# Transfer training from smaller language model

Han Zhang

(April 2021)

##  1 ABSTRACT

Large language models have led to state-of-the-art accuracies across a range
of tasks. However,training large language model needs massive computing
resource, as more and more open source pre-training models are available, it
is worthy to study how to take full advantage of available model. We find a
method to save training time and resource cost by changing the small well-
trained model to large model. We initialize a larger target model from a
smaller source model by copy weight values from source model and padding with
zeros or small initialization values on it to make the source and target model
have approximate outputs, which is valid due to block matrix multiplication
and residual connection in transformer structure. We test the target model on
several data sets and find it is still comparable with the source model. When
we continue training the target model, the training loss can start from a
smaller value.

##  2 INTRODUCTION

Transformer Vaswani et al. (2017) is a simple network architecture proposed by
Vaswani to solving machine translation problem, it is based solely on
attention mechanisms and soon become one of the most popular model structure
in deep learning community. The early applications of transformer is Natural
Language Processing(NLP) such as machine translation Zhang et al. (2018).
After Open-AI and Google successively propose large transformer-based
pretrained Language Models(LM) – GPT Radford et al. (2018) and BERT Devlin et
al. (2018), scholars study on all kinds of NLP tasks by using pretrained LM,
including Text Classification, Named Entity Recognition, Natural Language
Inference, Read Comprehension, Automated Abstracting and so on. Except NLP
tasks, transformers also be applied for many multi-model tasks to bridging
text and videoLi et al. (2020), time-series forecasting tasks Zhou et al.
(2020) and so on. As the success of the pre-training model, there are two
routes to enhance model’s performance. The first is Lightweighting the
pretrained model such as model compressing Tambe et al. (2020); Lan et al.
(2019) or accelerating the inference Lin et al. (2020), the second is
increasing model size, such as GPT3 Brown et al. (2020) and Switch Transformer
Fedus et al. (2021).

Megatron Shoeybi et al. (2019) is the first open-sourced project to increasing
transformer size by efficient intra-layer model parallel approach that enables
training transformer models with 8.3 billion of parameters on 512 GPUs. T5
Raffel et al. (2019) increasing the parameters number to 11 billion by Text-
to-Text Transfer Transformer, and treat every text processing problem as a
“text-to-text” problem, witch shows some general intelligence ability. The
study of GPT3 Brown et al. (2020) shows that large Language Model has strong
context learning ability, which without parameters updating but are compareble
with state of the art fine-tune style methods. And as the model size
increasing, there is still room for performerence improving. Lots of work
shows that larger language models showing potential to more intelligence
ablity Zhang et al. (2020a), however, training a large model form head costs
much resources for instance that training GPT3 from the beginning costs more
than ten millions of dollars.

There have been some studies on making pretraining more efficiency by
improving pretraining tasks. Roberta Liu et al. (2019) employs a dynamic mask
strategy to make the best of each sentence in corpus witch reduces the demand
for corpus volume. Electra Liello et al. (2021) uses several efficient pre-
training objectives for transformers based models by adversarial learning,
which requires smaller classification heads and exhibits a general superiority
over MLM. PMLM Liao et al. (2020) adopts a probabilistic masking scheme for
the MLM, and has a unique ability to autoregressively generate sequences in
arbitrary order, which improves training effctively by enhance the difficulty
of the pretraining task.

Efficient Transformers is a research focus to reducing transformer complexity
to saving training resource cost. The typical approachs are Memory method,
Low-Rank method and Kernel method. Memory method is to leverage a side memory
module that can access multiple tokens at once, for instance Longformer
Beltagy et al. (2020) and set transformer Lee et al. (2018). Low-Rank method
is to improve efficiency by leveraging low-rank approximations of the self-
attention matrix such as Linformer Wang et al. (2020). Kernel method to
improve the efficiency of Transformers is to view the attention mechanism
through kernelization such as Performer Choromanski et al. (2020), Linformer
Wang et al. (2020) and Linear Transformer Katharopoulos et al. (2020). Yi Tay
Tay et al. (2020) proposes a long range arena to evaluate the Efficient
Transformers’ performance, speed and memory footprint.

Vertical to improving pre-training tasks or optimizing transformer model, we
study on transfering model parameters and it can be compatible with all of
above methods. Our works focus on how to reduce the resource cost of training
large language model from the beginning. There are many open-resourced
pretrained models in transformers library Wolf et al. (2020) such as BERT,
BART, XLM, ALBERT, Electra, XLNet and so on. In the common situation, we fetch
the pretrained model and add task-specific layers on it then fine-tune the
whole model in different tasks. Continuing this idea, we could also add new
parameters to each layer to broaden the model or add new layers to deepen the
model. According to the rule of block matrix multiplication , when we padding
zeros at matrix in the same position, the result of matrix multiplication
remain unchanged. The details will be discussed in section 4.

##  3 RELATED WORK

In the following paragraphs we summarize the related work needed to introduce
our approach. Transformer is a sequence-to-sequence model and therefore
comprises an encoder and a decoder.

Encoder The Transformer’s encoder is a function defined as the composition of
L identical layers or blocks, each composed of two sub-layers. The first sub-
layer is the self-attention mechanism which allows the encoder to focus on the
relevant part of the sequence for each position, similarly to the inter
attention depicted in Figure 3. The second sub-layer is a simple fully
connected feed-forward network applied independently and identically to every
position (position-wise). The feedforward network increases the encoder’s
expressiveness and transforms the self-attention’s output for the next layer.

Stacked Encoder: BERT is a stacked encoder Transformer, which inputs a
sequence of tokens and applies position and token and token type embeddings.
Each layer applies multi-head self-attention in combination with a feedforward
network, layer normalization, and residual connections. The BERT base model
has 12 layers and 12 heads.

Decoder The decoder is also composed of L identical layers. Note that although
it is common for the decoder to have the same number of layers as the encoder,
one may adjust their depth independently. Each decoder’s layer comprises three
sub-layers. The first sub-layer is the self-attention mechanism as in the
encoder, except that future positions are masked. Indeed, the encoder is
allowed to look at future positions since the input sequence is entirely
available. The decoder, however, cannot look at future positions since they
have not yet been predicted. Therefore, the i-th position may only attend to
positions less than i. The second sub-layer is the inter-attention mechanism,
which helps the decoder focus on the relevant parts of the input, such as
depicted in Figure 3. Finally, the third sub-layer is a simple feed-forward
network. As for the encoder, a residual connection and a layer normalization
are applied to each sub-layer.

Stacked Decoder: GPT-2 is a stacked decoder Transformer, which inputs a
sequence of tokens and applies position and token embeddings followed by
several decoder layers. Each layer applies multi-head attention combination
with a feedforward network, layer normalization, and residual connections. And
the attention score is controlled by a lower triangular matrix which is
designed to prevent the current word from seeing the following words. The
GPT-2 small model has 12 layers and 12 heads.

To verify the universality of the method, we apply our method to transfer BERT
and GPT models, both of which are typical transformer based pretrained
language model.

##  4 Method: transformer-transfering

In this section, we present the implementation of Weight-transferring. Given a
well-trained transformer model such as GPT-small as our source model, there
are two kinds of variables in it — parameters and buffers. Parameters are
variables can be updated by optimizer, including embedding weight, dense
weight and bias, buffers are variables can’t be updated by optimizer, such as
bias values in attention layer which is used to control attention score in
masked position.

Embedding-transfer. Different to directly copying word embedding matrix like
word2vector, we need to transfer a smaller size matrix from source model to a
bigger matrix in target model. For computational equivalence, we split the
target embedding matrix into two blocks by rows, the first block has the same
size with source embedding matrix, so we can straightforward copy the weights
from source. The second block is initialized by zeros tensor (showed in Figure
1) so that the inner product operation results in the language model head (the
last layer for GPT) can be unchanged.

Figure 1: Embedding transfer

Dense-transfer. In order of calculation in transformer, there are several
mapping layers after embedding and LayerNorm layer. Such as in self-attention
layer, the hidden states matrix are map into three time width matrix for
query, key, and value to calculating attention score. If the dimension of
hidden states of target model is 1024, then there is a (3072,1024) shape
weight and a (3027,) shape bias parameters in the mapping layer. If the
dimension of source model is 768, so the matrix shape in mapping layer is
(2304,768) which both dimension are not same as matrix in target. Block Matrix
Multiplication ensure our transfer method is valid. As showed in Figure 2, by
padding zeros at the tail of source matrix, the result of Block Matrix
Multiplication is equal to directly do matrix multiplication on source matrix
and then padding zeros at the tail. And the bias transfer is analogical.

Figure 2:  Dense-transfer

Attention Layer transfer. As showed in Figure 3, attention layer calculation
is consist of several Matrix Multiplication, so we can use the same method as
Dense transfer. The multi-head attention splits the hidden states into multi-
heads and execute scaled dot-product attention respectively. But if we change
the dimension of hidden states or head number, each head will change and may
influence the attention score. There are two ways to ensure the results valid.
The first way is keep each head dimension and only increase the head numbers,
the second way is padding each head zeros at the tail respectively, both of
them are mathematically equivalent.

Figure 4 shows the scaled dot-product attention calculation process of one
head.

Figure 3: Attention structure Figure 4:  Attention-transfer

About LayerNorm. Layer normalization (LayerNorm) is a technique to normalize
the distributions of intermediate layers. It enables smoother gradients,
faster training, and better generalization accuracy, because of its capability
in handling re-centering and re-scaling of both inputs and weight matrix.
However, both re-centering and re-scaling opeartion are related to hidden
size, which is changed after transfering. It is not totally mathematics
equivalent for source and target model, but the influence is very small since
the scaling weight and bias can be updated fast after training several steps
to adjust new model parameters.

Deeper Layers Parameters. Deepening neural net is the the most commonly way to
increasing model size. When we transfer a source model with few layers to a
deeper target model, the parameters in deep layers need to be initialized in
smaller values. Because of residual connection, the inputs and outputs of
these layers are approximate. It makes the target model have similar output
distribution with the source model.

##  5 Experiments

In this section, we conduct extensive experiments on inference performance and
subsequent training. We test BERT and GPT on Cloze, Next Sentence Prediction
and Next Word Prediction task, which are pretraining objective in the stage of
pre-training. Then we continue train the target model find it can reach better
performance than source model.

###  5.1 Infertence ability

In this subsection, we test our target BERT model on LAMA Petroni et al.
(2020) dataset, which is a probe for analyzing the factual and commonsense
knowledge contained in pretrained language models. As can be seen in Table 1,
after transferring, the target model still remain the mask filling ability,
which shows that our transfer method is valid. The performance loss is due to
the LayerNorm part, which is not mathematically equivalent when transferring.

Table 1: Mask Filling Task Dataset | Source Model(108M) | Target Model(355M)
---|---|---
ConceptNet | 14.80 | 7.20
Squad | 15.89 | 10.26

###  5.2 Subsequent training

In this part, we train a 81 million parameters source dialog GPT Zhang et al.
(2020b) Zhang et al. (2020b) model in chat corpus, then we transfer the source
GPT model to a 165 million target model. Next, we conduct two experiments to
verify our method. In the first experiment, We train the transfered target
model and a random initialized 165 million GPT model on chat corpus. The
training loss is showed in Figure 5 left, the target model have smaller loss
at beginning steps. In second experiment, we continue train the source and
target model on douban corpus which is new to both models. As showed in Figure
5 right, the target model can achieve lower training loss.

Figure 5: Left is training loss compared on chat corpus, right is on douban
corpus

##  6 Conclusion

We propose a transfer strategy which can increase the model size with
acceptable performance decreasing by transferring the parameters of source
model. It is compute resource saving by avoid training a large model from the
beginning. The method is valid due to block matrix multiplication and residual
connection in transformer structure except the LayerNrom layer which causes a
little performance decreasing. Our feature work is optimizing the transfer
strategy to more compatibility and with less performance decreasing.

## References

  * Vaswani et al. [2017] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  2017\.
  * Zhang et al. [2018] Jiacheng Zhang, Huanbo Luan, Maosong Sun, FeiFei Zhai, Jingfang Xu, Min Zhang, and Yang Liu.  Improving the transformer translation model with document-level context.  2018\.
  * Radford et al. [2018] Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever.  Improving language understanding by generative pre-training.  2018\.
  * Devlin et al. [2018] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.  Bert: Pre-training of deep bidirectional transformers for language understanding.  2018\.
  * Li et al. [2020] Zekang Li, Zongjia Li, Jinchao Zhang, Yang Feng, Cheng Niu, and Jie Zhou.  Bridging text and video: A universal multimodal transformer for video-audio scene-aware dialog.  2020\.
  * Zhou et al. [2020] Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hui Xiong, and Wancai Zhang.  Informer: Beyond efficient transformer for long sequence time-series forecasting.  2020\.
  * Tambe et al. [2020] Thierry Tambe, Coleman Hooper, Lillian Pentecost, Tianyu Jia, En-Yu Yang, Marco Donato, Victor Sanh, Paul Whatmough, Alexander M. Rush, David Brooks, and Gu-Yeon Wei.  Edgebert: Sentence-level energy optimizations for latency-aware multi-task nlp inference, 2020.
  * Lan et al. [2019] Zhenzhong Lan, Mingda Chen, Sebastian Goodman, Kevin Gimpel, Piyush Sharma, and Radu Soricut.  Albert: A lite bert for self-supervised learning of language representations, 2019.
  * Lin et al. [2020] Ye Lin, Yanyang Li, Tengbo Liu, Tong Xiao, Tongran Liu, and Jingbo Zhu.  Towards fully 8-bit integer inference for the transformer model, 2020\.
  * Brown et al. [2020] Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei.  Language models are few-shot learners, 2020.
  * Fedus et al. [2021] William Fedus, Barret Zoph, and Noam Shazeer.  Switch transformers: Scaling to trillion parameter models with simple and efficient sparsity, 2021.
  * Shoeybi et al. [2019] Mohammad Shoeybi, Mostofa Patwary, Raul Puri, Patrick LeGresley, Jared Casper, and Bryan Catanzaro.  Megatron-lm: Training multi-billion parameter language models using model parallelism, 2019.
  * Raffel et al. [2019] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu.  Exploring the limits of transfer learning with a unified text-to-text transformer, 2019.
  * Zhang et al. [2020a] Zhengyan Zhang, Xu Han, Hao Zhou, Pei Ke, Yuxian Gu, Deming Ye, Yujia Qin, Yusheng Su, Haozhe Ji, Jian Guan, Fanchao Qi, Xiaozhi Wang, Yanan Zheng, Guoyang Zeng, Huanqi Cao, Shengqi Chen, Daixuan Li, Zhenbo Sun, Zhiyuan Liu, Minlie Huang, Wentao Han, Jie Tang, Juanzi Li, Xiaoyan Zhu, and Maosong Sun.  Cpm: A large-scale generative chinese pre-trained language model, 2020a.
  * Liu et al. [2019] Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov.  Roberta: A robustly optimized bert pretraining approach, 2019.
  * Liello et al. [2021] Luca Di Liello, Matteo Gabburo, and Alessandro Moschitti.  Efficient pre-training objectives for transformers, 2021.
  * Liao et al. [2020] Yi Liao, Xin Jiang, and Qun Liu.  Probabilistically masked language model capable of autoregressive generation in arbitrary word order, 2020.
  * Beltagy et al. [2020] Iz Beltagy, Matthew E. Peters, and Arman Cohan.  Longformer: The long-document transformer, 2020.
  * Lee et al. [2018] Juho Lee, Yoonho Lee, Jungtaek Kim, Adam R. Kosiorek, Seungjin Choi, and Yee Whye Teh.  Set transformer: A framework for attention-based permutation-invariant neural networks, 2018.
  * Wang et al. [2020] Sinong Wang, Belinda Z. Li, Madian Khabsa, Han Fang, and Hao Ma.  Linformer: Self-attention with linear complexity, 2020.
  * Choromanski et al. [2020] Krzysztof Choromanski, Valerii Likhosherstov, David Dohan, Xingyou Song, Andreea Gane, Tamas Sarlos, Peter Hawkins, Jared Davis, Afroz Mohiuddin, Lukasz Kaiser, David Belanger, Lucy Colwell, and Adrian Weller.  Rethinking attention with performers, 2020.
  * Katharopoulos et al. [2020] Angelos Katharopoulos, Apoorv Vyas, Nikolaos Pappas, and François Fleuret.  Transformers are rnns: Fast autoregressive transformers with linear attention, 2020.
  * Tay et al. [2020] Yi Tay, Mostafa Dehghani, Samira Abnar, Yikang Shen, Dara Bahri, Philip Pham, Jinfeng Rao, Liu Yang, Sebastian Ruder, and Donald Metzler.  Long range arena: A benchmark for efficient transformers, 2020.
  * Wolf et al. [2020] Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush.  Transformers: State-of-the-art natural language processing.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 38–45, Online, October 2020. Association for Computational Linguistics.  URL https://www.aclweb.org/anthology/2020.emnlp-demos.6.
  * Petroni et al. [2020] Fabio Petroni, Patrick Lewis, Aleksandra Piktus, Tim Rocktäschel, Yuxiang Wu, Alexander H. Miller, and Sebastian Riedel.  How context affects language models’ factual predictions.  In _Automated Knowledge Base Construction_ , 2020.  URL https://openreview.net/forum?id=025X0zPfn.
  * Zhang et al. [2020b] Yizhe Zhang, Siqi Sun, Michel Galley, Yen-Chun Chen, Chris Brockett, Xiang Gao, Jianfeng Gao, Jingjing Liu, and Bill Dolan.  Dialogpt: Large-scale generative pre-training for conversational response generation.  In _ACL, system demonstration_ , 2020b.

##  7 Appendix

The core codes of transferring gpt model is show in Figure 6

Figure 6: Code for transferring GPT


This can be readily performed by measuring the mean values of the relative
population $z(t)$ and phase $\varphi(t)$ for different times and initial
conditions. This would allow for the characterization of the effective
Hamiltonian on a “classical” level.

Ultracold atomic systems offer the possibility to access genuine quantum
properties, such as coherent spin squeezing esteve2008squeezing ;
gross2010nonlinear ; julia2012dynamic ; strobel2014fisher ; evrard2019enhanced
; kunkel_simultaneous_2019 . In particular, generalized measurements can be
used to evaluate non-commuting observables (such as the imbalance $z$ and
relative phase $\varphi$) within the same experimental realization
kunkel_simultaneous_2019 . Besides, the Husimi distribution [Fig. (12)] can be
reconstructed from projective measurements evrard2019enhanced .

Drive-induced pair-hopping processes are a striking feature of the effective
Hamiltonian in Eq. (34). To detect this effect, we propose to exploit an
additional spatial degree of freedom (“tube” geometry), as we illustrate in
Fig. 20(a). Specifically, we apply an energy offset $\Delta_{0}$ to one of the
wells (colored in blue), and assume that atoms are initially prepared at
momentum $k\\!\approx\\!0$. When activating the driving sequence, pair-hopping
processes are effectively generated, and atoms would then be allowed to hop by
pairs to the other well (colored in red), where they would acquire a finite
momentum $\pm k_{0}$; see Figs. 20(a)-(b). The momentum correlation could then
be revealed experimentally by letting the cloud expand for a long time-of-
flight borselli_two-particle_2021 : the finite momentum leads to a separation
of the atom pairs, such that counting the number of atoms at $\pm k_{0}$ would
reveal the correlation in the reduced variance (compared to a binomial
distribution) of the population imbalance.

Figure 20: Detecting pair-hopping processes in a driven double well potential.
(a) An additional spatial degree of freedom, combined with an energy offset
$\Delta_{0}$ between the two wells, allows for specific pair-hopping
processes, which result in pairs of atoms with opposite momentum $\pm k_{0}$.
(b) Dispersion relation associated with the two wells, which are shifted in
energy by an amount $\Delta_{0}$. A pair-hopping process converts two atoms in
the ‘blue’ well with momentum $k\\!=\\!0$, into a pair of atoms in the ‘red’
well with momentum $\pm k_{0}$. These finite-momentum pairs can be detected
after a long TOF, hence revealing the effective pair-hopping processes
generated by the driving sequence.

Finally, we note that the quench protocol introduced in Section VII.3 could be
directly implemented in a quantum-gas experiment, in view of revealing the
orbital order and TRS-broken nature of the chiral superfluids and Mott phases
analyzed in Section VII. As illustrated in Fig. 19, the finite orbital
polarization in the ground state can be unambiguously detected by monitoring
the time-evolving population imbalance $z_{n}(t)$, locally defined at the
level of each dimer, upon performing the quench protocol.

##### Acknowledgments

This work was initiated through discussions with J. Fatome and S. Coen, who
are warmly acknowledged. The authors also thank M. Bukov, I. Carusotto, N. R.
Cooper, J. Dalibard, A. Eckardt, N. Englebert, M. Jürgensen, Yun Li, B. Mera,
F. Petiziol, M. C. Rechtsman, J. Schmiedmayer, A. Schnell and H. Strobel for
various discussions. We are very grateful to S. Coen, N. Englebert, J. Fatome,
P. Kockaert and S. Mukherjee for their comments on an early version of this
manuscript.

N. G and L. P. G. are supported by the FRS-FNRS (Belgium), the ERC Starting
Grants TopoCold and LATIS and the EOS project CHEQS. O. K. D. acknowledges
funding from the International Max Planck Research School for Quantum Science
and Technology (IMPRS - QST). L. B. acknowledges funding from Politecnico di
Torino, starting package Grant No. 54 RSG21BL01. The work in Vienna was
performed under the QUANTERA project MENTA (FWF: I-6006) and M. P. has
received funding from the European Unions Horizon 2020 research and innovation
program under the Marie Sklodowska-Curie grant agreement No 101032523. M. D.
L. is supported by the Rita Levi Montalcini Program.

##  Appendix A Useful formulas

The Section III uses two families of operators: the bosonic operators
$\hat{a}_{1}^{(\dagger)}$ and $\hat{a}_{2}^{(\dagger)}$ associated with the
two modes, and which satisfy the canonical bosonic commutation relations,
$[\hat{a}_{s},\hat{a}_{s^{\prime}}^{\dagger}]=\delta_{s,s^{\prime}}$, where
$s\\!=\\!1,2$; and the angular momentum (Schwinger) operators, defined as

$\displaystyle\hat{J}_{x}=\frac{1}{2}\left(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1}\right),\quad\hat{J}_{y}=\frac{1}{2i}\left(\hat{a}_{2}^{\dagger}\hat{a}_{1}-\hat{a}_{1}^{\dagger}\hat{a}_{2}\right),$
$\displaystyle\hat{J}_{z}=\frac{1}{2}\left(\hat{a}_{2}^{\dagger}\hat{a}_{2}-\hat{a}_{1}^{\dagger}\hat{a}_{1}\right),\quad\hat{N}=\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2}.$
(99)

These operators satisfy the spin commutation relations
$[\hat{J}_{\mu},\hat{J}_{\nu}]=i\varepsilon_{\mu\nu\lambda}\hat{J}_{\lambda}$,
and the operator $\hat{N}$ counts the total number of bosons in the system
(assumed to be constant).

In view of expressing interaction processes with Schwinger operators, it is
useful to note that

$\displaystyle\hat{J}_{z}^{2}=\frac{1}{4}\left(\hat{a}_{1}^{\dagger}\hat{a}_{1}^{\dagger}\hat{a}_{1}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{2}\hat{a}_{2}-2\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{1}\hat{a}_{2}+\hat{N}\right),$
$\displaystyle\hat{J}_{y}^{2}=\frac{1}{4}\left(2\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{1}\hat{a}_{2}-\hat{a}_{1}^{\dagger}\hat{a}_{1}^{\dagger}\hat{a}_{2}\hat{a}_{2}-\hat{a}_{2}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{1}\hat{a}_{1}+\hat{N}\right),$
$\displaystyle\hat{N}^{2}=\left(\hat{a}_{1}^{\dagger}\hat{a}_{1}^{\dagger}\hat{a}_{1}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{2}\hat{a}_{2}+2\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{1}\hat{a}_{2}+\hat{N}\right).$
(100)

Hence, both $\hat{J}_{z}^{2}$ and $\hat{N}^{2}$ contain intra-mode (Hubbard)
and inter-mode (cross) interactions, while $\hat{J}_{y}^{2}$ contains a
combination of inter-mode interactions and pair-hopping processes [Fig. 4]. We
point out that $\hat{J}_{z}^{2}$ is related to $\hat{J}_{y}^{2}$ through a
unitary transformation; see Eq. (28).

From Eq. (100), we can express the intra-mode (Hubbard) interaction terms as

$\frac{1}{2}\left(\hat{a}_{1}^{\dagger}\hat{a}_{1}^{\dagger}\hat{a}_{1}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{2}\hat{a}_{2}\right)=\hat{J}_{z}^{2}+\text{constant},$
(101)

where the irrelevant constant term reads $\hat{N}(\hat{N}-2)/4$. Similarly,
the inter-mode (cross) interaction term reads

$\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{1}\hat{a}_{2}=-\hat{J}_{z}^{2}+\text{constant},$
(102)

with the irrelevant constant term $\hat{N}^{2}/4$. These expressions were used
to derive the Hamiltonian in Eq. (12) from Eq. (9).

Finally, it is useful to note that a combination of intra-mode (Hubbard)
interactions and pair-hopping processes can be expressed as

$\displaystyle\hat{J}_{z}^{2}+\hat{J}_{y}^{2}$
$\displaystyle=\frac{1}{4}\left(\hat{a}_{1}^{\dagger}\hat{a}_{1}^{\dagger}\hat{a}_{1}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{2}\hat{a}_{2}\right)$
$\displaystyle-\frac{1}{4}\left(\hat{a}_{1}^{\dagger}\hat{a}_{1}^{\dagger}\hat{a}_{2}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{1}\hat{a}_{1}\right)+\text{constant}.$
(103)

##  Appendix B Orbital order and phase transitions from a mean-field analysis

In this Appendix, we provide a detailed mean-field analysis of the extended
Bose-Hubbard model in Eq. (66).

Upon performing the mean-field substitution

$\hat{b}_{n,\sigma}\rightarrow\langle\hat{b}_{n,\sigma}\rangle\equiv\psi_{n,\sigma},$
(104)

we obtain the mean-field functional

$\displaystyle\mathcal{F}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{2}\sum_{n\sigma}\Bigg{[}U_{\xi}|\psi_{n,\sigma}|^{2}(|\psi_{n,\sigma}|^{2}-1)+W_{\xi}|\psi_{n,\sigma}|^{2}|\psi_{n,\,\overline{\sigma}}|^{2}\Bigg{]}$
$\displaystyle-$
$\displaystyle\\!\sum_{n\sigma}\Bigg{[}\left(\Omega+\frac{\Omega_{12}}{2}e^{i\sigma\pi/2}\right)\psi^{*}_{n+1,\sigma}\psi_{n,\sigma}+h.c.\Bigg{]}$
$\displaystyle-$
$\displaystyle\\!\frac{\Omega_{12}}{2}\sum_{n\sigma}\left(e^{-i\sigma\frac{\pi}{2}}\psi^{*}_{n+1,\sigma}\psi_{n,\,\overline{\sigma}}+h.c.\right)-\mu\sum_{n\sigma}|\psi_{n,\sigma}|^{2},$

from which we derive the coupled NLSE in Eq. (LABEL:DNLS).

When imposing periodic boundary conditions (ring geometry), and in the limit
of weak interactions $U\\!\ll\\!\Omega$, the mean-field ground state is
expected to have a uniform density distribution over the entire chain. We can
then propose Bloch states as stationary solutions of Eq. (LABEL:DNLS), namely

$\displaystyle\psi_{n,\sigma}(t)=e^{-i(\varepsilon(k)-\mu)t/\hbar}\,e^{ikn}\,\phi_{k,\sigma},$
$\displaystyle\phi_{k,\sigma}=\sqrt{\frac{\rho}{2}[1+\sigma
m(k)]}e^{i\Theta_{k,\sigma}}.$ (106)

In this way, the local density
$\rho_{n}\\!=\\!\rho\\!=\\!\sum_{\sigma}|\psi_{\sigma}|^{2}\\!=\\!N/N_{d}$ is
constant, with $N$ the total number of bosons and $N_{d}$ the number of unit
cells (dimers) in the ring. We note that the orbital polarization entering Eq.
(106) is given by

$m(k)=\frac{1}{\rho}(|\phi_{k,+}|^{2}-|\phi_{k,-}|^{2}).$ (107)

Inserting the ansatz (106) into the NLSE in Eq. (LABEL:DNLS), we find that
$\phi_{k}=(\phi_{k,+},\phi_{k,-})^{T}$ should be an eigenstate of the Gross-
Pitaevskii Hamiltonian written in Bloch representation,

$\displaystyle\hat{H}_{\textrm{GP}}(k)$ $\displaystyle=$
$\displaystyle\left[\frac{U_{\xi}}{2}(\rho-1)+\frac{W_{\xi}}{2}\rho-2\Omega\cos(k)\right]\hat{1}$
$\displaystyle-$
$\displaystyle\left[\Omega_{c}m(k)+\Omega_{12}\sin(k)\right]\hat{\sigma}_{z}-\Omega_{12}\cos(k)\hat{\sigma}_{y},$

where $m(k)$ should be determined self-consistently [Eq. (107)], and where
$\Omega_{c}$ is given by Eq. (76). Note that, for each value of $k$, this
Hamiltonian exactly maps to the mean-field Hamiltonian of a transverse
spin-$1/2$ Ising model with an additional longitudinal magnetic field. Indeed,
$m(k)$ represents the self-consistent magnetization along the $z$-direction
with $\Omega_{c}$ the magnitude of the ferromagnetic coupling constant. The
magnitudes of the transverse field along the $y$-direction and the
longitudinal field along the $z$-direction are respectively given by
$\Omega_{12}\cos(k)$ and $\Omega_{12}\sin(k)$.

The solution with lowest eigenenergy determines the mean-field energy
functional

$\displaystyle\varepsilon_{\textrm{MF}}(k)$ $\displaystyle=$
$\displaystyle-\sqrt{\Omega_{12}^{2}+\Omega_{c}^{2}m(k)^{2}+2m(k)\Omega_{12}\Omega_{c}\sin(k)}$
(109) $\displaystyle-$ $\displaystyle
2\Omega\cos(k)+\frac{U_{\xi}}{2}(\rho-1)+\frac{W_{\xi}}{2}\rho,$

which should be minimized. We will denote by $k_{0}$ the value of $k$ that
achieves this minimization (to be specified below).

By inserting the eigenstate of this low-energy branch into Eq. (107), we
obtain that the orbital polarization $m(k)$ should satisfy the self-consistent
condition

$m(k)=\frac{m(k)\Omega_{c}+\Omega_{12}\sin(k)}{\sqrt{\Omega_{12}^{2}+\Omega_{c}^{2}m(k)^{2}+2m(k)\Omega_{12}\Omega_{c}\sin(k)}}.$
(110)

Moreover, we find that the phases $\Theta_{k,\sigma}\\!=\\!\Theta_{\sigma}$
are independent of the wavevector, and that they are determined by

$\Theta=\Theta_{+}-\Theta_{-}=-(\pi/2)\textrm{sgn}(\Omega_{12}).$ (111)

It is insightful to analyze how these conditions on the nature of the ground-
state translate in the original basis of Eq. (63). In this representation, the
mean-fields are given by $\langle\hat{a}_{n,s}\rangle_{0}\equiv\psi_{n,s}$,
with $s\\!=\\!1,2$, and they read [Eqs. (65) and (106)]

$\displaystyle\psi_{n,1}$ $\displaystyle=$ $\displaystyle
e^{ik_{0}n}\sqrt{\rho}\left(\frac{\sqrt{1+m_{0}}e^{i\Theta_{+}}+\sqrt{1-m_{0}}e^{i\Theta_{-}}}{2}\right)$
$\displaystyle\psi_{n,2}$ $\displaystyle=$ $\displaystyle
ie^{ik_{0}n}\sqrt{\rho}\left(\\!\frac{\sqrt{1+m_{0}}e^{i\Theta_{+}}-\sqrt{1-m_{0}}e^{i\Theta_{-}}}{2}\right).$

Here, we explicitly evaluated the fields at $k\\!=\\!k_{0}$ and we introduced
the notation $m(k_{0})\\!=\\!m_{0}$; we also omitted the trivial dynamical
phase.

The condition in Eq. (111) then simply corresponds to having the $a_{1,2}$
orbitals equally populated in the ground state, i.e.
$|\psi_{n,1}|^{2}\\!=\\!|\psi_{n,2}|^{2}\\!=\\!\rho/2$. Without loss of
generality, we henceforth set $\Omega_{12}\\!>\\!0$, and we express the
relative phase $\varphi$ between the components $\psi_{n,2}$ and $\psi_{n,1}$
according to

$\frac{\psi_{n,2}}{\psi_{n,1}}=e^{i\varphi}=\sqrt{1-m_{0}^{2}}+im_{0},$ (113)

where we used Eqs. (111)-(LABEL:psi_original_basis). We thus obtain a simple
relation between the local relative phase (internal angle) and the ground-
state orbital polarization given in Eq. (74).

The interplay of local orbital polarization and hopping processes gives rise
to a ground-state current on a ring geometry. This can be obtained by
evaluating the current operator derived from Eq. (66),

$\displaystyle\hat{J}$ $\displaystyle=$ $\displaystyle i\frac{\Omega}{\hbar
N_{d}}\sum_{n\sigma}\left(\hat{b}^{\dagger}_{n+1,\sigma}\hat{b}_{n,\sigma}-\text{h.c.}\right)$
$\displaystyle+$ $\displaystyle\frac{\Omega_{12}}{2\hbar
N_{d}}\sum_{n\sigma}\left(\sigma\hat{b}^{\dagger}_{n+1,\sigma}\hat{b}_{n,\,\overline{\sigma}}-\sigma\hat{b}^{\dagger}_{n+1,\sigma}\hat{b}_{n,\sigma}+\text{h.c.}\right).$

In the mean-field solution, the current flowing through the ring is given by
Eq. (79), where $k_{0}$ and $m_{0}$ are still to be determined below.

Finding the minimum of Eq. (109) by imposing the self-consistent condition
given by Eq. (110) can be cumbersome from an analytical point of view. Hence,
it is useful to consider particular limits. When $\Omega_{12}\\!=\\!0$, the
minimum energy is precisely reached at $k_{0}\\!=\\!0$, which leads to two
possible degenerate polarizations $m_{0}\\!=\\!\pm 1$; this situation
corresponds to a relative phase $\varphi\\!=\\!\pm\pi/2$. A finite coupling
$\Omega_{12}\ll\Omega_{c}$ leads to a non-zero ground-state quasi-momentum
$k_{0}\approx(\Omega_{12}/2\Omega)\,\textrm{sgn}(m_{0})\approx\Phi\,\textrm{sgn}(m_{0})$
and to a small depolarization of the system [see Eq. (75)]. We recall that
$\Phi$ is the effective flux generated by the complex tunneling in Eq. (66);
see Eq. (71) and Fig. 16. Importantly, condensation at a finite quasi-momentum
activates an effective longitudinal field in the Ising picture—recall that
this field scales as $\Omega_{12}\sin(k)$—leading to a smoothing of the
transition to the unpolarized state and an eventual absence of critical
behavior. As a technical note, we remark that reaching this finite $k_{0}$
requires sufficiently large lattices satisfying
$N_{d}\\!>\\!2\pi\Omega/|\Omega_{12}|$. In any case, to lowest order in
$\Omega_{12}$, the ground-state polarization decreases according to Eq. (75).

The chiral current flowing through the ring is activated by $\Omega_{12}$; see
Eq. (B). In the mean-field ground state, the leading-order contribution is
given by

$J_{\textrm{MF}}(k_{0})\approx-\textrm{sgn}(m_{0})\rho\Omega_{12}.$ (115)

Importantly, the sign of the persistent current in Eq. (115) depends on the
orbital order that spontaneously emerges in the system. This emergent
chirality is a striking signature of the spontaneous breaking of TRS; see also
the main text.

##  Appendix C Orbital order in the strongly-correlated regime

In this Appendix, we derive the effective Ising spin model in Eq. (91) and we
obtain the chiral superfluid-to-Mott phase diagram in the strong-coupling
regime [Fig. 18]. In the absence of kinetic terms in the Hamiltonian (66),
there is a macroscopic degeneracy of $2^{N_{d}}$ possible ground-state
configurations $|\\{\sigma_{n}\\}\rangle$, which may be written as product
states

$|\\{\sigma_{n}\\}\rangle=\prod_{n=1}^{N_{d}}|\sigma_{n}\rangle,$ (116)

with $|\sigma_{n}\rangle$ the states having well defined angular momentum
along the $y$-direction, namely $J^{(n)}_{y}=\sigma_{n}\rho/2$ with
$\sigma_{n}=\pm 1$; see Eq. (89) in the main text. The corresponding ground-
state energy reads

$E^{(0)}_{N}(|\\{\sigma_{n}\\}\rangle)=U_{\xi}N(\rho-1)/2-\mu N.$ (117)

The tunneling terms in the Hamiltonian (66) do not couple these states at
first order, but they do lift their degeneracy in second-order perturbation
theory. Indeed, the first non-trivial correction to the energy of these
$N$-particle states is given by

$\Delta E(|\\{\sigma_{n}\\}\rangle)=\sum_{l}\frac{|\langle
l|\hat{H}_{T}|\\{\sigma_{n}\\}\rangle|^{2}}{E^{(0)}_{N}(|\\{\sigma_{n}\\}\rangle)-E^{(0)}_{N}(|l\rangle)},$
(118)

where $\hat{H}_{T}$ contains all the tunneling terms of Eq. (66), and where
$|l\rangle$ is an excited state. Since the Hamiltonian (66) only couples first
nearest neighbors, this expression can be further simplified as a sum of pair
contributions,

$\Delta E(|\\{\sigma_{n}\\}\rangle)=\sum_{n}\Delta
E(|\sigma_{n}\rangle|\sigma_{n+1}\rangle).$ (119)

The energy corrections for each pair are readily obtained as

$\displaystyle\Delta E(|+\rangle|+\rangle)=\Delta E(|-\rangle|-\rangle)$
$\displaystyle\hskip
28.45274pt=-\frac{2|t_{\sigma}|^{2}\rho(\rho+1)}{U_{\xi}}-\frac{2|t_{\sigma\overline{\sigma}}|^{2}\rho}{[\rho(W_{\xi}-U_{\xi})+U_{\xi}]},$
$\displaystyle\Delta E(|+\rangle|-\rangle)=\Delta E(|+\rangle|-\rangle)$
$\displaystyle\hskip
28.45274pt=-\frac{2|t_{\sigma\overline{\sigma}}|^{2}\rho(\rho+1)}{U_{\xi}}-\frac{2|t_{\sigma}|^{2}\rho}{[\rho(W_{\xi}-U_{\xi})+U_{\xi}]}.$

We note that this approach is valid whenever $\xi\\!<\\!4$, which ensures
repulsive intraspecies interactions $U_{\xi}>0$ in Eq. (66). The correction to
the energy of the manifold of states given by Eq. (116), up to second order,
can hence be expressed as a constant shift (which is independent of the
configuration $\\{\sigma_{n}\\}$) plus an orbital exchange interaction,

$\Delta
E(|\\{\sigma_{n}\\}\rangle)=\sum_{n}E_{0}+K_{yy}\sum_{n}J_{y}^{(n)}J^{(n+1)}_{y},$
(120)

with $J_{y}^{(n)}=\sigma_{n}\rho/2$. The shift and exchange coupling are given
by

$\displaystyle E_{0}$ $\displaystyle=$
$\displaystyle-\frac{\left(|t_{\sigma}|^{2}+|t_{\sigma\overline{\sigma}}|^{2}\right)\rho[W_{\xi}\rho(\rho+1)-U_{\xi}(\rho^{2}-2)]}{U_{\xi}[U_{\xi}+\rho(W_{\xi}-U_{\xi})]}$
$\displaystyle K_{yy}$ $\displaystyle=$
$\displaystyle-\frac{4\left(|t_{\sigma}|^{2}-|t_{\sigma\overline{\sigma}}|^{2}\right)[W_{\xi}+\rho(W_{\xi}-U_{\xi})]}{U_{\xi}[U_{\xi}+\rho(W_{\xi}-U_{\xi})]}<0.$

These results lead to the effective Ising spin model displayed in Eq. (91). We
remark that the exchange coupling at this order only depends on the tunneling
$\Omega$ and that it favors a uniform ‘ferromagnetic’ ordering. In a one-
dimensional ring geometry, the approximated ground-state energy can then be
expressed as

$E_{\textrm{GS}}(N)=E^{(0)}_{N}+N_{d}\left(E_{0}+K_{yy}\frac{\rho^{2}}{4}\right).$
(122)

With the aim of comparing this analytical model with a more accurate numerical
tool, we performed DMRG simulations to analyze the evolution of the ground-
state orbital polarization within the chiral-Mott phase. The results are
presented in Fig. 21 for both $\Omega_{12}=0$ and $\Omega_{12}/\Omega=1$. The
orbital order is practically unaltered by the existence of a finite
$\Omega_{12}$, in agreement with what we expect from the effective spin model:
the exchange coupling constant $K_{yy}$ in Eq.
(LABEL:coupling_constant_appendix) does not depend on $\Omega_{12}$. On the
other hand, as the system gets closer to the superfluid phase, the presence of
this hopping term destabilizes the angular momentum ordering.

Figure 21: Evolution of the averaged ground-state orbital polarization
$\overline{m}_{0}$, within the Mott regime, for $\rho\\!=\\!2$ and
$\Omega_{12}=0$ (red) and $\Omega_{12}/\Omega=1$ (blue). The points were
obtained using a DMRG algorithm in a chain with open boundary conditions and
$N_{d}=100$ dimers.

The perturbative expansion described above can be further used to elucidate
the boundaries between the (chiral) Mott and superfluid phases, by
additionally considering how the ground state energy changes by adding or
removing a particle in the system: the phase boundary precisely occurs when
this particle-hole excitation gap vanishes. Since we are interested in the
regime $W_{\xi}>U_{\xi}$, the relevant low-energy manifold to consider upon
adding one extra particle is given by $N_{d}$ states of the form

$|n_{N+1}\rangle=\frac{\left(\hat{b}^{\dagger}_{1,\sigma}\right)^{\rho}\ldots\left(\hat{b}^{\dagger}_{n,\sigma}\right)^{\rho+1}\ldots\left(\hat{b}^{\dagger}_{N_{d},\sigma}\right)^{\rho}}{\sqrt{(\rho!)^{N_{d}-1}(\rho+1)!}}|0\rangle,$
(123)

so that the state $|n_{N+1}\rangle$ has one more boson in the $n-$th dimer of
a ferromagnetic state with $\sigma$-order. The unperturbed energy of these
states is

$E_{N+1}^{(0)}(\\{|n_{N+1}\rangle\\})=U_{\xi}N(\rho-1)/2+U_{\xi}\rho-\mu(N+1).$
(124)

The ground-state energy can then be approximately found via a canonical
transformation procedure Slichter1990 , in which an effective Hamiltonian
$\hat{H}^{{}^{\prime}}$ that takes into account up to second order processes
in the tunneling amplitudes is defined within the manifold of the original set
of $\\{|n\rangle_{N+1}\\}$-states. The matrix elements of $\hat{H}^{\prime}$
are determined as

$\displaystyle\langle n_{N+1}|\hat{H}^{{}^{\prime}}|n^{\prime}_{N+1}\rangle$
$\displaystyle=$ $\displaystyle E^{(0)}_{N+1}+\langle
n_{N+1}|\hat{H}_{T}|n^{\prime}_{N+1}\rangle$ $\displaystyle+$
$\displaystyle\sum_{l}\frac{\langle n_{N+1}|\hat{H}_{T}|l\rangle\langle
l|\hat{H}_{T}|n^{\prime}_{N+1}\rangle}{E^{(0)}_{N+1}-E_{N+1}^{(0)}(|l\rangle)},$

with $|l\rangle$ the excited states and $E_{N+1}^{(0)}(|l\rangle)$ their
corresponding unperturbed energy. The $N_{d}\times N_{d}$ matrix has a
tridiagonal form and can be analytically diagonalized, revealing the splitting
of the degenerate states of Eq. (123) into a band described by

$\displaystyle E^{\prime}_{N+1}(k)$ $\displaystyle=$ $\displaystyle
E^{(0)}_{N+1}+\Sigma_{N+1}-2|t_{\sigma}|(\rho+1)\cos(k-\Phi_{\sigma})$ (126)
$\displaystyle-$
$\displaystyle\frac{2(\rho+1)|t_{\sigma\overline{\sigma}}|^{2}}{(W_{\xi}-U_{\xi})\rho}\cos(2k)$
$\displaystyle-$
$\displaystyle\frac{2\rho(\rho+1)|t_{\sigma}|^{2}}{U_{\xi}}\cos(2(k-\Phi_{\sigma})),$

with

$\displaystyle\Sigma_{N+1}$ $\displaystyle=$
$\displaystyle-\frac{|t_{\sigma}|^{2}\rho(\rho+2)}{U_{\xi}}-\frac{2\rho|t_{\sigma\overline{\sigma}}|^{2}}{\rho(W_{\xi}-U_{\xi})+W_{\xi}+U_{\xi}}$
(127) $\displaystyle-$
$\displaystyle\frac{2(\rho+1)|t_{\sigma\overline{\sigma}}|^{2}}{\rho(W_{\xi}-U_{\xi})}-\frac{2(N_{d}-2)\rho(\rho+1)|t_{\sigma}|^{2}}{U_{\xi}}$
$\displaystyle-$
$\displaystyle\frac{2(N_{d}-2)\rho|t_{\sigma\overline{\sigma}}|^{2}}{\rho(W_{\xi}-U_{\xi})+U_{\xi}}.$

In Eq. (126), we introduced the flux
$\Phi_{\sigma}\\!=\\!\sigma\arctan(\Omega_{12}/2\Omega)$ and the quasi-
momentum $k=2\pi j/N_{d}$, with $j=0,\ldots,N_{d}-1$. Interestingly, at this
order, the extra boson feels the presence of the flux $\Phi_{\sigma}$ in the
lattice via effective hopping processes at first and second nearest neighbors.

In the thermodynamic limit, the minimum energy of this band will be precisely
at $k=\Phi_{\sigma}$, so that the ground-state energy with one extra boson is
obtained as

$E_{\textrm{GS}}(N+1)=E^{\prime}_{N+1}(\Phi_{\sigma}).$ (128)

We follow a similar procedure to consider the effect of the hopping terms in
the low-energy manifold with one boson less. When removing one particle from
the $\sigma$-ordered $N$-particle ground state, the relevant manifold of
states where the perturbation theory should be applied will be given by

$|n_{N-1}\rangle=\frac{\left(\hat{b}^{\dagger}_{1,\sigma}\right)^{\rho}\ldots\left(\hat{b}^{\dagger}_{n,\sigma}\right)^{\rho-1}\ldots\left(\hat{b}^{\dagger}_{N_{d},\sigma}\right)^{\rho}}{\sqrt{(\rho!)^{N_{d}-1}(\rho-1)!}}|0\rangle,$
(129)

which have a zero-th order energy of

$E_{N-1}^{(0)}(\\{|n_{N-1}\rangle\\})=U_{\xi}N(\rho-1)/2-U_{\xi}(\rho-1)-\mu(N-1).$
(130)

By diagonalizing the corresponding canonically transformed Hamiltonian, we
obtain the broadening of these states into a band described by

$\displaystyle E^{\prime}_{N-1}(k)$ $\displaystyle=$ $\displaystyle
E^{(0)}_{N-1}+\Sigma_{N-1}-2|t_{\sigma}|\rho\cos(k+\Phi_{\sigma})$ (131)
$\displaystyle-$
$\displaystyle\frac{2\rho|t_{\sigma\overline{\sigma}}|^{2}}{(W_{\xi}-U_{\xi})\rho+U_{\xi}}\cos(2k)$
$\displaystyle-$
$\displaystyle\frac{2\rho(\rho+1)|t_{\sigma}|^{2}}{U_{\xi}}\cos(2(k+\Phi_{\sigma})),$

with

$\displaystyle\Sigma_{N-1}$ $\displaystyle=$
$\displaystyle-\frac{|t_{\sigma}|^{2}(\rho^{2}-1)}{U_{\xi}}-\frac{2\rho|t_{\sigma\overline{\sigma}}|^{2}}{\rho(W_{\xi}-U_{\xi})+U_{\xi}-W_{\xi}}$
(132) $\displaystyle-$
$\displaystyle\frac{2(\rho-1)|t_{\sigma\overline{\sigma}}|^{2}}{\rho(W_{\xi}-U_{\xi})+2U_{\xi}}-\frac{2(N_{d}-2)\rho(\rho+1)|t_{\sigma}|^{2}}{U_{\xi}}$
$\displaystyle-$
$\displaystyle\frac{2(N_{d}-2)\rho|t_{\sigma\overline{\sigma}}|^{2}}{\rho(W_{\xi}-U_{\xi})+U_{\xi}}.$

Note that the hole-like excitation feels the opposite flux ($-\Phi_{\sigma}$).
We then find that, in the thermodynamic limit, the ground-state energy with
one boson less is given by

$E_{\textrm{GS}}(N-1)=E^{\prime}_{N-1}(-\Phi_{\sigma}).$ (133)

The phase boundary between the (chiral) Mott insulator and superfluid phases
is determined by the conditions

$\displaystyle E_{\textrm{GS}}(N+1)\\!=\\!E_{\textrm{GS}}(N),$ $\displaystyle
E_{\textrm{GS}}(N)\\!=\\!E_{\textrm{GS}}(N-1).$

Solving these equations separately for $\mu$, we find the boundaries for the
particle sector ($\mu_{+}$) and hole sector ($\mu_{-}$), which are displayed
in Eq. (LABEL:Mott_lobe). The difference $\mu_{+}-\mu_{-}$ determines the
charge gap in the Mott phase. The resulting Mott-SF phase diagram is depicted
in Fig. 18, in terms of the chemical potential $\mu/U_{\xi}$ and ratio
$\Omega/U_{\xi}$.

##  Appendix D Strong-coupling expansion
and the effective Hamiltonian for filling $\rho=1$

At unit filling, the spin-states in Eq. (89) are coupled to each other via
spin-flip processes, which also scale as the square of the tunneling
amplitudes. In order to describe the corresponding low-energy manifold with an
effective theory, we must perform a canonical transformation procedure and
project the resulting effective Hamiltonian into the subspace of unit filling.
Since the Hamiltonian only couples nearest neighbors, we can focus on the
effective theory for just two dimers ($n$ and $n+1$) and then sum over all the
lattice links connecting them. For simplicity, we will work in the basis of
the original orbitals $a_{n,s}$ in each dimer (with $s=1,2$).

In the absence of hopping terms, there are four possible degenerate states
with 2 particles in the nearest neighbor dimer configuration that satisfy the
unit filling condition. Their corresponding energy is given by
$E^{(0)}_{2}=-2\mu$ and they can be expressed as:

$|s_{n}s^{\prime}_{n+1}\rangle=\hat{a}^{\dagger}_{n,s}\hat{a}^{\dagger}_{n+1,s^{\prime}}|0\rangle,$
(134)

where we introduced the notation
$|s_{n}\rangle=\hat{a}^{\dagger}_{n,s}|0\rangle$ with $s=1,2$.

The matrix elements of the canonically transformed Hamiltonian
$\hat{H}^{\prime}$ within this subspace are given by

$\displaystyle\langle
s_{n}s^{\prime}_{n+1}|\hat{H}^{\prime}|s^{\prime\prime}_{n}s^{\prime\prime\prime}_{n+1}\rangle\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!E_{2}^{(0)}+\langle
s_{n}s^{\prime}_{n+1}|\hat{H}_{T}|s^{\prime\prime}_{n}s^{\prime\prime\prime}_{n+1}\rangle$
$\displaystyle+$ $\displaystyle\\!\\!\\!\sum_{l}\frac{\langle
s_{n}s^{\prime}_{n+1}|\hat{H}_{T}|l\rangle\langle
l|\hat{H}_{T}|s^{\prime\prime}_{n}s^{\prime\prime\prime}_{n+1}\rangle}{E^{(0)}_{2}-E_{2}^{(0)}(|l\rangle)},$

with $|l\rangle$ the excited states and $E_{2}^{(0)}(|l\rangle)$ their
unperturbed energy. In particular, the states in Eq. (134) are coupled through
$\hat{H}_{T}$ to the following high-energy virtual states:

$\displaystyle|1\rangle\\!$
$\displaystyle=\\!\frac{1}{\sqrt{2}}\\!\left(\hat{a}^{\dagger
2}_{n,1}+\hat{a}^{\dagger 2}_{n,2}\right)\\!\\!|0\rangle$
$\displaystyle|2\rangle\\!$
$\displaystyle=\\!\frac{1}{\sqrt{2}}\\!\left(\hat{a}^{\dagger
2}_{n+1,1}+\hat{a}^{\dagger 2}_{n+1,2}\right)\\!\\!|0\rangle$
$\displaystyle|3\rangle\\!$
$\displaystyle=\\!\frac{1}{\sqrt{2}}\\!\left(\hat{a}^{\dagger
2}_{n,1}-\hat{a}^{\dagger 2}_{n,2}\right)\\!\\!|0\rangle$
$\displaystyle|4\rangle\\!$
$\displaystyle=\\!\frac{1}{\sqrt{2}}\\!\left(\hat{a}^{\dagger
2}_{n+1,1}-\hat{a}^{\dagger 2}_{n+1,2}\right)\\!\\!|0\rangle$
$\displaystyle|5\rangle\\!$
$\displaystyle=\\!\hat{a}^{\dagger}_{n,1}\hat{a}^{\dagger}_{n,2}|0\rangle$
$\displaystyle|6\rangle\\!$
$\displaystyle=\\!\hat{a}^{\dagger}_{n+1,1}\hat{a}^{\dagger}_{n+1,2}|0\rangle,$
(136)

where the first two states have energy $W_{\xi}-2\mu$ and the others
$U_{\xi}-2\mu$.

Neglecting irrelevant constant energy shifts, we can express the projected
canonically transformed Hamiltonian in the following form

$\displaystyle\hat{H}^{\prime}_{n,n+1}\\!\\!$
$\displaystyle=\\!\\!\\!\\!\sum_{\begin{subarray}{c}s,s^{\prime}\\\
s^{\prime\prime},s^{\prime\prime\prime}\end{subarray}}\\!\\!\langle
s_{n}s^{\prime}_{n+1}|\hat{H}^{\prime}|s^{\prime\prime}_{n}s^{\prime\prime\prime}_{n+1}\rangle\hat{a}^{\dagger}_{n,s}\hat{a}^{\dagger}_{n+1,s^{\prime}}\hat{a}_{n,s^{\prime\prime}}\hat{a}_{n+1,s^{\prime\prime\prime}}$
$\displaystyle=\sum_{s}\left(\gamma_{ss}\hat{n}_{n,s}\hat{n}_{n+1,s}+\gamma_{s\overline{s}}\hat{n}_{n,s}\hat{n}_{n+1,\overline{s}}\right)$
(137)
$\displaystyle+\Gamma_{1}\sum_{s}\hat{a}^{\dagger}_{n,s}\hat{a}^{\dagger}_{n+1,s}\hat{a}_{n,\overline{s}}\hat{a}_{n+1,\overline{s}}$
$\displaystyle+\Gamma_{2}\sum_{s}\hat{a}^{\dagger}_{n,s}\hat{a}^{\dagger}_{n+1,\overline{s}}\hat{a}_{n,\overline{s}}\hat{a}_{n+1,s}$
$\displaystyle+\\!\\!(\hat{a}^{\dagger}_{n,1}\hat{a}_{n,2}+\hat{a}^{\dagger}_{n,2}\hat{a}_{n,1})(\Gamma_{3}\hat{n}_{n+1,1}+\Gamma_{4}\hat{n}_{n+1,2})$
$\displaystyle+\\!\\!(\hat{a}^{\dagger}_{n+1,1}\hat{a}_{n+1,2}+\hat{a}^{\dagger}_{n+1,2}\hat{a}_{n+1,1})(\Gamma_{5}\hat{n}_{n,1}+\Gamma_{6}\hat{n}_{n,2}),$

where $\overline{1}=2$ and $\overline{2}=1$. We note that the tunneling
Hamiltonian effectively leads to the emergence of correlated hopping terms
within each dimer (the last two terms in Eq. (137)), which are absent in
p-band bosonic models; see Ref. Pinheiro2013 . The corresponding coupling
constants are explicitly given by

$\displaystyle\gamma_{11}$
$\displaystyle=-2\Omega^{2}\left(\frac{1}{U_{\xi}}+\frac{1}{W_{\xi}}\right)-\frac{\Omega_{12}^{2}}{U_{\xi}}$
$\displaystyle\gamma_{12}$ $\displaystyle=-\frac{2\Omega^{2}}{U_{\xi}}$ (138)
$\displaystyle\gamma_{21}$
$\displaystyle=-2\Omega_{12}^{2}\left(\frac{1}{W_{\xi}}+\frac{1}{U_{\xi}}\right)-\frac{2\Omega^{2}}{U_{\xi}}$
$\displaystyle\gamma_{22}$ $\displaystyle=\gamma_{11}$
$\displaystyle\Gamma_{1}$
$\displaystyle=2\Omega^{2}\left(\frac{1}{U_{\xi}}-\frac{1}{W_{\xi}}\right)$
$\displaystyle\Gamma_{2}$ $\displaystyle=-\frac{2\Omega^{2}}{U_{\xi}}$
$\displaystyle\Gamma_{3}$
$\displaystyle=-\Omega\Omega_{12}\left(\frac{2}{W_{\xi}}+\frac{1}{U_{\xi}}\right)$
$\displaystyle\Gamma_{4}$ $\displaystyle=-\frac{\Omega\Omega_{12}}{U_{\xi}}$
$\displaystyle\Gamma_{5}$ $\displaystyle=\Gamma_{4}$ $\displaystyle\Gamma_{6}$
$\displaystyle=\Gamma_{3}.$

By further employing the Schwinger angular momentum representation [see Eq.
(52) in the main text] and the constraint
$\hat{N}^{(n)}=\hat{n}_{n,1}+\hat{n}_{n,2}=1$, the effective Hamiltonian at
each link can be expressed as

$\displaystyle\hat{H}^{\prime}_{n,n+1}$ $\displaystyle=$
$\displaystyle\sum_{\nu=x,y,z}K_{\nu\nu}\hat{J}_{\nu}^{(n)}\hat{J}_{\nu}^{(n+1)}$
$\displaystyle+$
$\displaystyle\left(\frac{\gamma_{22}-\gamma_{11}}{2}\right)\left(\hat{J}_{z}^{(n)}+\hat{J}_{z}^{(n+1)}\right)$
$\displaystyle+$
$\displaystyle\left(\frac{\gamma_{21}-\gamma_{12}}{2}\right)\left(\hat{J}_{z}^{(n)}-\hat{J}_{z}^{(n+1)}\right)$
$\displaystyle-$ $\displaystyle
D\left(\hat{J}_{z}^{(n)}\hat{J}_{x}^{(n+1)}-\hat{J}_{x}^{(n)}\hat{J}_{z}^{(n+1)}\right)$
$\displaystyle+$
$\displaystyle(\Gamma_{3}+\Gamma_{4})\hat{J}_{x}^{(n)}+(\Gamma_{5}+\Gamma_{6})\hat{J}_{x}^{(n+1)},$

with $K_{xx}=2(\Gamma_{1}+\Gamma_{2})$, $K_{yy}=2(\Gamma_{2}-\Gamma_{1})$,
$K_{zz}=\gamma_{11}+\gamma_{22}-\gamma_{12}-\gamma_{21}$ and
$D=2(\Gamma_{4}-\Gamma_{3})$. The magnitudes of these couplings, expressed in
terms of the original hopping and interaction parameters of the model, are
provided in Eq. (95) in the main text.

The effective spin-$1/2$ Hamiltonian in the one-dimensional lattice is finally
written as

$\displaystyle\hat{H}^{\textrm{eff}}_{1/2}$
$\displaystyle=\sum_{n}\hat{H}^{\prime}_{n,n+1}$
$\displaystyle=\sum_{n}\sum_{\nu=x,y,z}K_{\nu\nu}\hat{J}_{\nu}^{(n)}\hat{J}_{\nu}^{(n+1)}+h_{x}\sum_{n}\hat{J}_{x}^{(n)}$
$\displaystyle\hskip
5.69046pt-\bm{D}\sum_{n}\left(\hat{\bm{J}}^{(n)}\times\hat{\bm{J}}^{(n+1)}\right),$
(140)

where $\bm{D}=(0,D,0)$ and
$h_{x}=\Gamma_{3}+\Gamma_{4}+\Gamma_{5}+\Gamma_{6}$.

## References

  * (1) S. Haroche, C. Cohen-Tannoudji, C. Audoin, and J. Schermann, “Modified zeeman hyperfine spectra observed in h 1 and rb 87 ground states interacting with a nonresonant rf field,” Physical Review Letters, vol. 24, no. 16, p. 861, 1970.
  * (2) P. Avan, C. Cohen-Tannoudji, J. Dupont-Roc, and C. Fabre, “Effect of high frequency irradiation on the dynamical properties of weakly bound electrons,” Journal de Physique, vol. 37, no. 9, pp. 993–1009, 1976.
  * (3) C. Raab, J. Eschner, J. Bolle, H. Oberst, F. Schmidt-Kaler, and R. Blatt, “Motional sidebands and direct measurement of the cooling rate in the resonance fluorescence of a single trapped ion,” Physical Review Letters, vol. 85, no. 3, p. 538, 2000.
  * (4) H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, “Dynamical control of matter-wave tunneling in periodic potentials,” Physical review letters, vol. 99, no. 22, p. 220403, 2007\.
  * (5) A. Eckardt, “Colloquium: Atomic quantum gases in periodically driven optical lattices,” Reviews of Modern Physics, vol. 89, no. 1, p. 011004, 2017.
  * (6) T. Oka and S. Kitamura, “Floquet engineering of quantum materials,” Annual Review of Condensed Matter Physics, vol. 10, pp. 387–408, 2019.
  * (7) M. S. Rudner and N. H. Lindner, “Band structure engineering and non-equilibrium dynamics in floquet topological insulators,” Nature reviews physics, vol. 2, no. 5, pp. 229–244, 2020.
  * (8) C. Weitenberg and J. Simonet, “Tailoring quantum gases by floquet engineering,” Nature Physics, vol. 17, no. 12, pp. 1342–1348, 2021.
  * (9) I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Reviews of Modern Physics, vol. 86, no. 1, p. 153, 2014.
  * (10) E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Demler, C. Chin, B. DeMarco, S. E. Economou, M. A. Eriksson, K.-M. C. Fu, et al., “Quantum simulators: Architectures and opportunities,” PRX Quantum, vol. 2, no. 1, p. 017003, 2021.
  * (11) G. Salerno, T. Ozawa, H. M. Price, and I. Carusotto, “Floquet topological system based on frequency-modulated classical coupled harmonic oscillators,” Physical Review B, vol. 93, no. 8, p. 085105, 2016.
  * (12) R. Fleury, A. B. Khanikaev, and A. Alu, “Floquet topological insulators for sound,” Nature communications, vol. 7, no. 1, pp. 1–11, 2016.
  * (13) M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature, vol. 496, no. 7444, pp. 196–200, 2013.
  * (14) N. Schine, A. Ryou, A. Gromov, A. Sommer, and J. Simon, “Synthetic landau levels for photons,” Nature, vol. 534, no. 7609, pp. 671–675, 2016.
  * (15) P. Roushan, C. Neill, A. Megrant, Y. Chen, R. Babbush, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, et al., “Chiral ground-state currents of interacting photons in a synthetic magnetic field,” Nature Physics, vol. 13, no. 2, pp. 146–151, 2017.
  * (16) T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, et al., “Topological photonics,” Reviews of Modern Physics, vol. 91, no. 1, p. 015006, 2019\.
  * (17) N. Goldman and J. Dalibard, “Periodically driven quantum systems: effective hamiltonians and engineered gauge fields,” Physical review X, vol. 4, no. 3, p. 031027, 2014.
  * (18) M. Aidelsburger, S. Nascimbene, and N. Goldman, “Artificial gauge fields in materials and engineered systems,” Comptes Rendus Physique, vol. 19, no. 6, pp. 394–432, 2018.
  * (19) P. Berman et al., “Manipulating the speed of sound in a two-component bose-einstein condensate,” Physical Review A, vol. 63, no. 4, p. 043612, 2001.
  * (20) H. Saito and M. Ueda, “Dynamically stabilized bright solitons in a two-dimensional bose-einstein condensate,” Physical review letters, vol. 90, no. 4, p. 040403, 2003.
  * (21) R. Ma, M. E. Tai, P. M. Preiss, W. S. Bakr, J. Simon, and M. Greiner, “Photon-assisted tunneling in a biased strongly correlated bose gas,” Physical Review Letters, vol. 107, no. 9, p. 095301, 2011.
  * (22) Á. Rapp, X. Deng, and L. Santos, “Ultracold lattice gases with periodically modulated interactions,” Physical review letters, vol. 109, no. 20, p. 203005, 2012.
  * (23) A. Ajoy and P. Cappellaro, “Quantum simulation via filtered hamiltonian engineering: Application to perfect quantum transport in spin networks,” Physical review letters, vol. 110, no. 22, p. 220503, 2013.
  * (24) M. Di Liberto, C. E. Creffield, G. Japaridze, and C. M. Smith, “Quantum simulation of correlated-hopping models with fermions in optical lattices,” Physical Review A, vol. 89, no. 1, p. 013624, 2014.
  * (25) A. J. Daley and J. Simon, “Effective three-body interactions via photon-assisted tunneling in an optical lattice,” Physical Review A, vol. 89, no. 5, p. 053619, 2014.
  * (26) M. Bukov, L. D’Alessio, and A. Polkovnikov, “Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to floquet engineering,” Advances in Physics, vol. 64, no. 2, pp. 139–226, 2015.
  * (27) A. Eckardt and E. Anisimovas, “High-frequency approximation for periodically driven quantum systems from a floquet-space perspective,” New journal of physics, vol. 17, no. 9, p. 093039, 2015.
  * (28) E. Anisimovas, G. Žlabys, B. M. Anderson, G. Juzeliūnas, and A. Eckardt, “Role of real-space micromotion for bosonic and fermionic floquet fractional chern insulators,” Physical Review B, vol. 91, no. 24, p. 245135, 2015.
  * (29) F. Meinert, M. J. Mark, K. Lauber, A. J. Daley, and H.-C. Nägerl, “Floquet engineering of correlated tunneling in the bose-hubbard model with ultracold atoms,” Physical review letters, vol. 116, no. 20, p. 205301, 2016.
  * (30) C.-L. Hung, A. González-Tudela, J. I. Cirac, and H. Kimble, “Quantum spin dynamics with pairwise-tunable, long-range interactions,” Proceedings of the National Academy of Sciences, vol. 113, no. 34, pp. E4946–E4955, 2016\.
  * (31) G. Pieplow, F. Sols, and C. E. Creffield, “Generation of atypical hopping and interactions by kinetic driving,” New Journal of Physics, vol. 20, no. 7, p. 073045, 2018.
  * (32) C. H. Lee, W. W. Ho, B. Yang, J. Gong, and Z. Papić, “Floquet mechanism for non-abelian fractional quantum hall states,” Physical review letters, vol. 121, no. 23, p. 237401, 2018.
  * (33) J. Choi, H. Zhou, H. S. Knowles, R. Landig, S. Choi, and M. D. Lukin, “Robust dynamic hamiltonian engineering of many-body spin systems,” Physical Review X, vol. 10, no. 3, p. 031002, 2020.
  * (34) L. Barbiero, L. Chomaz, S. Nascimbene, and N. Goldman, “Bose-hubbard physics in synthetic dimensions from interaction trotterization,” Physical Review Research, vol. 2, no. 4, p. 043340, 2020.
  * (35) H. Dehghani, M. Hafezi, and P. Ghaemi, “Light-induced topological superconductivity via floquet interaction engineering,” Physical Review Research, vol. 3, no. 2, p. 023039, 2021.
  * (36) S. Geier, N. Thaicharoen, C. Hainaut, T. Franz, A. Salzinger, A. Tebben, D. Grimshandl, G. Zürn, and M. Weidemüller, “Floquet hamiltonian engineering of an isolated many-body spin system,” Science, vol. 374, no. 6571, pp. 1149–1152, 2021.
  * (37) H. Zahn, V. Singh, M. Kosch, L. Asteria, L. Freystatzky, K. Sengstock, L. Mathey, and C. Weitenberg, “Formation of spontaneous density-wave patterns in dc driven lattices,” Physical Review X, vol. 12, no. 2, p. 021014, 2022.
  * (38) J. Sanz, A. Frölian, C. Chisholm, C. Cabrera, and L. Tarruell, “Interaction control and bright solitons in coherently coupled bose-einstein condensates,” Physical Review Letters, vol. 128, no. 1, p. 013201, 2022\.
  * (39) D. Fausti, R. Tobey, N. Dean, S. Kaiser, A. Dienst, M. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi, and A. Cavalleri, “Light-induced superconductivity in a stripe-ordered cuprate,” science, vol. 331, no. 6014, pp. 189–191, 2011.
  * (40) M. Mitrano, A. Cantaluppi, D. Nicoletti, S. Kaiser, A. Perucchi, S. Lupi, P. Di Pietro, D. Pontiroli, M. Riccò, S. R. Clark, et al., “Possible light-induced superconductivity in k3c60 at high temperature,” Nature, vol. 530, no. 7591, pp. 461–464, 2016.
  * (41) J. Struck, C. Ölschläger, R. Le Targat, P. Soltan-Panahi, A. Eckardt, M. Lewenstein, P. Windpassinger, and K. Sengstock, “Quantum simulation of frustrated classical magnetism in triangular optical lattices,” Science, vol. 333, no. 6045, pp. 996–999, 2011.
  * (42) J. Struck, M. Weinberg, C. Ölschläger, P. Windpassinger, J. Simonet, K. Sengstock, R. Höppner, P. Hauke, A. Eckardt, M. Lewenstein, et al., “Engineering ising-xy spin-models in a triangular lattice using tunable artificial gauge fields,” Nature Physics, vol. 9, no. 11, pp. 738–743, 2013.
  * (43) F. Görg, M. Messer, K. Sandholzer, G. Jotzu, R. Desbuquois, and T. Esslinger, “Enhancement and sign change of magnetic correlations in a driven quantum many-body system,” Nature, vol. 553, no. 7689, pp. 481–485, 2018.
  * (44) P. Ponte, Z. Papić, F. Huveneers, and D. A. Abanin, “Many-body localization in periodically driven systems,” Physical review letters, vol. 114, no. 14, p. 140401, 2015.
  * (45) D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, “Colloquium: Many-body localization, thermalization, and entanglement,” Reviews of Modern Physics, vol. 91, no. 2, p. 021001, 2019.
  * (46) W. K. Hensinger, H. Häffner, A. Browaeys, N. R. Heckenberg, K. Helmerson, C. McKenzie, G. J. Milburn, W. D. Phillips, S. L. Rolston, H. Rubinsztein-Dunlop, et al., “Dynamical tunnelling of ultracold atoms,” Nature, vol. 412, no. 6842, pp. 52–55, 2001.
  * (47) M. Arnal, G. Chatelain, M. Martinez, N. Dupont, O. Giraud, D. Ullmo, B. Georgeot, G. Lemarié, J. Billy, and D. Guéry-Odelin, “Chaos-assisted tunneling resonances in a synthetic floquet superlattice,” Science advances, vol. 6, no. 38, p. eabc4886, 2020.
  * (48) L. Barbiero, C. Schweizer, M. Aidelsburger, E. Demler, N. Goldman, and F. Grusdt, “Coupling ultracold matter to dynamical gauge fields in optical lattices: From flux attachment to $Z2$ lattice gauge theories,” Science advances, vol. 5, no. 10, p. eaav7444, 2019.
  * (49) C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero, E. Demler, N. Goldman, I. Bloch, and M. Aidelsburger, “Floquet approach to $Z_{2}$ lattice gauge theories with ultracold atoms in optical lattices,” Nature Physics, vol. 15, no. 11, pp. 1168–1173, 2019.
  * (50) A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 43, no. 16, p. 163001, 2010.
  * (51) S. Mukherjee, A. Spracklen, M. Valiente, E. Andersson, P. Öhberg, N. Goldman, and R. R. Thomson, “Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice,” Nature communications, vol. 8, no. 1, pp. 1–7, 2017.
  * (52) L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, “Observation of photonic anomalous floquet topological insulators,” Nature communications, vol. 8, no. 1, pp. 1–7, 2017.
  * (53) S. Mukherjee, H. K. Chandrasekharan, P. Öhberg, N. Goldman, and R. R. Thomson, “State-recycling and time-resolved imaging in topological photonic lattices,” Nature communications, vol. 9, no. 1, pp. 1–6, 2018.
  * (54) S. Mukherjee and M. C. Rechtsman, “Observation of floquet solitons in a topological bandgap,” Science, vol. 368, no. 6493, pp. 856–859, 2020.
  * (55) S. Mukherjee and M. C. Rechtsman, “Observation of unidirectional solitonlike edge states in nonlinear floquet topological insulators,” Physical Review X, vol. 11, no. 4, p. 041057, 2021.
  * (56) E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, “Photonic topological insulator in synthetic dimensions,” Nature, vol. 567, no. 7748, pp. 356–360, 2019.
  * (57) S. Mukherjee, M. Di Liberto, P. Öhberg, R. R. Thomson, and N. Goldman, “Experimental observation of aharonov-bohm cages in photonic lattices,” Physical review letters, vol. 121, no. 7, p. 075502, 2018.
  * (58) L. Yuan, Q. Lin, M. Xiao, and S. Fan, “Synthetic dimension in photonics,” Optica, vol. 5, no. 11, pp. 1396–1405, 2018.
  * (59) A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. Miller, and S. Fan, “Experimental band structure spectroscopy along a synthetic dimension,” Nature communications, vol. 10, no. 1, pp. 1–8, 2019.
  * (60) A. Dutt, Q. Lin, L. Yuan, M. Minkov, M. Xiao, and S. Fan, “A single photonic cavity with two independent physical synthetic dimensions,” Science, vol. 367, no. 6473, pp. 59–64, 2020.
  * (61) A. Balčytis, T. Ozawa, Y. Ota, S. Iwamoto, J. Maeda, and T. Baba, “Synthetic dimension band structures on a si cmos photonic platform,” Science Advances, vol. 8, no. 4, pp. eabk0468, 2022.
  * (62) N. Englebert, N. Goldman, M. Erkintalo, N. Mostaan, S.-P. Gorza, F. Leo, and J. Fatome, “Bloch oscillations of coherently driven dissipative solitons in a synthetic dimension,” Nature Physics, vol. 19, pp. 1014-1021 (2023).
  * (63) L. W. Clark, N. Jia, N. Schine, C. Baum, A. Georgakopoulos, and J. Simon, “Interacting floquet polaritons,” Nature, vol. 571, no. 7766, pp. 532–536, 2019.
  * (64) C. H. Johansen, J. Lang, A. Morales, A. Baumgärenter, T. Donner, and F. Piazza, “Multimode-polariton superradiance via Floquet engineering,” SciPost Physics, vol. 12, pp. 094, 2022.
  * (65) J.-Y. Shan, M. Ye, H. Chu, S. Lee, J.-G. Park, L. Balents, and D. Hsieh, “Giant modulation of optical nonlinearity by floquet engineering,” Nature, vol. 600, no. 7888, pp. 235–239, 2021.
  * (66) Y. Zhang, J. C. Curtis, C. S. Wang, R. Schoelkopf, and S. Girvin, “Drive-induced nonlinearities of cavity modes coupled to a transmon ancilla,” Physical Review A, vol. 105, no. 2, p. 022423, 2022.
  * (67) G. Agrawal, Applications of nonlinear fiber optics.  Elsevier, 2001.
  * (68) M. Delqué, G. Fanjoux, and T. Sylvestre, “Polarization dynamics of the fundamental vector soliton of isotropic kerr media,” Physical Review E, vol. 75, no. 1, p. 016611, 2007.
  * (69) G. Agrawal, Nonlinear fiber optics.  Elsevier, 2012.
  * (70) Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Physical Review Letters, vol. 118, no. 3, p. 033901, 2017.
  * (71) L. Hill, G.-L. Oppo, M. T. Woodley, and P. Del’Haye, “Effects of self-and cross-phase modulation on the spontaneous symmetry breaking of light in ring resonators,” Physical Review A, vol. 101, no. 1, p. 013823, 2020.
  * (72) B. Garbin, J. Fatome, G.-L. Oppo, M. Erkintalo, S. G. Murdoch, and S. Coen, “Asymmetric balance in symmetry breaking,” Physical Review Research, vol. 2, no. 2, p. 023244, 2020.
  * (73) S. Coen, B. Garbin, G. Xu, L. Quinn, N. Goldman, G.-L. Oppo, M. Erkintalo, S. G. Murdoch, and J. Fatome, “Nonlinear topological symmetry protection in a dissipative system,” arXiv preprint arXiv:2303.16197, 2023.
  * (74) A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, “Inhibition of light tunneling in waveguide arrays,” Physical review letters, vol. 102, no. 15, p. 153901, 2009.
  * (75) M. Andrews, C. Townsend, H.-J. Miesner, D. Durfee, D. Kurn, and W. Ketterle, “Observation of interference between two bose condensates,” Science, vol. 275, no. 5300, pp. 637–641, 1997.
  * (76) A. Smerzi, S. Fantoni, S. Giovanazzi, and S. Shenoy, “Quantum coherent atomic tunneling between two trapped bose-einstein condensates,” Physical Review Letters, vol. 79, no. 25, p. 4950, 1997.
  * (77) M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single bosonic josephson junction,” Physical review letters, vol. 95, no. 1, p. 010402, 2005.
  * (78) T. Schumm, Bose-Einstein condensates in magnetic double well potentials.  PhD thesis, 2005.
  * (79) S. Hofferberth, I. Lesanovsky, B. Fischer, J. Verdu, and J. Schmiedmayer, “Radio-frequency dressed state potentials for neutral atoms,” Nature Physics, vol. 2, pp. 710–716, Oct. 2006.
  * (80) T. Zibold, E. Nicklas, C. Gross, and M. K. Oberthaler, “Classical bifurcation at the transition from rabi to josephson dynamics,” Physical review letters, vol. 105, no. 20, p. 204101, 2010.
  * (81) C. Gross, T. Zibold, E. Nicklas, J. Esteve, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature, vol. 464, no. 7292, pp. 1165–1169, 2010.
  * (82) A. Recati and S. Stringari, “Coherently coupled mixtures of ultracold atomic gases,” Annual Review of Condensed Matter Physics, vol. 13, pp. 407–432, 2022.
  * (83) N. Goldman, J. Dalibard, M. Aidelsburger, and N. R. Cooper, “Periodically driven quantum matter: The case of resonant modulations,” Physical Review A, vol. 91, no. 3, p. 033632, 2015.
  * (84) T. Mikami, S. Kitamura, K. Yasuda, N. Tsuji, T. Oka, and H. Aoki, “Brillouin-wigner theory for high-frequency expansion in periodically driven systems: Application to floquet topological insulators,” Physical Review B, vol. 93, no. 14, p. 144307, 2016.
  * (85) L. Pitaevskii and S. Stringari, Bose-Einstein condensation and superfluidity, vol. 164.  Oxford University Press, 2016.
  * (86) I. Carusotto and C. Ciuti, “Quantum fluids of light,” Reviews of Modern Physics, vol. 85, no. 1, p. 299, 2013.
  * (87) M. Holthaus, “Towards coherent control of a bose-einstein condensate in a double well,” Physical Review A, vol. 64, no. 1, p. 011601, 2001.
  * (88) Q.-T. Cao, R. Liu, H. Wang, Y.-K. Lu, C.-W. Qiu, S. Rotter, Q. Gong, and Y.-F. Xiao, “Reconfigurable symmetry-broken laser in a symmetric microcavity,” Nature communications, vol. 11, no. 1, pp. 1–7, 2020.
  * (89) J. Fatome, G. Xu, B. Garbin, N. Berti, G.-L. Oppo, S. G. Murdoch, M. Erkintalo, and S. Coen, “Self-symmetrization of symmetry-breaking dynamics in passive kerr resonators,” arXiv preprint arXiv:2106.07642, 2021.
  * (90) P. Kockaert, C. Cambournac, M. Haelterman, G. Kozyreff, and T. Erneux, “Fast self-pulsing through nonlinear incoherent feedback,” Optics letters, vol. 31, no. 4, pp. 495–497, 2006.
  * (91) G. Kozyreff, T. Erneux, M. Haelterman, and P. Kockaert, “Fast optical self-pulsing in a temporal analog of the kerr-slice pattern-forming system,” Physical Review A, vol. 73, no. 6, p. 063815, 2006.
  * (92) C. Gardiner and P. Zoller, The Quantum World of Ultra-Cold Atoms and Light Book I: Foundations of Quantum Optics, vol. 2.  World Scientific Publishing Company, 2014.
  * (93) M. Kitagawa and M. Ueda, “Squeezed spin states,” Physical Review A, vol. 47, no. 6, p. 5138, 1993.
  * (94) Y. Liu, Z. Xu, G. Jin, and L. You, “Spin squeezing: transforming one-axis twisting into two-axis twisting,” Physical review letters, vol. 107, no. 1, p. 013601, 2011.
  * (95) The nonlinear Schrödinger equation in Eq. (1) assumes that the parameter $\gamma$ is the same for both modes. If this is not the case, then Eq. (1) would display a term of the form $\sim(-\partial^{2}/\partial x^{2})\hat{\sigma}_{z}$. Since the Pauli matrix $\hat{\sigma}_{z}$ does not commute with the mixing operator $\hat{\sigma}_{x}$ in Eq. (6), this kinetic term will be modified in the effective NLSE [Eq. (7)].
  * (96) T.-S. Yang and W. L. Kath, “Analysis of enhanced-power solitons in dispersion-managed optical fibers,” Optics Letters, vol. 22, no. 13, pp. 985–9897, 1997.
  * (97) I. Towers and B. A. Malomed, “Stable (2+ 1)-dimensional solitons in a layered medium with sign-alternating kerr nonlinearity,” JOSA B, vol. 19, no. 3, pp. 537–543, 2002.
  * (98) F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in bose-einstein condensates by temporal modulation of the scattering length,” Physical Review A, vol. 67, no. 1, p. 013605, 2003\.
  * (99) M. Matuszewski, E. Infeld, B. A. Malomed, and M. Trippenbach, “Fully three dimensional breather solitons can be created using feshbach resonances,” Physical review letters, vol. 95, no. 5, p. 050403, 2005.
  * (100) M. Krämer, C. Tozzo, and F. Dalfovo, “Parametric excitation of a bose-einstein condensate in a one-dimensional optical lattice,” Physical Review A, vol. 71, no. 6, p. 061602, 2005.
  * (101) H. Susanto, P. Kevrekidis, B. Malomed, and F. K. Abdullaev, “Effects of time-periodic linear coupling on two-component bose–einstein condensates in two dimensions,” Physics Letters A, vol. 372, no. 10, pp. 1631–1638, 2008\.
  * (102) S. Lellouch, M. Bukov, E. Demler, and N. Goldman, “Parametric instability rates in periodically driven band systems,” Physical Review X, vol. 7, no. 2, p. 021015, 2017.
  * (103) S. Higashikawa, H. Fujita, and M. Sato, “Floquet engineering of classical systems,” arXiv preprint arXiv:1810.01103, 2018.
  * (104) I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Reviews of modern physics, vol. 80, no. 3, p. 885, 2008.
  * (105) O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.-S. Lühmann, B. A. Malomed, T. Sowiński, and J. Zakrzewski, “Non-standard hubbard models in optical lattices: a review,” Reports on Progress in Physics, vol. 78, no. 6, p. 066001, 2015.
  * (106) A. Auerbach, Interacting electrons and quantum magnetism.  Springer Science & Business Media, 2012.
  * (107) S. V. Pereverzev, A. Loshak, S. Backhaus, J. Davis, and R. Packard, “Quantum oscillations between two weakly coupled reservoirs of superfluid 3he,” Nature, vol. 388, no. 6641, pp. 449–451, 1997.
  * (108) S. Raghavan, A. Smerzi, S. Fantoni, and S. Shenoy, “Coherent oscillations between two weakly coupled bose-einstein condensates: Josephson effects, $\pi$ oscillations, and macroscopic quantum self-trapping,” Physical Review A, vol. 59, no. 1, p. 620, 1999.
  * (109) S. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, “The ac and dc josephson effects in a bose–einstein condensate,” Nature, vol. 449, no. 7162, pp. 579–583, 2007.
  * (110) T. Zibold, Classical bifurcation and entanglement generation in an internal bosonic josephson junction.  PhD thesis, 2012.
  * (111) H. J. Lipkin, N. Meshkov, and A. Glick, “Validity of many-body approximation methods for a solvable model:(i). exact solutions and perturbation theory,” Nuclear Physics, vol. 62, no. 2, pp. 188–198, 1965.
  * (112) G.-S. Paraoanu, S. Kohler, F. Sols, and A. Leggett, “The josephson plasmon as a bogoliubov quasiparticle,” Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 34, no. 23, p. 4689, 2001.
  * (113) M. Di Liberto, S. Mukherjee, and N. Goldman, “Nonlinear dynamics of aharonov-bohm cages,” Physical Review A, vol. 100, no. 4, p. 043829, 2019\.
  * (114) T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems,” Physical Review B, vol. 82, no. 23, p. 235114, 2010.
  * (115) R. Kidd, M. Olsen, and J. Corney, “Quantum chaos in a bose-hubbard dimer with modulated tunneling,” Physical Review A, vol. 100, no. 1, p. 013625, 2019\.
  * (116) J. Ma, X. Wang, C.-P. Sun, and F. Nori, “Quantum spin squeezing,” Physics Reports, vol. 509, no. 2-3, pp. 89–165, 2011.
  * (117) W. Huang, Y.-L. Zhang, C.-L. Zou, X.-B. Zou, and G.-C. Guo, “Two-axis spin squeezing of two-component bose-einstein condensates via continuous driving,” Physical Review A, vol. 91, no. 4, p. 043642, 2015.
  * (118) D. Kajtoch and E. Witkowska, “Quantum dynamics generated by the two-axis countertwisting hamiltonian,” Physical Review A, vol. 92, no. 1, p. 013623, 2015.
  * (119) D. Kajtoch and E. Witkowska, “Spin squeezing in dipolar spinor condensates,” Physical Review A, vol. 93, no. 2, p. 023627, 2016.
  * (120) P. He, M. A. Perlin, S. R. Muleady, R. J. Lewis-Swan, R. B. Hutson, J. Ye, and A. M. Rey, “Engineering spin squeezing in a 3d optical lattice with interacting spin-orbit-coupled fermions,” Physical Review Research, vol. 1, no. 3, p. 033075, 2019.
  * (121) P. Groszkowski, H.-K. Lau, C. Leroux, L. Govia, and A. Clerk, “Heisenberg-limited spin squeezing via bosonic parametric driving,” Physical Review Letters, vol. 125, no. 20, p. 203601, 2020.
  * (122) T. Roscilde, F. Mezzacapo, and T. Comparin, “Spin squeezing from bilinear spin-spin interactions: Two simple theorems,” Physical Review A, vol. 104, no. 4, p. L040601, 2021.
  * (123) T. H. Yanes, M. Płodzień, M. M. Sinkevičienė, G. Žlabys, G. Juzeliūnas, and E. Witkowska, “One-and two-axis squeezing via laser coupling in an atomic fermi-hubbard model,” Physical Review Letters, vol. 129, no. 9, p. 090403, 2022.
  * (124) A. Kruckenhauser, R. van Bijnen, T. V. Zache, M. Di Liberto, and P. Zoller, “High-dimensional so (4)-symmetric rydberg manifolds for quantum simulation,” Quantum Science and Technology, vol. 8, no. 1, p. 015020, 2022\.
  * (125) In optics, this change of basis corresponds to a change of representation, from linear polarization modes to circular-polarization modes.
  * (126) J.-Q. Liang, J.-L. Liu, W.-D. Li, and Z.-J. Li, “Atom-pair tunneling and quantum phase transition in the strong-interaction regime,” Physical Review A, vol. 79, no. 3, p. 033617, 2009.
  * (127) O. Jürgensen, K. Sengstock, and D.-S. Lühmann, “Twisted complex superfluids in optical lattices,” Scientific reports, vol. 5, no. 1, pp. 1–11, 2015.
  * (128) D.-S. Lühmann, “Twisted superfluid phase in the extended one-dimensional bose-hubbard model,” Physical Review A, vol. 94, no. 1, p. 011603, 2016\.
  * (129) A. Hemmerich, “Bosons condensed in two modes with flavor-changing interaction,” Physical Review A, vol. 99, no. 1, p. 013623, 2019.
  * (130) Z. Lin, C. Liu, and Y. Chen, “Novel quantum phases of two-component bosons with pair hopping in synthetic dimension,” Physical Review Letters, vol. 125, no. 24, p. 245301, 2020.
  * (131) For $\Omega_{0}\\!=\\!0$, there is the additional discrete symmetry: $\varphi\to\varphi+\pi$. For $\alpha\\!=\\!1/2$, there is the additional continuous symmetry: $\sqrt{1-z^{2}}\to c\sqrt{1-z^{2}},\,\mathop{cos}\nolimits(\varphi)\to\mathop{cos}\nolimits(\varphi)/c$ for any real $c>0$.
  * (132) M. Pigneur and J. Schmiedmayer, “Analytical pendulum model for a bosonic josephson junction,” Physical Review A, vol. 98, no. 6, p. 063632, 2018\.
  * (133) M. Gajda and K. Rzazewski, “Fluctuations of bose-einstein condensate,” Physical review letters, vol. 78, no. 14, p. 2686, 1997.
  * (134) P.-É. Larré and I. Carusotto, “Propagation of a quantum fluid of light in a cavityless nonlinear optical medium: General theory and response to quantum quenches,” Physical Review A, vol. 92, no. 4, p. 043802, 2015.
  * (135) B. Julia-Diaz, T. Zibold, M. Oberthaler, M. Mele-Messeguer, J. Martorell, and A. Polls, “Dynamic generation of spin-squeezed states in bosonic josephson junctions,” Physical Review A, vol. 86, no. 2, p. 023615, 2012.
  * (136) P. Bruno, “Quantum geometric phase in majoranas stellar representation: mapping onto a many-body aharonov-bohm phase,” Physical Review Letters, vol. 108, no. 24, p. 240402, 2012.
  * (137) H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B. Hume, L. Pezzè, A. Smerzi, and M. K. Oberthaler, “Fisher information and entanglement of non-gaussian spin states,” Science, vol. 345, no. 6195, pp. 424–427, 2014\.
  * (138) A. Evrard, V. Makhalov, T. Chalopin, L. A. Sidorenkov, J. Dalibard, R. Lopes, and S. Nascimbene, “Enhanced magnetic sensitivity with non-gaussian quantum fluctuations,” Physical review letters, vol. 122, no. 17, p. 173601, 2019\.
  * (139) S. Nascimbene, “Quantum-enhanced sensing and topological matter with ultracold dysprosium atoms,” 2020.
  * (140) X. Li and W. V. Liu, “Physics of higher orbital bands in optical lattices: a review,” Reports on Progress in Physics, vol. 79, no. 11, p. 116401, 2016\.
  * (141) M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Physics, vol. 7, no. 11, pp. 907–912, 2011.
  * (142) B. M. Anderson, R. Ma, C. Owens, D. I. Schuster, and J. Simon, “Engineering topological many-body materials in microwave cavity arrays,” Physical Review X, vol. 6, no. 4, p. 041043, 2016.
  * (143) S. Trotzky, P. Cheinet, S. Folling, M. Feld, U. Schnorrberger, A. M. Rey, A. Polkovnikov, E. A. Demler, M. D. Lukin, and I. Bloch, “Time-resolved observation and control of superexchange interactions with ultracold atoms in optical lattices,” Science, vol. 319, no. 5861, pp. 295–299, 2008.
  * (144) L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T. Esslinger, “Creating, moving and merging dirac points with a fermi gas in a tunable honeycomb lattice,” Nature, vol. 483, no. 7389, pp. 302–305, 2012.
  * (145) A.-S. Walter, Z. Zhu, M. Gächter, J. Minguzzi, S. Roschinski, K. Sandholzer, K. Viebahn, and T. Esslinger, “Quantization and its breakdown in a HubbardThouless pump,” Nature Physics, 2023.
  * (146) M. Jürgensen and M. C. Rechtsman Private communications.
  * (147) P. G. Kevrekidis, The discrete nonlinear Schrödinger equation: mathematical analysis, numerical computations and physical perspectives, vol. 232.  Springer Science & Business Media, 2009.
  * (148) F. Hébert, Z. Cai, V. Rousseau, C. Wu, R. Scalettar, and G. Batrouni, “Exotic phases of interacting p-band bosons,” Physical Review B, vol. 87, no. 22, p. 224505, 2013.
  * (149) M. Di Liberto and N. Goldman, “Chiral orbital order of interacting bosons without higher bands,” Physical Review Research, vol. 5, p. 023064, 2023.
  * (150) M. Tovmasyan, E. P. van Nieuwenburg, and S. D. Huber, “Geometry-induced pair condensation,” Physical Review B, vol. 88, no. 22, p. 220510, 2013.
  * (151) S. Takayoshi, H. Katsura, N. Watanabe, and H. Aoki, “Phase diagram and pair tomonaga-luttinger liquid in a bose-hubbard model with flat bands,” Physical Review A, vol. 88, no. 6, p. 063613, 2013.
  * (152) J. Jünemann, A. Piga, S.-J. Ran, M. Lewenstein, M. Rizzi, and A. Bermúdez, “Exploring interacting topological insulators with ultracold atoms: The synthetic creutz-hubbard model,” Physical Review X, vol. 7, no. 3, p. 031057, 2017.
  * (153) S. R. White, “Density matrix formulation for quantum renormalization groups,” Physical review letters, vol. 69, no. 19, p. 2863, 1992.
  * (154) J. K. Freericks and H. Monien, “Phase diagram of the Bose-Hubbard Model,” Europhysics Letters, vol. 26, p. 545, jun 1994.
  * (155) M. Niemeyer, J. K. Freericks, and H. Monien, “Strong-coupling perturbation theory for the two-dimensional Bose-Hubbard model in a magnetic field,” Phys. Rev. B, vol. 60, pp. 2357–2362, Jul 1999.
  * (156) A. Keleş and M. O. Oktel, “Mott transition in a two-leg Bose-Hubbard ladder under an artificial magnetic field,” Phys. Rev. A, vol. 91, p. 013629, Jan 2015.
  * (157) I. Dzyaloshinsky, “A thermodynamic theory of weak ferromagnetism of antiferromagnetics,” Journal of physics and chemistry of solids, vol. 4, no. 4, pp. 241–255, 1958.
  * (158) T. Moriya, “New mechanism of anisotropic superexchange interaction,” Physical Review Letters, vol. 4, no. 5, p. 228, 1960.
  * (159) T. Moriya, “Anisotropic superexchange interaction and weak ferromagnetism,” Physical review, vol. 120, no. 1, p. 91, 1960.
  * (160) V. Dmitrienko, E. Ovchinnikova, S. Collins, G. Nisbet, G. Beutier, Y. Kvashnin, V. Mazurenko, A. Lichtenstein, and M. Katsnelson, “Measuring the dzyaloshinskii–moriya interaction in a weak ferromagnet,” Nature Physics, vol. 10, no. 3, pp. 202–206, 2014.
  * (161) B. Liu, P. Zhang, H. Gao, and F. Li, “Chiral orbital magnetism of p-orbital bosons in optical lattices,” Physical Review Letters, vol. 121, no. 1, p. 015303, 2018.
  * (162) F. Pinheiro, G. M. Bruun, J.-P. Martikainen, and J. Larson, “$xyz$ Quantum Heisenberg Models with $p$-Orbital Bosons,” Phys. Rev. Lett., vol. 111, p. 205302, Nov 2013.
  * (163) A. Y. Kitaev, “Unpaired majorana fermions in quantum wires,” Physics-uspekhi, vol. 44, no. 10S, p. 131, 2001.
  * (164) X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Reviews of Modern Physics, vol. 83, no. 4, p. 1057, 2011.
  * (165) C. W. Duncan, M. J. Hartmann, R. R. Thomson, and P. Öhberg, “Synthetic mean-field interactions in photonic lattices,” The European Physical Journal D, vol. 74, no. 5, pp. 1–7, 2020.
  * (166) A. E. Kraych, P. Suret, G. El, and S. Randoux, “Nonlinear evolution of the locally induced modulational instability in fiber optics,” Physical review letters, vol. 122, no. 5, p. 054101, 2019.
  * (167) A. E. Kraych, D. Agafontsev, S. Randoux, and P. Suret, “Statistical properties of the nonlinear stage of modulation instability in fiber optics,” Physical review letters, vol. 123, no. 9, p. 093902, 2019.
  * (168) A. Kraych, Instabilités modulationnelles dans un anneau de recirculation fibré.  PhD thesis, Lille, 2020.
  * (169) C. Cambournac, T. Sylvestre, H. Maillotte, B. Vanderlinden, P. Kockaert, P. Emplit, and M. Haelterman, “Symmetry-breaking instability of multimode vector solitons,” Physical review letters, vol. 89, no. 8, p. 083901, 2002\.
  * (170) A. Tikan, J. Riemensberger, K. Komagata, S. Hönl, M. Churaev, C. Skehan, H. Guo, R. N. Wang, J. Liu, P. Seidler, et al., “Emergent nonlinear phenomena in a driven dissipative photonic dimer,” Nature Physics, vol. 17, no. 5, pp. 604–610, 2021.
  * (171) K. Komagata, A. Tusnin, J. Riemensberger, M. Churaev, H. Guo, A. Tikan, and T. J. Kippenberg, “Dissipative kerr solitons in a photonic dimer on both sides of exceptional point,” Communications Physics, vol. 4, no. 1, p. 159, 2021.
  * (172) A. Tusnin, A. Tikan, K. Komagata, and T. J. Kippenberg, “Dissipative structures in topological lattices of nonlinear optical resonators,” arXiv preprint arXiv:2104.11731, 2021.
  * (173) A. Tikan, A. Tusnin, J. Riemensberger, M. Churaev, X. Ji, K. N. Komagata, R. N. Wang, J. Liu, and T. J. Kippenberg, “Protected generation of dissipative kerr solitons in supermodes of coupled optical microresonators,” Science Advances, vol. 8, no. 13, p. eabm6982, 2022.
  * (174) S. K. Ivanov, Y. V. Kartashov, and V. V. Konotop, “Four-wave mixing floquet topological solitons,” Optics Letters, vol. 46, no. 19, pp. 4710–4713, 2021.
  * (175) S. Chakram, A. E. Oriani, R. K. Naik, A. V. Dixit, K. He, A. Agrawal, H. Kwon, and D. I. Schuster, “Seamless high-q microwave cavities for multimode circuit quantum electrodynamics,” Physical review letters, vol. 127, no. 10, p. 107701, 2021.
  * (176) L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Physical review letters, vol. 58, no. 21, p. 2209, 1987\.
  * (177) M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Optics communications, vol. 91, no. 5-6, pp. 401–407, 1992.
  * (178) F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional kerr media as bits in an all-optical buffer,” Nature Photonics, vol. 4, no. 7, pp. 471–476, 2010\.
  * (179) S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning kerr frequency combs using a generalized mean-field lugiato–lefever model,” Optics letters, vol. 38, no. 1, pp. 37–39, 2013\.
  * (180) A. Schnell, A. Eckardt, and S. Denisov, “Is there a floquet lindbladian?,” Physical Review B, vol. 101, no. 10, p. 100301, 2020.
  * (181) A. Schnell, S. Denisov, and A. Eckardt, “High-frequency expansions for time-periodic lindblad generators,” Physical Review B, vol. 104, no. 16, p. 165414, 2021.
  * (182) M.-A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Physical review letters, vol. 111, no. 5, p. 053602, 2013.
  * (183) M.-A. Lemonde, N. Didier, and A. A. Clerk, “Enhanced nonlinear interactions in quantum optomechanics via mechanical amplification,” Nature communications, vol. 7, no. 1, p. 11338, 2016.
  * (184) T. Boulier, J. Maslek, M. Bukov, C. Bracamontes, E. Magnan, S. Lellouch, E. Demler, N. Goldman, and J. Porto, “Parametric heating in a 2d periodically driven bosonic system: Beyond the weakly interacting regime,” Physical Review X, vol. 9, no. 1, p. 011047, 2019.
  * (185) K. Wintersperger, M. Bukov, J. Näger, S. Lellouch, E. Demler, U. Schneider, I. Bloch, N. Goldman, and M. Aidelsburger, “Parametric instabilities of interacting bosons in periodically driven 1d optical lattices,” Physical Review X, vol. 10, no. 1, p. 011030, 2020.
  * (186) A. Di Carli, R. Cruickshank, M. Mitchell, A. La Rooij, S. Kuhr, C. E. Creffield, and E. Haller, “Instabilities of interacting matter waves in optical lattices with floquet driving,” Phys. Rev. Res., vol. 5, p. 033024, Jul 2023.
  * (187) M. Pigneur, T. Berrada, M. Bonneau, T. Schumm, E. Demler, and J. Schmiedmayer, “Relaxation to a phase-locked equilibrium state in a one-dimensional bosonic josephson junction,” Physical review letters, vol. 120, no. 17, p. 173601, 2018.
  * (188) C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach resonances in ultracold gases,” Reviews of Modern Physics, vol. 82, no. 2, p. 1225, 2010\.
  * (189) S.-C. Ji, T. Schweigler, M. Tajik, F. Cataldini, J. a. Sabino, F. S. Møller, S. Erne, and J. Schmiedmayer, “Floquet engineering a bosonic josephson junction,” Phys. Rev. Lett., vol. 129, p. 080402, Aug 2022.
  * (190) A. Qu, B. Evrard, J. Dalibard, and F. Gerbier, “Probing spin correlations in a bose-einstein condensate near the single-atom level,” Physical Review Letters, vol. 125, no. 3, p. 033401, 2020.
  * (191) S. van Frank, M. Bonneau, J. Schmiedmayer, S. Hild, C. Gross, M. Cheneau, I. Bloch, T. Pichler, A. Negretti, T. Calarco, et al., “Optimal control of complex atomic quantum systems,” Scientific reports, vol. 6, no. 1, p. 34187, 2016.
  * (192) F. Borselli, M. Maiwger, T. Zhang, P. Haslinger, V. Mukherjee, A. Negretti, S. Montangero, T. Calarco, I. Mazets, M. Bonneau, and J. Schmiedmayer, “Two-Particle Interference with Double Twin-Atom Beams,” Physical Review Letters, vol. 126, p. 083603, Feb. 2021.
  * (193) B. M. Spar, E. Guardado-Sanchez, S. Chi, Z. Z. Yan, and W. S. Bakr, “Realization of a fermi-hubbard optical tweezer array,” Phys. Rev. Lett., vol. 128, p. 223202, Jun 2022.
  * (194) F. Gerbier and J. Dalibard, “Gauge fields for ultracold atoms in optical superlattices,” New Journal of Physics, vol. 12, no. 3, p. 033007, 2010\.
  * (195) N. Goldman, F. Gerbier, and M. Lewenstein, “Realizing non-abelian gauge potentials in optical square lattices: an application to atomic chern insulators,” Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 46, no. 13, p. 134010, 2013.
  * (196) Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, “Realization of two-dimensional spin-orbit coupling for bose-einstein condensates,” Science, vol. 354, no. 6308, pp. 83–88, 2016.
  * (197) J. Esteve, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, “Squeezing and entanglement in a bose–einstein condensate,” Nature, vol. 455, no. 7217, pp. 1216–1219, 2008.
  * (198) P. Kunkel, M. Prfer, S. Lannig, R. Rosa-Medina, A. Bonnin, M. Grttner, H. Strobel, and M. K. Oberthaler, “Simultaneous Readout of Noncommuting Collective Spin Observables beyond the Standard Quantum Limit,” Physical Review Letters, vol. 123, p. 063603, Aug. 2019.
  * (199) C. P. Slichter, Principles of Magnetic Resonance.  Springer Berlin Heidelberg, 1990.