appendix A Effect of inter-valley scattering on quarter metal groundstates

The landscape of magnetic anisotropy energy can be affected by several
independent parameters. In the main text, we have examined four such
parameters: $n_{e}$, $U$, $\lambda$, and $B_{\parallel}$. In this section, we
study the impact of inter-valley scattering on quarter-metal ground states and
how it alters the $n_{e}$-$U$ phase diagram outlined in Fig. 2. We introduce
the the self-energy $\hat{\Sigma}_{\perp\bm{k}}$ into Eq. 1 of the maintext,
where

$\displaystyle\hat{\Sigma}_{\perp\bm{k}}=u_{\perp}\sum_{a=x,y}\left[\text{Tr}(\hat{\tau}_{a}\hat{\rho}_{\bm{k}}^{\theta_{v}})\hat{\tau}_{a}-\hat{\tau}_{a}\hat{\rho}_{\bm{k}}^{\theta_{v}}\hat{\tau}_{a}\right].$
(23)

Here $u_{\perp}$ represents the strength of the inter-valley Coulomb
scattering. In the above equation, first term denotes the Hartree
contribution, while the second one corresponds to the Fock contribution. Their
contribution to the total energy is given by the following:

$\displaystyle E_{\perp H}^{\theta_{v}}$
$\displaystyle=\frac{u_{\perp}}{2}\sum_{a=x,y}\int\frac{d^{2}k}{(2\pi)^{2}}\left[\text{Tr}(\hat{\tau}_{a}\hat{\rho}_{\bm{k}}^{\theta_{v}})\right]^{2},$
(24) $\displaystyle E_{\perp ex}^{\theta_{v}}$
$\displaystyle=-\frac{u_{\perp}}{2}\sum_{a=x,y}\int\frac{d^{2}k}{(2\pi)^{2}}\text{Tr}\left[\hat{\tau}_{a}\hat{\rho}_{\bm{k}}^{\theta_{v}}\hat{\tau}_{a}\hat{\rho}_{\bm{k}}^{\theta_{v}}\right],$
(25)

where $\hat{\rho}_{\bm{k}}^{\theta_{v}}$ is the density-matrix determined
self-consistently. As shown in Table 3, despite a notable exchange energy
shift related to the Dirac sea, the energy difference between them is very
small:$E_{\perp,H}^{\theta_{v}=0}+E_{\perp,ex}^{\theta_{v}=0}-E_{\perp,H}^{\theta_{v}=\pi/2}-E_{\perp,ex}^{\theta_{v}=\pi/2}\sim
4\mu$eV per-hole. Furthermore, Table 2 shows that the variations in both band
energy and (long-range Coulomb )exchange energy, due to $u_{\perp}$, are
similar for both the Ising and XY states. Consequently, the introduction of
inter-valley scattering doesn’t bring about significant alterations to the
quarter-metal phase diagram.

$\mathbf{\theta_{v}}$ | $\mathbf{0}$ | $\mathbf{\pi/2}$
---|---|---
$\mathbf{\Delta E_{b}/|n_{e}|}$ (meV) | $0.761$ | $0.753$
$\mathbf{\Delta E_{ex}/|n_{e}|}$ (meV) | $-0.713$ | $-0.703$
Table 2: Change in band and Fock exchange energy for valley-Ising and valley-XY phase due to change in $u_{\perp}$ from $0$ meV to $0.5$ meV at $(n_{e},U)=-2.9\times 10^{11}\leavevmode\nobreak\ \text{cm}^{-2},18\leavevmode\nobreak\ \text{meV}$. $\mathbf{\theta_{v}}$ | $\mathbf{0}$ | $\mathbf{\pi/2}$
---|---|---
$\mathbf{E_{\perp H}/|n_{e}|}$ (meV) | $0$ | $0.41$
$\mathbf{E_{\perp ex}/|n_{e}|}$ (meV) | $-55.814$ | $-56.228$
Table 3: Hartree and exchange energy component of intervalley scattering for
valley-Ising and valley-XY phase using $u_{\perp}=0.5$ meV at
$(n_{e},U)=-2.9\times 10^{11}\leavevmode\nobreak\
\text{cm}^{-2},18\leavevmode\nobreak\ \text{meV}$.

##  Appendix B Spin susceptibility of valley-Ising quarter metal

Figure 9:  In-plane spin susceptibility of valley-Ising quarter metal is
plotted against increasing SOC strength $\lambda$ for $n_{e},U=-2.9\times
10^{11}\leavevmode\nobreak\ \text{cm}^{-2},18\leavevmode\nobreak\ \text{meV}$.

We compute the in-plane spin susceptibility ($\chi_{{}_{\parallel}}$) of the
valley-Ising quarter metal for different values of spin-orbit coupling
($\lambda$). $\chi_{{}_{\parallel}}$ is defined as:

$\displaystyle\chi_{{}_{\parallel}}=\frac{\partial^{2}E^{\theta_{v}=0}}{\partial
B_{\parallel}^{2}}\left.\vphantom{\frac{A}{B}}\right|_{B_{\parallel}=0}=\frac{\partial
M_{\parallel}^{\theta_{v}=0}}{\partial
B_{\parallel}}\left.\vphantom{\frac{A}{B}}\right|_{B_{\parallel}=0}$ (26)

Fig. 9 shows $\chi_{{}_{\parallel}}$ $v.s.$ $\lambda$ at $n_{e},U=-2.9\times
10^{11}\leavevmode\nobreak\ \text{cm}^{-2},18\leavevmode\nobreak\ \text{meV}$.
Since the SOC pinned the spin-quantization axis of the valley-Ising state in
the direction normal to the applied in-plane field $B_{\parallel}$,
$\chi_{{}_{\parallel}}$ decreases with increasing $\lambda$.

##  Appendix C Energy landscape of quarter-metal in the presence of spin-orbit
coupling

Figure 10: The first column shows the total energy v.s. $\theta$ while the
second and third columns show the order parameters of the converged solution
$v.s.$ $\theta$. $\theta$ is defined in Eq. C1 and and we set the
$\lambda_{L}=1\leavevmode\nobreak\ \mu$eV, $\lambda=30\leavevmode\nobreak\
\mu$eV.

So far, we have identified only two quarter metal groundstate: valley-Ising
and valley-XY phases. They are separated by a first order phase transition
boundaries in the $n_{e}-U$ space. In this section, we used the following
Lagrange multipliers to further examine their energy landscape:

$\displaystyle\hat{H}_{L}(\bm{n})=$
$\displaystyle\lambda_{L}[\bm{n}\cdot\hat{\bm{s}}+(\bm{n}\times\bm{z})\cdot\hat{\bm{\tau}}]$
$\displaystyle=$
$\displaystyle\lambda_{L}(\text{sin}\theta\leavevmode\nobreak\
\hat{s}_{x}+\text{cos}\theta\leavevmode\nobreak\
\hat{s}_{z}+\text{sin}\theta\leavevmode\nobreak\
\hat{\tau}_{x}-\text{cos}\theta\leavevmode\nobreak\ \hat{\tau}_{z}).$ (27)

Here $\bm{n}=(\sin\theta,\cos\theta)$ represents the angle of spin-
polarization. We opted not to delve into the two-dimensional energy landscape
generated by independent valley and spin rotations as
$H_{KM}=\lambda\sigma_{z}s_{z}\tau_{z}$ favors the anti-alignment of the
$s_{z}$ with $\tau_{z}$. Note $\lambda\langle\sigma_{z}\rangle>0$.

In Fig. 10, we plot the evolution of both the energy and the associated spin
and valley order parameters for the quarter-metal phases as functions of hole
density. At a high density of $n_{e}=-3\times 10^{11}\leavevmode\nobreak\
\text{cm}^{-2}$, the valley-Ising state, denoted by
$\ket{\theta_{v}=\pi,\theta_{s}=0}$, establishes itself as the global minimum,
whereas the valley-XY state $\ket{\theta_{v}=\pi/2,\theta_{s}=\pi/2}$ appears
unstable. However, when the density is lowered to $n_{e}=-2.8\times
10^{11}\leavevmode\nobreak\ \text{cm}^{-2}$, the valley-XY state stabilizes
and becomes a local minimum. Reducing the density further to $n_{e}=-2.7\times
10^{11}\leavevmode\nobreak\ \text{cm}^{-2}$ prompts a shift in the global
minimum from $\ket{\theta_{v}=\pi,\theta_{s}=0}$ to
$\ket{\theta_{v}=\pi/2,\theta_{s}=\pi/2}$, signifying a first-order phase
transition. At $n_{e}=-2.1\times 10^{11}\leavevmode\nobreak\ \text{cm}^{-2}$,
$\ket{\theta_{v}=\pi,\theta_{s}=0}$ becomes unstable. Note that the Lagrange
multiplier $\lambda_{L}$ has a minimal impact on shifting the Ising-XY phase
boundary.

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