# Machine Learning-based Methods for Joint Detection-Channel Estimation in
OFDM Systems

Wilson de Souza Junior, Taufik Abrão
This work was partly supported by the National Council for Scientific and
Technological Development (CNPq) of Brazil under Grants 310681/2019-7, and in
part by the CAPES (Financial code 001), and by State University of Londrina
(UEL), PR, Brazil.Wilson de Souza Junior and T. Abrão are with the Electrical
Engineering Department, State University of Londrina, PR, Brazil. Rod. Celso
Garcia Cid - PR445, s/n, Campus Universitário, Po.Box 10.011. CEP: 86057-970.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, two machine learning (ML)-based structures for joint detection-
channel estimation in OFDM systems are proposed and extensively characterized.
Both ML architectures, namely Deep Neural Network (DNN) and Extreme Learning
Machine (ELM), are developed to provide improved data detection performance
and compared with the conventional matched filter (MF) detector equipped with
the minimum mean square error (MMSE) and least square (LS) channel estimators.
The bit-error-rate (BER) performance vs computational complexity trade-off is
analyzed, demonstrating the superiority of the proposed DNN-OFDM and ELM-OFDM
detectors methodologies.

###### Index Terms:

Machine learning, neural networks, OFDM, detection, deep learning, DNN, ELM,
MMSE, LS, BER.

##  I Introduction

The conventional orthogonal frequency-division multiplexing (OFDM) system is a
multi-carrier scheme widely utilized in communication systems due to its
capacity to combat frequency-selective fading in wireless channels. Besides,
Artificial intelligence (AI) and machine learning (ML) are relevant approaches
in the current complex, highly demanded radio access scenarios, combined with
the spectrum scarceness. The ML resources and techniques can be applied to
improve the performance-complexity trade-off of OFDM systems, specifically on
the receiver side. In this work, two AI-based methods, specifically a DNN-
based and an ELM-based jointly symbol detection and pilot-assisted channel
estimation are investigated; both techniques are compared with the
conventional linear estimation methods, such as least square (LS) and minimum
mean square error (MMSE) [1, 2].

ML techniques have been widely used in different telecommunication
applications as a satisfactory predictor in OFDM system [3, 4, 5], as a near-
optimal signal detection in OFDM with index modulation (OFDM-IM) [6], and as a
channel estimator for massive MIMO [7]. In [3], the authors discuss the deep
learning (DL) applicability for channel estimation and signal detection in
OFDM systems. The DL-based prediction technique is explored to implicitly
estimate the channel state information (CSI) and then detect the transmitted
symbols using the estimated CSI. For that, the deep learning model is first
trained offline using the data generated by simulation based on channel
statistics and then used for recovering the online transmitted data directly.

Deep neural network (DNN) is a type of artificial neural network presenting a
large number of hidden layers and hyper-parameters into its composition, i.e.,
DNN implies high computational operations in contrast to Extreme Learning
Machine (ELM) that has simply one hidden layer [8]. In intricate
telecommunication scenarios, specifically in the 5G and beyond systems,
authors in [4] propose a different architecture for the OFDM receiver aided by
ELM technique. Besides, a multi-ELM, i.e., a parallel multiple split complex
ELM structure is proposed in [5]. In [6], a DL-based detector structure for
OFDM with index modulation (OFDM-IM) is proposed, termed DeepIM. The authors
deploy a deep neural network with fully connected layers to recover data.
Aiming to enhance the DeepIM performance, the received signal and channel
vectors are pre-processed based on the domain knowledge before entering the
network. Data sets available by simulations are deployed to train offline the
DeepIM aiming at optimizing the bit error rate (BER) performance. After that,
the trained model is deployed for the online signal detection.

Contributions. We propose and analyze the deployment of promising ML tools
applied through jointly detection and channel estimation in OFDM systems. i)
First, we have adopted and analyzed two ML-based topologies for OFDM data
detection and channel estimation: a DNN-based and ELM-based OFDM joint
detector and channel estimator. ii) We have deployed and characterized
existing models by applying them to more complex scenarios, including multi-
user systems, realistic path-loss, and short-term fading wireless channel
configurations. iii) Extensive numerical results characterizing the
performance-complexity trade-off for both ML-based OFDM detectors,
demonstrating that such an approach is quite competitive.

Notations. Italic lowercase or capital letters are scalars, boldface capital
letters denote the frequency-domain vectors meanwhile boldface lowercase
letters are vectors in the time domain. Operators $\mathbb{E}[\cdot]$,
$(\cdot)^{T}$, $(\cdot)^{H}$ and $(\cdot)^{\dagger}$ denote the statistical
expectation, a vector or matrix transpose, Hermitian and Moore-Penrose pseudo-
inverse, respectively; $|\mathcal{A}|$ holds for the cardinality of the set
$\mathcal{A}$, $\odot$ denotes the element-wise multiplication, $\oslash$
denotes the element-wise division, and $\circledast$ represents the
convolution operation.

##  II System Model

Assuming an OFDM system with a set $\mathcal{U}=\\{1,\dots,U\\}$ users, a
number of $N_{c}$ sub-carriers and the duration of the cyclic prefix $T_{g}$,
a transmitted signal in frequency domain can be defined as
$\textbf{X}=[X_{1},\dots,X_{N_{c}}]^{T}$, leading to a received signal
$\textbf{Y}=[Y_{1},\dots,Y_{N_{c}}]^{T}$, with multi-path channel
$\textbf{H}=[H_{1},\dots,H_{N_{c}}]^{T}$, and zero-mean Gaussian noise samples
$\textbf{Z}=[{Z}_{1},\dots,Z_{N_{c}}]^{T}$ described by complex random
variables $Z_{i}\sim\mathcal{C}\mathcal{N}(0,\sigma^{2})$, where $\sigma^{2}$
is the noise power in each OFDM sub-channel. The received signal in the
frequency and time domain can be written, respectively:

$\textbf{Y}=\textbf{X}\odot\textbf{H}+{\textbf{Z}},\qquad\text{{and}}\qquad\textbf{y}=\textbf{x}\circledast\textbf{h}+{\textbf{z}}$
(1)

where y, x, h and z means IDFT of Y, X, H and Z respectively. In the
considered path-loss model the received signal power decays according to
$d_{k}^{\eta}$, where $d_{k}$ is the distance between BS and the $k$th user,
while $\eta$ represents the path-loss exponent. Hence, the transmitted power
per sub-carrier and the average receiver power per subcarrier ($P$) are
related by: $P=d_{u}^{-\eta}\cdot\frac{P_{\textsc{t}}}{N_{c}}$, where
$P_{\textsc{t}}$ is the total power available at transmitter side. Besides,
since more than one user sharing the same sub-channel is admitted, resulting
in an OFDM system operating under inter-user interference (IuI), in Section IV
we proceed with sub-carrier selection to know what sub-carriers sub-set
results in smaller IuI, aiming to maximize the SINR in the $k$th sub-carrier.

LS OFDM Channel Estimation. As aforementioned, firstly we assume that an OFDM
pilot symbol is transmitted (channel estimation mode) and then OFDM data
symbols can be transmitted (data mode) inside the channel coherence time
$(\Delta t)_{\textsc{c}}$ interval; this composition form an OFDM frame.
Inside an OFDM frame, the channel state information (CSI) is unchangeable but
it changes from one frame to another. One common technique for OFDM channel
estimation is the least-squares (LS) method [1]. This technique is the
simplest way to estimate the state of the channel; as a result, it is possible
to estimate the OFDM symbols inside the same $(\Delta t)_{\textsc{c}}$ time
interval. Once $\textbf{X}_{p}=[X_{1},\dots,X_{N_{c}^{\rm pilot}}]^{T}$ and
$\textbf{Y}_{p}=[Y_{1},\dots,Y_{N_{c}^{\rm pilot}}]^{T}$ are the transmitted
and received OFDM pilot vectors, respectively, with $N^{\rm pilot}_{c}$ the
number of sub-carriers reserved to the pilots in the OFDM frame, then the LS
channel estimation in the pilot OFDM sub-channels, and the data detection
based on LS channel estimator are obtained, respectively, by:

$\tilde{\textbf{H}}^{\rm{LS}}=\textbf{Y}_{p}\oslash\textbf{X}_{p},\qquad\text{and}\qquad\tilde{\textbf{X}}_{d}=\textbf{Y}_{d}\oslash\tilde{\textbf{H}}^{\rm{LS}},$
(2)

where $N^{\rm data}_{c}$ is the number of sub-carriers destined to data symbol
in the OFDM frame; $\tilde{\textbf{X}}_{d}$ and $\textbf{Y}_{d}$ are the
recovery data and received data vectors, respectively.

MMSE OFDM Channel Estimation. The MMSE channel estimator is considered a
better linear solution than the aforementioned LS channel estimation due to
the weight (regularization) channel matrix inversion, which is optimized in
the same way as the LS solution according to the minimum mean square error
problem. However, the development of the MMSE solution requires the knowledge
of the signal-to-noise ratio (SNR), being the channel estimate obtained from
the LS solution as [1]:

$\tilde{\textbf{H}}^{\rm{MMSE}}=\mathbf{R}_{\mathbf{H}\tilde{\textbf{H}}^{\rm{LS}}}\begin{bmatrix}\mathbf{R}_{\mathbf{HH}}+\mathbf{I}\frac{1}{\bar{\gamma}}\end{bmatrix}^{-1}\tilde{\textbf{H}}^{\rm{LS}},$
(3)

where $\mathbf{R_{AB}}$ denotes the cross-correlation matrix between matrices
A and B, $i.e$., $\mathbf{R_{AB}}=\mathbb{E}[\textbf{AB}^{H}]$; the pre-
processing SNR at the receiver side is defined as
$\bar{\gamma}\triangleq\frac{P}{\sigma^{2}}$, with $P$ the average power per
sub-channel at receiver side.

##  III ML-based OFDM Detection Schemes

In the context of machine learning, the training occurs by generating random
data communicating across a channel that arrives at the receiver, and then
that data is part of a data set containing labels and features. This paper
presents an analysis and comparison of two different ML-based detectors that
are promising for realistic OFDM system scenarios. The deployed OFDM system
model is depicted in Fig. 1; the DNN and ELM architectures are described in
the following.

DNN-based Detection. The DNN is an architecture composed by $2\cdot N_{c}$
input nodes being a real and imaginary part of OFDM frame, where $N_{c}=N^{\rm
pilot}_{c}+N^{\rm data}_{c}$. This model has exclusively 3 hidden layers, Fig.
2, in which each layer is composed of 500, 250, and 120 neurons, respectively,
in the same way as adopted in [3].

The proposed DNN-based OFDM detector is exclusively inspired in offline
training strategy, i.e., for the training stage, the DNN inputs (features) are
composed by the real and imaginary part of OFDM symbols that arrives at the
receiver, while the outputs (labels) are estimates for the transmitted bits.
The DNN outputs are obtained from a non-linear function of input nodes:

$\mathbf{\hat{X}}_{\textsc{dnn}}=f(\textbf{Y},\mathbf{\theta})=f^{L-1}(f^{L-2}(...f^{1}(\mathit{\mathbf{Y}})))$
(4)

where $\mathbf{\theta}$ is the set of bias and weights and $L$ means the
number of layers. The bias and weight coefficients are optimized in the
training stage. The DNN model has the goal of minimizing the mean squared
error (MSE) loss function, defined by:

$\mathcal{F}_{\rm
loss}=\frac{1}{L}\sum_{k=1}^{L}\left[\mathit{\mathbf{\hat{X}}}(k)-\mathit{\mathbf{X}}(k)\right]^{2}$
(5)

where $\mathit{\mathbf{\hat{X}}}(k)$ denotes the predictions,
$\mathit{\mathbf{X}}(k)$ the data symbol, and $L$ is the number of data
samples in the estimation data set.

DNN Training. The training step is responsible for the DNN to learn the
channel characteristics; hence the data set must be known and sufficiently
large, beyond it should be transmitted in a fraction of the channel coherence
time $(\Delta t)_{\textsc{c}}$ interval to allow the system attains suitable
accuracy in the channel estimate process. Once trained, the network may be
utilized to decode data to any online transmission scheme (test stage), also
assuming the same parameters utilized previously at the training stage.

Figure 1: OFDM & DNN training.

Figure 2: DNN architecture.

Figure 3: ELM architecture.

Figure 4: BER for DNN, LS and MMSE CE under 4-QAM: (a) 64 pilots and (b)
Reducing #pilots

Extreme Learning Machine (ELM) based Detection. The ELM network has other
important feature that differs from DNN, its architecture is subdivided into
$N_{c}$ sub-networks, i.e., in the OFDM context, a sub-network is deployed to
treat the signal of each sub-carrier. Fig. 3 depicts an ELM topology for each
OFDM sub-carrier. The parameters of hidden nodes of ELM must be randomly
generated, and afterward, they should be fixed to determine the output layer
weights according to [4, 5]. ELM architecture has 2 input nodes, the real and
imaginary parts of the $n$th sub-channel. In OFDM systems, the ELM
architecture assumes that the data and pilots are time-division multiplexed,
i.e., inside a channel coherence time $(\Delta t)_{\textsc{c}}$ interval,
there are $I$ transmitted pilots and $K$ transmitted data symbols. This is a
primordial feature once the channel can be assumed invariant into the $(\Delta
t)_{\textsc{c}}$ interval. Still, the noise and possibly co-channel
interference at the receiver side are variant in time. The data set provided
by $I$ pilots is helpful for the network to learn about statistics from noise
plus interference.

The hidden layer from ELM has an activation function applied to the data from
the input layer, Fig. 3, where the output of $\ell$th hidden node is given by:

${\bf o}_{i,L}=g(\textbf{{a}}_{i}^{T}\cdot\textbf{{Y}}_{i}+b_{L})$ (6)

where $L$ means the number of hidden neurons, $\textbf{{a}}_{i}$ is a column
vector of weights with dimension 2 $\times$ 1, while $b_{L}$ means the bias
from $\ell$th hidden node and $\textbf{{Y}}_{i}\in\mathbb{R}^{2\times 1}$ is
the transmitted data. The ${j}$th hidden layer matrix ${\bf
O}\in\mathbb{R}^{I\times L}$ is given by:

${\bf
O}_{j}=\begin{bmatrix}g(\textbf{{a}}_{1}^{T}\cdot\textbf{{Y}}_{1}+b_{1})\cdots
g(\textbf{{a}}_{L}^{T}\cdot\textbf{{Y}}_{1}+b_{L})\\\
\vdots\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cdots\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vdots\\\
g(\textbf{{a}}_{1}^{T}\cdot\textbf{{Y}}_{I}+b_{1})\cdots
g(\textbf{{a}}_{L}^{T}\cdot\textbf{{Y}}_{I}+b_{L})\\\ \end{bmatrix}$ (7)

Different from DNN, the ELM topology can be trained in a non-iterative mode in
order to minimize a training error function:

$\hat{\bf B}=\underset{{\bf B}\in\mathbb{R}^{L\times
2}}{\min}\begin{Vmatrix}\mathbf{O}\mathbf{B}-\mathbf{X}_{\rm
pilot}\end{Vmatrix}\,\,=\,\,\mathbf{O}^{\dagger}\mathbf{X}_{\rm pilot}$ (8)

where $\mathbf{\hat{B}}$ is a $L\times 2$ dimensional matrix denominated
output weight matrix. Once the training stage is completed for each sub-
network (sub-carrier), then ELM can operate, i.e. the ELM-OFDM detector can
estimate the data by:

$\mathbf{\hat{X}}_{\textsc{elm}}=\mathbf{O}{\hat{\mathbf{B}}}$ (9)

Eqs. (4) and (9) explicitly provide estimates of the transmitted signal (OFDM
data detection). Although the OFDM channel estimation intrinsically occurs
inside the ML detection step, explicitly, we do not proceed with the channel
estimation step. In this sense, the proposed ML-based techniques are also
suitable for non-coherent signal detection since such schemes explicitly
neither require channel state information knowledge at the transmitter nor the
receiver.

##  IV Sub-Carrier Selection for maximizing SINR

We describe a method for sub-carrier selection and allocation aiming at
maximizing the signal-to-interference plus noise ratio (SINR) across the
users. Hence, in this work, we fix a pre-defined value for the maximum number
of sub-carriers per user, $N_{\rm cpu}$, and the number of sub-carriers that
can support two OFDM users simultaneously subject to equal interference level
in those subcarriers. Notice that in the adopted system model, we have
admitted more than one user sharing the same sub-channel, resulting in an OFDM
system operating under inter-user interference. Also, to facilitate the
analysis, but without loss of generality, we have defined and selected an
equal number of sub-carriers per user that generate the lesser interference
over other users, $N_{\rm cpu}^{\rm eq}$. Our principal goal is to allocate as
best as possible the sub-carriers among the users in such a way that maximizes
the SINR (optimization metric) in the $k$th sub-carrier subject to
interference in a given OFDM frame, which can be written as:

$\displaystyle{\rm SINR}_{u}(k)=\,\,$
$\displaystyle\frac{P_{u}(k)|H_{u}(k)|^{2}}{P_{j}(k)|H_{j}(k)|^{2}+\sigma^{2}},\qquad
j\neq u;\qquad\forall k=1,\dots,N^{\rm data}_{c},\quad\forall
u\in\mathcal{U};$ (10)

$\displaystyle\text{with}\quad P_{u}(k)=\frac{P_{\textsc{t}}}{N^{\rm
data}_{c}},\,\,\,\text{and}\,\,\,P_{j}(k)=\left\\{\begin{matrix}\frac{{P_{\textsc{t}}}}{N^{\rm
data}_{c}},&\text{if}\,\,\,j\in\mathcal{J}_{u}\\\ 0&\rm
otherwise\end{matrix}\right.$

where $\mathcal{J}_{u}$ represents the sub-set of users interfering in the
$k$th subcarrier of user $u$. Notice that
$P_{u}(k)=\frac{{P_{\textsc{t}}}}{N^{\rm data}_{c}}$ indicates equal power
allocation (EPA) policy across the users, where
$P_{j}{(k)}=\frac{{P_{\textsc{t}}}}{N^{\rm data}_{c}}$ means the EPA policy
also for the $j$th interfering user in the $k$th sub-carrier;
$H_{u}(k),H_{j}(k)$ mean the channel response for the $u$th and $j$th user in
the $k$th sub-carrier, respectively. A pseudo-code for subcarriers selection
aiming at maximizing the SINR, eq. (10), is depicted in Algorithm 1.
$\mathcal{U}\setminus\mathcal{J}_{u}$ is the set difference; i.e., it is the
set of all those elements that are in $\mathcal{U}$ but not in
$\mathcal{J}_{u}$.

Algorithm 1 Sub-carrier Selection for SINR maximization

1:for $u=1,2,\dots,{|\mathcal{U}\setminus\mathcal{J}_{u}|}$ do

2: for $k$= $N_{\rm cpu}(u-1)+1$ $:$$N_{\rm cpu}(u-1)+N_{\rm cpu}$ do

3: Evaluate the $k$th SINR for $u$th user as in (10);

4: end for;

5: Sort all SINR’s for $u$th user according to the descending order;

6: Select the first $N_{\rm cpu}^{\rm eq}$ SINR’s;

7:end for

##  V Simulation Results

Numerical results in terms of performance vs complexity trade-off for DNN and
ELM are analyzed. The parameters for learning DNN-OFDM and ELM-OFDM
architectures are summarized as follows, Fig. I.
OFDM System – Transmitter Antenna: $n_{T}=1$; Receiver Antenna: $n_{R}=1$;
Modulation Order ($M$-QAM): $M=4,16,32$; Number of Users: $U=4$; Sub-carriers:
$N_{c}=64$; Pilots Number: $N_{\rm pilot}=64,32,16$ and $8$; Cyclic Prefix:
$T_{g}$ = 25% and 0%, (Fig. 6); Estimation Methods: LS e MMSE.
Sub-carrier Interference & Power Allocation – Equal Power Allocation (EPA):
$P_{u}(k)=\frac{P_{\textsc{t}}}{N_{c}}$; Max. #sub-carriers/user: $N_{\rm
cpu}=$ 16; Interfering sub-carriers/user: $N_{\rm cpu}^{\rm eq}$ = 4.
NLoS Channel – Cell radius: 500 m; Total Power: $P_{\textsc{t}}=1$ mW; SNR
range: $\bar{\gamma}\in[5;\,\,25]$ dB; LoS Channel Model: Rayleigh; Path Loss
coefficient: $\eta=-3$; Path Loss model: $d_{u}^{-\eta}$; Coherence Time:
$(\Delta t)_{\textsc{c}}$ = 5 ms;
DNN Architecture – Hidden Layer: 3; Input Neuron: 256; Hidden Neurons: 500,
250 and 120; Output Neurons: 64; Activation Function: 3 Relu and 1 Sigmoid;
Optimizer: Adam; Loss Function: MSE; Epochs: $10^{3}$; MCS realizations:
$10^{4}$;
ELM Architecture – Hidden Layer: 1; Input Neurons: $2$ ; Hidden Neurons:
$L=50$; Output Neurons: 2; Sub-Network: 64; Activation Function: Radbas;
Pilots symbols: $I=50,100,200$; Data symbols: $K=400$; and MCS realizations:
$\mathcal{T}=10^{3}$.

We analyze the influence of several parameters, such as the number of pilots,
the number of users, and the SNR training on the BER performance is analyzed.
The analytical average BER performance is given by [9]:
$\textrm{BER}^{\textrm{theo}}=\frac{\alpha_{\textsc{m}}}{2}\begin{bmatrix}1-\sqrt{\frac{0.5\beta_{\textsc{m}}{\bar{\gamma}}}{1+0.5\beta_{\textsc{m}}{\bar{\gamma}}}}\end{bmatrix},$
where $\alpha_{\textsc{m}}$ and $\beta_{\textsc{m}}$ are constants that depend
on the modulation type, i.e. $\alpha_{\textsc{m}}$ is the number of nearest
neighbors to a constellation at the minimum distance, and $\beta_{\textsc{m}}$
is a factor relating the minimum distance to the average symbol energy [9].
For 4-QAM results
$\textrm{BER}^{\textrm{theo}}=\frac{1}{2}\left[1-\sqrt{\frac{\gamma_{b}}{1+\gamma_{b}}}\right]$,
where $\gamma_{b}=\frac{{\bar{\gamma}}}{2}$.

Channel Estimation Task. In this sub-section, we have analyzed the numerical
results related to both channel estimation methods through the normalized mean
square error (NMSE) metric, which can be defined as

${\rm
NMSE}=\frac{\sum_{i=1}^{{S}}\left|\mathbf{H}(i)-\tilde{\mathbf{H}}(i)\right|^{2}}{{S}\,\mathbf{H}_{\textsc{v}}\mathbf{H}_{\textsc{v}}^{H}}$
(11)

where ${\bf H}_{\textsc{v}}={\rm
vec}\left(\left[\mathbf{H}(1)\,\mathbf{H}(2)\,\ldots\,\mathbf{H}(S)\right]\right)$,
with vec($\cdot$) operator indicating the vectorization of a matrix, which
converts the $(N_{c}\times S)$ OFDM channel samples matrix into the ${\bf
H}_{\textsc{v}}$ channel frequency domain column vector ($N_{c}S\times 1$);
finally, $S$ is the number of channel estimate samples performed in the
channel coherence time interval. Under medium to high SNR, both linear channel
estimators methods are suitable, e.g., for $\bar{\gamma}=20$ dB, the
NMSE${}_{\textsc{ls}}=52\times 10^{-4}$, with an order of magnitude in favor
or the MMSE method.

Number of Pilots. The DNN architecture has the advantage of attaining a good
performance when compared with the classical channel estimators, such as MMSE
and LS, the DNN make-up for the information lack made by interpolation in LS
and MMSE methods as depicted in Fig. 4. The attainable BER for the DNN-OFDM
detector is comparable to the BER from MMSE and LS for all SNR regions. Such a
BER is similar to that achieved by the MMSE receiver assuming the same block-
type size and arrangement, evidencing that the DNN approach is promising.
Also, Fig. 4.(b) reveals that DNN-detector under a reduced number of pilots
(8-pilots for DNN vs. 16-pilots MMSE vs. 8-pilots LS) outperforms the
conventional estimation methods, i.e., the DNN-OFDM detector presents
desirable robustness against the incomplete pilot’s size.

Table I: Simulation Parameters – Channel, System and DNN and ELM parameters.

Parameter | Value
---|---
OFDM System
# Transmitter Antenna | $n_{T}=1$
# Receiver Antenna | $n_{R}=1$
# Modulation Order ($M$-QAM) | $M=4,16,32$
# Number of Users | $U=4$
# Sub-carriers | $N_{c}=64$
# Pilots Number |  $N_{\rm pilot}=64,32,16$ and $8$
# Cyclic Prefix |  $T_{g}$ = 25% & 0%, (Fig. 6)
Estimation Methods | LS e MMSE
Sub-carrier Interference & Power Allocation
Equal Power Allocation (EPA) | $P_{u}(k)=\frac{P_{\textsc{t}}}{N_{c}}$
Max. # sub-carriers/user |  $N_{\rm cpu}=$ 16
# Interfering sub-carriers/user |  $N_{\rm cpu}^{\rm eq}$ = 4
NLoS Channel
# Cell radius | 500 m
Total Power |  $P_{\textsc{t}}=1$ mW
SNR range |  $\bar{\gamma}\in[5;\,\,25]$ dB
Channel Model | Rayleigh
Path Loss coef. | $\eta=-3$
Path Loss model | $d_{u}^{-\eta}$
Coherence Time |  $(\Delta t)_{\textsc{c}}$ = 5 ms

Parameter | Value
---|---
DNN Architecture
# Hidden Layer | 3
# Input Neuron | 256
# Hidden Neurons | 500, 250 and 120
# Output Neurons | 64
Activation Function | 3 Relu and 1 Sigmoid
Optimizer | Adam
Loss Function | MSE
# Epochs | $10^{3}$
# MCS realizations | $10^{4}$
ELM Architecture
# Hidden Layer | 1
# Input Neurons | $2$
# Hidden Neurons | $L=50$
# Output Neurons | 2
# Sub-Network | 64
Activation Function | Radbas
# Pilots symbols | $I=50,100,200$
# Data symbols | $K=400$
# MCS realizations | $\mathcal{T}=10^{3}$

Figure 5: BER for DNN with the number of users varying from $U=4$ to $8$ and
${\mathcal{J}_{u}}=\\{5,6,7,8\\}$. Modulation: 4-QAM.

Figure 6: Cyclic prefix influence on the BER performance for DNN, MMSE and LS
channel estimators for CP = 25%.

Figure 7: BER performance for different range of $\textsc{snr}_{\rm train}$
and CP = 25%.

Figure 8: DNN detector with CP = 25%: a) BER $\times$ SNR curves parameterized
on the size of Data Set and Epochs; b) training time period; data-set size is
$10^{6}$.

Impact of the Number of Users. So far, we have considered 4 users in the cell;
hence, we equally spread $N_{\rm cpu}=N_{c}/U$ sub-carriers per user. However,
this model does not challenge the DNN, since, in this scenario, there is no
inter-carrier interference. In this way, we have increased the number of
active users inside the cell, aiming at verifying how DNN deals with the
interference i.e., when two users share the same subcarrier. Fig. 5 depicts
the BER for a cell with $|\mathcal{U}|=5,\dots,8$ users, where eight users
mean the maximum co-subchannel interference in a system with 64 sub-carriers
where for each user it has been allocated 16 sub-carriers, which results in a
maximal number of user per sub-carrier equal to 2.

Cyclic Prefix Influence on the Estimation Quality. In OFDM systems, the cyclic
prefix (CP) is paramount to mitigate inter-symbol interference (ISI); however,
it has a cost in terms of power, spectrum, and time. As depicted in Fig. 6,
the DNN-based detector holds certain robustness against the absence of CP in
low and medium SNR regions due to better BER performance compared to the
linear MMSE and LS detectors with no use of CP. This indicates that the
analyzed DNN architecture can learn the characteristics of channels and tends
to better estimate the channel w.r.t. channel inversion strategy implemented
in the LS and MMSE. The non-use of cyclic prefix is a great advantage offered
by ML-based OFDM detectors, resulting in a substantial increment in the
overall energy and spectral efficiencies.

The SNRtrain Effect on the BER Performance. the SNR was set for the training
stage, $\textsc{snr}_{\rm train}$ influences notably the BER performance, Fig.
7; e.g., at the low SNR data transmission regime, $\bar{\gamma}=5$ dB, the
DNN-OFDM detector trained under $\textsc{snr}_{\rm train}=5$ dB attains better
performance; similarly, at high SNR regime ($\bar{\gamma}=25$ dB), the better
performance is reached by the DNN topology trained under $\textsc{snr}_{\rm
train}=25$ dB.

Pilot, Data Set and Modulation sizes. In the training phase, the DNN deals
with the batch size ($\Psi$), and epochs ($\Omega$); such influence on the BER
performance is shown in Fig. 8.a), while the training time is depicted in Fig.
8.b). The better performance is obtained with $\Psi=10000$ and $\Omega=500$,
although this demands a high time for training. It is notable that under a low
number of batch sizes and epochs the performance is suitable yet, e.g., for
$\Psi=250$ and $\Omega=1000$, result in a reasonable performance with low
training time.

Testing Time. In contrast with the training time of Fig. 8.b), the testing
time, $i.e.$, the time necessary for each ML-based OFDM detector operate in
real scenarios for realizing all detection steps completion. The operation
time for both detectors have resulted in $T_{\textsc{dnn}}=4.7$ ms and
$T_{\textsc{elm}}=4.2$ ms. Such an operation time has been measured based on
the transmission of a single OFDM frame for DNN, while for ELM, we have
considered $I$ = 100 pilots and $K$ = 1 symbol, for all the 64 sub-carriers.
As expected, the ELM-OFDM detector presents a lower execution time to perform
all steps, since its architecture comprises just one single layer.

Computational Complexity for LS-OFDM, MMSE-OFDM, and ELM-OFDM detectors, in
terms of the number of operations parameterized on the number of subcarrier
$N_{c}$, pilots $I$, and neurons $L$ is given by $\mathcal{O}(I\cdot N_{c})$,
$\mathcal{O}(4\cdot I\cdot N_{c}^{3})$ and $\mathcal{O}(2\cdot N_{c}\cdot
L^{3})$ respectively. The number of flops can be computed directly from the
implemented code in Matlab using [10].

##  VI Conclusions

Two ML-based topologies of OFDM detectors have been extensively analyzed and
compared with the conventional LS and MMSE detectors, evidencing that such
topologies can be more advantageous in terms of performance-complexity trade-
off perspective. The DNN-based OFDM detector presented a) a promising BER
performance when compared to the classical linear high-complexity inversion
matrix-based OFDM detection techniques; b) robustness against the incomplete
data training, implying in several pilots reduction; and c) absence of cyclic
prefix requirement, increasing the energy and spectral efficiencies of the
OFDM system. However, the DNN-based detector architecture results in
relatively high computational complexity. The ELM-based OFDM detector has been
extensively characterized to overcome this limitation, presenting a superior
performance-complexity tradeoff.

## References

  * [1] Cho YS, Kim J, Yang WY, Kang CG. MIMO-OFDM Wireless Communications with MATLAB.  Wiley Publishing .  2010\.
  * [2] James AR, Benjamin RS, John S, Joseph TM, Mathai V, Pillai SS. Channel estimation for OFDM systems. In: International Conference on Signal Processing, Communication, Computing and Networking Technologies; 2011: 587–591.
  * [3] Ye H, Li GY, Juang B. Power of Deep Learning for Channel Estimation and Signal Detection in OFDM Systems. IEEE Wireless Communications Letters 2018; 7(1): 114-117.
  * [4] Liu J, Mei K, Zhang X, Ma D, Wei J. Online Extreme Learning Machine-Based Channel Estimation and Equalization for OFDM Systems. IEEE Communications Letters 2019; 23(7): 1276-1279.
  * [5] Yang L, Zhao Q, Jing Y. Channel Equalization and Detection With ELM-Based Regressors for OFDM Systems. IEEE Communications Letters 2020; 24(1): 86-89.
  * [6] Luong TV, Ko Y, Vien NA, Nguyen DHN, Matthaiou M. Deep Learning-Based Detector for OFDM-IM. IEEE Wireless Communications Letters 2019; 8(4): 1159-1162.
  * [7] Balevi E, Doshi A, Andrews JG. Massive MIMO Channel Estimation With an Untrained Deep Neural Network. IEEE Transactions on Wireless Communications 2020; 19(3): 2079-2090.
  * [8] Huang GB, Zhu QY, Siew C. Extreme Learning Machine: Theory and Applications. Neurocomputing 2006; 70: 489-501.  doi: 10.1016/j.neucom.2005.12.126
  * [9] Goldsmith A. Wireless Communications.  Cambridge University Press .  2005
  * [10] Qian H. Counting the Floating Point Operations (FLOPS) (https://www.mathworks.com/matlabcentral/fileexchange/50608-counting-the-floating-point-operations-flops). 2020\.


# Cubic* criticality emerging from quantum loop model on triangular lattice

Xiaoxue Ran  Department of Physics and HKU-UCAS Joint Institute of Theoretical
and Computational Physics,The University of Hong Kong, Pokfulam Road, Hong
Kong SAR, China  Zheng Yan  Department of Physics, School of Science, Westlake
University, Hangzhou 310030, China  Institute of Natural Sciences, Westlake
Institute for Advanced Study, Hangzhou 310024, China  Yan-Cheng Wang  Zhongfa
Aviation Institute of Beihang University, Hangzhou 311115, China  Tianmushan
Laboratory, Hangzhou 310023, China  Junchen Rong  Institut des Hautes Études
Scientifiques, 91440 Bures-sur-Yvette, France  Yang Qi  State Key Laboratory
of Surface Physics, Fudan University, Shanghai 200433, China  Center for Field
Theory and Particle Physics, Department of Physics, Fudan University, Shanghai
200433, China  Collaborative Innovation Center of Advanced Microstructures,
Nanjing 210093, China  Zi Yang Meng<EMAIL_ADDRESS>Department of Physics and
HKU-UCAS Joint Institute of Theoretical and Computational Physics,The
University of Hong Kong, Pokfulam Road, Hong Kong SAR, China

###### Abstract

Quantum loop and dimer models are archetypal examples of correlated systems
with local constraints, whose generic solutions for different lattice
geometries and parameter regimes are difficult to obtain due to the lack of
controlled methods to solve them in the thermodynamic limit, yet their
solutions are of immediate relevance towards both statistical and quantum
field theories and the fast-growing experiments in Rydberg atom arrays and
quantum moiré materials where the interplay between correlation and local
constraints give rise to a plethora of novel phenomena. In a recent work Yan
_et al._ , it was found via the sweeping cluster quantum Monte Carlo (QMC)
simulations and field theory analysis that the triangular lattice quantum loop
model (QLM) host a rich ground state phase diagram with lattice nematic (LN),
vison plaquette (VP) crystals and the $\mathbb{Z}_{2}$ quantum spin liquid
(QSL) close to the Rokhsar-Kivelson (RK) point. Here, we focus on the
continuous quantum critical point separating the VP and QSL phases, and
demonstrate via both the static and dynamic probes in QMC simulations that
this transition is of the (2+1)d Cubic* universality in which the
fractionalized visons in QSL condense to give rise to the crystalline VP
phase, meanwhile leaves its trace in the anomalously large anomalous dimension
exponent and pronounced continua in the dimer and vison spectra compared with
those at the conventional Cubic or O(3) quantum critical points. The
experiment proposal of the detection of such phenomena is discussed.

Introduction.— In our recent work Yan  _et al._ , the ground state phase
diagram of quantum loop model (QLM) on triangular lattice Moessner and Sondhi
(2001a); Moessner  _et al._ (2001a); Plat  _et al._ (2015); Roychowdhury  _et
al._ (2015) is mapped out with the sweeping cluster quantum Monte Carlo (QMC)
algorithm Yan  _et al._ (2019, 2021a); Yan (2022); Yan  _et al._ (2021b, 2022,
2023) (shown in Fig. 1). The physics questions revealed therein are profound
and manifolded, such as the nature of the hidden vison plaquette (VP) phase –
invisible from dimer correlations – sandwiched between the lattice nematic
order and $\mathbb{Z}_{2}$ quantum spin liquid (QSL) close to the RK point
Rokhsar and Kivelson (1988); Moessner and Sondhi (2001b); Henley (2004), and
the structure of the phase diagram when connected to the finite temperature –
expected to be richer compared with its square lattice loop or dimer model
cousins Alet  _et al._ (2005a); Dabholkar  _et al._ (2022, 2023), etc, but
perhaps the most interesting one related with the quantum critical aspect of
the model, is the (2+1)d Cubic* transition from the VP phase to the
$\mathbb{Z}_{2}$ QSL, where the VP order parameter emerged from the underlying
resonance of the dimer pairs fractionalizes into vison order parameter which
are the O(3)/Cubic CFT primary field operator, and the condensation of such
fractionalized excitations, in return, gives rise to a strong enhancement of
the scaling dimension of the VP order parameter at the transition, in a
unconventional manner Wang  _et al._ ; Sun  _et al._ (2018); Wang  _et al._
(2018); Isakov  _et al._ (2012); Wang  _et al._ (2021, 2021).

It is therefore our motivation in this work, to elucidate the precise nature
of such intriguing unconventional Cubic* quantum critical point (QCP), both
from static and dynamic probes, and achieve its full understanding in a
combined manner of field-theoretical interpretation and the state-of-the-art
QMC simulations. Here we show via the large-scale sweeping cluster QMC, the
Cubic* transition acquires anomalously large anomalous dimension if one view
the critical scaling from the $t$-term (the resonance term in the QLM
Hamiltonian, will be explained below) correlation functions, as they are the
composite object of the fractionalized vison and correspond to the rank-2
tensor (or tensorial magnetization) of the (2+1)d O(3)/Cubic universality with
large scaling dimension $\eta_{T}\approx 1.42$ Ballesteros  _et al._ (1996);
Aharony (1973); Hasenbusch and Vicari (2011); Adzhemyan  _et al._ (2019);
Aharony  _et al._ (2022); Pelissetto and Vicari (2002); Chester  _et al._
(2021). On the other hand, if one measures the correlation of vison operator,
the observed anomalous dimension is consistent with the conventional small
values of the $\eta\approx 0.04$ for (2+1)d O(3)/Cubic universality
Ballesteros  _et al._ (1996); Aharony (1973); Hasenbusch and Vicari (2011);
Adzhemyan  _et al._ (2019); Aharony  _et al._ (2022); Pelissetto and Vicari
(2002); Chester  _et al._ (2021). Such sharp contrast clearly reveal the
unconventional nature of the Cubic* transition separating the unconventional
VP phase hidden from the dimer measurement and the $\mathbb{Z}_{2}$ QSL in
which the visons are the anyonic particles of the underlying topological
order.

Figure 1: Quantum loop model on the triangular lattice and its phase diagram.
(a) $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ are the triangular lattice primitive
vectors. The $t$ and $V$ terms are the kinetic and potential terms in the
Hamiltonian Eq. (1), respectively. (b) The transition between the LN and VP
crystals is first order Yan  _et al._ . The transition between VP and the
$\mathbb{Z}_{2}$ QSL is continuous and of (2+1)d Cubic* universality Yan  _et
al._ . The correlation functions of the VP and vison order parameters around
the Cubic* QCP ($V_{c}=0.59(2)$) are shown in Figs. 2 and 3. The schematic
plot in the VP phase is the real-space vison correlations with the red (grey)
color conveys its positive (negative) value in each triangle. The schematic
plot in the QSL phase shows two visons connected by a string (the green dashed
line), which represents the path $P$ of the vison-vison correlation function
$v_{\gamma}(0)v_{\gamma}(r)$ $=$ $(-1)^{N_{P}}$, with $N_{P}$ the number of
dimers cut along $P$. Here we set the vison in the lower triangle $v_{1}(0)=1$
as the reference to fix the gauge.

Besides these purely theoretical motivations, the QLM we studied here has been
widely treated as the low-energy effective model for many frustrated magnets
and blockaded cold-atom arrays Roychowdhury  _et al._ (2015); Plat  _et al._
(2015); Wang  _et al._ (2017, ); Zhang  _et al._ (2018); Feng  _et al._
(2017); Wen (2017); Wei  _et al._ ; Li  _et al._ (2020); Hu  _et al._ (2020);
Wen  _et al._ (2019); Feng  _et al._ (2018); Wen and Lee (2019); Zhou  _et
al._ (2017); Broholm  _et al._ (2020); Da Liao  _et al._ (2021); Zhou  _et
al._ (2022); Samajdar  _et al._ (2021); Verresen  _et al._ (2021); Semeghini
_et al._ (2021); Yan  _et al._ (2022); Zhou  _et al._ ; Samajdar  _et al._
(2023); Yan  _et al._ (2023); Moessner and Sondhi (2001a); Moessner  _et al._
(2001b, a); Ivanov (2004); Ralko  _et al._ (2005); Misguich and Mila (2008);
Yan  _et al._ (2021b); Ran  _et al._ (2023) in condensed matter and cold atom
experiments. In the Rydberg array, the static characteristics can be obtained
easily via snapshoot technique Scholl  _et al._ (2021); Semeghini  _et al._
(2021); Tran  _et al._ (2023); Wurtz  _et al._ , while the dynamic information
can be measured through the real-time evolution Surace  _et al._ (2020);
Notarnicola  _et al._ (2020); Guardado-Sanchez  _et al._ (2018); Turner  _et
al._ (2018). The similar static and dynamic information for quantum magnets
can be detected by neutron scattering or nuclear magnetic resonance
experiments Feng  _et al._ (2017); Wei  _et al._ ; Hu  _et al._ (2020);
Paddison  _et al._ (2017); Zorko  _et al._ (2017); Wen  _et al._ (2019), and
our computational scheme of the QMC + stochastic analytic continuation (SAC)
Sandvik (1998a); Beach ; Syljuåsen (2008); Shao and Sandvik (2023) for the
frustrated spin model, QDM and QLM models have provided well consistent static
and dynamic information that have been used to explain experiments Shao  _et
al._ (2017); Li  _et al._ (2020); Hu  _et al._ (2020); Sun  _et al._ (2018);
Ma  _et al._ (2018); Wang  _et al._ (2021, 2021); Zhou  _et al._ (2021); Yan
_et al._ (2021b, 2022, 2023). Based on these previous experiences, in this
work, our QMC static correlations reveal the different scaling dimensions at
the Cubic* QCP corresponding to the different constituent operators in the CFT
data for $\eta_{T}$ and $\eta$, and at the same time our QMC+SAC dynamic
measurements exhibit the continua of the dimer and vison spectra as the
dynamic signature of the $\mathbb{Z}_{2}$ topological order and its associated
vison condensation, in the vicinity of the Cubic* QCP, in a experimentally
detectable manner.

Model and Methods.— The Hamiltonian of the QLM on a triangular lattice is
defined as

$\displaystyle H=$ $\displaystyle-t$
$\displaystyle\sum_{\alpha}\left(\left|\begin{picture}(22.0,10.0)(-8.0,-6.0)\put(-6.0,-10.0){\line(1,0){12.0}}\put(5.2,-10.0){\rotatebox{60.0}{\rule[-1.0pt]{11.38092pt}{1.99997pt}}}\put(-6.8,-10.0){\rotatebox{60.0}{\rule[-1.0pt]{11.38092pt}{1.99997pt}}}\put(12.0,0.0){\line(-1,0){12.0}}
\put(-6.0,-10.0){\circle*{4.0}}\put(6.0,-10.0){\circle*{4.0}}\put(0.0,0.0){\circle*{4.0}}\put(12.0,0.0){\circle*{4.0}}
\end{picture}\right>\left<\begin{picture}(22.0,10.0)(-8.0,-6.0)\put(-6.0,-10.0){\rule[-1.0pt]{11.38092pt}{1.99997pt}}\put(6.0,-10.0){\line(3,5){6.0}}\put(0.0,0.0){\line(-3,-5){6.0}}\put(0.0,0.0){\rule[-1.0pt]{11.38092pt}{1.99997pt}}
\put(-6.0,-10.0){\circle*{4.0}}\put(6.0,-10.0){\circle*{4.0}}\put(0.0,0.0){\circle*{4.0}}\put(12.0,0.0){\circle*{4.0}}
\end{picture}\right|+\mathrm{h.c.}\right)$ (1) $\displaystyle+V$
$\displaystyle\sum_{\alpha}\left(\left|\begin{picture}(22.0,10.0)(-8.0,-6.0)\put(-6.0,-10.0){\line(1,0){12.0}}\put(5.2,-10.0){\rotatebox{60.0}{\rule[-1.0pt]{11.38092pt}{1.99997pt}}}\put(-6.8,-10.0){\rotatebox{60.0}{\rule[-1.0pt]{11.38092pt}{1.99997pt}}}\put(12.0,0.0){\line(-1,0){12.0}}
\put(-6.0,-10.0){\circle*{4.0}}\put(6.0,-10.0){\circle*{4.0}}\put(0.0,0.0){\circle*{4.0}}\put(12.0,0.0){\circle*{4.0}}
\end{picture}\right>\left<\begin{picture}(22.0,10.0)(-8.0,-6.0)\put(-6.0,-10.0){\line(1,0){12.0}}\put(5.2,-10.0){\rotatebox{60.0}{\rule[-1.0pt]{11.38092pt}{1.99997pt}}}\put(-6.8,-10.0){\rotatebox{60.0}{\rule[-1.0pt]{11.38092pt}{1.99997pt}}}\put(12.0,0.0){\line(-1,0){12.0}}
\put(-6.0,-10.0){\circle*{4.0}}\put(6.0,-10.0){\circle*{4.0}}\put(0.0,0.0){\circle*{4.0}}\put(12.0,0.0){\circle*{4.0}}
\end{picture}\right|+\left|\begin{picture}(22.0,10.0)(-8.0,-6.0)\put(-6.0,-10.0){\rule[-1.0pt]{11.38092pt}{1.99997pt}}\put(6.0,-10.0){\line(3,5){6.0}}\put(0.0,0.0){\line(-3,-5){6.0}}\put(0.0,0.0){\rule[-1.0pt]{11.38092pt}{1.99997pt}}
\put(-6.0,-10.0){\circle*{4.0}}\put(6.0,-10.0){\circle*{4.0}}\put(0.0,0.0){\circle*{4.0}}\put(12.0,0.0){\circle*{4.0}}
\end{picture}\right>\left<\begin{picture}(22.0,10.0)(-8.0,-6.0)\put(-6.0,-10.0){\rule[-1.0pt]{11.38092pt}{1.99997pt}}\put(6.0,-10.0){\line(3,5){6.0}}\put(0.0,0.0){\line(-3,-5){6.0}}\put(0.0,0.0){\rule[-1.0pt]{11.38092pt}{1.99997pt}}
\put(-6.0,-10.0){\circle*{4.0}}\put(6.0,-10.0){\circle*{4.0}}\put(0.0,0.0){\circle*{4.0}}\put(12.0,0.0){\circle*{4.0}}
\end{picture}\right|\right),$

where $\alpha$ denotes all the rhombi (with three orientations) on the
triangular lattice as shown in Fig. 1(a). The local constraint of the fully
packed QLM requires two dimers to touch every site in any configuration. The
kinetic term is controlled by $t$, which generates the dimer pair resonance on
every flippable plaquette while respecting the local constraint, and $V$ is
the repulsion ($V>0$) or attraction ($V<0$) between dimers facing each other.
The RK point is located at $V=t=1$ and has an exact $\mathbb{Z}_{2}$ QSL
solution Moessner and Sondhi (2001a). We set $t=1$ as the energy unit and
perform the simulations for system sizes $L=8,12,16,20,24$ with the inverse
temperature $\beta=\frac{1}{T}=L$ via the sweeping cluser QMC methods Yan  _et
al._ (2019); Yan (2022); Yan  _et al._ (2021b, 2022, 2022), and utilize the
SAC scheme Sandvik (1998b); Shao  _et al._ (2017); Sun  _et al._ (2018); Ma
_et al._ (2018); Yan  _et al._ (2021b); Shao and Sandvik (2023); Li  _et al._
(2020); Hu  _et al._ (2020); Wang  _et al._ (2021); Zhou  _et al._ (2021); Yan
_et al._ (2022) to obtain both the dimer and vison spectral functions in real
frequency for $L=12$ system from imaginary time correlation functions with
$\tau\in[0,\beta=100]$.

According to Ref. Yan  _et al._ , the order parameter of the VP phase is given
by the real space $t$-term correlation function

$\langle T(0)T(\mathbf{r})\rangle=\frac{1}{3}[\langle
t_{1}(0)t_{1}(\mathbf{r})\rangle+\langle
t_{2}(0)t_{2}(\mathbf{r})\rangle+\langle t_{3}(0)t_{3}(\mathbf{r})\rangle],$
(2)

where $\langle t_{\alpha}(0)t_{\alpha}(\mathbf{r})\rangle$ $(\alpha=1,2,3)$
represent correlators on the three rhombus directions in our triangular
lattice with distance $\mathbf{r}$ between two rhombi. The vison correlation
function, constructed from the dimer configurations, is

$\langle V(0)V(\mathbf{r})\rangle=\frac{1}{2}[\langle
v_{1}(0)v_{1}(\mathbf{r})\rangle+\langle v_{2}(0)v_{2}(\mathbf{r})\rangle],$
(3)

where $v_{\gamma}$ $(\gamma=1,2)$ for the A (lower triangle) and B (upper
triangle) sublattices in one rhombus. To obtain the vison configuration from
dimer configuration, one needs to fix a gauge with the reference vison in the
plaquette $(0,0)$ and sublattice A as $v_{1}(0)=1$, as shown in the schematic
plot of Fig. 1 (b). Then we map the dimer pattern to the vison configuration
through $v_{1}(0)v_{\gamma}(\mathbf{r})$ $=$ $(-1)^{N_{P}}$, with $N_{P}$
being the number of dimers cut along the path $P$ between triangle at $0$ and
$\mathbf{r}$, which refer to the green dashed line in the Fig. 1 (b).
Therefore, the vison in each triangle holds the value $\pm 1$, as denoted by
the red ($+1$) and grey ($-1$) triangles in the schematic plots of Fig. 1 (b).

In the field theoretical description Yan  _et al._ , the Cubic* CFT of the VP-
QSL transition can be described with three scalars coupled together. The
Lagrangian is

$\mathcal{L}_{int}=m^{2}(\sum_{i}\phi_{i}^{2})++u(\sum_{i}\phi_{i}^{2})^{2}+v(\sum_{i}\phi_{i}^{4})+\cdots,$
(4)

together with kinetic terms for the scalars, where the scalar order parameter
describing the vison modes Blankschtein  _et al._ (1984a, b); Huh  _et al._
(2011); Roychowdhury  _et al._ (2015) is given by

$\phi_{j}=\sum_{\mathbf{r}}(v_{1}(\mathbf{r}),v_{2}(\mathbf{r}))\cdot\mathbf{u}_{j}e^{i\mathbf{M}_{j}\cdot\mathbf{r}},\quad
j=1,2,3,$ (5)

with $\mathbf{M}_{j=1,2,3}$ the three $\mathbf{M}$ points of the Brillouin
zone as shown in the inset of Fig. 4 (b) and $v_{1,2}(\mathbf{r})$ the vison
fields in Eq. (3). The vector $\bm{\phi}$ $=$ $(\phi_{1},\phi_{2},\phi_{3})$
encapsulates the (2+1)d Cubic order parameters of the visons. The mass term
can be roughly identified as $m^{2}\sim V-V_{c}$, and the phase transition
happens at $m^{2}=0$. Conformal field theory tells us the correlation of
$\phi$ fields follows a power law behavior. At the phase transition, the
quantum fluctuation of the vison field is dominated by their modes at the
$\mathbf{M}$ points. The vison correlation in Eq. (3) therefore will follow
the same power law (with spatial modulation). Similarly, the $t$-term operator
$t_{i}$ can be identified as the field theory operators. In particular,
$\\{t_{1},t_{2},t_{3}\\}\sim\\{\phi_{1}\phi_{2},\phi_{2}\phi_{3},-\phi_{1}\phi_{3}\\}$.
The symmetry group of the CFT is the Cubic(3)=$S_{3}\rtimes(Z_{2})^{3}$ group,
the group elements of Cubic(3) can be identified with lattice symmetries. The
precise identification of $t$-operators is fixed by the symmetries that they
break. Here we are following the convention of Ref. Yan  _et al._ . The
Cubic(3) group is a subgroup of O(3). It is known, based on various
theoretical works Ballesteros  _et al._ (1996); Calabrese  _et al._ (2003);
Chester  _et al._ (2021), that the O(3) CFT and the Cubic(3) are connected by
a very short renormalizations group flow, therefore their operators have
similar anomalous dimensions. In particular, the O(3) group has a rank-2
symmetric traceless tensor representation, formed by
$\\{\phi_{1}\phi_{2},\phi_{2}\phi_{3},-\phi_{1}\phi_{3}\\}$ and
$\\{\phi_{1}^{2}-\phi_{2}^{2},\phi_{2}^{2}-\phi_{3}^{2}\\}$, which is five
dimensional. In the view of the subgroup Cubic(3), the triple
$\\{\phi_{1}\phi_{2},\phi_{2}\phi_{3},-\phi_{1}\phi_{3}\\}$ forms a three
dimensional irreducible representation of the Cubic(3) group. We can safely
use the well-known value of the critical exponents $\eta_{T}\approx 1.42$ of
O(3) CFT to approximate its value at the Cubic CFT. The subscript “$T$”
reminds us that it corresponds to the rank-2 tensor of O(3). Interestingly,
the off-diagonal correlator $\langle t_{1}(0)t_{2}(\mathbf{r})\rangle$ decay
much faster than the diagonal ones $\langle t_{1}(0)t_{1}(\mathbf{r})\rangle$,
which is a also a CFT prediction and we show these results in the Supplemental
Materials (SM) sup . The anomalous dimension of scalar
$\\{\phi_{1},\phi_{2},\phi_{3}\\}$ for O(3) CFT, i.e. the vison $v_{1,2}$
correlation in Eq. (3), on the other hand, is of very small value $\eta\approx
0.04$ Ballesteros  _et al._ (1996); Chester  _et al._ (2021).

Figure 2: The static $t$-term correlation. (a) $\langle
T(0)T(\mathbf{r})\rangle$ as a function of the distance $\mathbf{r}=(x,0)$
with the system size $L=24$, the largest system size achieved. The log-log
plot for the absolute values of the data in (a) is shown in (b). We also show
the log-log plot of the $t$-term correlators with different system sizes in
(c) to demonstrate the finite-size effect of the decay behavior. In (a) and
(b), we show the correlators in the VP phase with $V=0.3$, at the transition
point with $V_{c}$ (we use $V=0.6$ here), and at the RK point when $V=1$. The
dark solid line shown in (b) and (c) is proportional to $1/x^{1+\eta_{T}}$
with the large anomalous dimension $\eta_{T}=1.42$, which corresponds to the
rank-2 tensor field of the (2+1)d O(3)/Cubic universality Ballesteros  _et
al._ (1996); Chester  _et al._ (2021). Figure 3: The static vison correlation.
(a) $\langle V(0)V(\mathbf{r})\rangle$ as a function of the distance
$\mathbf{r}=(x,0)$ with the fixed system size $L=24$. (b) The log-log plot of
only the odd value of $x$ in (a). Similar with the $t$-term correlators, We
show the vison correlators in the VP phase ($V=0.3$), at the transition point
$V_{c}$ (use $V=0.61$ here), and at the RK point when $V=1$. Panel (c) shows
the outstanding critical decay behavior with different system sizes. Different
from the $t$-term correlators in Figs. 2 (b) and (c), the dark solid line
shown in (b) and (c) here is proportional to $1/x^{1+\eta}$ with the anomalous
dimension $\eta=0.04$ for the (2+1)d O(3)/Cubic scalar order parameter
Ballesteros  _et al._ (1996); Chester  _et al._ (2021).

We also compute the dynamic dimer correlation function

$D(\mathbf{k},\tau)=\frac{1}{3N}\sum_{\begin{subarray}{c}i,j\\\
\alpha=1,2,3\end{subarray}}^{L^{2}}e^{i\mathbf{k}\cdot\mathbf{r}_{ij}}\left(\langle
n_{i,\alpha}(\tau)n_{j,\alpha}(0)\rangle-\langle n_{i,\alpha}\rangle\langle
n_{j,\alpha}\rangle\right),$ (6)

where $n_{i,\alpha}$ is the dimer number operator on bond $i$ and $\alpha$
stands for the three bond orientations, and the vison dynamic correlation
function

$v(\mathbf{k},\tau)=\frac{1}{2N}\sum_{\begin{subarray}{c}i,j\\\
\gamma=1,2\end{subarray}}^{L^{2}}e^{i\mathbf{k}\cdot\mathbf{r}_{ij}}\left(\langle
v_{i,\gamma}(\tau)v_{j,\gamma}(0)\rangle-\langle v_{i,\gamma}\rangle\langle
v_{j,\gamma}\rangle\right),$ (7)

which average the correlation functions of visons in A, B sublattices. Since
the value of vison in each triangle is $\pm 1$, the second term $\langle
v_{i,\gamma}\rangle$ in Eq. (7) is expected to be zero, i.e., no background
needs to be subtracted.

Figure 4: The dynamic dimer and vison spectra. We display the spectra at the
Cubic* QCP $V_{c}$ with $V=0.6$, and at the RK point with $V=1$ for $L=12$
system. The $\beta$ used in the simulations is $100$ and we employ QMC+SAC
scheme to generate the real frequency data. The inset in (b) shows the high-
symmetry path in the Brillouin zone along which the spectra are presented. For
dimer spectra in the upper row (a) and (b), all the spectra exhibit broad
continua due to the fractionalization. At the $\mathbf{M}$ point, the spectra
are gapped, indicating dimer correlator is unable to detect the transition
between the VP and QSL phase. While for vison spectra, the gap at $\mathbf{M}$
point closes at the Cubic* QCP in panel (c), and reopens at the RK point in
panel (d). The vison gap at the RK point in (d) is smaller than that of the
dimer gap in (b).

Numerical results.— Figs. 2 and 3 show the $\langle T(0)T(\mathbf{r})\rangle$
and $\langle V(0)V(\mathbf{r})\rangle$ across the Cubic* QCP with system size
up to $L=24$. The distance is along the $\mathbf{r}=(x,0)$ with $x$ up to $12$
for the periodic boundary condition. The real-space decay behaviors is
observed for both correlators in three regions: (i) VP phase with $V=0.3$.
(ii) The Cubic* QCP $V_{c}=0.59(2)$. (iii) The RK point $V=1$.

In the VP phase, both $\langle T(0)T(\mathbf{r})\rangle$ and $\langle
V(0)V(\mathbf{r})\rangle$ exhibit strong even-odd oscillations and with
amplitude decaying with the distance $x$. The oscillations derive from the
hidden vison order and eventually vanish as $V$ goes to 1 as shown in Figs. 2
(a) and 3 (a). We note the even-odd oscillation still exists at the transition
point due to the finite-size effect. The oscillations of all $V$ are symmetric
with respect to $\langle T(0)T(\mathbf{r})\rangle=0$, therefore, we illustrate
$|\langle T(0)T(\mathbf{r})\rangle|$ in log-log scale in Fig. 2 (b) and (c).
Moreover, due to the gauge choice we set manually to construct the vison
configuration, the oscillations of the vison correlation are asymmetrical with
respect to $\langle V(0)V(\mathbf{r})\rangle=0$ for different values of $V$.
Thus we only use the odd value of the distance to fit the data of $|\langle
V(0)V(\mathbf{r})\rangle|$ in log-log scale, as shown in Fig. 3 (b) and (c).

We found in the VP phase both correlation functions decay to a constant value,
while exhibit power-law decay at the Cubic* QCP. Interestingly, these two
correlators decay with obvious different exponents. For the $t$-term
correlation $\langle T(0)T(\mathbf{r})\rangle\sim 1/x^{1+\eta_{T}}$ is
consistent with an anomalously large anomalous dimension of the rank-2 tensor
of Cubic CFT with $\eta_{T}=1.42$; and for the vison correlation $\langle
V(0)V(\mathbf{r})\rangle\sim 1/x^{1+\eta}$ is consistent with $\eta=0.04$
which is the (2+1)d O(3)/Cubic value of $\eta$ for the order parameter. To
access the thermodynamic limit, we depict correlators with different system
sizes at the Cubic* QCP in Figs. 2 (c) and 3 (c), and put the small system
sizes data of other values of $V$ in SM sup , all these results reveal the
$\eta_{T}=1.42$ for $\langle T(0)T(\mathbf{r})\rangle$ and the $\eta=0.04$ for
the $\langle V(0)V(\mathbf{r})\rangle$. On the other hand, inside the QSL
phase such as $V=1$ the RK point, both the correlators decay exponentially as
shown in Figs. 2 (b) and 3 (b).

Large anomalous dimension means a large scaling dimension as
$\Delta_{T}=\frac{1+\eta_{T}}{2}$ for the rank-2 tensor and
$\Delta=\frac{1+\eta}{2}$ for the scalar operators of the Cubic/O(3) CFT, our
results therefore mean that at the Cubic* QCP, the $t_{1},2,3$-term is a
composite of the fractionalized visons $v_{1,2}$, instead of a well-defined
critical mode, and it is the proliferated visons $v$ give rise to the large
anomalous dimension of $t$, which serves as a defining signature of the Cubic*
transition, different from the conventional Cubic/O(3) QCPs. Similar behavior
have been observed in the (2+1)d XY* transition between $\mathbb{Z}_{2}$ QSL
and U(1) symmetry-breaking superfluid phase Wang  _et al._ ; Isakov  _et al._
(2012); Wang  _et al._ (2018, 2021).

Such fractionalization signature is also vividly seen from the dynamic probes.
We measure the dynamic correlation functions in Eqs. (6) and Eq. (7) and
obtain the dimer and vison spectra via QMC+SAC (details of the scheme is given
in SM sup ). Fig. 4 show the obtained spectra across the Cubic* transition.
Inside the QSL phase denoted by panels (b) and (d), both spectra exhibit
gapped behavior and substantial continua in a large fraction of the momenta
along the high-symmetry path. It is interesting to note that the minimal dimer
gap is larger than the minimal vison gap due to the fact that dimer is the
composite of a pair of visons Feldner  _et al._ (2011); Yan  _et al._ (2021b).

At the Cubic* QCP, the dimer spectra remains gapped as shown in Fig. 4 (a),
especially at the $\mathbf{M}_{j=1,2,3}$ points (the seemingly small gap at
$\Gamma$ point is due to the finite-size effect), but as shown in Fig. 4 (c),
the vison spectra developed a clear gapless mode close to the $\mathbf{M}$
points, as $\mathbf{M}$ points are the ordered wavevector of the VP phase (as
explained in Eq. (5)), this critical and gapless vison mode is the dynamic
signature of the vison condensation at the Cubic* transition. The contrast
between Fig. 4 (a) and (c) explaines the reason why the dimer correlation
cannot see the "hidden" VP order and only the vison spectra reveal the
translational symmetry breaking of the VP phase. Similar dynamic signature of
the $\mathbb{Z}_{2}$ topological order in QSL and the condensation of
fractionalized anyons has also been shown in the (2+1)d XY* transition Wang
_et al._ ; Wang  _et al._ (2018); Sun  _et al._ (2018); Becker and Wessel
(2018); Wang  _et al._ (2021, 2021).

Discussions.— Via the combined numeric and analytic approaches, we find static
and dynamic signatures of the Cubic* transition from $\mathbb{Z}_{2}$ QSL to
VP crystal in the QLM on triangular lattice. Both correlations and spectra
show that at the transition the fractionalized vison in QSL condenses to give
rise to the crystalline VP phase, meanwhile leaves its trace in the
anomalously large anomalous dimension exponent and pronounced continua in the
dimer and vison spectra compared with those at the conventional Cubic or O(3)
quantum critical points. Many new directions are opened from here such as what
is the underlying reason that the $t$-term correlation is exactly the rank-2
symmetric traceless tensor of the Cubic/O(3) CFT and why the VP phase is
invisible from dimer after the vison condensation process, etc. But our static
and dynamic signatures discovered here can already be used to guide further
experiments in frustrated quantum magnets and the blocked cold-atom arrays, in
which the unconventional quantum matter and quantum phase transitions are
being realized in an astonishing speed Feng  _et al._ (2017); Wen (2017); Wei
_et al._ ; Li  _et al._ (2020); Hu  _et al._ (2020); Wen  _et al._ (2019);
Feng  _et al._ (2018); Wen and Lee (2019); Zhou  _et al._ (2017); Broholm  _et
al._ (2020); Da Liao  _et al._ (2021); Zhou  _et al._ (2022); Samajdar  _et
al._ (2021); Verresen  _et al._ (2021); Semeghini  _et al._ (2021); Yan  _et
al._ (2022); Samajdar  _et al._ (2023); Yan  _et al._ (2023).

Acknowledgements—We thank Ning Su and Fabien Alet for insightful discussions
on the O($3$) and cubic fixed points and the phase diagram of triangular
lattice QLM. XXR, ZY and ZYM acknowledge support from the Research Grants
Council (RGC) of Hong Kong Special Administrative Region (SAR) of China
(Projects Nos. 17301420, 17301721, AoE/P-701/20, 17309822 and HKU C7037-22G),
the ANR/RGC Joint Research Scheme sponsored by the RGC of Hong Kong SAR of
China and French National Research Agency (Project No. A_HKU703/22). The K. C.
Wong Education Foundation (Grant No. GJTD-2020-01). YQ acknowledges support
from the the National Natural Science Foundation of China (Grant Nos. 11874115
and 12174068). JR is supported by Huawei Young Talents Program at IHES. Y.C.W.
acknowledges support from Zhejiang Provincial Natural Science Foundation of
China (Grant Nos. LZ23A040003). We thank the High performance Computing Centre
of Beihang Hangzhou Innovation Institute Yuhang, HPC2021 system under the
Information Technology Services and the Blackbody HPC system at the Department
of Physics, University of Hong Kong for providing computational resources that
have contributed to the research results in this paper.

## References

  * (1) Z. Yan, X. Ran, Y.-C. Wang, R. Samajdar, J. Rong, S. Sachdev, Y. Qi, and Z. Y. Meng, arXiv e-prints arXiv:2205.04472 (2022) [cond-mat.str-el] .
  * Moessner and Sondhi (2001a) R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001a).
  * Moessner  _et al._ (2001a) R. Moessner, S. L. Sondhi, and E. Fradkin, Phys. Rev. B 65, 024504 (2001a).
  * Plat  _et al._ (2015) X. Plat, F. Alet, S. Capponi, and K. Totsuka, Phys. Rev. B 92, 174402 (2015).
  * Roychowdhury  _et al._ (2015) K. Roychowdhury, S. Bhattacharjee, and F. Pollmann, Phys. Rev. B 92, 075141 (2015).
  * Yan  _et al._ (2019) Z. Yan, Y. Wu, C. Liu, O. F. Syljuasen, J. Lou, and Y. Chen, Phys. Rev. B 99, 165135 (2019).
  * Yan  _et al._ (2021a) Z. Yan, Z. Zhou, O. F. Syljuasen, J. Zhang, T. Yuan, J. Lou, and Y. Chen, Phys. Rev. B 103, 094421 (2021a).
  * Yan (2022) Z. Yan, Phys. Rev. B 105, 184432 (2022).
  * Yan  _et al._ (2021b) Z. Yan, Y.-C. Wang, N. Ma, Y. Qi, and Z. Y. Meng, npj Quantum Mater. 6, 39 (2021b).
  * Yan  _et al._ (2022) Z. Yan, R. Samajdar, Y.-C. Wang, S. Sachdev, and Z. Y. Meng, Nat. Commun. 13, 5799 (2022).
  * Yan  _et al._ (2023) Z. Yan, Y.-C. Wang, R. Samajdar, S. Sachdev, and Z. Y. Meng, Phys. Rev. Lett. 130, 206501 (2023).
  * Rokhsar and Kivelson (1988) D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988).
  * Moessner and Sondhi (2001b) R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001b).
  * Henley (2004) C. L. Henley, Journal of Physics: Condensed Matter 16, S891 (2004).
  * Alet  _et al._ (2005a) F. Alet, J. L. Jacobsen, G. Misguich, V. Pasquier, F. Mila, and M. Troyer, Phys. Rev. Lett. 94, 235702 (2005a).
  * Dabholkar  _et al._ (2022) B. Dabholkar, G. J. Sreejith, and F. Alet, Phys. Rev. B 106, 205121 (2022).
  * Dabholkar  _et al._ (2023) B. Dabholkar, X. Ran, J. Rong, Z. Yan, G. J. Sreejith, Z. Y. Meng, and F. Alet, Phys. Rev. B 108, 125112 (2023).
  * (18) Y.-C. Wang, C. Fang, M. Cheng, Y. Qi, and Z. Y. Meng, arXiv e-prints arXiv:1701.01552 (2017) [cond-mat.str-el] .
  * Sun  _et al._ (2018) G.-Y. Sun, Y.-C. Wang, C. Fang, Y. Qi, M. Cheng, and Z. Y. Meng, Phys. Rev. Lett. 121, 077201 (2018).
  * Wang  _et al._ (2018) Y.-C. Wang, X.-F. Zhang, F. Pollmann, M. Cheng, and Z. Y. Meng, Phys. Rev. Lett. 121, 057202 (2018).
  * Isakov  _et al._ (2012) S. V. Isakov, R. G. Melko, and M. B. Hastings, Science 335, 193 (2012).
  * Wang  _et al._ (2021) Y.-C. Wang, Z. Yan, C. Wang, Y. Qi, and Z. Y. Meng, Phys. Rev. B 103, 014408 (2021).
  * Wang  _et al._ (2021) Y.-C. Wang, M. Cheng, W. Witczak-Krempa, and Z. Y. Meng, Nat. Commun. 12, 5347 (2021).
  * Ballesteros  _et al._ (1996) H. Ballesteros, L. Fernández, V. Martín-Mayor, and A. Muñoz Sudupe, Physics Letters B 387, 125 (1996).
  * Aharony (1973) A. Aharony, Phys. Rev. B 8, 4270 (1973).
  * Hasenbusch and Vicari (2011) M. Hasenbusch and E. Vicari, Phys. Rev. B 84, 125136 (2011).
  * Adzhemyan  _et al._ (2019) L. T. Adzhemyan, E. V. Ivanova, M. V. Kompaniets, A. Kudlis, and A. I. Sokolov, Nucl. Phys. B 940, 332 (2019).
  * Aharony  _et al._ (2022) A. Aharony, O. Entin-Wohlman, and A. Kudlis, Phys. Rev. B 105, 104101 (2022).
  * Pelissetto and Vicari (2002) A. Pelissetto and E. Vicari, Physics Reports 368, 549 (2002).
  * Chester  _et al._ (2021) S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su, and A. Vichi, Phys. Rev. D 104, 105013 (2021).
  * Wang  _et al._ (2017) Y.-C. Wang, Y. Qi, S. Chen, and Z. Y. Meng, Phys. Rev. B 96, 115160 (2017).
  * Zhang  _et al._ (2018) X.-F. Zhang, Y.-C. He, S. Eggert, R. Moessner, and F. Pollmann, Phys. Rev. Lett. 120, 115702 (2018).
  * Feng  _et al._ (2017) Z. Feng, Z. Li, X. Meng, W. Yi, Y. Wei, J. Zhang, Y.-C. Wang, W. Jiang, Z. Liu, S. Li, F. Liu, J. Luo, S. Li, G. qing Zheng, Z. Y. Meng, J.-W. Mei, and Y. Shi, Chin. Phys. Lett. 34, 077502 (2017).
  * Wen (2017) X.-G. Wen, Chin. Phys. Lett. 34, 90101 (2017).
  * (35) Y. Wei, Z. Feng, W. Lohstroh, D. H. Yu, D. Le, C. dela Cruz, W. Yi, Z. F. Ding, J. Zhang, C. Tan, L. Shu, Y.-C. Wang, H.-Q. Wu, J. Luo, J.-W. Mei, F. Yang, X.-L. Sheng, W. Li, Y. Qi, Z. Y. Meng, Y. Shi, and S. Li, arXiv e-prints arXiv:1710.02991 (2017) [cond-mat.str-el] .
  * Li  _et al._ (2020) H. Li, Y. D. Liao, B.-B. Chen, X.-T. Zeng, X.-L. Sheng, Y. Qi, Z. Y. Meng, and W. Li, Nature Communications 11, 1111 (2020).
  * Hu  _et al._ (2020) Z. Hu, Z. Ma, Y.-D. Liao, H. Li, C. Ma, Y. Cui, Y. Shangguan, Z. Huang, Y. Qi, W. Li, Z. Y. Meng, J. Wen, and W. Yu, Nature Communications 11, 5631 (2020).
  * Wen  _et al._ (2019) J. Wen, S.-L. Yu, S. Li, W. Yu, and J.-X. Li, npj Quantum Materials 4, 12 (2019).
  * Feng  _et al._ (2018) Z. Feng, W. Yi, K. Zhu, Y. Wei, S. Miao, J. Ma, J. Luo, S. Li, Z. Y. Meng, and Y. Shi, Chin. Phys. Lett. 36, 017502 (2018).
  * Wen and Lee (2019) J. J. Wen and Y. S. Lee, Chin. Phys. Lett. 36, 50101 (2019).
  * Zhou  _et al._ (2017) Y. Zhou, K. Kanoda, and T.-K. Ng, Rev. Mod. Phys. 89, 025003 (2017).
  * Broholm  _et al._ (2020) C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Science 367, eaay0668 (2020).
  * Da Liao  _et al._ (2021) Y. Da Liao, H. Li, Z. Yan, H.-T. Wei, W. Li, Y. Qi, and Z. Y. Meng, Phys. Rev. B 103, 104416 (2021).
  * Zhou  _et al._ (2022) Z. Zhou, C. Liu, Z. Yan, Y. Chen, and X.-F. Zhang, npj Quantum Materials 7, 60 (2022).
  * Samajdar  _et al._ (2021) R. Samajdar, W. W. Ho, H. Pichler, M. D. Lukin, and S. Sachdev, Proc. Nat. Acad. Sci. 118, e2015785118 (2021).
  * Verresen  _et al._ (2021) R. Verresen, M. D. Lukin, and A. Vishwanath, Phys. Rev. X 11, 031005 (2021).
  * Semeghini  _et al._ (2021) G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuletić, and M. D. Lukin, Science 374, 1242 (2021).
  * (48) Z. Zhou, Z. Yan, C. Liu, Y. Chen, and X.-F. Zhang, arXiv e-prints arXiv:2212.10863 (2022) [quant-ph] .
  * Samajdar  _et al._ (2023) R. Samajdar, D. G. Joshi, Y. Teng, and S. Sachdev, Phys. Rev. Lett. 130, 043601 (2023).
  * Moessner  _et al._ (2001b) R. Moessner, S. L. Sondhi, and P. Chandra, Phys. Rev. B 64, 144416 (2001b).
  * Ivanov (2004) D. A. Ivanov, Phys. Rev. B 70, 094430 (2004).
  * Ralko  _et al._ (2005) A. Ralko, M. Ferrero, F. Becca, D. Ivanov, and F. Mila, Phys. Rev. B 71, 224109 (2005).
  * Misguich and Mila (2008) G. Misguich and F. Mila, Phys. Rev. B 77, 134421 (2008).
  * Ran  _et al._ (2023) X. Ran, Z. Yan, Y.-C. Wang, J. Rong, Y. Qi, and Z. Y. Meng, Phys. Rev. B 107, 125134 (2023).
  * Scholl  _et al._ (2021) P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, _et al._ , Nature 595, 233 (2021).
  * Tran  _et al._ (2023) M. C. Tran, D. K. Mark, W. W. Ho, and S. Choi, Phys. Rev. X 13, 011049 (2023).
  * (57) J. Wurtz, A. Bylinskii, B. Braverman, J. Amato-Grill, S. H. Cantu, F. Huber, A. Lukin, F. Liu, P. Weinberg, J. Long, _et al._ , arXiv e-prints arXiv:2306.11727 (2023) [cond-mat.str-el] .
  * Surace  _et al._ (2020) F. M. Surace, P. P. Mazza, G. Giudici, A. Lerose, A. Gambassi, and M. Dalmonte, Phys. Rev. X 10, 021041 (2020).
  * Notarnicola  _et al._ (2020) S. Notarnicola, M. Collura, and S. Montangero, Phys. Rev. Res. 2, 013288 (2020).
  * Guardado-Sanchez  _et al._ (2018) E. Guardado-Sanchez, P. T. Brown, D. Mitra, T. Devakul, D. A. Huse, P. Schauß, and W. S. Bakr, Phys. Rev. X 8, 021069 (2018).
  * Turner  _et al._ (2018) C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Nature Physics 14, 745 (2018).
  * Paddison  _et al._ (2017) J. A. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu, M. B. Stone, H. Zhou, and M. Mourigal, Nature Physics 13, 117 (2017).
  * Zorko  _et al._ (2017) A. Zorko, M. Herak, M. Gomilšek, J. van Tol, M. Velázquez, P. Khuntia, F. Bert, and P. Mendels, Phys. Rev. Lett. 118, 017202 (2017).
  * Sandvik (1998a) A. W. Sandvik, Phys. Rev. B 57, 10287 (1998a).
  * (65) K. S. D. Beach, arXiv e-prints arXiv:cond-mat/0403055 (2004) [cond-mat.stat-mech] .
  * Syljuåsen (2008) O. F. Syljuåsen, Phys. Rev. B 78, 174429 (2008).
  * Shao and Sandvik (2023) H. Shao and A. W. Sandvik, Physics Reports 1003, 1 (2023).
  * Shao  _et al._ (2017) H. Shao, Y. Q. Qin, S. Capponi, S. Chesi, Z. Y. Meng, and A. W. Sandvik, Phys. Rev. X 7, 041072 (2017).
  * Ma  _et al._ (2018) N. Ma, G.-Y. Sun, Y.-Z. You, C. Xu, A. Vishwanath, A. W. Sandvik, and Z. Y. Meng, Phys. Rev. B 98, 174421 (2018).
  * Zhou  _et al._ (2021) C. Zhou, Z. Yan, H.-Q. Wu, K. Sun, O. A. Starykh, and Z. Y. Meng, Phys. Rev. Lett. 126, 227201 (2021).
  * Yan  _et al._ (2022) Z. Yan, Z. Y. Meng, D. A. Huse, and A. Chan, Phys. Rev. B 106, L041115 (2022).
  * Sandvik (1998b) A. W. Sandvik, Phys. Rev. B 57, 10287 (1998b).
  * Blankschtein  _et al._ (1984a) D. Blankschtein, M. Ma, A. N. Berker, G. S. Grest, and C. M. Soukoulis, Phys. Rev. B 29, 5250 (1984a).
  * Blankschtein  _et al._ (1984b) D. Blankschtein, M. Ma, and A. N. Berker, Phys. Rev. B 30, 1362 (1984b).
  * Huh  _et al._ (2011) Y. Huh, M. Punk, and S. Sachdev, Phys. Rev. B 84, 094419 (2011).
  * Calabrese  _et al._ (2003) P. Calabrese, A. Pelissetto, and E. Vicari, Phys. Rev. B 67, 024418 (2003), arXiv:cond-mat/0202292 .
  * (77) Implementation of the real-space $t$-term and vison correlation functions for different system sizes and across the Cubic* transition, the QMC+SAC scheme and the representative data of the imaginary time correlations and their corresponding spectra, are given in this Supplemental Material .
  * Feldner  _et al._ (2011) H. Feldner, Z. Y. Meng, T. C. Lang, F. F. Assaad, S. Wessel, and A. Honecker, Phys. Rev. Lett. 106, 226401 (2011).
  * Becker and Wessel (2018) J. Becker and S. Wessel, Phys. Rev. Lett. 121, 077202 (2018).
  * Syljuasen and Sandvik (2002) O. F. Syljuasen and A. W. Sandvik, Phys. Rev. E 66, 046701 (2002).
  * Alet  _et al._ (2005b) F. Alet, S. Wessel, and M. Troyer, Phys. Rev. E 71, 036706 (2005b).

Supplemental Material for “Cubic* criticality emerging from quantum loop model
on triangular lattice”

In this Supplemental Material, we first present the average $t$-term
correlation functions $\langle T(0)T(\mathbf{r})\rangle$, the average vison
correlations $\langle V(0)V(\mathbf{r})\rangle$, and the the off-diagonal
$t$-term correlations $\langle t_{1}(0)t_{2}(\mathbf{r})\rangle$ with
different system sizes in Fig. S1. All this correlation functions, including
the dynamic correlators which will be introduced later, are measured in the VP
phase with $V=0.3$, at the transition point $V_{c}$, and at the RK point with
$V=1$. Then, we discuss the details of the QMC+SAC scheme in Sec. II,
including a brief introduction to the sweeping cluster QMC method. To
demonstrate the SAC process, we utilize the dimer spectra as an example. The
dimer and vison dynamic correlation functions as a function of the imaginary
time $\tau$ for $k$ points along the high-symmetry path are shown in Fig. S2.
The frequency dependence of the dimer and vison spectrum functions are
presented in Fig. S3. Besides, we show the dimer and vison spectra in the VP
phase in Fig. S4 as a supplement of Fig. 4 in the main text. QMC simulations
for all the dynamic correlators are conducted with a fixed system size of
$L=12$ and $\beta=100$.

##  I The $t$-term and vison correlation functions

In this section, we show the raw data for the average $t$-term and vison
correlation functions with different system sizes in Fig. S1. In the VP phase
with $V=0.3$, both correlations exhibits strong even-odd oscillations and with
amplitude decaying with the distance $x$ as shown in Fig. S1 (a) and (c). At
the transition point, the oscillations become weak but not vanish due to the
finite-size effect in Fig. S1 (g) and (i). At the RK point, which in the
$\mathbb{Z}_{2}$ QSL phase, both correlations oscillate weakly along the $y=0$
horizontal line.

We also measured the off-diagonal $t$-term correlations $\langle
t_{1}(0)t_{2}(\mathbf{r})\rangle$ as shown in the right hand side of Fig. S1.
The dark solid line shown in Fig. S1(l) is proportional to $x^{-2.42}$, which
indicates $\langle t_{1}(0)t_{2}(\mathbf{r})\rangle$ decays faster than
$\langle T(0)T(\mathbf{r})\rangle$ at the Cubic* transitions point.

Figure S1: The diagonal $t$-term correlation function $\langle
T(0)T(\mathbf{r})\rangle$ (left), the diagonal vison correlation function
$\langle V(0)V(\mathbf{r})\rangle$ (middle), and the off-diagonal $t$-term
correlation function $\langle t_{1}(0)t_{2}(\mathbf{r})\rangle$ (right) as a
function of the distance $\mathbf{r}=(x,0)$ with different system sizes. We
show all the correlators in the VP phase with $V=0.3$ in the upper row, at the
transition point with $V_{c}$ in the middle row, and at the RK point when
$V=1$ in the lower row. The log-log plots for the absolute values of the raw
data are presented in the even column as well. For vison correlations, only
odd $x$ data is used for log-log plots in (d), (j) and (p). The dark solid
lines shown in (h) and (l) are proportional to $x^{-2.42}$, and the dashed one
in (i) is proportional to $x^{-1.04}$.

##  II QMC+SAC SCHEME

To obtain the dynamic spectral functions of the dimer and vison operators, we
perform the SAC process to the dynamic correlations of the Eq. 6 and Eq. 7
obtained in the QMC simulations. The sweeping cluster QMC approach employed in
our work is a new method, whose main idea is to record the local constraints
through sweeping and updating operators layer by layer along the imaginary
time direction, thus all the samplings are performed in the restricted Hilbert
space. This method is based on the world-line Monte Carlo scheme Syljuasen and
Sandvik (2002); Alet  _et al._ (2005b, a) and works well in constrained
quantum lattice models Yan  _et al._ (2019); Yan (2022); Yan  _et al._ (2021b,
2022, ).

Here we take the dimer spectra as an example to demonstrate this QMC-SAC
scheme. The dimer dynamic correlation functions $D(\mathbf{k},\tau)$, where
$\mathbf{k}$ is the momentum point along the high-symmetry path as shown in
Fig. 4(c) in our simulations, relates to the real-frequency spectral function
$S(\mathbf{k},\omega)$ through the equation

$D(\mathbf{k},\tau)=\int_{-\infty}^{+\infty}S(\mathbf{k},\omega)e^{-\tau\omega}d\omega.$
(S1)

With the relation between the positive and negative frequencies
$S(\mathbf{k},-\omega)=e^{-\beta\omega}S(\mathbf{k},\omega)$, Eq. S1 can be
written only with positive frequencies

$D(\mathbf{k},\tau)=\int_{0}^{\infty}K(\tau,\omega)S(\mathbf{k},\omega)d\omega,$
(S2)

where the kernel is defined as

$K(\tau,\omega)=\frac{1}{\pi}(e^{-\tau\omega}+e^{-(\beta-\tau)\omega}).$ (S3)

Note that $D(\mathbf{k},\beta-\tau)=D(\mathbf{k},\tau)$ under this relation as
well, thus we only need $\tau$ in the range of $[0,\beta/2]$. For the
normalization $D(\mathbf{k},0)=1$, we therefore define
$D(\mathbf{k},\omega)=(1+e^{-\beta\omega})S(\mathbf{k},\omega)/\pi$, where
$\int_{0}^{\infty}D(\mathbf{k},\omega)d\omega=1$. Then, Eq. S2 is replaced
with

$D(\mathbf{k},\tau)=\int_{0}^{\infty}\frac{e^{-\tau\omega}+e^{-(\beta-\tau)\omega}}{1+e^{-\beta\omega}}D(\mathbf{k},\omega)d\omega.$
(S4)

The main idea of the SAC is to obtain the estimate spectrum functions by means
of the Monte Carlo sampling of the $\delta$-functions Shao  _et al._ (2017);
Shao and Sandvik (2023). The weight function is $e^{-\chi^{2}/2\Theta}$, with
$\chi^{2}$ the goodness of fit and $\Theta$ the sampling temperature.
$\chi^{2}$ quantifies the difference between the average QMC data
$\bar{D}(\mathbf{k},\tau)$ and the estimate correlations
$D^{\prime}(\mathbf{k},\tau)$, which is given by

$\chi^{2}=\sum_{i,j}(D^{\prime}(\mathbf{k},\tau_{i})-\bar{D}(\mathbf{k},\tau_{i}))C_{ij}^{-1}(D^{\prime}(\mathbf{k},\tau_{j})-\bar{D}(\mathbf{k},\tau_{j})),$
(S5)

which sum over all imaginary time $\tau_{i},i=1,...,N_{\tau}$, a quadratic
grid of $\tau_{i}$ is used with $\tau_{i}\propto i^{2}$Shao and Sandvik
(2023). $C_{ij}$ is the covariance matrix element

$C_{ij}=\frac{1}{N_{b}(N_{b}-1)}\sum^{N_{b}}_{b=1}(D^{b}(\mathbf{k},\tau_{i})-\bar{D}(\mathbf{k},\tau_{i}))(D^{b}(\mathbf{k},\tau_{j})-\bar{D}(\mathbf{k},\tau_{j})),$
(S6)

where $b=1,2,..,N_{b}$ denotes bins of the QMC data,
$\bar{D}=\frac{1}{N_{b}}\sum_{b}D^{b}$. In the sampling process, we adjust
$\Theta$ to find the minimal $\chi^{2}$ to satisfy
$\langle\chi^{2}\rangle\approx\chi^{2}_{min}+\sqrt{2N_{\tau}}$, which finally
generate a smooth average spectral function.

Our QMC simulation results for the dynamic dimer and vison correlations as a
function of the imaginary time $\tau$ for the fixed system size $L=12$ with
$\beta=100$ are shown in Fig. S2. $D(\mathbf{k},\tau)$ and
$v(\mathbf{k},\tau)$ with different high-symmetry momentum points are observed
in the VP phase with $V=0.3$, at the VP-QSL transition point with $V=0.6$, and
at the RK point where $V=1$. Besides, we also demonstrate the semi-log plots
in the even column of Fig. S2 to show the exponential decay of the correlation
data. For the dynamic dimer correlations, we found all $k$ points in the three
phase exhibit the exponential decay with non-zero exponents, which indicates
the spectra are all gapped in these momentum points. However, the decay
exponents are very small when $V=0.3$ in Fig. S2(h) and $V=0.6$ in Fig. S2(j)
for the dynamic vison correlations.

To further confirm our prediction, the SAC method is performed to obtain the
dimer spectra in the three phases. The frequency dependence of the dimer and
vison spectrum functions $D(\mathbf{k},\omega)$ and $v(\mathbf{k},\omega)$ are
shown in the first and second row of Fig. S3 respectively, in which the peak
of the $k$ point represents the strength of the dimer spectra in Fig. 4 in the
main text. We also present the dimer and vison spectra in the VP phase with
$V=0.3$ in Fig. S4. The gap at the $M$ point is a non-zero value in dimer
spectra but seems very close to zero in viosn spectra. The spectrum at the
$\Gamma$ point in the dimer spectra seems gapless may due to the finite-size
effect.

Figure S2: The semi-log plots of dimer (upper row) and vison (lower row)
dynamic correlation functions obtained in QMC simulations with system size
$L=12$ and $\beta=100$. Similar to the real-time correlation functions,
$D(\mathbf{k},\tau)$ and $v(\mathbf{k},\tau)$ are measured in the VP phase
with $V=0.3$ (left), at the VP-QSL transition point with $V=0.6$ (middle), and
at the RK point where $V=1$ (right). Selected $k$ points along the high-
symmetry path shown in Fig. 4(c) in the main text are presented to make the
figures more clear. Figure S3: The dimer spectrum functions
$D(\mathbf{k},\omega)$ and the vison spectrum functions $v(\mathbf{k},\omega)$
as a function of frequency, which generated through SAC process from dynamic
dimer and vison correlations data obtained from QMC, are shown in the first
and second row respectively. $k$ points along the high-symmetry path are
illustrated in the VP phase with $V=0.3$ in (a) and (d), at the VP-QSL
transition point with $V=0.6$ in (b) and (e), and at the RK point where $V=1$
in (c) and (f). The peaks of each $k$ point corresponding to the strength of
the spectra in Fig. 4 and Fig. S4. Insets in (d) and (e) zoom in the data to
display $k$ points with small peaks. Note that the maximum values of the
spectrum functions show different magnitudes for dimer and vison spectra, thus
we have used multi colorbars for demonstration in Fig. 4 and Fig. S4.  Figure
S4: (a) The dimer and (b) vison spectra for $L=12$ system with $\beta=100$ in
the VP phase with $V=0.3$. At the $M$ point, the dimer spectra is gapped while
the vison spectra seems gapless.