# Semantic Role Aware Correlation Transformer
for Text to Video Retrieval

###### Abstract

With the emergence of social media, voluminous video clips are uploaded every
day, and retrieving the most relevant visual content with a language query
becomes critical. Most approaches aim to learn a joint embedding space for
plain textual and visual contents without adequately exploiting their intra-
modality structures and inter-modality correlations. This paper proposes a
novel transformer that explicitly disentangles the text and video into
semantic roles of objects, spatial contexts and temporal contexts with an
attention scheme to learn the intra- and inter-role correlations among the
three roles to discover discriminative features for matching at different
levels. The preliminary results on popular YouCook2 indicate that our approach
surpasses a current state-of-the-art method, with a high margin in all
metrics. It also overpasses two SOTA methods in terms of two metrics.

Index Terms—  Video understanding, text-to-video retrieval, transformer,
multi-modal, hierarchical, cross-modal

##  1 Introduction

With the popularity of TikTok, Youtube and Instagram, millions of videos are
uploaded every minute, making video retrieval an important function for users
to find relevant content. Conventional models are based on keywords query [1,
2]. However, unstructured keywords are limited and insufficient to retrieve
fine-grained and compositional events. Therefore, communities are shifting
attention to cross-modal video text retrieval [3, 4, 5, 6, 7] which can
retrieve videos using natural language descriptions that includes more
structured details.

Most existing cross-modal text video retrieval maps each modality into a joint
embedding space to measure their similarities. Some works [8, 4, 9] directly
embeds whole videos and texts into flat vectors for matching, however losing
fine-grained details in texts and videos. To avoid losing those details, some
other works [10, 11] align a sequence of frames and words in texts to compute
overall similarities. Although these approaches have achieved certain
progress, aligning video and text is still an open problem, given a huge
semantic gap between video and text.

Recently, there are different attempts to resolve the video-text semantic gap.
[3] propose decomposing text into three semantic roles (events, actions and
entities) and then embedding 2D video features into these three spaces
accordingly for matching. Another line of research uses a BERT-like
transformer [12, 13, 14] to learn the text-video correspondence, based on
recent mixture-of-expert embedding [5] which requires large-scale dataset for
pre-training.

Fig. 1: Overview of our model on text-to-video retrieval, where the textual
input is a caption of a video clip, and the visual input are the visual expert
features from RoI, 2D Frames and 3D clips. It computes the similarity between
the caption and a candidate video clip in the three embedding spaces
correspond to object contexts, spatial contexts and temporal contexts by
embracing transformer with self-attention and cross-modal attention that can
capture the specific and complement information within three semantic role
modalities. In the cross-modal matching unit, a matching score is calculated
for three-level of embeddings. Then, we average the similarities and utilize
contrastive ranking loss as a training objective.

We propose a novel transformer architecture for video-text matching inspired
by [3] and [12, 13, 14]. Different from [3], which only considers multi-head
embedding of the spatial frame and ignores the interaction between different
visual contexts, our method explicitly considers more fine-grained visual
encoding of object contexts, spatial contexts and temporal contexts by
embedding RoI regions, 2D frames and video sequences into the corresponding
space with their interactions. Different from [12, 13, 14], which only uses
self-attention to discover modality-specific information, our method uses a
self-attention scheme to discover modality-specific discriminative features.
Moreover, our model utilizes cross-modal attention to consider the
interactions between object, spatial and temporal contexts to discover
modality-complement features for better align video and text.

We experimented with the YouCook2 dataset for the text-to-video retrieval
task. The results show that our approach surpasses a recent SOTA with a high
margin. While our approach gives better results over other approaches in terms
of two parameters(R@1, R@5), it falls a close behind them in terms of the
other two parameters(R@10, MedR).

##  2 Related Work

The text-video retrieval task is challenging because there are significant
semantic and structure gap between videos and text. Mithun et al. [4] employ
multimodal image, motion, audio modalities in video. Liu et al. [8] further
utilize all modalities that can be extracted from videos such as speech
contents and scene texts for video encoding. Dong et al. [9] use biGRU and CNN
to encode sequential videos and texts. Yu et al. [11] propose to fuse the
sequential interaction of videos and texts for video text retrieval. Song et
al. [10] employ to align encoded video and text elements for matching. Tan et
al. [15] use LSTM and Graph Neural Networks to capture cross-modal relation.
Chen et al. [3] disentangle phrases into different part-of-speech such as
verbs and nouns for fine-grained retrieval, and also it uses GNNs to encode
the interactions between the roles in text modalities. Our work complements
Chen et al. [3] by focusing on video encoding parts using a transformer to
learn interactions between object contexts, spatial contexts and video
contexts.

Some methods [16, 17] extend BERT-like models with self-supervised learning to
have better video-text representation. For example, while Sun et al. [16]
build upon the BERT model to learn bidirectional joint distributions over
sequences of visual and linguistic tokens. Zhu et al. [17] aim to encode
global actions, local regions, objects, and linguistic descriptions with self-
supervised learning without considering the linguistic and visual structures.
Our work aims to exploit transformer like architecture to learn interactions
for better object contexts, spatial context and video context encoding.

Most recent papers [5, 13] use complex datasets such as [6] for pre-training
to have a more general textual and visual embeddings. Alayrac et al. [18] even
use an audio-based dataset [19] to have audio embedding. In this current work,
we have yet to exploit pre-training on a huge dataset motivated by studying
the computational time and the effects of pre-training absence, although our
work can incorporate pre-trained backbones.

##  3 Method

Figure 1 is an illustration of our semantic role aware correlation transformer
model, which includes three main blocks: textual encoding, visual encoding,
and cross-modal matching.

###  3.1 Textual Encoding

Text description internally has a hierarchical structure. For example, the
whole sentence can describe global contexts; nouns and verbs can define object
and actions. We use an off-the-shelf toolkit [20] for semantic role labelling
to parse these components and the semantic relationship between them in a
graph structure. Verb nodes are directly connected with the global node by
showing temporal relations of various actions. Noun nodes are connected with
verb nodes by defining the objects. $r_{ij}$ edges show the semantic
relationship between verb nodes $i$ and object nodes $j$. For graph
representation, we follow the method proposed by Chen et al. [3]. This
approach, which is shown in Eq. 1, uses factorized weights in GCN, where
$W_{t}$ is a transformation matrix and shared among all relationship types.
$W_{t}$ denotes a unique matrix for different semantic roles. $g_{a}$ and
$g_{b}$ denote node embeddings which can be for sentences, verbs or objects.
$\beta$ is the outcome after attention is applied at the nodes.

$g^{l}_{i}=g^{l-1}_{i}+\sum_{j\varepsilon N_{i}}(\beta_{ij}(W^{l}_{t}\odot
W_{r}r_{ij})g_{j})$ (1)

###  3.2 Visual Encoding

Parsing videos into hierarchical semantic features is more challenging than
texts since it includes detection, action segmentation and so on. We extract
spatial context features $F_{S}$ with 2D CNNs by ResNet-152 [21], pre-trained
on Imagenet [22]. The temporal context features $F_{T}$ are extracted with 3D
CNNs by ResNeXt-101 [23] pre-trained on Kinetics [24], then we perform
temporal max-pooling on feature maps. We extract object context features
$F_{O}$ by using Faster R-CNN [25], pre-trained on MS COCO [26], and the
backbone is ResNet-101 [21]. We follow the settings of Zhu et al. [17] by
extracting the features at 1 FPS after RoI-pooling happens. Confidence
threshold is set as 0.4, while each frame could contain up to ten boxes.
Dimension for all three expert embedding is 2048.

To capture the modality-complement and specific information, we adapt a
transformer with attention modules [12, 13, 14, 27]. For example, to generate
spatial context-specific feature $s_{e}$, the temporal context features
$F_{T}$, and object context features $F_{O}$ are concatenated in feature
dimension and fed into an encoder with a multi-head self-attention layer and a
feed-forward linear layer FF, as shown in Eq. LABEL:encoder.

$\displaystyle f_{e}=\textrm{Concat}(F_{T},F_{O})$ (2) $\displaystyle
z_{e}=\textrm{Norm}\big{(}\textrm{MultiHead}(f_{e},f_{e},f_{e})+f_{e}\big{)}$
$\displaystyle s_{e}=\textrm{Norm}\big{(}\textrm{FF}(z_{e})+z_{e}\big{)}$

The attention layers are multi-headed as Vaswani et al. [12]’s dot-product
attention, and each layer also follows layer normalization and residual
connection, as shown in Eq. 3. All the W matrices are trainable parameters.

$\displaystyle\textrm{MultiHead}(Q,K,V)=\textrm{Concat}(\textrm{Head}_{1},...,\textrm{Head}_{h})W^{O}$
(3)
$\displaystyle\textrm{Head}_{i}=\textrm{Attention}(QW_{i}^{Q},KW_{i}^{K},VW_{i}^{V})$
$\displaystyle\textrm{Attention}(Q,K,V)=\sigma\Big{(}\frac{QK^{T}}{\sqrt{d}}V\Big{)}$

Eq. LABEL:decoder shows the formula of generating spatial context-complement
feature. The spatial context feature $F_{S}$ is first fed into the self-
attention layer to encode into $z_{s}$, then with cross-modal attention is
conditioned on $s_{e}$ and $z_{s}$ to generate complement feature $c_{e}$.
Next, it is encoded through feed-forward linear layer FF and together with the
modality-specific feature $s_{e}$ to deliver final embeddings for the spatial
context $E_{S}$. ’Norm’ refers to layer normalization.

$\displaystyle
z_{s}=\textrm{Norm}\big{(}\textrm{MultiHead}(F_{S},F_{S},F_{S})+F_{S}\big{)}$
(4) $\displaystyle
c_{e}=\textrm{Norm}\big{(}\textrm{MultiHead}(s_{e},z_{s},z_{s})+z_{s}\big{)}$
$\displaystyle E_{S}=\textrm{Norm}\big{(}\textrm{FF}(c_{e})+c_{e}\big{)}$

The same process is applied to generate the final embeddings $E_{T}$ and
$E_{O}$ for temporal context feature $F_{T}$ and object context feature
$F_{O}$, respectively, for matching.

###  3.3 Cross-modal Matching

We utilize cosine similarity to calculate the cross-modal matching score for
each level by corresponding visual and textual embeddings.

$s(V,C)=\frac{<v,c>}{||v||_{2}||c||_{2}}$ (5)

Then, we average the similarities and utilize contrastive ranking loss as a
training objective. Our aim is to have such positive pair $(v_{p},c_{p})$
pushed away from the negative pairs $(v_{p},c_{n})$ and $(v_{n},c_{p})$ than a
set margin. $v$ and $c$ refer to the video and textual representation,
respectively. This matching applies to three levels of embeddings. $\Delta$
represents the pre-defined margin, which is for training with contrastive
loss.

$\displaystyle L(v_{p},c_{p})=[\Delta+s(v_{p},c_{n})-s(v_{p},c_{p})]+$ (6)
$\displaystyle[\Delta+s(v_{n},c_{p})-s(v_{p},c_{p})]$

Table 1: Text-to-video retrieval comparison with SOTA approaches on YouCook2 validation set. ’Visual Backbone’ only refers to 3D CNNs Features. Our method surpasses the SOTA methods in the first two parameters when without pre-training. Method | Pre-training | Visual Backbone | Batch Size | R@1↑ | R@5↑ | R@10↑ | MedR↓
---|---|---|---|---|---|---|---
Random | No | - | - | 0.03 | 0.15 | 0.3 | 1675
Miech et al [6] | No | ResNeXt-101 | - | 4.2 | 13.7 | 21.5 | 65
HGLMM [28] | No | - | - | 4.6 | 14.3 | 21.6 | 75
HGR [3] | No | ResNeXt-101 | 32 | 4.7 | 14.1 | 20.0 | 87
Ours | No | ResNeXt-101 | 32 | 5.3 | 14.5 | 20.8 | 77
Miech et al+FT [6] | HowTo100M | ResNeXt-101 | - | 8.2 | 24.5 | 35.3 | 24
ActBert [17] | HowTo100M | ResNet-3D | - | 9.6 | 26.7 | 38.0 | 19
MMV FAC [18] | HowTo100M+AudioSet | TSM-50 | 4096 | 11.5 | 30.2 | 41.5 | 16
MIL-NCE [7] | HowTo100M | S3D | 8192 | 15.1 | 38.0 | 51.2 | 10

##  4 Experiments

We compare our model with state-of-the-art methods on text-to-video retrieval
task, shown in Table 1. We also share an ablation study using various expert
embeddings in different hierarchical levels, shown in Table 2.

###  4.1 Dataset and Metrics

Dataset. We evaluate our model on YouCook2 [29]. It is a video dataset on
cooking gathered from YouTube. The videos have a diverse number of cooking
styles and methods. It includes 89 recipe types and 14k video clips correlated
by imperative English captions defining the action. Note that since
annotations for the test set is not published yet, we evaluate the task on
validation clips which is around 3.5k totally.

Evaluation metrics. The task is retrieving video clips based on text queries.
We evaluate our model using common metrics on recall at various sets and
median rank. R@1, R@5, R@10 gives the number of accurately retrieved clips in
the ranking list’s top associated rates. MedR gives the median rank of correct
clips in the ranking list. While higher is better for recall metrics, lower is
better for the median rank.

Table 2: Ablation studies on YouCook2 dataset to investigate the contributions of various feature experts at different levels. The same ablation is also done on HGR method [3] since it is a strong baseline. On 2D + 3D visual features setting, when the feature dimension is 4096, concatenation is done on dimension one; otherwise is done on dimension zero. Our model surpasses HGR with the same hierarchical features with a high margin by using cross-modal attention. Method | Visual Features |  Feature Dimension | R@1↑ | R@5↑ | R@10↑ | MedR↓
---|---|---|---|---|---|---
Appearance | Action | Object
HGR [3] : Ours | 2D | 2D | 2D | 2048 | 4.7 : 4.2 | 13.8 : 13.7 | 19.7 : 19.4 | 86 : 86
HGR [3] : Ours | 2D + 3D | 2D + 3D | 2D + 3D | 2048 | 4.8 : 4.5 | 14.0 : 13.2 | 20.3 : 20.0 | 85 : 85
HGR [3] : Ours | 2D + 3D | 2D + 3D | 2D + 3D | 4096 | 4.8 : 4.5 | 14.0 : 13.2 | 20.3 : 20.0 | 85 : 85
HGR [3] : Ours | 2D | 3D | RoI | 2048 | 4.7 : 5.3 | 14.1 : 14.5 | 20.0 : 20.8 | 87 : 77

###  4.2 Implementation Details

Training. We use Glove embeddings [30] by setting the word embedding size as
300 for the text encoding. We adopt the text encoding approach of [3] to
disentangle the text embeddings. Graph convolutions have two layers, and it
outputs features with 1024 dimensions. A linear layer is applied to each
visual expert embeddings to transform their dimensions into 1024. We use a
cross-modal attention mechanism for each level in visual encoding. For
training, the margin is set to $\Delta=0.2$; the epoch is 100 for each
experiment. The mini-batch size is 32.

###  4.3 Comparison

Table 1 shows the comparison with SOTA methods. While ’Visual Backbone’ in the
table only refers to 3D CNNs Features, ResNet-based models are also used in
all the models as 2D CNNs backbone. We focus on training only with YouCook2
dataset; however, we also add the methods when using pre-training for
reference. ’FT’ denotes fine-tuning on YouCook2 dataset. We see that the
accelerator type, which directly defines the maximum batch size, and pre-
training usage affects the result sharply. There are even differences with the
same model when training on different accelerators, epoch, and batch sizes;
which can be found on corresponding papers. For example, the MIL-NCE method
reaches 50% of its accuracy when trained with less batch size and epochs. Our
method surpasses the HGR method [3] with a high margin for all metrics. We
also outperform the other two SOTA methods: Miech et al. [6], and HGLMM [28],
in terms of the first two parameters. We think that modality-specific and
modality-complement features improve accuracy at R@1 and R@5, which are more
demanding and useful for real-world applications.

###  4.4 Ablation Studies

Since we adopt our model from HGR [3], we implement extensive ablation to show
the improvement in our approach on the visual encoding part. We aim to find
how the feature expert combination affects the result. Our model falls short
when we feed three levels with only 2D features. The same decrease continues
with the concatenated 2D and 3D features at dimension zero as well as the
concatenated 2D and 3D features at dimension one. However, when we feed them
with 2D, 3D, and RoI features respectively, while the HGR model shows a slight
decrease, our model reaches a better result with a high margin. This confirms
our insight that inter-modal correlation can be exploited with our proposed
cross-modal attention mechanism to achieve better results.

##  5 Conclusion

Retrieving related video on a textual query gets harder since the number of
videos on the internet increases. Most works use one joint embedding space for
text-to-video retrieval task without fully exploiting cross-modal features. We
propose a hierarchical model representing complex textual and visual features
with three joint embedding spaces by utilizing self-attention and cross-modal
attention to exploit the modality-specific and modality-complement visual
embeddings. Our model surpasses a strong baseline with a high margin, and it
also overpasses other SOTA methods in R@1, R@5 metrics.

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# Thermopower across the phase diagram of the cuprate La1.6-xNd0.4SrxCuO4 :
signatures of the pseudogap and charge-density-wave phases

C. Collignon⋆⋆ Present address: Department of Physics, Massachusetts Institute
of Technology, Cambridge, Massachusetts 02139, USA Institut Quantique,
Département de physique & RQMP, Université de Sherbrooke, Sherbrooke, Québec
J1K 2R1, Canada  A. Ataei⋆⋆ Institut Quantique, Département de physique &
RQMP, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada  A.
Gourgout  Institut Quantique, Département de physique & RQMP, Université de
Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada  S. Badoux  Institut Quantique,
Département de physique & RQMP, Université de Sherbrooke, Sherbrooke, Québec
J1K 2R1, Canada  M. Lizaire  Institut Quantique, Département de physique &
RQMP, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada  A. Legros
Institut Quantique, Département de physique & RQMP, Université de Sherbrooke,
Sherbrooke, Québec J1K 2R1, Canada  SPEC, CEA, CNRS-UMR 3680, Université
Paris-Saclay, Gif sur Yvette Cedex, France  S. Licciardello  High Field Magnet
Laboratory (HFML-EMFL) & Institute for Molecules and Materials, Radboud
University, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands  S. Wiedmann
High Field Magnet Laboratory (HFML-EMFL) & Institute for Molecules and
Materials, Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The
Netherlands  J.-Q. Yan  Materials Science and Engineering Program, Mechanical
Engineering, University of Texas at Austin, Austin, Texas 78712, USA  J.-S.
Zhou  Materials Science and Engineering Program, Mechanical Engineering,
University of Texas at Austin, Austin, Texas 78712, USA  Q. Ma  Department of
Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada
B. D. Gaulin  Department of Physics and Astronomy, McMaster University,
Hamilton, Ontario L8S 4M1, Canada  Canadian Institute for Advanced Research,
Toronto, Ontario M5G 1M1, Canada  Nicolas Doiron-Leyraud  nicolas.doiron-
<EMAIL_ADDRESS>Institut Quantique, Département de physique & RQMP,
Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada  Louis Taillefer
<EMAIL_ADDRESS>Institut Quantique, Département de physique &
RQMP, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada  Canadian
Institute for Advanced Research, Toronto, Ontario M5G 1M1, Canada

###### Abstract

The Seebeck coefficient (thermopower) $S$ of the cuprate superconductor
La1.6-xNd0.4SrxCuO4 was measured across its doping phase diagram (from
$p=0.12$ to $p=0.25$), at various temperatures down to $T\simeq 2$ K, in the
normal state accessed by suppressing superconductivity with a magnetic field
up to $H=37.5$ T. The magnitude of $S/T$ in the $T=0$ limit is found to
suddenly increase, by a factor $\simeq 5$, when the doping is reduced below
$p^{\star}$ $=0.23$, the critical doping for the onset of the pseudogap phase.
This confirms that the pseudogap phase causes a large reduction of the carrier
density $n$, consistent with a drop from $n=1+p$ above $p^{\star}$ to $n=p$
below $p^{\star}$, as previously inferred from measurements of the Hall
coefficient, resistivity and thermal conductivity. When the doping is reduced
below $p=0.19$, a qualitative change is observed whereby $S/T$ decreases as
$T\to 0$, eventually to reach negative values at $T=0$. In prior work on other
cuprates, negative values of $S/T$ at $T\to 0$ were shown to result from a
reconstruction of the Fermi surface caused by charge-density-wave (CDW) order.
We therefore identify $p_{\rm CDW}$ $\simeq 0.19$ as the critical doping
beyond which there is no CDW-induced Fermi surface reconstruction. The fact
that $p_{\rm CDW}$ is well separated from $p^{\star}$ reveals that there is a
doping range below $p^{\star}$ where the transport signatures of the pseudogap
phase are unaffected by CDW correlations, as previously found in YBa2Cu3Oy and
La2-xSrxCuO4.

##  I Introduction

The chief mystery in our understanding of cuprate superconductors is the
pseudogap phase Keimer  _et al._ (2015), a region in the temperature-doping
phase diagram of hole-doped cuprates bounded by the crossover temperature
$T^{\star}$ and the critical doping $p^{\star}$, where the pseudogap phase
ends at $T=0$ Cyr-Choinière  _et al._ (2018). Some of the sharpest
experimental signatures of this phase have recently been found in the $T=0$
limit, as the doping $p$ is tuned across $p^{\star}$, in the absence of
superconductivity, removed by applying a large magnetic field Proust and
Taillefer (2019). A key signature is the drop in the Hall number $n_{\rm H}$,
from $n_{\rm H}$ $\simeq 1+p$ above $p^{\star}$ to $n_{\rm H}$ $\simeq p$
below $p^{\star}$, as revealed by measurements of the Hall effect in YBa2Cu3Oy
(YBCO) Badoux  _et al._ (2016a), La1.6-xNd0.4SrxCuO4 (Nd-LSCO) Collignon  _et
al._ (2017) and Bi2Sr2CuO6 (Bi2201) Lizaire  _et al._ (2020). It is tempting
to equate the Hall number $n_{\rm H}$ with the actual carrier density $n$, as
in the case of simple single-band metals. If this were correct for cuprates,
then a key property of the pseudogap phase is elucidated: it causes the
carrier density to change from $n=1+p$ to $n=p$. The natural explanation for
such a change is the onset of a state that breaks translational symmetry by
introducing a new periodicity – e.g. with wavevector ${\bf Q}=(\pi,\pi)$ –
which causes a transformation of the Fermi surface from a large cylinder (of
volume proportional to $1+p$) to small nodal hole pockets (of volume
proportional to $p$) Storey (2016).

However, before we can assert that these are amongst the defining properties
of the pseudogap phase, further evidence for a drop in carrier density is
needed, because other mechanisms can also cause $n_{\rm H}$ to drop. For
example, a change in the curvature of the Fermi surface will affect $n_{\rm
H}$, as shown in the case of a nematic transition, for which $n_{\rm H}$ can
mimic the behavior found in cuprates across $p^{\star}$ Maharaj  _et al._
(2017). The loss of quasiparticle coherence (or the advent of additional
scattering) has also been invoked to explain the drop in $n_{\rm H}$, as
opposed to a transformation from large Fermi surface to small Putzke  _et al._
(2019).

In Nd-LSCO, the observation of a large increase in the electrical resistivity
Collignon  _et al._ (2017); Daou  _et al._ (2009a) upon entering the pseudogap
phase, along with a correspondingly large decrease in the electronic thermal
conductivity Michon  _et al._ (2018), rules out a simple change in the
curvature of the Fermi surface as the explanation for the change in $n_{\rm
H}$. Indeed, for the conductivity to drop, either the carrier density has to
drop or the scattering rate has to increase (or both).

Figure 1:  Temperature-doping phase diagram of Nd-LSCO showing the pseudogap
temperature $T^{\star}$ extracted from resistivity (red dots, from Collignon
_et al._ (2017) and Fig. S1 in the Supplementary Material) and ARPES
measurements (red square, from Matt  _et al._ (2015)), the CDW ordering
temperature $T_{\rm CDW}$ as seen in x-ray diffraction measurements (green
squares, from Zimmermann  _et al._ (1998); Niemöller  _et al._ (1999)), the
temperature $T_{\rm max}$ of the maximum in $S/T$ vs $T$ (blue dots, see main
text and Fig. 6), and a schematic of the zero-field superconducting transition
temperature $T_{\rm c}$ (black line). Corresponding values for $T_{\rm max}$
Laliberté  _et al._ (2011) and $T_{\rm CDW}$ Fink  _et al._ (2011) in
La1.8-xEu0.2SrxCuO4 (Eu-LSCO) are shown as open symbols. The red and blue full
lines are guides to the eye. The red dashed line marks the end of the
pseudogap phase, at the critical doping $p^{\star}$ $=0.23\pm 0.01$ Collignon
_et al._ (2017). The blue dashed line is a linear extension of the full blue
line, extrapolating to $p=$ $p_{\rm CDW}$ $\simeq 0.19$ at $T=0$.

Here, we use a new experimental probe to investigate the pseudogap critical
point, namely the thermopower, or Seebeck coefficient $S$. Unlike the Hall
coefficient, $S$ does not depend on the curvature or shape of the Fermi
surface. Unlike the conductivity (electrical or thermal), it does not depend,
at least to first order, on the scattering rate. Fundamentally, $S$ is the
specific heat per carrier. In the simple case of a free electron gas, the
Seebeck coefficient is given by Behnia  _et al._ (2004); Miyake and Kohno
(2005):

$\frac{S}{T}=\pm\frac{\pi^{2}}{3}\frac{k_{B}^{2}}{e}\frac{1}{n}N(\epsilon_{F})$
(1)

where $e$ is the electron charge and $k_{B}$ Boltzmann’s constant,
$N(\epsilon_{F})$ is the density of states at the Fermi energy, and $n$ is the
carrier density (the negative sign is for electrons, the positive sign is for
holes). The analogous expression for the electronic specific heat is:

$\frac{C_{el}}{T}=\frac{\pi^{2}k_{B}^{2}}{3}N(\epsilon_{F}).$ (2)

Combining the two expressions yields:

$\frac{S}{T}=\pm\frac{1}{ne}\frac{C_{el}}{T}.$ (3)

While seemingly over-simplistic, this relation was shown to hold in the $T=0$
limit for a large variety of materials, even in the presence of strong
electronic correlations, including heavy-fermion metals, organic conductors
and cuprates Behnia  _et al._ (2004). In the cuprate YBCO, measurements in the
field-induced normal state yield a large Seebeck coefficient, namely
$S/T=-0.8~{}\mu$V/K2 Laliberté  _et al._ (2011), and a small specific heat,
namely $C_{el}/T=\gamma=2.4$ mJ/K2mol-Cu Kaǎmarčík  _et al._ (2018), in the
$T=0$ limit, for $p=0.11$. From Eq. 3, this implies a very small carrier
density, namely $n=0.032$ per (planar) Cu atom. In YBCO at $p=0.11$, the Fermi
surface is well known from quantum oscillations and the volume of its tiny
electron pocket is precisely given by the oscillation frequency, $F=530$ T,
which yields $n=0.038$ Doiron-Leyraud  _et al._ (2007). This excellent
agreement confirms that the Seebeck coefficient can be used as a measure of
the carrier density in cuprates.

Here we focus on Nd-LSCO, a single-layer, tetragonal cuprate superconductor
with a low critical temperature $T_{\rm c}$ and critical field $H_{\rm c2}$,
such that superconductivity can be readily suppressed with static fields down
to $T\to 0$. In Fig. 1, we display the temperature-doping phase diagram of Nd-
LSCO, showing the pseudogap temperature $T^{\star}$, defined as the
temperature at which the resistivity $\rho(T)$ departs from its high-
temperature $T$-linear behaviour Collignon  _et al._ (2017); Daou  _et al._
(2009a) (Fig. 7(a)), as a function of doping. At $p=0.24$, $\rho(T)$ remains
$T$-linear down to $T\to 0$ (Fig. 7(a); Collignon  _et al._ (2017); Daou  _et
al._ (2009a); Michon  _et al._ (2018)), implying that the pseudogap phase ends
at $p^{\star}$ $=0.23\pm 0.01$. This agrees well with angle-resolved
photoemission spectroscopy (ARPES) measurements on Nd-LSCO Matt  _et al._
(2015), which find the opening of an anti-nodal pseudogap below $T^{\star}$
$=75$ K at $p=0.20$ (red square, Fig. 1) and no pseudogap at $p=0.24$ (down to
$T\simeq$ $T_{\rm c}$). Hall effect measurements in Nd-LSCO find that the Hall
number $n_{\rm H}$ $\simeq 1+p$ at $p=0.24>$ $p^{\star}$ and $n_{\rm H}$
$\simeq p$ at $p=0.20<$ $p^{\star}$ Collignon  _et al._ (2017), in good
agreement with YBCO Badoux  _et al._ (2016a) and, as reported recently, with
Bi2201 Lizaire  _et al._ (2020), pointing to a universal signature.

At $p=0.12$, charge-density-wave (CDW) order appears below a temperature
$T_{\rm CDW}$ $=70\pm 5$ K (Fig. 1), as measured by neutron Tranquada  _et
al._ (1995) and x-ray Zimmermann  _et al._ (1998) diffraction. At $p=0.15$,
$T_{\rm CDW}$ $=62\pm 5$ K Niemöller  _et al._ (1999). How the boundary of the
CDW phase in Nd-LSCO evolves beyond $p=0.15$, and in particular at what doping
$p_{\rm CDW}$ ends, is not known.

In the present Letter, we report measurements of the Seebeck coefficient of
Nd-LSCO in the field-induced normal state down to low temperature ($T\simeq 2$
K), accessed by applying magnetic fields up to 37.5 T, for dopings ranging
from $p=0.12$ to $p=0.25$, thereby spanning both $p^{\star}$ and the likely
termination of the CDW phase at $p_{\rm CDW}$.

We find that $S/T$ at $T\to 0$ exhibits a sharp five-fold increase in
magnitude upon crossing below $p^{\star}$, which confirms – via Eq. 3 – that
the carrier density drops upon entering the pseudogap phase (given that
$C_{el}/T$ decreases below $p^{\star}$ Michon  _et al._ (2019)). With
decreasing $p$, $S/T$ remains high until $p=0.19$, at which point it starts to
decrease towards negative values. We therefore find that the usual effect of
CDW order on the transport properties of Nd-LSCO are observed up to at most
$p_{\rm CDW}$ $=0.19$, which is well below $p^{\star}$.

Figure 2:  Isotherms of the Seebeck coefficient in Nd-LSCO, plotted as $S/T$
vs magnetic field $H$, at temperatures and dopings as indicated. Isotherms for
$p=0.23$ and 0.25 are shown in Fig. S2 of the Supplementary Material. Note the
change of scale for $S/T$ – by a factor of about 4 – when going from $p=0.20$,
0.21, 0.22 $<$ $p^{\star}$ to $p=0.24>$ $p^{\star}$.

##  II Methods

The thermopower was measured in ten single-crystalline samples of Nd-LSCO,
each with a different doping $p$: 0.12, 0.15, 0.17, 0.19, 0.20, 0.21, 0.22,
0.23, 0.24 and 0.25. Five of those samples – with $p=0.20$, 0.21, 0.22, 0.23
and 0.24 – were previously studied with measurements of resistivity and Hall
effect Collignon  _et al._ (2017). Details of the sample and contact
preparation can be found there. The thermal conductivity of those same 5
samples was reported in Michon  _et al._ (2018), along with another two – with
$p=0.12$ and 0.15. Structural details on some of our Nd-LSCO samples, in
particular where they lie on the structural phase diagram, can be found in
ref. Dragomir  _et al._ (2020). The value of $T_{\rm c}$ (defined as the point
of zero resistance in zero field) varies from a local minimum of $T_{\rm c}$
$=5$ K at $p=0.12$ to a local maximum of $T_{\rm c}$ $\simeq 15$ K at $p\simeq
0.18$ back down to $T_{\rm c}$ $=5$ K at $p=0.25$ (see Extended Data Fig. 2b
in Michon  _et al._ (2019)). Samples were cut from large single crystals of
Nd-LSCO grown by a traveling float-zone technique, into small rectangular
platelets of typical dimensions 1 mm x 0.5 mm x 0.2 mm, with the shortest
dimension along the $c$ axis.

Figure 3:  (a) Isotherms of $S/T$ versus $H$ for Nd-LSCO at $p=0.22$, at
temperatures as indicated. The dashed lines are linear-$H$ fits to $S/T$ in
the normal state above the superconducting transition, highlighting the
negative field dependence. In order to remove this magneto-Seebeck effect, we
back extrapolate the fits to $H=0$ and thereby extract the zero-field values
of $S/T$ in the absence of superconductivity. Data and fits for all other
dopings are shown in Fig. S2 of the Supplementary Material. (b) $S/T$ versus
$T$ for Nd-LSCO at $p=0.22$, showing data taken in zero field above $T_{c}$
(black dots), at $H$ = 35 T (turquoise dots), and the values extrapolated to
$H\rightarrow 0$ (blue dots, obtained from fits in panel (a)). All lines are a
guide to the eye. (Data at $H=33$ T for all dopings are shown in Fig. S3 of
the Supplementary Material.)

We employ a standard one-heater-two-thermometers steady-state method to apply
and measure a temperature difference $\Delta T_{x}$ along the length of our
samples. At their cold end, samples were anchored to a copper block which
provides both an electrical and thermal ground. The heat current was applied
in the $ab$ plane of the low-temperature tetragonal structure of Nd-LSCO. The
longitudinal (Seebeck) voltage $\Delta V_{x}$ was measured on the very same
contacts as used for measuring $\Delta T_{x}$, which removes the uncertainty
on the geometrical factor. The Seebeck coefficient $S$ is then simply given by
$S=\Delta V_{x}/\Delta T_{x}$. For measurements in a magnetic field, the field
was applied along the $c$ axis of the crystal structure of Nd-LSCO. The
$ab$-plane Seebeck voltage was measured at positive and negative polarities of
the field, and was symmetrized with respect to field in order to obtain the
pure Seebeck signal, free of contamination from the Nernst effect.
Measurements in fields up to $H$ = 18 T were performed at Sherbrooke, and up
to 37.5 T at the High Field Magnet Laboratory (HFML) in Nijmegen.

The thermopower of Nd-LSCO was previously measured on polycrystalline samples
of La2-y-xNdySrxCuO4 with $y=0.6$, in zero field, for $p=0.08$, 0.10, 0.11,
0.12, 0.13, 0.15, 0.17, 0.19 and 0.21, as reported in ref. Hucker  _et al._ ,
1998, and on single-crystal samples of La2-y-xNdySrxCuO4 with $y=0.4$, in
fields up to 15 T, for $p=0.20$ and 0.24, as reported in ref. Daou  _et al._ ,
2009b. The data reported here are in excellent agreement with these prior low-
field data.

##  III Results

In Fig. 2, we show isothermal field sweeps of the Seebeck coefficient $S$,
expressed as $S/T$ vs $H$, at $p=0.20$, 0.21, 0.22, and 0.24. For all dopings,
$S/T$ is null in the superconducting state at low field, rises quickly upon
crossing the vortex-solid transition and then reaches the normal-state value
at $H_{c2}$ and above. At all dopings we observe an increase of the normal-
state $S/T$ with decreasing temperature. For a plot of $S/T$ vs $T$ at $H=35$
T, for $p=0.22$, see Fig. 3(b). (For a similar plot at other dopings, see Fig.
S2 of the Supplementary Material.) In Fig. 2, we readily observe from the raw
data at 5 K and at the highest field that the amplitude of $S/T$ remains
roughly constant as a function of doping for $p=0.20,0.21,0.22<$ $p^{\star}$,
but then drops by a factor of about 4 when going to $p=0.23,0.24,0.25>$
$p^{\star}$. This is one of the main findings of our study: the pseudogap
phase causes a large increase in $S/T$ in the $T=0$ limit, in the ground state
without superconductivity. This increase is strong evidence for a drop in
carrier density, given the fundamental relation $S/T\propto(C_{\rm el}/T)/n$
(Eq. 3), since $C_{\rm el}/T$ decreases below $p^{\star}$ Michon  _et al._
(2019).

Figure 4:  $S/T$ versus $T$ extrapolated to $H\rightarrow 0$, at dopings as
indicated. The data points are based on the extrapolations shown in Fig. 3(a)
and in Fig. S2 of the Supplementary Material. There is a clear and abrupt
change in the magnitude of $S/T$ at $T\simeq 0$ upon crossing into the
pseudogap phase ($p<0.23$). All lines are a guide to the eye.

In Fig. 2, we see that the normal-state $S/T$ displays a negative field
dependence. This is a bit like the magneto-resistance seen in the same samples
Collignon  _et al._ (2017), which causes the conductivity to decrease with
increasing field. This magneto-Seebeck effect is more pronounced in our sample
with $p=0.20$ than in our sample with $p=0.24$, and it is nearly absent in our
samples with $p=0.23$ and $p=0.25$ (see Fig. S2 of the Supplementary
Material.). In order to correct for this effect, and obtain the value of $S/T$
free from this field-induced decrease, we back-extrapolate the isotherms to
$H=0$ as illustrated in Fig. 3(a). As shown by the straight dashed lines for
$p=0.22$ (and in Fig. S2 of the Supplementary Material for other dopings), the
field dependence of $S/T$ in the normal state is well described by a
linear-$H$ dependence. We apply linear fits to our data, extend them below
$H_{\rm c2}$, and take the $y$-intercept of those fits at $H=0$ to obtain the
value of $S/T(H\rightarrow 0)$. In Fig. 3(b), we plot the resulting curve of
$S/T(H\rightarrow 0)$ vs $T$ and also our raw curve of $S/T(H=35~{}T)$ vs $T$.
Both continuously extend the zero-field curve taken above $T_{c}$, but they
clearly separate at low temperature, where $S/T(H\rightarrow 0)$ saturates at
a significantly higher value, revealing the magnitude of the magneto-Seebeck
effect in that sample.

Applying the same analysis to $p=0.20$, 0.21, 0.23, 0.24, and 0.25 (see in
Fig. S2 of the Supplementary Material), we plot the resulting curves of
$S/T(H\rightarrow 0)$ vs $T$ in Fig. 4. This immediately reveals a clear
thermopower signature of the pseudogap phase: upon crossing below $p^{\star}$
$=0.23$, $S/T(H\rightarrow 0)$ at low temperature undergoes a large and sudden
increase. The curves of $S/T(H\rightarrow 0)$ essentially appear to belong to
one of two groups depending on whether they are above or below $p^{\star}$,
with little variation within each group, clear evidence that the sudden jump
is caused by the onset of the pseudogap phase at $p^{\star}$. At low
temperature, the value of $S/T(H\rightarrow 0)$ increases by a factor of about
5 in going from $p>$ $p^{\star}$ to $p<$ $p^{\star}$. The same factor is
observed in the raw data at high fields (see Fig. 6(b) and Fig. S3 of the
Supplementary Material).

Figure 5:  Isotherms of the Seebeck coefficient plotted as $S/T$ vs $H$ for
Nd-LSCO at $p=0.17$ (a) and $p=0.19$ (b), at temperatures as indicated. Note
that there is a reversal of the temperature dependence between the two
dopings, with $S/T$ now decreasing with decreasing temperature for $p=0.17$.
Figure 6:  (a) $S/T$ as a function of temperature for Nd-LSCO at six dopings
as indicated, in a field of $H=16$ T (squares) and 33 T (dots). We observe a
clear evolution of $S/T$ vs $T$ with doping: for $p=0.24>$ $p^{\star}$, $S/T$
has a weak, nearly featureless temperature dependence; then for $p=0.19$ and
$0.20<$ $p^{\star}$, $S/T$ exhibits a pronounced growth at low $T$, in the
pseudogap phase below $T^{\star}$; and finally for $p<0.19$, $S/T$ shows a
clear drop at low temperature, which results in a negative $S/T$ at $p=0.12$
and 0.15. The arrows indicate the temperature of the maximum in $S/T$ vs $T$,
labeled $T_{\rm max}$ and plotted as a function of doping in Fig. 1. (b) $S/T$
taken in the normal state at $T=5$ K, in a field $H=16$ T (red squares) and
$H=33$ T (blue dots), as a function of doping. This plot of $S/T$ vs $p$ in
the $T=0$ limit shows clearly the sudden (5-fold) jump in $S/T$ upon entry
into the pseudogap phase at $p^{\star}$ $=0.23$, followed by the gradual fall
to negative values, a transport signature of entry into the CDW phase at
$p_{\rm CDW}$ $\simeq 0.19$, or slightly below.

We now turn to dopings $p<0.20$ and in Fig. 5 we show isotherms of $S/T$ for
Nd-LSCO at $p=0.19$ and 0.17. For $p=0.19$, the data are consistent with those
at $p=0.20$, having a similar amplitude and showing an increase with
decreasing temperature. At $p=0.17$, however, one important change occurs: at
low temperatures, $S/T$ decreases with decreasing temperature. The full
temperature dependence of $S/T$ is displayed in Fig. 6(a), where we see that
$S/T$ vs $T$ at $p=0.17$ goes through a maximum at $T\simeq 28$ K, before
falling at low temperature. This is a clear departure from the low-temperature
enhancement seen for $0.19\leq p<0.23$. As shown in Fig. 6(a), reducing the
doping further to $p=0.15$ and 0.12 shifts the maximum in $S/T$ to higher
temperatures and makes the downturn more pronounced, resulting in negative
values of $S/T$ as $T\rightarrow 0$. When plotted as a function of doping
(Fig. 6(b)), the low-temperature value of $S/T$ is seen to start its drop
towards negative values at $p\simeq 0.19$, or slightly below.

A negative Seebeck coefficient was previously reported for a number of
cuprates near $p\simeq 1/8$, namely La2-xSrxCuO4 (LSCO) Badoux  _et al._
(2016b), La2-xBaxCuO4 (LBCO) Li  _et al._ (2007), Nd-LSCO Nakamura and Uchida
(1992); Hucker  _et al._ (1998), Eu-LSCO Hucker  _et al._ (1998); Chang  _et
al._ (2010), YBCO Chang  _et al._ (2010); Laliberté  _et al._ (2011), and
HgBa2CuO4+δ(Hg1201) Doiron-Leyraud  _et al._ (2013). These studies showed that
a negative Seebeck coefficient is a consequence of the Fermi surface
reconstruction caused by the CDW order. Our extensive data on Nd-LSCO now
reveal over what doping range this reconstruction is present in that material.

Figure 7:  (a) Resistivity $\rho(T)$ as a function of $T$ in the field-induced
normal state of Nd-LSCO at $H=33$ T, at dopings as indicated (from Collignon
_et al._ (2017)). At $p=0.24$ (blue), there is no pseudogap and the
resistivity remains linear down to $T\rightarrow 0$. At $p=0.22$ (red), the
pseudogap opening causes an upward deviation from linearity (dash-dotted line)
below $T^{\star}$, and a large upturn in resistivity at low temperature. The
red dashed line is a linear extrapolation to $T=0$ of the data at $p=0.22$ in
33 T; it yields the zero-temperature estimate of $\rho(0)$. The dash-dotted
line yields the residual resistivity $\rho_{0}$ in the absence of the
pseudogap (see text), a measure of the disorder level in the sample. (b) $S/T$
as a function of $T$ based on extrapolated $H\rightarrow 0$ data, at the same
two dopings. In analogy to the resistivity, the pseudogap at $p=0.22$ causes a
large enhancement of $S/T$ below $T^{\star}$ with respect to the otherwise
slowly varying curve seen in the absence of the pseudogap at $p=0.24$.

##  IV Discussion

In Fig. 6, we summarize our Seebeck data in Nd-LSCO. In Fig. 6(a), we show the
evolution of the temperature dependence with doping and identify three
regions: for $p\geq p^{\star}$, $S/T$ varies weakly with temperature; for
$p<p^{\star}$, it exhibits a large increase at low temperature; and for
$p<0.19$, it goes through a maximum before falling at low temperature. In Fig.
6(b), we plot the normal-state value of $S/T$ measured at $T=5$ K, as a
function of doping, and again define three regions: for $p\geq p^{\star}$,
$S/T$ is small, positive and weakly doping-dependent; for $p<p^{\star}$, $S/T$
is initially large, positive and roughly constant; and for $p<0.19$, $S/T$
drops with decreasing $p$ to eventually become negative. In the following, we
discuss in more detail this evolution of $S/T$ across the phase diagram of Nd-
LSCO.

###  IV.1 Overdoped metal ($p>$ $p^{\star}$)

At $p=0.24$, a doping just above $p^{\star}$, the resistivity displays a
perfect $T$-linear dependence down to $T\rightarrow 0$ Daou  _et al._ (2009a);
Collignon  _et al._ (2017), as reproduced in Fig. 7(a). This is a classic
signature of quantum criticality (specifically for an antiferromagnetic QCP in
2D) Löhneysen  _et al._ (2007), suggesting that $p^{\star}$ is a quantum
critical point. This inference is strongly supported by the logarithmic
dependence of the electronic specific heat in Nd-LSCO at $p^{\star}$, whereby
$C_{\rm el}/T\propto$ log$(1/T)$ Michon  _et al._ (2019), a second classic
signature of the same type of quantum criticality Löhneysen  _et al._ (2007).
Note that the slope of the $T$-linear resistivity is consistent with the so-
called Planckian limit for the scattering rate, a universal property of
cuprates Legros  _et al._ (2019) and other materials near a quantum critical
point Bruin  _et al._ (2013).

At $p=0.24$, the Seebeck coefficient also displays a logarithmic dependence,
whereby $S/T\propto$ log$(1/T)$, as reported previously for Nd-LSCO Daou  _et
al._ (2009b) and Eu-LSCO Laliberté  _et al._ (2011). Recently, a log$(1/T)$
dependence was also observed in Bi2201 near $p^{\star}$ Lizaire  _et al._
(2020). This is also considered to be a signature of the same type of quantum
criticality Paul and Kotliar (2001).

###  IV.2 Pseudogap phase ($p<$ $p^{\star}$)

In Nd-LSCO, the entry into the pseudodap phase at $p^{\star}$, in the normal
state at $T\to 0$, has been examined via a number of transport measurements.
The Hall coefficient $R_{\rm H}$ and the electrical resistivity $\rho$, which
in a simple model would both be inversely proportional to the carrier density
$n$, show a large increase when the doping is reduced below $p^{\star}$
Collignon  _et al._ (2017), by a factor of about 5. The thermal conductivity
$\kappa$, which would be proportional to $n$, shows a drop by the same factor
upon crossing $p^{\star}$, in $\kappa/T$ in the $T=0$ limit (ref. Michon  _et
al._ , 2018). (In other words, the Wiedemann-Franz law is satisfied.) The
present paper now adds the Seebeck coefficient, inversely proportional to $n$
(Eq. 3), which shows a large enhancement at $p^{\star}$, by a factor of $\sim
5$ between $p=0.24$-0.25 and $p=0.19$-0.22 (Figs. 4 and 6(b)).

In Fig. 7, we compare $S/T$ and the resistivity $\rho$ in Nd-LSCO at two
dopings, above and below $p^{\star}$. At $p=0.22$, $\rho(T)$ displays a
pronounced upturn at low temperature Collignon  _et al._ (2017) whose onset,
taken as the initial upward deviation of $\rho(T)$ from the high-temperature
$T$-linear regime (Fig. 7(a)), is a standard definition of the pseudogap
temperature $T^{\star}$. This signature matches the actual pseudogap opening
observed in ARPES measurements Matt  _et al._ (2015) (Fig. 1), and is absent
at $p=0.24$ (Fig. 7(a)), where there is no pseudogap Matt  _et al._ (2015). As
seen in Fig. 7(b), an upturn in $S/T$ is observed at low temperature at
$p=0.22$, relative to the slow growth seen at $p=0.24$, and it starts roughly
at $T^{\star}$. The two upturns, in $\rho$ and in $S/T$, are two signatures of
the pseudogap phase that develops below $T^{\star}$.

Figure 8:  Seebeck coefficient $S/T$ (red squares) and resistivity ratio
$\rho(0)/\rho_{0}$ (blue dots, see main text) as a function of doping in Nd-
LSCO. Both quantities are taken in the $H\rightarrow 0$ and $T\rightarrow 0$
limits, and both exhibit a sudden jump at the pseudogap critical point
$p^{\star}$, a signature of the underlying change in carrier density.  Figure
9:  Magnitude of $S/T$ in the normal state at $T=25$ K for three different
cuprates as a function of doping: Nd-LSCO ($y=0.4$, solid red squares, this
work; $y=0.6$, open red squares Hucker  _et al._ (1998)); Eu-LSCO ($y=0.15$,
open blue circles Hucker  _et al._ (1998); $y=0.2$, solid blue circles
Laliberté  _et al._ (2011)); YBCO (solid green diamonds Laliberté  _et al._
(2011)). The grey band is a guide to the eye. In addition to the sudden jump
in $S/T$ upon entering the pseudogap phase below $p^{\star}$, this shows the
universal drop in $S/T$ towards negative values upon entering the CDW phase at
$p_{\rm CDW}$ ($\simeq 0.19$ in Nd-LSCO and Eu-LSCO, $\simeq 0.16$ in YBCO).

In the $T=0$ limit, the normal-state values of $\rho$ and $S/T$ display a
comparable 5-fold increase between $p=0.24$ and $p=0.22$. Since (in a simple
model) $S/T\propto m^{\star}/n$, $\rho\propto m^{\star}/(n\tau)$, and $R_{\rm
H}$ $\propto 1/n$, where $1/\tau$ is the scattering rate, this suggests that
it is a change in carrier density that causes the jump of comparable magnitude
in the three quantities across $p^{\star}$, with negligible change in
$1/\tau$. Magnetoresistance measurements provide independent evidence that
$1/\tau$ does not change significantly across $p^{\star}$, since the amplitude
of the orbital magnetoresistance (controlled by $\omega_{c}\tau$) is very
similar on both sides of $p^{\star}$, whether for $\rho_{a}$ Collignon  _et
al._ (2017) or for $\rho_{c}$ Fang  _et al._ (2020), despite the large change
in the absolute value of $\rho_{a}(T=0)$ Collignon  _et al._ (2017) and
$\rho_{c}(T=0)$ Cyr-Choinière  _et al._ (2010).

For a more quantitative comparison between the two coefficients, the effect of
the magnetic field should be accounted for in both $S/T$ and $\rho$. An
estimate of the zero-field resistivity below $T_{\rm c}$, from field sweeps
and temperature sweeps, was detailed in ref. Collignon  _et al._ , 2017 and
here we use those values. In Fig. 8, we plot the values of $S/T$ in the
$H\rightarrow 0$ and $T\rightarrow 0$ limits as a function of doping across
$p^{\star}$. For the resistivity, we plot the ratio $\rho(0)/\rho_{0}$ as a
function of doping, where $\rho(0)$ is the value of $\rho$ in the
$H\rightarrow 0$ and $T\rightarrow 0$ limits and $\rho_{0}$ is the value that
the resistivity would have in the absence of the pseudogap. As revealed by a
study where the pseudogap is turned on and off by pressure in Nd-LSCO Doiron-
Leyraud  _et al._ (2017), $\rho_{0}$ can be extracted by extending the high-
temperature $T$-linear regime to $T=0$, as illustrated for $p=0.22$ by the
dashed-dotted line in Fig. 7(a). The rationale for taking this ratio is that
it removes the effect of disorder, which is different for different samples.
For example, we can see that $\rho_{0}$ is slightly larger in our sample with
$p=0.22$ than it is in our sample with $p=0.24$. (By contrast, the Seebeck
coefficient does not depend on the disorder level of a particular sample.) As
seen in Fig. 8, both $S/T$ and $\rho(0)/\rho_{0}$ display the same sudden jump
at $p^{\star}$ and track each other below $p^{\star}$. Given that in a simple
model both $\rho$ and $S/T$ vary as $1/n$, the parallel jump in $\rho$ and
$S/T$ below $p^{\star}$ is further evidence for a drop in carrier density,
initially inferred from the Hall coefficient Badoux  _et al._ (2016a);
Collignon  _et al._ (2017). A large positive $S/T$ at low $T$ is also observed
in other cuprates below $p^{\star}$, such as Bi2201 and Bi2212 Kondo  _et al._
(2005).

Upon entering the pseudogap phase, the electronic specific heat $C_{\rm el}$
of cuprates drops. In Nd-LSCO, the normal-state $C_{\rm el}/T$ at $T=2$ K
decreases by a factor $\sim 2$ when going from $p=0.24$ to $p=0.20$ Michon
_et al._ (2019). So according to Eq. 3, this only reinforces our inference
that the carrier density must drop. Of course, Eq. 3 is an oversimplification
and a proper theory of the Seebeck coefficient in cuprates is needed to
understand in detail its behavior, whether above or below $p^{\star}$. To
highlight this need, let us mention that a calculation of $S$ based on the
band structure of Nd-LSCO yields a negative value above $p^{\star}$ Verret
_et al._ (2017), in contrast with the measured positive value.

###  IV.3 Charge-density-wave phase ($p<$ $p_{\rm CDW}$)

Early Seebeck measurements on Nd-LSCO and Eu-LSCO revealed that $S$ is
negative at low $T$ for dopings close to $p=0.12$ Nakamura and Uchida (1992);
Hucker  _et al._ (1998) (see open symbols in Fig. 9). Later studies showed
that a negative $S$ at low $T$ is also observed at $p\simeq 0.12$ in LBCO Li
_et al._ (2007), YBCO Chang  _et al._ (2010), Hg1201 Doiron-Leyraud  _et al._
(2013), and LSCO Badoux  _et al._ (2016b) showing that it is a universal
property of hole-doped cuprates. The negative $S$ at low $T$ came to be
understood as a consequence of a Fermi-surface reconstruction caused by the
onset of CDW order Taillefer (2009); Laliberté  _et al._ (2011), whose other
signatures are a negative Hall coefficient LeBoeuf  _et al._ (2007); Doiron-
Leyraud  _et al._ (2013) and low-frequency quantum oscillations Doiron-Leyraud
_et al._ (2007); Barišić  _et al._ (2013). In Fig. 9, we gather together
Seebeck data on Nd-LSCO, Eu-LSCO and YBCO by plotting the value of $S/T$ at
$T=25$ K vs doping. The agreement between early data on Nd-LSCO and our own
data is excellent, as is the agreement between early and later data on Eu-
LSCO. The fact that $S/T$ has its minimum (most negative) value at $p=0.12$ is
consistent with the fact that CDW order is strongest at $p=0.12$ in Nd-LSCO,
Eu-LSCO and YBCO (and LBCO and LSCO).

In Fig. 6, we define $T_{\rm max}$ to be the temperature below which $S/T$
starts to drop towards negative values. The values of $T_{\rm max}$ are
plotted vs $p$ in the phase diagram of Fig. 1, where we also show the
corresponding values for the closely related material Eu-LSCO (taken from
Laliberté  _et al._ (2011)). The fact that $T_{\rm max}$ matches $T_{\rm
CDW}$, the onset temperature for CDW order detected by x-ray diffraction in
Nd-LSCO Zimmermann  _et al._ (1998); Niemöller  _et al._ (1999)) and Eu-LSCO
Fink  _et al._ (2011), confirms that the drop in $S/T$ is caused by CDW order
(Fig. 1). We see that $T_{\rm max}$ decreases with doping, and vanishes close
to $p=0.19$. We therefore find no evidence of the usual effect of CDW order on
the transport properties of Nd-LSCO and Eu-LSCO beyond $p_{\rm CDW}$ $=0.19$
(Figs. 1, 6(b) and 9).

Note that the start of the CDW phase on the low doping side can also be
detected using Seebeck data. In Eu-LSCO and YBCO, there is no trace of a drop
in $S/T$ as $T\to 0$ for $p=0.08$ Laliberté  _et al._ (2011). The same is true
for Nd-LSCO Hucker  _et al._ (1998). We infer that the critical doping for the
onset of the CDW phase in these three materials is $p=0.08$ (Fig. 9). This is
nicely consistent with x-ray diffraction studies that show no CDW modulations
at $p<0.08$ in these same cuprates Fink  _et al._ (2011); Blanco-Canosa  _et
al._ (2014).

##  V Summary

We have used measurements of the Seebeck coefficient $S$ to investigate the
phase diagram of the cuprate superconductor Nd-LSCO, in the normal state as
$T\to 0$, reached by applying a magnetic field large enough to suppress
superconductivity. We identify three regions (Figs. 6(b) and 9): 1) $p>$
$p^{\star}$, outside the pseudogap phase, where $S$ is positive and $S/T$ is
small; 2) $p_{\rm CDW}$ $<p<$ $p^{\star}$, inside the pseudogap phase but
without CDW order, where $S$ is positive and $S/T$ is large; 3) $p<$ $p_{\rm
CDW}$, inside the CDW phase, where $S/T$ decreases as $T\to 0$. The large and
sudden increase in $S/T$ that occurs upon crossing below the pseudogap
critical doping $p^{\star}$ is further evidence for a drop in carrier density,
as also inferred from the drop in Hall number and in conductivity. The sudden
change from increasing to decreasing $S/T$ as $T\to 0$ in going from $p=0.19$
to $p=0.17$ allows us to identify $p_{\rm CDW}$ $=0.19$ as the upper bound for
the end of the CDW phase as seen by transport. This reveals that $p_{\rm CDW}$
is well separated from $p^{\star}$, as in the case of YBCO and LSCO.

##  VI Acknowledgements

We thank S. Fortier for his assistance with the experiments. L.T. and B.D.G.
acknowledge support from the Canadian Institute for Advanced Research (CIFAR)
as CIFAR Fellows and funding from the Natural Sciences and Engineering
Research Council of Canada. L.T. acknowledges funding from the Institut
Quantique, the Fonds de Recherche du Québec – Nature et Technologies (FRQNT),
the Canada Foundation for Innovation (CFI), and a Canada Research Chair. This
research was undertaken thanks in part to funding from the Canada First
Research Excellence Fund and the Gordon and Betty Moore Foundation’s EPiQS
Initiative (Grant GBMF5306 to L.T.). J.S.Z. was supported by NSF MRSEC under
Cooperative Agreement No. DMR-1720595. This work was supported by HFML-RU/NWO,
a member of the European Magnetic Field Laboratory (EMFL).

⋆⋆ These authors contributed equally to this work.

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Supplementary Material for
“Thermopower across the phase diagram of the cuprate La1.6-xNd0.4SrxCuO4 :
signatures of the pseudogap and charge-density-wave phases”

Figure S1:  Electrical resistivity $\rho$ as a function of temperature for Nd-
LSCO at $p=0.17$ (a) and $p=0.19$ (b), showing the linear-$T$ behavior at high
temperature (black dashed lines) and the departure from that regime at the
pseudogap temperature $T^{\star}$. The values of $T^{\star}$ are 100 $\pm 10$
K at $p=0.17$ and 90 $\pm 10$ K at $p=0.19$.  Figure S2:  Seebeck coefficient
$S$ plotted as $S/T$ as a function of magnetic field for Nd-LSCO at dopings as
labeled, at temperatures as indicated. The dashed lines are linear fits to the
normal state $S/T(H)$, whose extrapolation to $H=0$ provide values of $S/T$ in
the absence of a magneto-Seebeck effect (see main text).  Figure S3:  Seebeck
coefficient $S/T$ as a function of temperature in Nd-LSCO at dopings as
indicated, in $H=33$ T.