# Fusing Information Streams in
End-to-End Audio-visual Speech Recognition

###### Abstract

End-to-end acoustic speech recognition has quickly gained widespread
popularity and shows promising results in many studies. Specifically the joint
transformer/CTC model provides very good performance in many tasks. However,
under noisy and distorted conditions, the performance still degrades notably.
While audio-visual speech recognition can significantly improve the
recognition rate of end-to-end models in such poor conditions, it is not
obvious how to best utilize any available information on acoustic and visual
signal quality and reliability in these models. We thus consider the question
of how to optimally inform the transformer/CTC model of any time-variant
reliability of the acoustic and visual information streams. We propose a new
fusion strategy, incorporating reliability information in a decision fusion
net that considers the temporal effects of the attention mechanism. This
approach yields significant improvements compared to a state-of-the-art
baseline model on the Lip Reading Sentences 2 and 3 (LRS2 and LRS3) corpus. On
average, the new system achieves a relative word error rate reduction of 43%
compared to the audio-only setup and 31% compared to the audio-visual end-to-
end baseline.

Index Terms—  Audio-visual Speech Recognition, Joint CTC-Transformer, Decision
Fusion Net

##  1 Introduction

In recent years, end-to-end (E2E) automatic speech recognition (ASR) has
attracted a great amount of attention. E2E models have a much simpler
structure than conventional hybrid ASR, yet the results are often not as
satisfying [1]. The reason for this is two-fold. First, the language model in
a hybrid model is typically trained on large amounts of external text data. In
contrast, most E2E models do not use an explicit language model. Second, as
reported in [2], the E2E model—especially the transformer [3]—is more likely
to overfit. In this work, we use the E2E speech processing toolkit ESPnet [4]
for all experiments. ESPnet uses SpecAugment [5] to overcome the overfitting
problem, and it allows us to decode an E2E model using an external language
model, hence addressing both issues.

The sequence-to-sequence (S2S) transformer model with connectionist temporal
classification (CTC), denoted by TM-CTC, is used in our experiments. This
joint model shows high performance in many different tasks [6, 7]. As
described in [8], the CTC learns to align features and transcription
explicitly, which helps the model to converge faster. Therefore, during the
training stage, we choose the linear combination of CTC and S2S objectives as
the objective function

$L=\alpha\cdot\textrm{log}\
p_{ctc}(\textbf{s}|\textbf{o})+(1-\alpha)\textrm{log}\
p_{s2s}(\textbf{s}|\textbf{o}),$ (1)

with s as the states and the constant hyper-parameter $\alpha$. During
decoding, an RNN-language model $p_{LM}(\textbf{s})$ is also used:

$\textrm{log}\ p^{\ast}(\textbf{s}|\textbf{o})=\alpha\ \textrm{log}\
p_{ctc}(\textbf{s}|\textbf{o})+(1-\alpha)\ \textrm{log}\
p_{s2s}(\textbf{s}|\textbf{o})+\\\ \theta\ \textrm{log}\ p_{LM}(\textbf{s}),$
(2)

where $\theta$ controls the contribution of the language model.

Usually, when people are listening to speech in a noisy environment, they read
each other’s lips unconsciously for more supplementary information, which is
of great benefit for human speech perception [9]. Even in clean speech, seeing
the lips of the speaker influences perception, as demonstrated by the McGurk
effect [10]. Machine audio-visual speech recognition (AVSR) has been developed
for many years [11]. However, current AVSR, whether hybrid or E2E, still does
not seem to make optimal use of this secondary information stream, because
word error rates (WERs) are still clearly diminished in noisy conditions [7,
12]. We combine the advantages of the TM-CTC model and AVSR to alleviate this
issue. For this purpose, we propose to use additional reliability indicators,
which has proven beneficial for multimodal integration in many studies [13,
14, 15]. Due to the temporal re-alignment that occurs in the attention module
of the TM-CTC, it is not obvious how to achieve this combination. We suggest a
new approach, which aligns reliability information to match the focus of the
attention decoder at any given point. This approach is effective for audio-
visual speech recognition and we consider it as an interesting topology for a
wider array of attention-based time series models as well.

The paper is organized as follows: Section 2 shows the system framework, for
which Section 3 explains our core contribution. Section 4 introduces the
experimental setup, with results analyzed in Section 5. Finally, Section 6
discusses the overall outcomes and gives an outlook on future work.

##  2 System Overview

In this section, we describe the model structure for our AVSR system and give
a brief introduction to the baseline models.

###  2.1 Recognizer framework

Preliminary experiments on the audio-only task of our considered LRS 2 corpus
show that the word error rate of 3.7% for TM-CTC framework is clearly superior
to that of a hybrid CTC/attention model without transformer [16] at 8.3%, and
of a Kaldi-trained hybrid model at 11.28%, achieved with the nnet2 p-norm
network recipe [17]. This makes the TM-CTC model the architecture of choice
for targeting optimal performance in AVSR. In the following, we first recap
its architecture and then detail the modifications we took to endow it with
the benefits of time-variant stream reliability information.

####  2.1.1 Encoders

Fig. 1: Audio encoder (left), video encoder (middle) and reliability measures
encoder (right) for both modalities $i\in{a,v}$

As shown in Fig. 1, there are three encoders—an audio encoder, a video
encoder, and a reliability encoder—which are all shared between the CTC, the
transformer and the TM-CTC model. The audio encoder is a conventional
transformer encoder consisting of two parts [7]. First, two 2D convolution
layers used as a sub-sampling block to lower the computational complexity by
reducing the sequence length from $N_{F}$ (the number of frames) to $N_{F}/4$
and project the features to a common dimension $d_{att}=256$. Subsampling is
followed by a stack of 12 encoder blocks, each consisting of a multi-head
self-attention and a fully connected feed-forward layer.

As in [7], the video features are extracted by a pretrained spatio-temporal
visual front-end (the 3D/2D ResNet in Fig. 1) which is based on [18] and also
contains stacks of multi-head self-attention layers. As the audio and video
features have different frame rates, we use Bresenham’s algorithm [19] to
roughly align the video features.

In the multi-head self-attention layers of audio and video encoders, the
queries Q, keys K, and values V are identical. The attention transform matrix
of every attention head with index $j$ is computed via

$\textbf{T}_{j}=\textrm{softmax}\left(\frac{\left(\textbf{W}_{j}^{Q}\textbf{Q}^{T}\right)^{T}\left(\textbf{W}_{j}^{K}\textbf{K}^{T}\right)}{\sqrt{d_{k}}}\right).$
(3)

The attention is

$\bm{\alpha}_{j}=\textrm{attention}_{j}(\textbf{Q},\textbf{K},\textbf{V})=\textbf{T}_{j}\left(\textbf{W}_{j}^{V}\textbf{V}^{T}\right)^{T},$
(4)

where $\textbf{W}_{j}^{\ast}$ are learned parameters, superscript $T$ denotes
the transpose, $d_{k}=\frac{d_{att}}{h}$ and $h$ is the number of attention
heads. The attention transform matrix $\textbf{T}_{j}$ shows the relevance of
the current keys for the current queries. $\textbf{T}_{j}$ is of
size$N_{Q}\times N_{K}$, where $N_{Q}$ and $N_{K}$ are the length of Q and K,
respectively. Finally, the outputs of all heads $\bm{\alpha}_{j}$ are
concatenated and projected by a fully connected layer. This output of the
self-attention block is passed through a feed-forward layer to obtain the
encoder output $\textbf{h}^{i}$.

####  2.1.2 Decoders

Fig. 2: Transformer decoder (left) and CTC decoder (right) for both modalities
$i\in{a,v}$

The TM-CTC model has two decoders for each stream, see Fig. 2. As in [7], the
CTC decoder contains a stack of 6 multi-head self-attention blocks. The
transformer decoder is also formed by a stack of 6 decoder blocks. In these,
the keys (K) and values (V) are the encoder outputs $\textbf{h}^{i}$, both of
size $(N_{F}/4)\times 256$. The queries (Q) come from the previous decoder
block and are transformed by a multi-head self-attention block. Q is of size
$N_{T}$ $\times$ 256, where $N_{T}$ is the length, or number of tokens, of the
transcription.

####  2.1.3 Reliability measures

We additionally embed signal-based reliability measures R to inform our system
of the time-variant reliability of the information streams. As we do not
attempt to extract speech information from the reliability measures, they are
only subsampled, cf. Fig. 1, with Bresenham’s algorithm additionally applied
to the visual reliability measures $\textbf{R}^{v}$.

To obtain visual reliability measures, we use OpenFace [20] for face detection
and facial landmark extraction. The confidence of the face detector in each
frame is used as a visual feature quality indicator. Facial Action Units (AUs)
[21, 20] are also useful, of which we include AU12, AU15, AU17, AU23, AU25,
AU26 in $\textbf{R}^{v}$.

Our audio-based reliability measures $\textbf{R}^{a}$ comprise the first 5
MFCC coefficients, as in [15]. The estimated Signal-to-Noise Ratio (SNR) is a
proxy for the quality of the audio signal. In this work, we apply DeepXi [22]
to estimate the SNR per frame. The pitch $f_{0}$ and its first temporal
derivative, $\Delta f_{0}$ are also indicative of reliability, as high pitch
can negatively affect the MFCC quality due to insufficient smoothing of the
pitch harmonics [23, 24]. The probability of voicing (POV), $f_{0}$, and
$\Delta f_{0}$ are computed as described in [25].

##  3 Fusion strategy

We would like to combine stream-wise posteriors over symbols into joint
posteriors with the help of stream reliability measures. For the CTC, this is
straightforward, since the stream-wise posteriors
$p_{ctc}(\textbf{s}|\textbf{o}^{i})$ are temporally aligned with
$\bm{\xi}^{i}$, both of length $N_{F}/4$.

The S2S part of the TM-CTC is a challenge, however: Whereas the embeddings of
our reliability metrics, $\bm{\xi}^{i}$ in Fig. 1, are of length $N_{F}/4$, we
need $\bm{\xi}^{i}$ to be available per output token, to temporally match the
token-by-token log-posteriors of $p_{s2s}(\textbf{s}|\textbf{o}^{i})$. Hence,
we need to find a transform from the linear time domain of length $N_{F}/4$ to
length $N_{T}$, the number of tokens. To address this, we reconsider Eq. (4),
showing that each of the 6 multi-head attention blocks in the transformer
decoder has its own attention transform matrix $\textbf{T}_{j}^{i}$. We use
the transform matrix in the final block of modality $i$ to convert
$\bm{\xi}^{i}$ into one reliability embedding vector per token:

$\tilde{\bm{\xi}}_{j}^{i}=\textbf{T}_{j}^{i}\cdot\left(\textbf{W}_{j}^{i\xi}({\bm{\xi}^{i}})^{T}\right)^{T},$
(5)

$\tilde{\bm{\xi}}^{i}$ is obtained by projecting a concatenation of all
$\tilde{\bm{\xi}}_{j}^{i}$ via a fully connected layer.

Fig. 3: Topology of the fusion networks for CTC (left) and sequence-to-
sequence recognition (right)

Now, having temporally matching reliability embeddings, we can proceed to find
a simple and effective decision fusion net (DFN) to combine all modalities
dynamically, which is shown in Fig. 3. Here, the posterior probabilities from
all modalities are the inputs, and the corresponding reliability embeddings
are used to estimating the multi-modal log-posteriors $\textrm{log}\
p(\textbf{s}|\textbf{o})$, for both the CTC and the S2S model.

In the DFN for the CTC decoder, the first three hidden layers have 8,192,
4,096, and 512 units, respectively, each using the ReLU activation function
and layer normalization (LN). The dropout rate is 0.15, followed by 3 BLSTM
layers with 512 memory cells for each direction, using tanh as the activation
function. Finally, a fully connected (FC) layer projects the data to the
output dimension. A log-softmax function is applied to obtain the estimated
log-posteriors.

We have also tested BLSTM layers for the $\textrm{DFN}_{s2s}$, but this
results in overfitting, so the sequence-to-sequence posteriors $\textrm{log}\
p_{s2s}(\textbf{s}|\textbf{o})$ are obtained in a purely feed-forward, non-
recurrent architecture as shown in Fig. 3.

##  4 Experimental setup

###  4.1 Dataset

The Oxford-BBC Lip Reading Sentences 2 and 3 corpora [7, 26] are used for all
experiments, cf. Tab. 1 for their statistics.

Table 1: Characteristics of the utilized datasets Subset | Utterances | Vocabulary | Duration[hh:mm]
---|---|---|---
LRS2 pretrain | 96,318 | 41,427 | 196:25
LRS2 train | 45,839 | 17,660 | 28:33
LRS2 test | 1,243 | 1,698 | 00:35
LRS3 pretrain | 118,516 | 51k | 409:10

To train a robust AVSR system, we artificially apply audio data augmentation
to the raw audio signal for the training data. The ambient noise part of the
MUSAN corpus [27] is used as the noise data. Each audio signal is augmented
with a random SNR chosen from seven SNRs, between -9 dB and 9 dB in steps of 3
dB. Data augmentation is also performed for video sequences as in [18], i.e.
random cropping and horizontal flips with a 50% probability. To analyze the
performance in different acoustic noise environments and check the robustness
of the system, we perform a similar acoustic augmentation, but adding new
conditions that are unseen in the training data, to the test data. Ambient and
music sounds are used, from -12 dB to 12 dB in steps of 3 dB. Similarly, we
also apply Gaussian blur and salt-and-pepper noise to the visual data for the
test set.

###  4.2 Feature extraction

The audio model uses 80 log Mel features together with pitch, delta pitch, and
the probability of voicing. These 83-dimensional features are extracted with
25 ms frame size and 10 ms frame shift. The 96 $\times$ 96 pixel grayscale
mouth region of interest (ROI) is detected via OpenFace at 25 frames per
second and then fed into a pre-trained 3D/2D ResNet.

###  4.3 Pretrained model

All models are pretrained on the LRS2 and LRS3 pretrain sets. To save
computational effort, as described in [7], during pretraining, the parameters
of the ResNet feature extraction are frozen. In the second training phase, all
parameters, including those of the ResNet, are fine-tuned on the LRS2 training
set. In our proposed TM-CTC AVSR model, the model parameters are initialized
with the audio-only and video-only model, which were trained separately.

###  4.4 Language model

The language model is trained by a unidirectional 4-layer recurrent network.
Each layer has 2048 units. Here, we use the language model, which is trained
on the LibriSpeech corpus [28]. The language model predicts one character at a
time and receives the previous character as the input.

The language model parameter $\theta$ is 0.5; $\alpha$ in Eqs. (1) and (2) is
set to 0.3. We use $h=4$ attention heads. In ESPnet, the transformer-learning
factor controls the learning rate. We set it to 5.0 for the training stage,
and to 0.05 for fine-tuning.

All models are trained using NVIDIA’s Volta-based DGX-1 multi-GPU system. We
use 7 Tesla V100 GPUs, each with 32GB memory. The audio and video models were
trained for 100 epochs. The audio-visual model baseline and our proposed model
are pre-trained for 65 epochs and fine-tuned for 10 epochs.

##  5 results

The transformer in [7] fuses audio and video streams by combining context
vectors, and a CTC fuses the audio and video streams by concatenating the
audio and video encoder outputs. Here we implement the same structure, but
with the joint CTC/transformer, as an audio-visual model baseline,
outperforming the original baseline model in [7].

Table 2: Performance of audio-visual and uni-modal speech recognition (WER
[%]). AO: audio only. VO: video only. AV: AV baseline [7]. DFN: proposed DFN
fusion.
m: music noise. a: ambient noise. vc: clean visual data.
gb: visual Gaussian blur. sp: visual salt-and-pepper noise. models dB | -12 | -9 | -6 | -3 | 0 | 3 | 6 | 9 | 12 | clean | avg.
---|---|---|---|---|---|---|---|---|---|---|---
AO(m) | 18.9 | 13.7 | 11.2 | 8.4 | 6.3 | 6.8 | 4.5 | 4.1 | 4.3 | 4.2 | 8.24
AO(a) | 25.7 | 23.4 | 18.5 | 11.6 | 8.2 | 9.0 | 5.9 | 3.8 | 4.4 | 4.2 | 11.47
VO(vc) | 58.7 | 61.0 | 61.7 | 69.6 | 69.6 | 63.5 | 64.6 | 63.6 | 66.6 | 61.9 | 64.08
VO(gb) | 66.6 | 69.2 | 71.0 | 68.5 | 68.5 | 71.1 | 62.7 | 69.4 | 67.6 | 66.9 | 68.15
VO(sp) | 68.5 | 72.5 | 73.7 | 70.1 | 70.1 | 70.6 | 68.3 | 69.1 | 73.1 | 67.9 | 70.39
AV(m.vc) | 14.6 | 11.8 | 6.4 | 7.9 | 7.9 | 6.3 | 5.2 | 4.4 | 3.4 | 4.0 | 7.19
DFN(m.vc) | 11.1 | 8.7 | 5.5 | 4.8 | 4.8 | 4.5 | 3.6 | 3.3 | 2.2 | 2.4 | 5.09
AV(a.vc) | 19.1 | 19.0 | 14.3 | 7.3 | 6.3 | 6.0 | 5.7 | 4.5 | 4.9 | 4.0 | 9.11
DFN(a.vc) | 14.3 | 11.9 | 8.1 | 4.8 | 4.0 | 5.4 | 3.7 | 2.8 | 3.6 | 2.4 | 6.10
AV(a.gb) | 20.6 | 18.9 | 15.0 | 7.7 | 6.8 | 7.5 | 5.9 | 3.9 | 4.8 | 4.0 | 9.51
DFN(a.gb) | 14.9 | 12.8 | 9.4 | 5.2 | 4.2 | 5.5 | 3.8 | 3.0 | 4.1 | 2.6 | 6.55
AV(a.sp) | 19.5 | 19.9 | 15.3 | 7.7 | 7.2 | 6.3 | 5.6 | 4.4 | 4.6 | 4.3 | 9.48
DFN(a.sp) | 15.4 | 12.8 | 9.9 | 5.2 | 4.7 | 5.5 | 3.4 | 2.6 | 4.0 | 2.5 | 6.60

Table 2 shows the results for all experiments. As expected, the audio-only
model has a far better performance than the video-only model. Comparing the
performance between the baseline fusion model and the proposed DFN fusion
structure in different noisy environments, the proposed DFN fusion is clearly
preferable, both in clean and in all noisy conditions. Even in clean acoustic
conditions, the proposed model can clearly reduce the WER. On average, the new
system achieves a relative word error rate reduction of 43% compared to the
audio-only setup and 31% compared to the audio-visual end-to-end baseline.

##  6 Conclusion

In noisy conditions, large-vocabulary end-to-end speech recognition remains a
difficult task. It can be made significantly easier by adding video
information, i.e., an extent of lip-reading, to the system. This paper has
addressed the question of how acoustic and visual information can be
integrated optimally in such multi-modal end-to-end models. For this purpose,
we have proposed the decision fusion net (DFN), which explicitly combines
posterior token probabilities of acoustic and visual models based on time-
variant reliability information. The approach is applicable both for CTC-
trained models and for sequence-to-sequence recognition, allowing us to
notably improve recognition rates of the highly effective transformer/CTC-
system, not only in noisy conditions but also in clean acoustic environments.

It will be interesting to analyze the contribution of such reliability-guided
fusion in more detail and for other tasks, specifically considering the
question of how attention models can derive benefit from uncertainty
information and higher-order statistics.

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#  Tensor Network Renormalization Study on the Crossover in
Classical Heisenberg and $\mathrm{RP^{2}}$ Models in Two Dimensions

Atsushi Ueda<EMAIL_ADDRESS>Institute for Solid State Physics,
University of Tokyo, Kashiwa 277-8581, Japan  Masaki Oshikawa  Institute for
Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan  Kavli
Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo, Kashiwa, Chiba 277-8583, Japan  Trans-scale Quantum
Science Institute, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan

###### Abstract

We study the classical two-dimensional $\mathrm{RP^{2}}$ and Heisenberg
models, using the Tensor-Network Renormalization (TNR) method. The
determination of the phase diagram of these models has been challenging and
controversial due to the very large correlation lengths at low temperatures.
The finite-size spectrum of the transfer matrix obtained by TNR is useful in
identifying the conformal field theory describing a possible critical point.
Our results indicate that the ultraviolet fixed point for the Heisenberg model
and the ferromagnetic $\mathrm{RP^{2}}$ model in the zero temperature limit
corresponds to a conformal field theory with central charge $c=2$, in
agreement with two independent would-be Nambu-Goldstone modes. On the other
hand, the ultraviolet fixed point in the zero temperature limit for the
antiferromagnetic Lebwohl-Lasher model, which is a variant of the
$\mathrm{RP^{2}}$ model, seems to have a larger central charge. This is
consistent with $c=4$ expected from the effective SO(5) symmetry. At $T>0$,
the convergence of the spectrum is not good in both the Heisenberg and
ferromagnetic $\mathrm{RP^{2}}$ models. Moreover, there seems to be no
appropriate candidate of conformal field theory matching the spectrum, which
shows the effective central charge $c\sim 1.9$. These suggest that both models
have a single disordered phase at finite temperatures, although the
ferromagnetic $\mathrm{RP^{2}}$ model exhibits a strong crossover at the
temperature where the dissociation of $\mathbb{Z}_{2}$ vortices has been
reported.

##  I Introduction

The Berezinskii-Kosterlitz-Thouless (BKT) transition in the classical XY model
is a milestone for studying topological phase transitions [1, 2, 3, 4]. The
$\mathrm{U(1)}$ spin configuration allows the topological defects (vortex-
antivortex) with integer charge, and its phase transition mechanism is
currently well understood [4]. In contrast to the remarkable success in the XY
model, the nature of the topological dynamics with the other types of
topological defects remains elusive. The Lebwohl-Lasher (LL) [5] and the
antiferromagnetic Heisenberg model on the triangular lattice [6, 7, 8] host
$\mathbb{Z}_{2}$ vortices thanks to the $\mathbb{Z}_{2}$-valued fundamental
group of the target space. In analogy to the BKT transition, it is tempting to
expect a topological phase transition driven by the $\mathbb{Z}_{2}$ vortices
with a quasi-long range order in the low-temperature phase. In fact, such a
phase transition has been advocated and also supported by several numerical
studies [9, 10, 11, 12, 13, 14].

However, even after more than 30 years, the existence of the phase transition
is still inconclusive. From the standpoint of the continuum effective field
theory, the LL model for example would be described by a non-linear sigma
model with the $\mathrm{RP^{2}}$ as the target space, in $1+1$ dimensions.
$\mathrm{RP^{2}}$ is simply the two-dimensional sphere $S^{2}$ with the
antipodal points identified. Thus, as far as the local geometry of the target
space is concerned, the $\mathrm{RP^{2}}$ nonlinear sigma model is the same as
the more standard O(3) nonlinear sigma model (with $S^{2}$ as the target
space). Since the lowest-order term in the Renormalization Group (RG) beta
function is given by the curvature of the target space [15], both the
$\mathrm{RP^{2}}$ and the O(3) nonlinear sigma models are asymptotic-free [16,
17] (an infinitesimal coupling will be eventually renormalized to a strong
coupling at large enough lengthscale). In the context of statistical
mechanics, the asymptotic freedom of the nonlinear sigma model suggests that
the corresponding lattice model is disordered for any non-zero temperature. In
fact, this is perhaps the standard picture on the phase diagram of the
classical Heisenberg model[18, 19, 20, 21] defined by

$\displaystyle H=-J\sum_{\langle i,j\rangle}\vec{S}_{i}\cdot\vec{S}_{j},$ (1)

where $\langle i,j\rangle$ runs over the pairs of nearest-neighbor sites $i$
and $j$, and $\vec{S}_{i}$ denotes a three-component unit vector. This is one
of the most fundamental models in statistical mechanics. However, even for the
classical Heisenberg model, there is a controversy on the phase diagram, since
a phase transition at finite temperature was suggested based on numerical
simulations [22, 23, 24, 25, 26, 27, 28] and high temperature expansions [29,
30]. While the nonlinear sigma model description looks reasonable, its
applicability to the lattice model is not rigorously established and thus it
is possible that the lattice model behaves differently from the nonlinear
sigma model prediction. The phase diagram of the $\mathrm{RP^{2}}$ model [31,
32, 33, 34, 35, 36, 37] is even more controversial than that of the Heisenberg
model. While the Monte Carlo simulations are very powerful for those classical
spin systems, it is difficult to distinguish a crossover from a phase
transition because of the extremely large correlation length [37, 38, 34, 35,
39, 40, 25, 26, 27, 28].

In this paper, we reinvestigate the classical Heisenberg and $\mathrm{RP^{2}}$
models on the square lattice using the tensor network renormalization (TNR)
scheme [41, 42, 43, 44, 45, 46]. Contrary to the conventional Monte-Carlo
methods, TNR offers direct access to the spectrum of the transfer matrix and
central charge of the conformal field theory (CFT), and it proves to be a
powerful tool to study BKT transitions [47, 48, 49]. First we consider the
$T\rightarrow 0$ fixed point, where spontaneous symmetry breaking is not
prohibited by the Mermin-Wagner theorem [50].

Taking advantage of the TNR, we directly calculate the UV central charge of
the Heisenberg model in the $T\sim 0$ limit using TNR, to find that the
central charge is $c=2$, which is strong evidence of the asymptotic freedom.
We also analyze the $T\rightarrow 0$ limit of the $\mathrm{RP^{2}}$ model in a
similar manner. Interestingly, our numerical result indicates that the
ultraviolet $T\rightarrow 0$ $\mathrm{RP^{2}}$ fixed point also has the
central charge $c=2$. On the other hand, the transfer matrix spectra of these
two models have different structures, suggesting that the UV limits are
distinct. Moreover, the possibility of the phase transition at finite
temperature is also discussed. Our results support a crossover rather than a
true phase transition. However, the observed strong crossover corresponds to a
sudden change of the RG flow of the $\mathrm{RP^{2}}$ model in the vicinity of
the reported transition temperature, suggesting the existence of a repulsive
fixed point near the actual RG trajectory.

This paper is organized as follows. In Sec. II, we first explain our models
and their implementations in the tensor network in details. Then, we provide
the methodology on how to extract the universal information from the
renormalized tensors in Sec. III. As our main results, the numerically
obtained data for the zero and finite temperature fixed points are displayed
in Sec. IV and V, respectively. The interpretations and discussion thereof are
given in Sec.4, and we summarize and refer to our future issues in Sec. VII.

##  II Two-dimensional $\mathrm{RP^{2}}$ models

As discussed in the Introduction, we are interested in classical spin models
with a spin “target space” $M$ for which $\pi_{1}(M)=\mathbb{Z}_{2}$. This
allows $\mathbb{Z}_{2}$ vortices as topological defects that are argued to
induce a phase transition. There are actually many different models which can
host the $\mathbb{Z}_{2}$ vortices. The simplest choice of the target space is
$M=\mathrm{RP^{2}}$. We refer to such a spin model in general as a
“$\mathrm{RP^{2}}$ model”. There are a few variations within the
$\mathrm{RP^{2}}$ model. The first is to consider the square of the Heisenberg
interaction, and it is called the Lebwohl-Lasher (LL) model [5],

$\displaystyle H=-J\sum_{\langle
i,j\rangle}(\vec{S}_{i}\cdot\vec{S}_{j})^{2},$ (2)

where $\vec{S}_{i}$ is a three-component classical spin with
$|\vec{S}_{i}|=1$, as in the Heisenberg model. One can immediately check that
this model is invariant to the local head-tail flip
$\vec{S}_{i}\rightarrow-\vec{S}_{i}$ simultaneously at all the sites $i$.
There is plenty of research on the $J>0$ model, but less is found for $J<0$ in
the literature. We refer to these cases, respectively, as “ferromagnetic” and
“antiferromagnetic” LL models. In the case of the Heisenberg model, the
ferromagnetic and antiferromagnetic models on the square lattice are
equivalent, being mapped to each other by a redefinition of the spin variables
$\vec{S}_{i}\to-\vec{S}_{i}$ on the odd sites. In contrast, ferromagnetic and
antiferromagnetic LL models are not equivalent to each other. In the
antiferromagnetic LL model, the orthogonal alignment of neighboring “spins” is
favored; because of the freedom in choosing the orthogonal direction, the
dimension of the ground-state manifold is larger in the antiferromagnetic LL
model.

Another realization of the $\mathrm{RP^{2}}$ model is by introducing the Ising
spins $\sigma_{ij}=\pm 1$ on the bonds [51, 52], which act as a
$\mathbb{Z}_{2}$ gauge field on the lattice. The Hamiltonian is given by

$\displaystyle H=-J\sum_{\langle
i,j\rangle}\sigma_{ij}\vec{S}_{i}\cdot\vec{S}_{j},$ (3)

which we call the $\mathrm{RP^{2}}$ gauge model.

To compute the physical quantities by a tensor network, we decompose the local
Boltzmann weights by the spherical harmonic and Legendre functions [53, 54]
as,

$\displaystyle e^{(\beta\cos(\gamma_{ij}))^{2}}$
$\displaystyle=\sum_{l}f_{l}(\beta)\sum_{m=-l}^{l}\bar{Y}_{lm}(\theta_{i},\phi_{i})Y_{lm}(\theta_{j},\phi_{j})$
(4) $\displaystyle f_{l}(\beta)$
$\displaystyle=2\pi\int_{-1}^{+1}dxP_{l}(x)e^{\beta x^{2}}$ (5)

where $(\theta_{i},\phi_{i})$ is the position of the spin on $S^{2}$ and
$\gamma_{ij}$ is the angle between two spins. This allows to rewrite the
partition function in terms of the transfer matrix on each site.

$\displaystyle Z$ $\displaystyle={\rm
Tr}\prod_{i}\int\frac{d\Omega}{4\pi}\prod e^{(\beta\cos(\gamma_{ij}))^{2}}$
(6) $\displaystyle={\rm
Tr}\prod_{i\in\mathcal{L}}\sum_{l_{i},m_{i}}f_{l_{i}}(\beta)\prod_{s}F_{l_{1}m_{1},l_{2}m_{2}}^{l_{3}m_{3},l_{4}m_{4}}$
(7)

where

$\displaystyle F_{l_{1}m_{1},l_{2}m_{2}}^{l_{3}m_{3},l_{4}m_{4}}=$
$\displaystyle\frac{1}{4\pi}\sum_{l,m}G(l_{1},l_{2},l,m_{1},m_{2},m)G(l_{3},l_{4},l,m_{3},m_{4},-m)$

with $G$ being the gaunt function [55]. Finally, one can express the local
Boltzmann weights with a four-leg tensor as follows [54].

$\;\vbox{\hbox{\includegraphics[width=68.2866pt,page=1]{./transfermatrix.pdf}}}\;=\left(\prod_{i=1}^{4}f_{l_{i}}(\beta)\right)^{\frac{1}{2}}F_{l_{1},m_{1},l_{2},m_{2}}^{l_{3},m_{3},l_{4},m_{4}}$

The index $m_{i}$ takes the values in the range of $-l_{i}<m_{i}<l_{i}$ just
like the magnetic quantum numbers. In practice, we introduce the cutoff for
$l_{i}$ as $-l_{max}<l_{i}<l_{max}$ with its bond dimensions $(l_{max}+1)^{2}$
because the convergence of transfer matrix takes place for a large $l_{max}$.
Moreover, as the head-tail symmetry prohibits $l_{i}$ from being odd, the bond
dimension decreases to $\frac{1}{2}(l_{max}+1)(l_{max}+2)$. In the same way,
one can construct the tensor network representation of the $\mathrm{RP^{2}}$
gauge model by changing $f_{i}(\beta)$ to

$\displaystyle f_{l}(\beta)$
$\displaystyle=4\pi\int_{-1}^{+1}dxP_{l}(x)\cosh{(\beta x)}.$ (8)

As the angular momentum cutoff, we use $l_{max}=4$ for the Heisenberg model
and $l_{max}=6$ for both $\mathrm{RP^{2}}$ models. We note that the cutoffs
above are not sufficient to calculate the accurate observables such as the
free-energy density. Nonetheless, the truncation preserves the symmetry of the
models, and thus we suspect that the universal information at criticality such
as the central charge should remain intact under this truncation.

##  III Methodology

Critical points of statistical mechanical models correspond to RG fixed
points. In two dimensions, those RG fixed points are generally described by
CFT in 1+1 dimensions. Even if the system is off-critical, it could be
described by a perturbed CFT when the correlation length is large. Therefore,
it is important to identify the CFT describing a possible critical
point/phase, or underlying the region with a large correlation length. The
central charge $c$ is the most fundamental quantity characterizing a CFT.
Intuitively, it gives the “number of degrees of freedom,” so that free
massless boson field theory has a central charge $1$ for each independent
component. Under certain general assumptions, Zamolodchikov proved that the
central charge can only decrease along the RG flow [56].

In general, a two-dimensional statistical mechanical model, such as the
Heisenberg model or the $\mathrm{RP^{2}}$ model studied in this paper, can be
mapped to a quantum many-body system in one spatial dimension. The transfer
matrix for the former corresponds to (the exponential of) the Hamiltonian of
the latter. This leads to many useful observations. In particular, the TNR
approach we employ naturally produces the spectrum of the transfer matrix,
which corresponds to the energy spectrum of the quantum many-body system in
one spatial dimension. When the system is critical and described by a CFT, the
energy spectrum is dictated by the CFT. In particular, it contains information
on the central charge and scaling dimensions of operators. The finite-size
energy spectra of one-dimensional quantum systems have been extensively used
to study quantum phases. While it had been difficult to apply a similar method
to two-dimensional statistical mechanical models, recently a combination of
the TNR and the finite-size scaling of the spectrum was demonstrated
successfully for the two-dimensional XY model [49]. In this paper, we extend
the application of the TNR-based finite size scaling to the classical
Heisenberg model and the $\mathrm{RP^{2}}$ models. Since the analysis of the
finite-size spectrum has been largely developed for one-dimensional quantum
systems, we will refer to the (the logarithm of) the eigenvalue spectrum of
the finite-size transfer matrix as “energy levels,” referring to the quantum
counterparts. The $n$-th lowest energy eigenvalue $E_{n}(L)$
($n=0,1,2,\ldots$) is determined as

$E_{n}(L)=-\frac{1}{L}\ln{\lambda_{n}(L)},$ (9)

where $\lambda_{n}(L)$ is the $n$-th largest eigenvalue of the renormalized
tensor representing a $L\times L$ block contracted in the $x$-direction. The
(effective) central charge can be extracted from the scaling of the “ground-
state energy” at each scale [57]

$E_{0}(L)\sim\epsilon_{0}L-\frac{\pi c}{6L}.$ (10)

Although the central charge is defined only for CFTs, we may define the
effective central charge in terms of the scaling behavior of the “ground-state
energy” $E_{0}$ as above in a given range of the scale. Moreover, at a RG
fixed point described by a CFT, the scaling dimensions $x_{n}$ of operators in
the CFT can be extracted from the scaling of the finite-size “excitation
energies” [49, 57] as

$\displaystyle\frac{2\pi x_{n}}{L}=E_{n}(L)-E_{0}(L).$ (11)

Away from the RG fixed point, $x_{n}$ extracted from numerical data using this
relation does not give a scaling dimension of a CFT. However, again we often
consider such a quantity $x_{n}$ as an effective scaling dimension, as it is
useful in visualizing the RG flow.

In particular, $x_{1}(L)$ contains useful information about the correlation
length $\xi(L)$. Using the eigenvalues of a $L\times L$ transfer matrix, the
correlation length $\xi(L)$ can be represented as

$\displaystyle\xi(L)=\frac{L}{\ln(\lambda_{0}(L)/\lambda_{1}(L))}.$ (12)

Combining it with Eqs. (9) and (11), we find the following relation between
$x_{1}(L)$ and $\xi(L)$:

$\displaystyle x_{1}(L)=\frac{1}{2\pi}\frac{L}{\xi(L)}.$ (13)

At criticality, $\xi(L)$ diverges as $L$ increases, and $x_{1}(L)$ converges
to a finite value, corresponding to the scaling dimension of its theory. On
the other hand, $\xi(L)$ saturates for finitely-correlated systems, leading to
the divergence of $x_{1}(L)$. Thus, it is a sign of a finite correlation
length if $x_{1}(L)$ increases with RG steps.(Given the ground state is unique
in the IR limit.) We carry out this analysis for finding the finite
correlation length of the ${\rm RP^{2}}$ models shown in Figs. 3 and 5.

##  IV Ultraviolet Fixed Point for $T\rightarrow 0$

The ground states of the Heisenberg or $\mathrm{RP^{2}}$ model are given by
spins aligned in the same direction. This means that we have a spontaneous
symmetry breaking (SSB) at $T=0$. On the other hand, since these models have
continuous SO(3) symmetry, the Mermin-Wagner theorem [50] prohibits SSB at any
finite temperature $T>0$.

It is interesting to consider the limit of $T\to+0$ from the finite
temperature. This has been studied for the O(3) nonlinear sigma model, which
is believed to be the effective field theory for the classical Heisenberg
model. The SSB of continuous SO(3) symmetry in the $T\to 0$ limit implies the
existence of the Nambu-Goldstone modes. In the case of the O(3) nonlinear
sigma model, the “spin” can fluctuate in two orthogonal directions around the
ordered state. Thus, the $T\to 0$ limit may be identified with the two-
component free boson field theory with the central charge $c=2$. The
asymptotic freedom of the O(3) nonlinear sigma model implies that, at any
$T>0$, the theory is described by the $c=2$ conformal field theory with a
relevant operator. This means that the $c=2$ CFT is the ultraviolet (UV) fixed
point, and the O(3) nonlinear sigma model corresponds to the RG flow emanating
from it. It is believed that the RG flow from the UV fixed point ends up in a
massive theory, implying the absence of any intermediate phase or phase
transitions in the Heisenberg model at $T>0$. This picture has been confirmed
by the exact Bethe ansatz calculation of the free energy using the
factorizable $S$-matrix for the nonlinear sigma model [58, 59] though the
phase diagram for the Heisenberg model has yet to be identified [40, 25, 26,
27, 28].

On the other hand, the RG picture has not been clarified well for the
$\mathrm{RP^{2}}$ model. Since the target spaces of the $\mathrm{RP^{2}}$ and
Heisenberg models are locally isomorphic to each other, we may also expect
two-component $c=2$ free boson CFT as the effective theory of the would-be
Nambu-Goldstone modes in $T\to 0$. However, to the best of our knowledge, this
has not been studied for the $\mathrm{RP^{2}}$ model in Eqs. (2) and (3).
Thus, here we present our TNR study on the “effective central charge” of the
$\mathrm{RP^{2}}$ model.

Practically, it is challenging to simulate the limit $T\to 0$ due to the
diverging Boltzmann weights. Nevertheless, since the zero temperature fixed
points should be RG unstable, observing the UV behavior(finite-size behavior)
of $T\sim 0$ allows us to access the information of the zero-temperature fixed
point. Figure. 1(a) shows the central charge $c$ estimated with Eq. (10) from
Loop-TNR [44] at $D=40$ and $T=0.005$. We see that the effective central
charge for both the Heisenberg model and the ferromagnetic LL model is stable
at $c\sim 2$ at small RG steps. As representative values, at $L=16$ (the
fourth RG step), we estimate $c=2.003$ and $c=1.998$, respectively, for the
Heisenberg and the ferromagnetic LL model. This suggests that each of the
systems behaves as a $c=2$ CFT at short distances. In other words, the UV
fixed point of these models is the $c=2$ CFT, as expected.

Figure 1: (a)Calculated central charge of the ferromagnetic LL, Heisenberg,
and anti-ferromagnetic LL models at $T=0.005$. The ultra-violet central charge
for the first two models is $c=2$, whereas that of the AF-LL model is $c=4$.
The numerically obtained data are $c=2.003$ and 1.998 for the Heisenberg and
LL model, respectively, at the fourth RG step. The cusp around ten RG steps
appears because we derive the central charge by a derivative of $L$ as
$E_{0}(L)\sim\epsilon_{0}L-\frac{\pi c}{6L}$.[We only know $E_{0}(L)$ for
$L=\sqrt{2}^{n}$, so linear fitting is needed.] The appearance of the cusps is
a common feature when the system size reaches the correlation length. It could
be the gap due to the finite bond dimensions. (b) The finite-size scaling
dimension of the Heisenberg and LL model at $T=0.01$. The lowest excitation
corresponds to spin 1 and 2, respectively. The degeneracy is slightly lifted
due to the non-negligible finite-bond dimension effect at the large inverse
temperature. To avoid unnecessary divergence of the Boltzmann weights, we
shifted the energy in the computation, e.g., $H=-J\sum_{\langle
i,j\rangle}[(\vec{S}_{i}\cdot\vec{S}_{j})^{2}-1]$ for the $\mathrm{RP^{2}}$ LL
model. Figure 2: The effective central charge of the Heisenberg and
$\mathrm{RP^{2}}$ gauge models at finite temperature. The numerically obtained
central charge of the Heisenberg model is slightly smaller than 2, $c\sim
1.94$ at $T=0.14$ and the eighth RG step, implying that the correlation is
large, but finite. Similarly, the central charge of the LL model is not
converged and is $c\sim 1.91$ at $T=0.22$ and the eighth RG step. The cusps
after the 12 RG steps are a numerical artifact due to the finite-bond
dimension used in our calculation. On the other hand, at $T=0.27$ the
effective central charge quickly drops to zero, which is a clear signature of
the finite correlation length. Figure 3: (a) The finite-size scaling dimension
of the lowest excitation in the Heisenberg model. The scaling dimension grows
monotonically along the RG steps, which is a consequence of the RG flow from
the unstable $T=0$ fixed point. (b) The finite-size scaling dimension of the
lowest excitation in the $\mathrm{RP^{2}}$ gauge model. The monotonic increase
of $x_{1}(L)$ along the RG steps is observed in the whole range of
temperature. The vertical dotted line denotes the reported transition
temperature from previous studies, $T\sim 0.25$.

While the finite bond dimension effects are a limitation of the tensor-network
type calculations in general, the results are very accurate within
lengthscales smaller than the bond-dimension induced correlation length. The
comparison with the CFT can be carried out for these lengthscales, as has been
demonstrated in other contexts [49]. The detailed results of the finite bond
dimension effects are summarized in Fig. 8. Therefore, we believe that our
identification of the $c=2$ CFT as the UV fixed point for the low-temperature
regime of the Heisenberg and ferromagnetic $\mathrm{RP^{2}}$ models stands, in
spite of the challenging nature of the problem. On the other hand, this
observation does not directly answer the question about the nature of the
“transition” to the disordered phase at higher temperature.

It is also interesting to study the antiferromagnetic LL model for comparison.
As mentioned in Sec. II, the antiferromagnetic LL model has a higher-
dimensional ground-state manifold. It turns out that it corresponds to a O(5)
“spin” model, as discussed in detail in Appendix B. It is then expected that
the UV fixed point is a $c=4$ free boson CFT, corresponding to four
independent Nambu-Goldstone modes. As shown in Fig. 1, we indeed find that the
effective central charge of the antiferromagnetic LL model is significantly
larger than the other models. Although we do not observe the complete
convergence, we think that the numerical result is consistent with our
expectation of $c=4$ CFT representing the UV fixed point representing the
$T\to 0$ limit.

We also note that, Kawamura and Miyashita studied an antiferromagnetic
Heisenberg model on the triangular lattice. Effectively, this model would be
described by a non-linear sigma model with the $\rm SO(3)$ group element as
the field (principal chiral model). Topologically, the $\rm SO(3)$ group is
homeomorphic to $\mathrm{RP^{3}}$ and has the $\mathbb{Z}_{2}$ vortices. We
expect that the UV fixed point representing the $T\to 0$ limit of these models
is a $c=3$ CFT. It would be interesting to confirm this expectation
numerically.

Meanwhile, it is noteworthy that the spectra of these two models are different
in spite of the same central charge. One can immediately see this in the
effective scaling dimensions defined by Eq. (11). In the field theoretical
description, the scaling dimensions correspond to the energy levels of the
collective excitation. In particular, the lowest ones are three degenerate
spin-1 excitations for the Heisenberg model. In contrast, the $\rm O(3)$ spin
is no longer the order parameter for the LL model under the local
$\mathbb{Z}_{2}$ flip symmetry. Instead, we have nematic order parameter $Q$
defined as

$\displaystyle
Q^{\alpha\beta}=s^{\alpha}s^{\beta}-\frac{1}{3}\delta^{\alpha\beta},$ (14)

where $\alpha$ and $\beta$ runs through $(x,y,z)$. Thus, the lowest excitation
of the LL model is spin 2 with its multiplicity 5. The numerically obtained
finite-size scaling dimensions in Fig. 1(b) are consistent with the argument
above. This suggests that, although both are characterized by the same central
charge $c=2$, the UV limits for the Heisenberg and $\mathrm{RP^{2}}$ models
are distinct.

##  V $\mathbb{Z}_{2}$ vortex dissociation at $T\sim T_{*}$

The main interest in the $\mathrm{RP^{2}}$ model is the possible topological
phase transition due to the dissociation of the $\mathbb{Z}_{2}$ vortices,
similarly to the well-established BKT transition in the two-dimensional XY
model [1, 2, 3, 4]. While there have been numerous studies supporting the
existence of the transition, its nature has not been clarified. Moreover, the
very existence of the transition is still a question.

If the purported $\mathbb{Z}_{2}$ vortex dissociation transition is a second-
order phase transition, it would be described by a CFT. Likewise, if the
system has a critical low-temperature phase similar to the low-temperature
phase of the BKT transition, that phase would be also described by a CFT.
Hence, we have numerically estimated the effective central charge and scaling
dimensions in the temperature region where the $\mathbb{Z}_{2}$ vortex
dissociation is expected.

As a reference, we have also applied the same methodology to the classical
Heisenberg model. Figure. 2 shows the effective central charge of the
Heisenberg and $\mathrm{RP^{2}}$ gauge models at finite temperature,
calculated from the TNR. First, the effective central charge of the Heisenberg
model seems to have a plateau around $c\sim 2$ at $T=0.14$, which seems to
suggest a critical phase. However, the convergence of the effective central
charge is not as good as in a similar TNR study of known critical
points/phases in the Ising and XY model [49, 44]. (Compare also with Fig. 1.)
In fact, a detailed analysis shows that the central charge is smaller than
$c=2$ as is indicated with a black dotted line. For example, the best estimate
at $T=0.14$ for the Heisenberg model is $c=1.94(1)$, obtained at the third
through eighth RG steps. Moreover, the “energy levels” which would give the
scaling dimensions of the operators in the CFT change gradually, as shown in
Fig. 3. As we will discuss in the next section, we believe that our results
are more consistent with the standard “asymptotic freedom” picture, which
predicts the single disordered phase for $T>0$.

Now let us discuss the $\mathrm{RP^{2}}$ model around the purported
$\mathbb{Z}_{2}$ vortex dissociation transition temperature $T_{*}=0.25$, in a
similar manner. The effective central charge of the $\mathrm{RP^{2}}$ model
also seems to have a plateau at $c\sim 2$ for $T=0.22$ $(T<T_{*})$, which
seems to suggest a critical phase. On the other hand, for $T=0.27$
$(T>T_{*})$, we find a quick drop of the effective central charge to zero,
indicating a “massive” phase with a finite correlation length. This implies a
phase transition or a strong crossover around $T\sim T_{*}=0.25$, in agreement
with previous studies on the same model [51, 52].

On the other hand, we find that the convergence of the effective central
charge is again not good. The best estimates of the central charge is
$c=1.915(3)$ obtained at the fifth through eleventh RG steps for $T=0.22$ for
the $\mathrm{RP^{2}}$ gauge model. It is important to note that these values
are smaller than $c=2$, which is expected for the UV fixed point, beyond
numerical errors. Moreover, the energy levels that would give the scaling
dimensions of the operators in the CFT do not converge. Below the “transition
temperature” $T<T_{*}$, the change of the energy levels is certainly slower,
but it is still significant as shown in Fig. 4. In addition, the effective
central charge and scaling dimensions of the LL model shown in Figs. 5 and 7
turn out to behave in a similar manner. Thus, we conclude that the both the
$\rm RP^{2}$ gauge and LL models exhibit crossover rather than a true phase
transition.

##  VI Discussions

Figure 4: An enlargement of Fig. 3 (b) for the low-temperature regime
$T\lesssim T_{*}$. The scaling dimension from the lowest excitation does not
converge even here.

As noted in Sec. V, even though the effective central charge exhibits a
“plateau” at $c\sim 1.9$ for both the Heisenberg and $\mathrm{RP^{2}}$ models
at sufficiently low temperatures, the convergence is not as good as in the
similar TNR calculation for known critical points/phases. If we still take the
position that either of these models has a critical phase/point in those
temperature regimes, it would be described by a CFT with the central charge
$c\sim 1.9<2$. However, it is rather difficult to conceive such a CFT.

Considering that the model has a global SO(3) symmetry, it is tempting to
consider the SO(3) Wess-Zumino-Witten (WZW) models. SO(3) is locally
isomorphic to SU(2), and the SU(2) WZW models are classified by the level $k$,
which is a natural number. The level-$k$ SU(2) WZW model has the central
charge

$c=\frac{3k}{k+2},$ (15)

and the primary fields (with zero conformal spin) which have the scaling
dimension

$x_{j}=2\frac{j(j+1)}{k+2},$ (16)

where $j$ is the SU(2) spin quantum number (half-odd-integer or integer),
which satisfies $0\leq j\leq k/2$. An SO(3) WZW model corresponds to an SU(2)
WZW model at an even level $k$. The first nontrivial one is $k=2$ with
$c=3/2$, which is too small for the observed value $c\sim 1.9$ in the
Heisenberg and the $\mathrm{RP^{2}}$ models. The next one is $k=4$ with $c=2$,
which is too large. One might be tempted to interpret the numerical estimate
$c\sim 1.9$ (and poor convergence) as a result of some numerical error, and
postulate that the system is actually described by the level-4 SU(2) model
with $c=2$. However, this does not seem to hold.

The “energy levels” of the lowest excited states [an SU(2) triplet] found for
the Heisenberg model do not converge up to the eighth RG step, as shown in
Fig. 3(a). This also indicates that the system does not reach a RG fixed point
yet. Moreover, the observed energy level of the lowest excited state
corresponds to the scaling dimension $x<0.1$. This is significantly smaller
than the scaling dimension $x=2/3$ of the corresponding triplet operator
($j=1$) in the level-4 SU(2) WZW model, so that the identification is
difficult. Since our system does not contain a spinor representation of SU(2),
we do not expect an SU(2) WZW model with an odd level $k$ to appears as an
effective model for the classical Heisenberg model. Even if we disregard this
issue and consider the level-3 SU(2) WZW model as a candidate CFT, it also
does not match the numerical observation both in terms of the central charge
($c=9/5=1.8$) and the spectrum (the lowest triplet operator $j=1$ having the
scaling dimension $x=4/5$). With these observations, we conclude instead that
our numerical data support the standard “asymptotic freedom” scenario on the
Heisenberg model in which there is a single disordered phase for $T>0$. The
observed effective central charge $c\sim 1.9$ may reflect the RG flow from the
$c=2$ UV fixed point. This is consistent with the known exact RG flow [58, 59]
of the O(3) nonlinear sigma model from $c=2$ to $c=0$ (“massive” theory with a
finite correlation length), although its applicability to the Heisenberg model
on the lattice could be questioned in principle.

We find a similar difficulty in interpreting the low-temperature regime
$T\lesssim T_{*}$ of the $\mathrm{RP^{2}}$ model as a critical point/phase
described by a CFT. The lowest excited states in the $\mathrm{RP^{2}}$ form an
SU(2) quintet, and their energy level corresponds to the scaling dimension
$x\lesssim 0.2$. This is again significantly smaller than the scaling
dimension $x=2$ of the corresponding quintet operator ($j=2$) in the level-4
SU(2) WZW model. In fact, the convergence of the energy levels is much faster
below $T\sim T_{*}$ in the $\mathrm{RP^{2}}$ model, compared to the high-
temperature regime $T\gtrsim T_{*}$, as shown in Fig. 5(b). This indicates a
remarkable change in the system around $T\sim T_{*}$, which is consistent with
the $\mathbb{Z}_{2}$ vortex dissociation found in other studies. However, even
in the low-temperature regime $T\lesssim T_{*}$, the energy levels do not show
a true convergence as shown in Fig. 7, in contrast to the known critical
point/phase. Given the poor convergence of the effective central charge and
the scaling dimensions, and also the lack of an appropriate CFT, we conclude
that it is more likely that there is no true phase transition at $T\sim T_{*}$
and that the region $T\lesssim T_{*}$ is “massive,” albeit with a large
correlation length. That is, similarly to what is widely believed for the
Heisenberg model, the $\mathrm{RP^{2}}$ model at finite temperature consists
of the single disordered phase, which could be understood as an RG flow from
the $c=2$ UV fixed point. While we did observe a remarkable change in the
behavior around $T\sim T_{*}$ where the dissociation of $\mathbb{Z}_{2}$
vortices has been reported, it would be understood as a strong crossover
rather than a true phase transition in this scenario.

Recently, the issue of the $\mathrm{RP^{2}}$ (and more generally
$\mathrm{RP^{N-1}}$) model in two dimensions was studied in the framework of
the scale-invariant scattering theory [60] The conclusion of those authors
that there is no quasi-long-range ordered phase with varying exponents (RG
fixed line) seems consistent with our findings. On the other hand, their
statement that the zero temperature fixed point for the $\mathrm{RP^{N-1}}$
model has an enlarged O[$M_{N}=N(N+1)/2-1$] symmetry may not be consistent
with our finding of $c=2$ on the $T\to 0$ UV fixed point for the
$\mathrm{RP^{2}}$ model. We note that $M_{N}=5$ for the $\mathrm{RP^{2}}$
model ($N=3$), and we indeed found an emergent O(5) symmetry for the ground
states in the _antiferromagnetic_ LL model (see Appendix B), although we are
not sure if our finding is related to the statement in Ref. [60]. More
analysis is needed to clarify the relation between Ref. [60] and our results.
In particular, it is desirable to identify the CFT corresponding to the
“additional solution” for $N=3$ [Eq. (32) of Ref. [60]].

##  VII Conclusions

In this paper, we studied the low-temperature regimes of the Heisenberg and
$\mathrm{RP^{2}}$ models using TNR, and we analyzed the spectrum of the
finite-size transfer matrix. When the system is critical and can be described
by a CFT, the spectrum gives an estimate of the central charge and scaling
dimensions of operators. In the low-temperature limit $T\to 0$, we identify
the UV fixed point of the Heisenberg and the ferromagnetic $\mathrm{RP^{2}}$
models as a $c=2$ CFT, consistently with the standard expectation. On the
other hand, the antiferromagnetic LL model indicates a larger value of the
central charge of the UV fixed point. This is also consistent with the
effective SO(5) symmetry of the ground states, which predicts $c=4$ for the UV
fixed point.

At moderately low temperatures of the Heisenberg model, we find the effective
central charge $c\sim 1.9$ which looks somewhat stable over a range of the
lengthscales up to $L=128$. However, the finite-size energy levels, which give
the scaling dimensions of operators in the underlying CFT, do not show
convergence. This and the lack of candidate CFTs supports the standard picture
that the Heisenberg model has a single disordered phase with a finite
correlation length for $T>0$.

The $\mathrm{RP^{2}}$ model in $T\lesssim T_{*}$ shows a similar behavior with
the effective central charge $c\sim 1.9$. However, the convergence is also not
as good as in the spectrum of known critical points/phases, although it is
faster than that in the $T\gtrsim T_{*}$ regime. We conclude the it is more
likely that the $\mathrm{RP^{2}}$ model also has a single disordered phase
with a finite correlation length for $T>0$, based on the slow convergence and
the apparent lack of the candidate CFTs.

The phase diagrams of the Heisenberg and $\mathrm{RP^{2}}$ models have been
controversial, chiefly due to the very large correlation length at low
temperatures. It is very difficult to distinguish directly a diverging
correlation length from a large but finite one, in any numerical calculation.
On the other hand, analysis of the finite-size spectrum comparing with CFT has
proved very powerful in quantum many-body systems in one spatial dimension;
the phase diagram can be determined precisely with an exact numerical
diagonalization of the Hamiltonian of rather small systems. This has been
successfully extended to the two-dimensional classical XY model using the TNR
[49]. The present study demonstrates that, the finite-size spectrum from the
TNR provides useful information also on the controversial problems of
Heisenberg and $\mathrm{RP^{2}}$ models despite the difficulties associated
with the very large correlation lengths.

We do not completely rule out the possibility of the critical point at $T\sim
T_{*}$ or the critical phase in $T\lesssim T_{*}$ suggested in several papers.
However, to pursue this viewpoint, one would need to explain the effective
central charge and the spectra obtained in the present TNR study, which is
perhaps more difficult than simply discussing whether the correlation length
diverges or not. On the other hand, we do not have a complete explanation of
what we believe is a strong crossover around $T\sim T_{*}$. It is an
interesting problem for the future to describe the crossover in terms of field
theory, for example with a CFT representing an RG fixed point in the vicinity
and a relevant perturbation.

Note added. Recently, we learned that Burgelman, Vanderstraeten, and
Verstraete have also studied the Heisenberg and LL model using the variational
uniform MPS(VUMPS) method [61]. In that paper, they computed the high-accuracy
correlation length from the transfer matrix and the effective central charge
from the entanglement scaling. Detecting the deviation of the correlation
length from the BKT scaling, they also concluded that this model does not go
through a true phase transition. This is consistent with our diverging
$x_{1}(L)$, which is a sign of a finite correlation length. Moreover, the
effective central charge that they estimated is $c\sim 1.8$, which is
consistent with our results of $c\sim 1.9$. The slight difference originates
from the difference in the method for estimating the central charge, where we
used the scaling of the ground-state energy. Nevertheless, this small
discrepancy can be interpreted as additional evidence of a crossover because
both methods should yield the same central charge at true criticality.
Concerning the nature of the fixed point that causes the crossover, they
mention the similarity of the entanglement spectrum with the quantum bilinear-
biquadratic spin-1 Heisenberg chain. As future work, it would be interesting
to compare it with the scaling dimensions found in our study.

## Acknowledgements

We thank Lander Burgelman, Laurens Vanderstraeten, and Frank Verstraete for
sharing their draft prior to publication, and for useful discussion on related
subjects. A. U. is supported by the MERIT-WINGS Program at the University of
Tokyo, and by a JSPS Fellowship. A part of the computation in this work has
been done using the facilities of the Supercomputer Center, the Institute for
Solid State Physics, the University of Tokyo. This work was supported in part
by MEXT/JSPS KAKENHI Grants No. JP17H06462, No. JP19H01808, and No.
JP21J20523, and JST CREST Grant No. JPMJCR19T2

Figure 5: (a) The effective central charge of the Lebwohl-Lasher model at
$T=0.3(T<T_{*})$, $0.35(T\sim T_{*})$, and $0.375(T>T_{*})$. The numerically
obtained central charge of the Heisenberg model is slightly smaller than 2,
$c\sim 1.942(1)$ at $T=0.35$ between the seventh and ninth RG steps, implying
that the correlation is large, but finite. (b) The finite-size scaling
dimension of the lowest excitation in the LL model. The monotonic increase of
$x_{1}(L)$ along the RG steps is observed in the whole range of temperature.
In particular, the RG flow becomes stronger around $T=0.35$. This phenomenon
might be attributed to a repulsive fixed point in the vicinity of the
trajectory of this model. The vertical dotted line denotes the transition
temperature from previous studies $T=0.356$.  Figure 6: A schematic picture of
the procedure to extract the CFT information. It is possible to see the
spectrum by making the $L\times L$ transfer matrix a cylinder. Figure 7: An
enlargement of Fig. 5 (b) for the low-temperature regime $T\lesssim T_{*}$.
The scaling dimension from the lowest excitation does not converge even here.
Figure 8: The ultraviolet central charge of (a) the Heisenberg and (b) LL
model at $T=0.02$. The enormous finite bond dimension effect at $D=16$
decreases and the values converges at $D=44$.

##  Appendix A Effective central charge

Ground state energy and excitation energies are written in the language of CFT
as

$\displaystyle E_{n}-E_{0}$ $\displaystyle=\frac{2\pi}{L}x_{n},$ (17)
$\displaystyle E_{0}$ $\displaystyle=\epsilon_{0}L-\frac{\pi c}{6L},$ (18)

where $x_{n}$ and $c$ are the scaling dimension and central charge,
respectively. With real-space renormalization, the effective size of the
renormalized tensor becomes $b$ times larger at each step. Thus, starting from
$L=1$, the effective scale ends up being $L=b^{n}$ after $n$ RG steps, and we
denote the partition function at this scale as $Z(n)$. This partition function
can be expressed with the generator along the $y$-axis using Eqs. (17) and
(18).

$\displaystyle Z(n)$ $\displaystyle={\rm Tr}\exp(-b^{n}H_{p})$
$\displaystyle={\rm Tr}_{x_{i}}\exp\left[-b^{n}(\frac{2\pi}{L}x_{i}-\frac{\pi
c}{6L}+\epsilon_{0}L)\right]$ $\displaystyle={\rm
Tr}_{x_{i}}\exp(-2\pi(x_{i}-\frac{c}{12})-\epsilon_{0}b^{2n})$ (19)

This spectrum can be computed from the eigenvalues of the $L\times L$ transfer
matrix constructed from the renormalized tensor $T^{(n)}$ as depicted in Fig.
6. Assuming $x_{0}=0$, which corresponds to the identity operator, one is
ready to recognize the relation between $\lambda_{n}$ and $x_{n}$.

$\displaystyle\frac{\lambda_{n}}{\lambda_{0}}=\exp(-2\pi x_{n})$ (20)

There is a pathological problem determining the central charge because we lack
one equation. In order to eliminate the contribution from the bulk energy, or
to normalize in other words, one needs to employ the partition function from
previous step $Z(n-1)$.

$\displaystyle Z(n-1)={\rm
Tr}_{x_{i}}\exp(-2\pi(x_{i}-\frac{c}{12})-\epsilon_{0}b^{2n-2})$ (21)

For the fixed point tensor, it is reasonable to assume $c$ and $\epsilon_{0}$
to be constant. Then we get the central charge as below.

$\displaystyle
c=\frac{6}{\pi}\frac{1}{b^{2}-1}(b^{2}\ln\lambda_{0}^{(n-1)}-\ln\lambda_{0}^{(n)})$
(22)

The formula widely used [62]is

$\displaystyle c=\frac{6}{\pi}\left[\frac{b^{2}}{b^{2}-1}\left(\ln
Z(n-1)-\frac{\ln Z(n)}{b^{2}}\right)+\ln\frac{\lambda_{0}^{(n)}}{Z(n)}\right]$
(23)

Eqs. (22) and (23) are equivalent in the critical case because
$\frac{\lambda_{0}^{(n)}}{Z(n)}=\frac{\lambda_{0}^{(n-1)}}{Z(n-1)}$, though
Eq. (23) seems to be unstable when the system size exceeds the correlation
length. Throughout this paper, we calculated the effective central charge with
Eq. (23).

##  Appendix B Emergent $O(5)$ universality in the AF-$\mathrm{RP^{2}}$ model

The emergence of the $\mathrm{O(5)}$ universality class seems peculiar.
However, this phenomena is observed in three dimensions where the SSB of
continuous symmetry is not prohibited [50]. The key idea is that the order
parameter is no longer the $\mathrm{O(3)}$ spin, but the nematic order
parameter $Q$ as

$\displaystyle
Q^{\alpha\beta}=s^{\alpha}s^{\beta}-\frac{1}{3}\delta^{\alpha\beta}.$ (24)

As $Q^{\alpha\beta}$ is a symmetric trace-less rank-3 tensor, the number of
independent components is five. This five components behave like a five
component vector in Landau-Ginzburg-Wilson (LGW) theory, leading to the
emergence of the $\mathrm{O(5)}$ universality [63, 64].

The ground state itself is different in the first place between $J>0$ and
$J<0$. For ferromagnetic cases, the GS is fully aligned state $s_{i}\cdot
s_{j}=\pm 1$. Except for the massive degeneracy due to the local
$\mathbb{Z}_{2}$ gauge, it resembles that of the Heisenberg model. On the
other hand, the ground state of the AF-$\mathrm{RP^{2}}$ model is non-trivial
$(s_{i}\cdot s_{j})=0$. Due this break in the symmetry between even and odd
sites (sub-lattice), the order parameter might become a staggered one,

$\displaystyle A_{i}=(-1)^{x_{i}+y_{i}}Q_{i}.$ (25)

Given the even-odd symmetry of $A_{i}\rightarrow-A_{i}$, LGW action for the
AF-$\mathrm{RP^{2}}$ model become

$\displaystyle F(A)=\int dr[(\nabla\mathrm{Tr}A)^{2}+a(\mathrm{Tr}A^{2})+$
$\displaystyle b(\mathrm{Tr}A^{2})^{2}+c(\mathrm{Tr}A^{4})].$ (26)

Let A be

$\displaystyle A=\begin{pmatrix}a&c&d\\\ c&b&e\\\ d&e&-a-b\\\ \end{pmatrix}.$
(27)

Then $\mathrm{Tr}A^{2}=2(\Phi\cdot\Phi)$ and
$\mathrm{Tr}A^{2}=2(\Phi\cdot\Phi)^{2}$, where
$\Phi=(a+b/2,\frac{\sqrt{3}}{2}b,c,d,e)$. This is simply the $\mathrm{O(5)}$
model. Unlike the three dimensional model, $\phi^{6}$ is also relevant in 2D.
However, the most stable fixed point(smallest central charge) of $\Phi^{2n}$
theory is that of $\Phi^{4}$, so our argument might be correct. Note that this
LGW approach is justifiable only at $T=0$, where SSB is allowed under
continuous symmetry. The verification of $c=4$ with numerical simulation is
almost impossible due to the enormous bond dimension effect. Nonetheless, the
central charge is larger than 3, implying the shadow of $c=4$. In fact, our
conjecture is consistent with the recent study of the $\mathrm{RP^{2}}$ model
based on the conformal scattering theory, and this $\mathrm{O(5)}$
universality class corresponds to the solution $\mathrm{A_{+}}$ of Ref. [60],
while the remaining solution $\mathrm{A_{-}}$ corresponds to the
$c=\frac{5}{2}$ $\mathrm{SO(5)_{1}}$ WZW [65, 66]. Interestingly, the central
charge $c=2$ of the ferromagnetic $\mathrm{RP^{2}}$ model at zero temperature
is none of them above, leaving the possibility for it to become a candidate
for the solution $\mathrm{B_{\pm 3}}$.

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