# On the contribution of pre-trained models to accuracy and utility in
modeling distributed energy resources

Hussain Kazmi<EMAIL_ADDRESS>KU LeuvenBelgium and  Pierre
Pinson  Imperial College LondonUK

(2023)

###### Abstract.

Despite their growing popularity, data-driven models of real-world dynamical
systems require lots of data. However, due to sensing limitations as well as
privacy concerns, this data is not always available, especially in domains
such as energy. Pre-trained models using data gathered in similar contexts
have shown enormous potential in addressing these concerns: they can improve
predictive accuracy at a much lower observational data expense. Theoretically,
due to the risk posed by negative transfer, this improvement is however
neither uniform for all agents nor is it guaranteed. In this paper, using data
from several distributed energy resources, we investigate and report
preliminary findings on several key questions in this regard. First, we
evaluate the improvement in predictive accuracy due to pre-trained models,
both with and without fine-tuning. Subsequently, we consider the question of
fairness: do pre-trained models create equal improvements for heterogeneous
agents, and how does this translate to downstream utility? Answering these
questions can help enable improvements in the creation, fine-tuning, and
adoption of such pre-trained models.

Pre-trained models, distributed energy resources, dynamics models, accuracy,
fairness

††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: On the utility and fairness of pre-trained models
in modeling distributed energy resources; June 20–23, 2023; Orlando,
FL††booktitle: E-Energy ’23: 14th ACM International Conference on Future
Energy Systems, June 20–23, 2023, Orlando, FL††price: 15.00††isbn:
978-1-4503-XXXX-X/18/06††ccs: Computing methodologies Distributed artificial
intelligence††ccs: Hardware Power and energy

##  1\. Introduction

Data-driven modelling of complex dynamical systems typically requires access
to large amounts of data to generalize well (han2021pre, ). In the absence of
large datasets, model performance tends to be sub-par in regions of state-
space that have been poorly explored (chung2019slice, ). Additionally, models
trained with limited data may learn relationships that are not causal
(lapuschkin2019unmasking, ). Using such models for control can therefore lead
to undesirable outcomes in high stakes decision-making
(rudin2022interpretable, ), such as in the operation of critical
infrastructure or decisions that directly affect human well-being.

However, in many domains, outside of computer vision and natural language
processing, collecting large amounts of data is often infeasible
(kazmi2021towards, ). This is frequently the case with predicting demand for
new products or services, where data-driven models are not feasible due to the
cold-start problem (moon2020solving, ). Another case where observational data
tends to be limited is due to expensive sensing or communication
infrastructure, as well as privacy concerns (mcdaniel2009security, ;
morel2022your, ). This is frequently the case in the energy domain, where data
on both demand and generation side is typically gathered in the form of time
series, which need to be modelled and predicted. However, installing and
maintaining sensors to continuously monitor electricity demand and generation
at high spatiotemporal resolutions is an expensive and often privacy-
compromising operation.

As a result, several solutions have been proposed in recent years to reduce
the data requirements or sample complexity of learning algorithms, including
via the use of pre-trained models (PTMs), which rely on simulated or
historically observed data, thereby eschewing the need for observational data
(han2021pre, ). More recently, this has also seen significant interest in the
modeling and optimization of distributed energy resources
(peirelinck2022transfer, ; didden2021sample, ). In this paper, we dive deeper
into the topic to consider how (or whether) PTMs can accelerate the data-
driven modelling of distributed energy resources, e.g. hot water storage
systems. More concretely, we attempt to answer the following questions: (1) do
PTMs outperform purely data-driven dynamics models; (2) (how) does the size of
the source and target corpus, and fine-tuning affect the PTM’s performance;
and (3) do potential improvements induced by PTMs create uniform utility for
heterogeneous agents. The remainder of this paper is organized as follows. In
section 2, we describe the methodology we follow subsequently. Section 3
presents key results, while Section 4 concludes the paper with an eye to
future research directions.

##  2\. Methodology

###  2.1. Data and setup

To answer the afore-mentioned questions, we utilize detailed (quarter-hourly)
data spanning over one year from several identical heat pump hot water storage
systems in a net-zero energy neighborhood, constituting a corpus of 24
buildings. The general idea is to construct a model for state estimation of
the hot water storage system in a way that can then used for downstream
control (e.g. to reduce energy demand, maximize user comfort or minimize grid
load etc.). Of the entire corpus, 8 systems were partitioned into a separate
source set, $S$, for which data was assumed to be available before the start
of the experiment, i.e. this data can be used off-the-shelf to create PTMs.
The remaining 16 buildings are marked as the target set, $T$, which can be
used to construct ’local’ models for the hot water systems. A local model
identifies that only observational data from a particular building is used
during the modeling process. However, for each individual building, even after
an entire year of acquisition, this dataset is rather small at a few thousand
training examples. Even ignoring the effects of serial and autocorrelation on
the information content, this corpus is smaller than even a single high-
resolution image.

Even though the systems are identical, the users and operating conditions are
not. Consequently, each system generates data which is distributed
differently, due to the sampling bias these two factors induce. This means the
datasets observed at different systems, and used to train (and evaluate) the
local models can be significantly different.

Figure 1. Neural architecture for the trained models

###  2.2. Local models

We pose the problem as a standard one-step ahead regression problem, $y=f(x)$.
Here, the model, $f(.)$, predicts the temperature at the mid-point, $T_{m}$,
in the storage vessel. To do so, it uses three fields as input: (1) the time
since the last reheat cycle, $t$, (2) the hot water demand since the last
reheat cycle, $w$ and (3) the initial temperature of the vessel at its mid-
point, $T_{0}$. These features are extracted from three observational time
series respectively: user hot water demand, heat pump operation (i.e. active
or idle), and the observed temperature at the mid-point in the storage vessel.
This means that at any given time point, the entire history of the three time
series can be used to construct the input feature matrix, which is then
ingested by the function approximation model. Note that, by virtue of
including time since the last reheat cycle in the input features, arbitrary
future time horizon predictions can be obtained.

The model, $f(.)$ is chosen to be a neural network with two fully connected
hidden layers and a normalization layer (shown in Fig. 1). It utilizes the
ReLU activation function and L2 regularization, and is fit using the mean
absolute error (MAE) loss function (i.e. it minimizes the error when
predicting the mid-point temperature in the vessel). The reason for choosing a
neural network as the function approximation technique is motivated by its
prevalence in PTM literature. While theoretically linear models can be used in
a similar fashion, they are incapable of modelling complex functions and non-
linear dynamics, as is required here (heat losses are inherently non-linear).

The sampling bias in the observational data manifests as a distributional
shift in the three features for different households, and predominantly
affects the local models, which are trained only using data collected at each
site. These local models are trained at four distinct time points, namely
after 4, 8, 16, and 32 weeks. The holdout test dataset for each site is
derived locally by sampling the last three months of observations, and
consequently does not overlap with the training set. As such, it is still
subject to many of the same sampling biases (e.g., a high accuracy score for a
local model on a local test set does not guarantee broad generalization).

###  2.3. Pre-trained models

Instead of relying solely on observational data to fit model parameters, PTMs
utilize what is essentially a sequential two-stage process. First, a model is
trained on (abundant) simulated or prior observational data. Second, it is
fine-tuned using (sparse) observational local data. The pre-training relies
primarily on the fact that the processes being modelled are identical or, at
the very least, share some similarities. In addition to similarity, the amount
of the pre-collected training data (i.e. source corpus size) is also
important. Where both the quantity and quality of pre-collected data is
sufficient, PTMs can lead to large speed-ups in the modelling of dynamical
systems (typically observed as better initial accuracy, learning rate, and
asymptotic accuracy). To make the comparison fair, we employ the same model
architecture as before for creating the initial PTM, and investigate the
effect of several variations, including: (1) the source corpus size (i.e. how
much data was available to train the PTM), (2) fine-tuning (i.e. whether the
PTM employs fine-tuning using local data or not), and (3) the target corpus
size (i.e. the local dataset size, which is only relevant when we utilize
fine-tuning).

More concretely, to investigate the effect of the source corpus size, we
created two models. The first model used a large training corpus, including
data from eight systems (these systems were not included in the local models
discussed above) for eight months. The second model utilized a small corpus,
referring to data from only a single system for a duration of eight months.
After training to convergence, these two PTMs were then deployed in the
households, where their performance was evaluated using the same local holdout
datasets and methodology as described previously.

In the case where these two PTMs were not fine-tuned using local data, their
performance on the hold-out test set for any given system did not change over
time (the holdout set is kept fixed to avoid any reporting bias). On the
contrary, to investigate the impact of fine-tuning, given varying amounts of
target data, we followed the same logic as that of local models. Both the
small and large corpus PTMs were fine-tuned at four distinct points in time:
i.e. at 4, 8, 16, and 32 weeks. The fine-tuning process is carried out at a
lower learning rate and for fewer iterations, when compared with the training
of the PTMs or local models. The models are evaluated using the same method as
that described previously.

##  3\. Results

In this section, we present the most important results comparing local models
against different variants of PTMs. Subsequently, we take a look at the
implications this has on donwstream operations.

###  3.1. The importance of corpus size and fine-tuning

Fig. 2 shows the predictive performance of the different model configurations,
using training corpus size (or observation period) as the x-axis. As expected,
increasing amounts of data, either during the pre-training phase or the local
acquisition phase, improve performance. This holds for both local models, as
well as small and large fine-tuned PTMs. Asymptotically, the small training
corpus PTM performs the worst, but is still comparable to a local model during
the initial phase, when the local models have seen only four weeks of on-site
training data. Increasing amounts of observational data lead to improvements
in the local models, with the average error falling from around 0.5°C after 4
weeks to 0.33°C after 32 weeks. Fine-tuning both PTMs shows considerable
potential to improve this performance further, especially when only limited
amounts of locally observed data has been collected. Even so, the local and
fine-tuned small corpus PTM perform at roughly the same accuracy level as the
large corpus PTM without any fine-tuning. With fine-tuning, the large corpus
PTM significantly outperforms the local models with the average error falling
as low as 0.28°C after 32 weeks.

A curious trend emerges here. While the PTM with fine-tuning ends up being the
best performing model at the end of 32 weeks, the best model in the low
observational data regime is actually the large PTM without any fine-tuning.
This makes sense since the PTM has arguably already learnt the correct system
dynamics, and only a small amount of training data does not allow it to
further reduce its error on the (biased) evaluation test dataset for each
specific system. This also shows that perhaps the fine-tuning might be too
aggressive in the low data regime.

Figure 2. Mean absolute error (MAE) on holdout test dataset for different
model configurations with increasing amounts of training data, averaged over
16 systems

###  3.2. The importance of choosing the correct PTM

Figure 3. The cross-performance of local models evaluated on holdout test set
for each individual building; the large markers represent the performance of
the model trained using data from the system

The small corpus PTM (without fine-tuning) exhibits significant deviation in
the improvements it can bring to different systems (note the wide uncertainty
bands plotted in Fig. 2). In some instances, the performance of these models
can even begin to rival that of much better performing models (e.g. the large
corpus PTM with fine-tuning). To investigate this effect further, we treat
each individual local model trained for 8 months as a small corpus PTM. This
leads to 16 additional candidate small source corpus PTMs. As there is no
overlap in training or evaluation data, this does not pose any data leakage
concerns. Fig. 3 shows results for evaluating these models without fine-
tuning, and presents two key insights. The first is the enormous amount of
diversity in accuracy for most systems when data from a different system is
used to learn their dynamics. This is all the more surprising because the
systems are identical, and the only difference is the sampling bias due to
occupant and controller behavior. Second, we see that the model trained on the
system’s data itself is usually, but not always the best performing model.
This seems counter-intuitive at first but there are several explanations for
it, including (1) non-stationarity in the training and evaluation data, i.e.
the local dataset underwent some distributional shifts, and (2) data quality,
i.e. the data (training and/or validation) was rather noisy to begin with.

##  4\. Conclusions and future work

Fig. 3 has already demonstrated that there is a large difference in the use of
one source model vs. a different one, even when they are sourced from similar
data sources. This has profound implications in terms of downstream utility.
In our downstream experiments, we found that this caused significant
differences while optimizing heating demand for the storage vessel,
constrained on occupant comfort. Local models were, for most houses (14 out of
16), unusable after the initial 4 week period, to the extent that they led to
control policies that significantly violated user comfort due to not reheating
the storage vessel at all or at incorrect times. After 32 weeks, the situation
was reversed, as most local models (12 out of 16) were able to control the
vessel in a near optimal manner. Surprisingly enough, the large corpus PTM,
with no fine-tuning, led to the optimal control strategy for all systems. The
small corpus PTM performed roughly on par with the local model with only 4
weeks of data (i.e. it was largely unusable).

This obviously begs several question. First, if a large source corpus model
with no fine-tuning can be used to control distributed energy resources, then
local data can be used solely for validation (or fine-tuning). This means
local data can stay on the user’s premises, leading to a simultaneous
improvement in both downstream utility and reduction in privacy leakage. This
would not be possible otherwise with classical machine learning based
solutions. Furthermore, where large pre-trained models are not available, Fig.
3 shows that it is possible to use a carefully chosen small source corpus PTM
to achieve comparable, if not better, performance as using a local model. In
this paper, we have not explored in greater depth the drivers for this
heterogeniety although similarities in feature space certainly seem to play a
role here. This is an important avenue for future research.

The results presented in the paper also have important implications for future
electricity and data markets design. While scarce at the moment, movements to
collect more behind-the-meter data are gathering momentum, raising several
privacy concerns. By utilizing PTMs, this data can stay on-premises, but
valuing their contribution in monetary terms, especially tying in with
downstream utility, is an open and important direction for future research.

###### Acknowledgements.

Hussain Kazmi acknowledges support from FWO, Belgium in the preparation of
this manuscript.

## References

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# Note on the nonrelativistic $T\bar{T}$ deformation

Bin Chen1,2,3<EMAIL_ADDRESS>Jue Hou1<EMAIL_ADDRESS>Jia Tian4
<EMAIL_ADDRESS>1School of Physics, Peking University, No.5 Yiheyuan
Rd, Beijing 100871, P. R. China
2Center for High Energy Physics, Peking University, No.5 Yiheyuan Rd, Beijing
100871, P. R. China
3Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China
4Kavli Institute for Theoretical Sciences (KITS),
University of Chinese Academy of Science, 100190 Beijing, P. R. China

###### Abstract

In this paper, we present our study on the $T\bar{T}$-deformation of non-
relativistic complex scalar field theory. We find the closed form of the
deformed Lagrangian by using the perturbation and the method of
characteristics. Furthermore we compute the exact energy spectrum of the
deformed free theory by using the Brillouin-Wigner perturbation theory in an
appropriate regularization scheme.

##  I Introduction

Solvable irrelevant deformations have attracted many interests in recent years
due to their novel features that provide analytical control on the ultraviolet
(UV) physics regardless of the difficulties from strong coupling or
nonlocality. $T\bar{T}$-deformation is the most studied example of such
deformations111For a pedagogical review see LectureJiang , and for recent
progress see the workshop TTbar2020 . Smirnov:2016lqw ; Cavaglia:2016oda . It
is originally defined in two-dimensional relativistic quantum field theories
where $T\bar{T}$ is a well defined composite operator that can be written as
the determinant of the energy momentum tensor operator. Many interesting
physical quantities such as the energy spectrum, the partition function, the
entanglement entropy and the S-matrix have been computed exactly in the
deformed theories, see LectureJiang  and the references within. These exact
results reveal fruitful and meanwhile unexpected structures of solvable
irrelevant deformations which we have not fully understood. For example Hage1
; Hage2 ; Hage3 , the deformed density of states of a deformed field theory
shows the Hagedorn behavior which typically happens in string theory. On the
other hand, the $T\bar{T}$-deformation to holographic conformal field theory
opens a new window to study AdS/CFT correspondence verlinde ; Hage1 ; kutasov2
; dubovsky1 ; dubovsky2 ; bihdtott ; ttandcf ; cutoffads ; commentsontt ;
Chen:2018eqk ; Hartman:2018tkw ; Caputa:2019pam ; Murdia:2019fax ;
Jeong:2019ylz ; Guica:2019nzm ; Chen:2019mis ; Grieninger:2019zts ;
Lewkowycz:2019xse ; Apolo:2019zai ; Hirano:2020nwq ; Jafari:2019qns ;
He:2020hhm ; Kruthoff:2020hsi .

Recently similar integrable deformations have been considered in other types
of models including integrable lattice models Lattice1 ; Lattice2  and non-
relativistic integrable field theories NonRe1 ; NonRe2 ; BoseJ . Interestingly
$T\bar{T}$ (like) deformation can be interpreted as a special type of algebra-
preserving deformations studied in the context of integrable spin chains Long1
. The standard integrability technique of solving these integrable deformed
models is first to compute the deformed S-matrix in the infinite volume limit
where the deformation simply modifies the S-matrix by multiplying a 1.
Castillejo-Dalitz-Dyson-like factor Smirnov:2016lqw ; CDD ; CDD2 , and then to
substitute the deformed S-matrix into the Bethe equation to solve the theory
in the finite volume. Using this integrability technique the deformed one-
dimensional Bose gas known as the Lieb-Liniger model was carefully studied in
BoseJ  where the author showed that the deformed one-dimensional Bose gas
shares many qualitative features with the $T\bar{T}$-deformed relativistic
quantum field theories. For example, the spectrum can also be given by a flow
equation and the density of states exhibits the Hagedorn behavior in the
thermodynamic limit.

In this work we will use another method, the quantum perturbation theory, to
study the spectrum of the deformed non-relativistic field theories which are
not necessarily integrable. The model we focus on is the non-relativistic
Schrödinger model which can be viewed as the free limit of the Lieb-Liniger
model. We find that for models with a degenerate Legendre transformation such
as non-relativistic Schrödinger or Lieb-Liniger model, the Lagrangian and
Hamiltonian definition of the $T\bar{T}$-deformation may not be equivalent. In
particular, within the Hamiltonian definition of $T\bar{T}$-deformation only
the flow equation of the Hamiltonian is not enough to define the deformation,
one also needs to specify how the constraints change under the flow. In the
first-order deformation, the Legendre transformation is degenerate so one can
use the Dirac-Bergmann algorithm to quantize the system and compute the
spectrum perturbatively. We find that the perturbative expansion of the
deformed spectrum is superficially divergent as expected for the models under
an irrelevant deformation. After imposing the Dirichlet regularization and
dropping off the transcendental terms which are not supposed to appear in the
$T\bar{T}$-deformation, the deformed spectrum match the results derived from
the flow equation. We will also show that after including the second-order
deformation the theory either does not admit a perturbative description or has
ambiguities.

The paper is organized as follows: in Sect. II, we derive the closed form of
$T\bar{T}$-deformed Lagrangian in non-relativistic complex scalar theory; in
Sect. III, we compute the deformed energy spectrum perturbatively, and manage
to find an exact form by using Brillouin-Wigner perturbation; in Sect. IV, we
end with conclusions; we put some technical details into four appendices.

Note added: When we were finishing this project and had obtained the main
results in this paper, two interesting papers Italy ; JiangNew  appeared on
arXiv in the same day. The same closed form of a $T\bar{T}$-deformed
Lagrangian of non-relativistic models have also been derived. In Italy , the
Lagrangian is derived from the dynamical change of coordinates, and in
JiangNew  it is derived using a geometric method. Our method in this work is
similar to the one used in Close , and is somewhat complementary to the ones
in222We would like to mention that the closed form of the $T\bar{T}$-deformed
Lagrangian of non-relativistic models was also found by S. Frolov et.al. from
the light-cone gauge interpretation LG . Italy ; JiangNew .

##  II Non-relativistic $T\bar{T}$-deformed Lagrangian

In this section, we consider the $T\bar{T}$-deformation of a two-dimensional
(2D) non-relativistic field theory whose Lagrangian satisfies333We will work
in the convention that  $\begin{split}g_{\mu\nu}=\mbox{diag}(1,-1),\hskip
12.91663ptx^{\mu}=(x^{0},x^{1})=(t,x).\\\ \end{split}$

$\begin{split}\frac{\partial\mathcal{L}^{(\lambda)}}{\partial\lambda}=\operatorname{det}\left(T_{\mu\nu}^{(\lambda)}\right),\end{split}$
(1)

where $T_{\mu\nu}$ is the energy momentum tensor of this theory. Our goal is
to solve this flow equation (1) for a non-relativistic complex scalar theory
with a potential $V(|\phi|)$.

###  II.1 One real scalar case

To get some insights about (1) we warm up with a toy model involved with only
a free real scalar, whose Lagrangian density reads

$\begin{split}\mathcal{L}_{0}=\phi\partial_{0}\phi+(\partial_{1}\phi)^{2}.\end{split}$
(2)

To solve (1) we expand the Lagrangian with respect to $\lambda$:

$\begin{split}\mathcal{L}=\sum_{n=0}\lambda^{n}\mathcal{L}_{n}.\end{split}$
(3)

Substituting this expansion into the flow equation (1) one can read off
$\mathcal{L}_{n}$ term by term, and the first several orders of the Lagrangian
are of the forms

$\begin{split}&\mathcal{L}_{0}=X+Y,\\\ &\mathcal{L}_{1}=Y(X+Y),\\\
&\mathcal{L}_{2}=Y(X^{2}+3XY+2Y^{2}),\\\
&\mathcal{L}_{3}=Y(X^{3}+6X^{2}Y+10XY^{2}+5Y^{3}),\\\
&\mathcal{L}_{4}=Y(X^{4}+10X^{3}Y+30X^{2}Y^{2}+35XY^{3}+14Y^{4}),\\\
\end{split}$ (4)

where we have defined the convenient variables:

$\begin{split}X\equiv\phi\partial_{0}\phi,\hskip
12.91663ptY\equiv(\partial_{1}\phi)^{2}.\end{split}$ (5)

We observe that these terms can be cast into a general form as

$\begin{split}\mathcal{L}_{n}&=Y\sum_{k=0}^{n}\frac{1}{k}X^{n-k}Y^{k}C_{n-1}^{k-1}C_{n+k}^{n-1}\\\
&=X^{n}Y{}_{2}F_{1}\left(-n,n+1,2,-\frac{Y}{X}\right),\hskip
12.91663ptn=1,2,3,....\end{split}$ (6)

Performing the summation (3) directly gives the closed form of the Lagrangian

$\begin{split}\mathcal{L}&=X+\sum_{k=0}^{\infty}\sum_{n=k}^{n}\lambda^{n}Y\frac{1}{k}X^{n-k}Y^{k}C_{n-1}^{k-1}C_{n+k}^{n-1}\\\
&=-\frac{1}{2\lambda}\left(-1-\lambda X+\sqrt{(1-\lambda X)^{2}-4\lambda
Y}\right),\end{split}$ (7)

which indeed solves the flow equation (1).

Next we add in a potential $V$ which is just a function of $\phi$ but does not
contain the terms of the derivative of $\phi$. As before we expand the
Lagrangian with respect to $\lambda$. The first few terms in the deformed
Lagrangian are

$\begin{split}\mathcal{L}_{0}=&-V+X+Y\\\ \mathcal{L}_{1}=&-V^{2}+VX+Y(X+Y)\\\
\mathcal{L}_{2}=&-V^{3}+V^{2}X-VY^{2}+Y(X^{2}+3XY+2Y^{2})\\\
\mathcal{L}_{3}=&-V^{4}+V^{3}X-VY^{2}(3X+4Y)\\\
&+Y(X^{3}+6X^{2}Y+10XY^{2}+5Y^{3})\\\
\mathcal{L}_{4}=&-V^{5}+V^{4}X+2V^{2}Y^{3}-VY^{2}(6X^{2}+20XY+15Y^{2})\\\
&+Y(X^{4}+10X^{3}Y+30X^{2}Y^{2}+35XY^{3}+14Y^{4}),\\\ \end{split}$ (8)

which can be rewritten in a general form as

$\begin{split}\mathcal{L}_{n}&=-V^{n+1}+V^{n}X+\sum_{j=0}^{n}(-V)^{j}Y^{j+1}\sum_{k=0}^{n-2j}K(n,j,k),\\\
K(n,j,k)&=X^{n-2j-k}Y^{k}C_{n+k}^{n-j-1}C_{n-j-1}^{k-1+j}C_{k-1-j}^{j}\frac{1}{k}.\end{split}$
(9)

Substituting (9) into (3) and performing the summation we end up with the
closed form of the deformed Lagrangian

$\begin{split}\mathcal{L}=&-\frac{V}{1-\lambda V}\\\
&-\frac{1}{2\lambda(1-\lambda V))}\left[-1-\lambda X+\sqrt{(1-\lambda
X)^{2}-4\lambda(1-\lambda V)Y}\right].\end{split}$ (10)

Alternatively the deformed Lagrangian can also be obtained by using the method
of characteristics with the initial condition

$\begin{split}\mathcal{L}_{0}=\phi\alpha+\beta^{2}-V,\quad\mbox{with}\hskip
8.61108pt\alpha=\partial_{0}\phi,\ \beta=\partial_{1}\phi.\end{split}$ (11)

In terms of these new variables $\alpha$ and $\beta$ the flow equation (1)
could be rewritten as

$\begin{split}\frac{\partial\mathcal{L}}{\partial\lambda}=-\mathcal{L}^{2}+\beta\mathcal{L}\frac{\partial\mathcal{L}}{\partial\beta}+\alpha\mathcal{L}\frac{\partial\mathcal{L}}{\partial\alpha},\end{split}$
(12)

which is equivalent to a set of equations:

$\left\\{\begin{aligned} &\frac{d\lambda}{ds}=1\\\
&\frac{d\alpha}{ds}=-\alpha\mathcal{L}\\\
&\frac{d\beta}{ds}=-\beta\mathcal{L}\\\
&\frac{d\mathcal{L}}{ds}=-\mathcal{L}^{2}\\\ &\mathcal{L}(s=0)\\\
&=\phi\alpha(0)+\beta(0)^{2}-V\end{aligned}\right.\Rightarrow\hskip
8.61108pt\left\\{\begin{aligned} &\lambda=s\\\
&\alpha=\frac{C_{3}}{-s+C_{2}}\\\ &\beta=\frac{C_{4}}{-s+C_{2}}\\\
&\mathcal{L}=\frac{1}{s-C_{2}}\\\
&-\frac{1}{C_{2}}=\phi\frac{C_{3}}{C_{2}}+(\frac{C_{4}}{C_{2}})^{2}-V.\end{aligned}\right.$
(13)

Canceling the constants $C_{i}$’s in (13), we get

$\begin{split}\mathcal{L}&=-\frac{V}{1-\lambda V}\\\
&-\frac{1}{2\lambda(1-\lambda V))}\left[-1-\lambda X+\sqrt{(1-\lambda
X)^{2}-4\lambda(1-\lambda V)Y}\right],\end{split}$ (14)

which coincides with (10).

###  II.2 One complex scalar case

Now let us turn to the complex scalar case. The Lagrangian density of the free
complex scalar theory is

$\begin{split}\mathcal{L}_{0}=\frac{i}{2}(\phi^{\ast}\partial_{0}\phi-\phi\partial_{0}\phi^{\ast})-\partial_{1}\phi\partial_{1}\phi^{\ast}\end{split}$
(15)

Define
$\vec{\alpha}\equiv\left(\partial_{0}\phi,\partial_{0}\phi^{\ast}\right)$ and
$\vec{\beta}\equiv\left(\partial_{1}\phi,\partial_{1}\phi^{\ast}\right)$. The
$T\bar{T}$-flow equation is given by

$\begin{split}\frac{\partial\mathcal{L}}{\partial\lambda}=&-\mathcal{L}^{2}+\mathcal{L}\vec{\beta}\cdot\frac{\partial\mathcal{L}}{\partial\vec{\beta}}+\mathcal{L}\vec{\alpha}\cdot\frac{\partial\mathcal{L}}{\partial\vec{\alpha}}\\\
&-\left(\frac{\partial\mathcal{L}}{\partial\vec{\alpha}}\cdot\vec{\alpha}\right)\left(\frac{\partial\mathcal{L}}{\partial\vec{\beta}}\cdot\vec{\beta}\right)+\left(\frac{\partial\mathcal{L}}{\partial\vec{\alpha}}\cdot\vec{\beta}\right)\left(\frac{\partial\mathcal{L}}{\partial\vec{\beta}}\cdot\vec{\alpha}\right).\end{split}$
(16)

Without the last two terms this flow equation (16) can be solved by using the
method of characteristics with the initial condition

$\begin{split}\mathcal{L}(s=0)=\frac{i}{2}\left(\phi^{\ast}(\vec{\alpha})_{1}-\phi(\vec{\alpha})_{2}\right)-(\vec{\alpha})_{1}(\vec{\beta})_{2}\end{split}$
(17)

in exactly the same way as we described in last section. The solution is

$\begin{split}\mathcal{L}&=-\frac{1}{2\lambda}\left[-1-\lambda
X+\sqrt{(1-\lambda X)^{2}-4\lambda Y}\right],\end{split}$ (18)

where $X=\frac{i}{2}(\phi^{\ast}\partial_{0}\phi-\phi\partial_{0}\phi^{\ast})$
and $Y=-\partial_{1}\phi\partial_{1}\phi^{\ast}$.

To include the modifications of the last two terms in (16), we make an ansatz
on the exact Lagrangian and expand it with respect to $\lambda$ . By matching
the expanded terms444The explicit forms of the Lagrangians $\mathcal{L}_{i}$
are quite involved, and we list them in Appendix A.  to the first few orders
we can fix our ansatz completely, and in the end we check our result with the
flow equation. The final result we get is

$\begin{split}\mathcal{L}&=-\frac{1}{2\lambda}\left[-1-\lambda
X+\sqrt{(1-\lambda X)^{2}-4\lambda
Y+8i\lambda^{2}A-4\lambda^{3}B}\right]\end{split}$ (19)

where

$\begin{split}&A=(\phi^{\ast}\partial_{1}\phi+\phi\partial_{1}\phi^{\ast})(\partial_{1}\phi\partial_{0}\phi^{\ast}-\partial_{0}\phi\partial_{1}\phi^{\ast}),\\\
&B=\phi\phi^{\ast}(\partial_{1}\phi\partial_{0}\phi^{\ast}-\partial_{0}\phi\partial_{1}\phi^{\ast})^{2}.\\\
\end{split}$ (20)

In a similar way, we can find the deformed Lagrangian in the case that the
complex scalar has a potential $V$,

$\begin{split}&\mathcal{L}=-\frac{V}{1-\lambda V}-\frac{1}{2\lambda(1-\lambda
V)}(-1-\lambda X+\\\ &\sqrt{(1-\lambda X)^{2}-4\lambda(1-\lambda
V)Y+8i\lambda^{2}(1-\lambda V)A-4\lambda^{3}(1-\lambda V)B}).\end{split}$ (21)

The detailed derivation is straightforward but too tedious to be presented
here.

##  III Deformed energy from quantum perturbative theory

In this section, we will compute the spectrum of the $T\bar{T}$-deformed non-
relativistic free boson (an effective Schrödinger model). The model we are
considering is the free complex scalar defined on a compact region
$x\in[0,R]$, whose Lagrangian density and Hamiltonian are respectively

$\begin{split}&\mathcal{L}=\frac{i}{2}\left(\phi^{\ast}\partial_{t}\phi-\partial_{t}\phi^{\ast}\phi\right)-\partial_{x}\phi^{\ast}\partial_{x}\phi,\\\
&H=\int\mathrm{d}x\,\partial_{x}\phi^{\ast}(x)\partial_{x}\phi(x).\end{split}$
(22)

This simple free theory can be though as the free boson limit of the Lieb-
Liniger model whose $T\bar{T}$-deformation has been well studied in BoseJ .
The momentum operator is

$P=-\frac{i}{2}\int\left[\phi^{\ast}(x)\partial_{x}\phi(x)-\partial_{x}\phi^{\ast}(x)\phi(x)\right].$
(23)

The complex scalar field satisfies the equal-time commutation relations,

$\begin{split}&[\phi(x,t),\phi(y,t)]=0,\quad\left[\phi^{\ast}(x,t),\phi^{\ast}(y,t)\right]=0,\\\
&\left[\phi(x,t),\phi^{\ast}(y,t)\right]=\delta(x-y).\end{split}$ (24)

The Hilbert space is spanned by $N$-particle states,

$|\vec{x}\rangle=\phi^{\ast}\left(x_{1}\right)\ldots\phi^{\ast}\left(x_{N}\right)|0\rangle.$
(25)

Here, $\vec{x}=(x_{1},x_{2},...,x_{N})$ and $|0\rangle$ is the vacuum of Fock
space, which is annihilated by $\phi(x)$. The $N$-particle eigenfunction of
this free model is simply given by the plane waves

$\begin{split}&\langle\vec{x}|\vec{u}\rangle=\psi_{N}(\vec{u}\mid\vec{x})=\frac{1}{\sqrt{N!}}\sum_{\sigma\in\mathrm{S}_{N}}\exp\left[i\sum_{j=1}^{N}x_{j}u_{\sigma_{j}}\right],\\\
&\left|\vec{u}_{N}\right\rangle=\frac{1}{\sqrt{N!}}\int\mathrm{d}^{N}x\psi_{N}(\vec{u}\mid\vec{x})|\vec{x}\rangle,\end{split}$
(26)

with the eigenvalues

$E_{N}(\vec{u})=\sum_{j=1}^{N}u_{j}^{2},\quad
P_{N}(\vec{u})=\sum_{j=1}^{N}u_{j},$ (27)

and

$\langle\vec{x}|\vec{y}\rangle=\sum_{\sigma\in\mathrm{S}_{N}}\delta(x_{i}-y_{\sigma_{i}}).$

###  III.1 Lagrangian vs. Hamiltonian definition

The $T\bar{T}$-deformation is formally defined as a flow of the Lagrangian
density of the theory:

$\begin{split}\partial_{\lambda}\mathcal{L}=-\epsilon^{ab}\epsilon^{cd}T_{ac}^{(\lambda)}T_{bd}^{(\lambda)}\equiv-\mathcal{O}^{(\lambda)}_{T\bar{T}}\end{split}$
(28)

where $T_{ab}^{(\lambda)}$ is the stress tensor of the theory with the flow
parameter $\lambda$. It is generally believed that at least in the classical
level the deformation can be equivalently defined as a flow of the Hamiltonian
density of theory:

$\begin{split}\partial_{\lambda}\mathcal{H}=\mathcal{O}^{(\lambda)}_{T\bar{T}}.\end{split}$
(29)

The classical equivalence has been examined in the leading order in
Kruthoff:2020hsi  and more generally in Jorjadze:2020ili . However the
equivalence is built on an implicit assumption that the Lagrangian is not
singular or equivalently the Legendre transformation is invertible such that
the $T\bar{T}$-deformation and the Legendre transformation are commutative.
The subtlety of a singular Lagrangian is that it leads to a constrained
Hamiltonian. In this section we want to show that in the Hamiltonian formalism
the flow equation (29) itself is not enough to defined the
$T\bar{T}$-deformation.

To illustrate our point, let us consider the simple model (22) with the
Lagrangian

$\begin{split}\mathcal{L}=&\mathcal{L}_{0}+\lambda\mathcal{L}_{1},\end{split}$
(30)

where the leading-order deformation of the Lagrangian is

$\begin{split}\mathcal{L}_{1}=\frac{i}{2}\left(\phi\partial_{t}\phi(\partial_{x}\phi^{\ast})^{2}-\phi^{\ast}\partial_{t}\phi^{\ast}(\partial_{x}\phi)^{2}\right)+\left(\partial_{x}\phi^{\ast}\partial_{x}\phi\right)^{2}.\end{split}$
(31)

The energy density is now:

$\begin{split}T_{00}=&\frac{\partial\mathcal{L}}{\partial(\partial_{t}\phi)}\partial_{t}\phi+\frac{\partial\mathcal{L}}{\partial(\partial_{t}\phi^{\ast})}\partial_{t}\phi^{\ast}-\mathcal{L}\\\
=&\partial_{x}\phi\partial_{x}\phi^{\ast}-\lambda\left(\partial_{x}\phi^{\ast}\partial_{x}\phi\right)^{2}.\end{split}$
(32)

This energy density is supposed to be identified with Hamiltonian density
$\mathcal{H}$ after rewriting $\partial_{t}\phi,~{}\partial_{t}\phi^{\ast}$ in
terms of $\phi,~{}\phi^{\ast}$ and the canonical momenta $\Pi,\Pi^{\ast}$,
which are given by

$\begin{split}&\Pi=\frac{\partial\mathcal{L}}{\partial(\partial_{t}\phi)}=\frac{i}{2}\phi^{\ast}+\frac{i\lambda}{2}\phi(\partial_{x}\phi^{\ast})^{2},\\\
&\Pi^{\ast}=\frac{\partial\mathcal{L}}{\partial(\partial_{t}\phi^{\ast})}=-\frac{i}{2}\phi-\frac{i\lambda}{2}\phi^{\ast}(\partial_{x}\phi)^{2}.\end{split}$
(33)

The problem is that the Lagrangian density $\mathcal{L}$ is singular so that
the rewriting can not be performed. Since the energy density (32) does not
depend on $\partial_{t}\phi,~{}\partial_{t}\phi^{\ast}$ explicitly, one tends
to claim

$\begin{split}\mathcal{H}=T_{00}=\partial_{x}\phi\partial_{x}\phi^{\ast}-\lambda\left(\partial_{x}\phi^{\ast}\partial_{x}\phi\right)^{2}.\end{split}$
(34)

However clearly (34) is not consistent with the definition (29) even in the
leading order

$\begin{split}&\mathcal{H}=\mathcal{H}_{0}+\lambda\mathcal{O}^{(\lambda)}_{T\bar{T}}\\\
&=\partial_{x}\phi\partial_{x}\phi^{\ast}-\lambda\left(\partial_{x}\phi^{\ast}\partial_{x}\phi\right)^{2}-\frac{i\lambda}{2}\left(\phi\partial_{t}\phi(\partial_{x}\phi^{\ast})^{2}-\phi^{\ast}\partial_{t}\phi^{\ast}(\partial_{x}\phi)^{2}\right).\end{split}$
(35)

Strictly speaking, this is not the expression of Hamiltonian density because
it contains the time-derivative terms
$\partial_{t}\phi,~{}\partial_{t}\phi^{\ast}$ which come from $T_{ab}$. Using
the equations of motion

$\begin{split}i\partial_{t}\phi+\partial_{x}^{2}\phi=i\partial_{t}\phi^{\ast}-\partial_{x}^{2}\phi^{\ast}=0+\mathcal{O}(\lambda)\end{split}$
(36)

one can rewrite (LABEL:HTt) as

$\begin{split}&\mathcal{H}=\partial_{x}\phi\partial_{x}\phi^{\ast}-\lambda\left(\partial_{x}\phi^{\ast}\partial_{x}\phi\right)^{2}\\\
&+\frac{\lambda}{2}\left((\partial_{x}\phi^{\ast})^{2}\partial_{x}^{2}\phi\phi+(\partial_{x}\phi)^{2}\partial_{x}^{2}\phi^{\ast}\phi^{\ast}\right).\end{split}$
(37)

It is obvious that (34) and (LABEL:HT001) are in mismatch. This mismatch is
problematic when one tries to compute the spectrum of the theory
perturbatively. This problem does not appear in a relativistic theory, for
example in a theory of the relativistic free boson. In the relativistic case,
the canonical momentum depends on the flow parameter as well such that the
deformed Hamiltonian densities are the same at the classical level in two
different ways, as we show in the Appendix B. A more severe problem about the
definition (29) is that it is not enough to define the deformation. For
example if want to obtain the second-order correction of (LABEL:HT001) we need
to first introduce the Lagrangian density

$\begin{split}\mathcal{L}=\partial_{t}\phi\Pi+\partial_{t}^{\ast}\phi\Pi^{\ast}-\mathcal{H},\end{split}$
(38)

with the primary constraints

$\begin{split}\chi_{1}=\Pi-\frac{i}{2}\phi^{\ast}=0,\quad\chi_{2}=\Pi^{\ast}+\frac{i}{2}\phi=0,\end{split}$
(39)

then derive the first-order canonical stress tensor through

$\begin{split}T^{a}_{b}=\frac{\partial\mathcal{L}}{\partial(\partial_{a}\phi)}\partial_{a}\phi+\frac{\partial\mathcal{L}}{\partial(\partial_{a}\phi^{\ast})}\partial_{a}\phi^{\ast}-\delta^{a}_{b}\mathcal{L},\end{split}$
(40)

and in the end substitute it into (29). Therefore besides the flow equation
(29) one also needs to know how the constraints $\chi_{1}^{(\lambda)}$ and
$\chi_{2}^{(\lambda)}$ change under the flow.

Assuming that we use (29) together with $\chi_{1}^{(\lambda)}$ and
$\chi_{2}^{(\lambda)}$ to define the $T\bar{T}$-deformation of the non-
relativistic free boson gas, we can use the Hellmann-Feynman theorem and
factorization formula to derive the flow equation of the spectrum

$\begin{split}\frac{d\langle n|H|n\rangle}{d\lambda}=\langle
n|T_{00}|n\rangle\langle n|T_{11}|n\rangle-\langle n|T_{01}|n\rangle\langle
n|T_{10}|n\rangle.\end{split}$ (41)

But because the mismatch of (34) and (LABEL:HT001) we can not identify
$\langle n|T_{00}|n\rangle$ with $E_{n}/R$ and $\langle n|H|n\rangle=E_{n}$ at
the same time if we use (LABEL:HT001) as the Hamiltonian density, where $R$ is
the size of the system.

It is not hard to check that the two constraints (39) are the second-class
constraints so one may wonder how about to solve them to reduce the dimensions
of the phase space first and then perform the $T\bar{T}$-deformation. The
reduced Hamiltonian density is given in Gergely:2003zy

$\begin{split}\mathcal{H}_{r}=-\frac{i}{2}\partial_{x}\tilde{\phi}\partial_{x}\tilde{\Pi},\quad\tilde{\phi}=\frac{\phi}{2}+i\Pi^{\ast},\quad\tilde{\Pi}=\Pi+i\frac{\phi^{\ast}}{2}.\end{split}$
(42)

However this does not solve the problem: we still can not transform the
Hamiltonian density into a Lagrangian density which only depends on
$\tilde{\phi}$ and $\partial_{t}\tilde{\phi}$ so that the
$T\bar{T}$-deformation can not be defined. A caveat of our analysis is that
there may exist other ways to describe the reduced Hamiltonian system such
that the velocity $\partial_{t}\phi$ could be solved in terms of the momentum.

The singular Lagrangian density can be cured by including the higher-order
deformations. In particular the exact Lagrangian is not singular so in
principle we can use the Legendre transformation to obtain a Hamiltonian
density without any constraints. But it turns out the resulting Hamiltonian
has a singularity at $\lambda=0$, indicating that it can not be expanded in a
Taylor series of the flow parameter $\lambda$. Requiring the vanishing of
these singularities introduces the constraints. Since there are different ways
to introduce the constraints, there will be ambiguities in the constrained
Hamiltonian. In short, the mismatch (34) and (LABEL:HT001) is not just a
problem in the first-order deformation, instead it is a problem of the
definition of the $T\bar{T}$-deformation in a system with singular Lagrangian
density or equivalently in a constrained Hamiltonian system.

We conclude this section by stressing that the above argument is purely
classical, without considering the problem of Feynman path integral and the
ambiguities in the operator ordering.

###  III.2 Dirac-Bergmann Algorithm

The standard technique to solve the constrained Hamiltonian system is through
the Dirac-Bergman Algorithm (DBA). In this section, we will focus on the
first-order deformation and study how the deformation affects the commutation
relations using DBA. First let us demonstrate the algorithm in the free theory
with the action (22). The conjugate momentum of $\phi$ and $\phi^{\ast}$ are

$\begin{split}\Pi=\frac{i}{2}\phi^{\ast},\quad\Pi^{\ast}=-\frac{i}{2}\phi,\end{split}$
(43)

satisfying the standard equal-time commutation relation

$\begin{split}[\phi(x),\Pi(y)]=i\delta(x-y),\quad[\phi^{\ast}(x),\Pi^{\ast}(y)]=i\delta(x-y).\end{split}$
(44)

The Hamiltonian system has two constraints (39) satisfying

$\begin{split}[\chi_{1},\chi_{2}]\equiv
C_{1_{x},2_{y}}=\delta(x,y).\end{split}$ (45)

These two constraints are the second class, so we can introduce the Dirac
bracket which is defined by

$\begin{split}&[\phi(x),\phi^{\ast}(y)]_{D}\\\ &=[\phi(x),\phi^{\ast}(y)]-\int
dx^{\prime}dy^{\prime}[\phi(x),\chi_{a}(x^{\prime})]C_{a_{x}^{\prime},b_{y}^{\prime}}^{-1}[\chi_{b}(y^{\prime}),\phi^{\ast}(y)]\\\
&=0-\int
dx^{\prime}dy^{\prime}\left(i\delta(x-x^{\prime})(-\delta(x^{\prime}-y^{\prime}))(-i\delta(y^{\prime}-y))\right)\\\
&=\delta(x-y)\end{split}$ (46)

as expected, recalling that the inverse of the matrix $C^{-1}_{a,b}$ is
defined as

$\begin{split}\int
dx^{\prime}C_{a_{x},b_{x}^{\prime}}C^{-1}_{b_{x}^{\prime},c_{y}}=\delta_{ac}\delta(x-y).\end{split}$
(47)

Now we consider the first-order deformed theory with the action

$\begin{split}&\mathcal{L}=\mathcal{L}_{0}+\lambda\mathcal{L}_{1},\\\
&\mathcal{L}_{1}=(\partial_{x}\phi^{\ast}\phi_{x}\phi)^{2}+\frac{1}{2}\left(i\phi\partial_{t}\phi(\partial_{x}\phi^{\ast})^{2}-i\phi^{\ast}\partial_{t}\phi^{\ast}(\partial_{x}\phi)^{2}\right).\end{split}$
(48)

Correspondingly the two constraints become

$\begin{split}&\chi_{1}=\Pi-\frac{i}{2}\phi^{\ast}-\frac{i\lambda}{2}\phi(\partial_{x}\phi^{\ast})^{2},\\\
&\chi_{2}=\Pi^{\ast}+\frac{i}{2}\phi+\frac{i\lambda}{2}(\phi^{\ast}(\partial_{x}\phi)^{2}).\end{split}$
(49)

From them we can compute

$\begin{split}[\chi_{1}(x),\chi_{2}(y)]&\equiv
C_{1_{x},2_{y}}=-C_{2_{y},1_{x}}\\\ &=\delta(x-y)+\lambda G(x,y).\end{split}$
(50)

where

$\begin{split}G(x,y)=\partial_{y}\delta(x-y)\phi^{\ast}(y)\partial_{y}\phi(y)+\partial_{x}\delta(x-y)\partial_{x}\phi^{\ast}(x)\phi(x).\end{split}$
(51)

The inverse of the matrix $C_{1_{x},2_{y}}$ up to the first order of $\lambda$
is simply

$\begin{split}C_{1_{x},2_{y}}^{-1}=-\delta(x-y)+\lambda G(y,x).\end{split}$
(52)

Then at the first order of $\lambda$ the non-vanishing commutation relations
between $\phi$ and $\phi^{\ast}$ is

$\begin{split}[\phi(x),\phi^{\ast}(y)]=0-C_{1_{x},2_{y}}^{-1}=\delta(x-y)-\lambda
G(y,x).\end{split}$ (53)

This implies that $\phi$ and $\phi^{\ast}$ are not canonical variables so that
$\phi^{\ast}$ is not the creation operator to generate the Fock space. This
fact is a reflection of the fact that $T\bar{T}$-deformation is equivalent to
a dynamical coordinate transformation.

At the leading order we can introduce the canonical variables

$\begin{split}\varphi=\phi+\frac{\lambda}{2}\phi^{\ast}(\partial_{x}\phi)^{2},\quad\varphi^{\ast}=\phi^{\ast}+\frac{\lambda}{2}(\partial_{x}\phi^{\ast})^{2}\phi,\end{split}$
(54)

and conversely

$\begin{split}\phi=\varphi-\frac{\lambda}{2}\varphi^{\ast}(\partial_{x}\varphi)^{2},\quad\phi^{\ast}=\varphi^{\ast}-\frac{\lambda}{2}\varphi(\partial_{x}\varphi^{\ast})^{2},\end{split}$
(55)

such that

$\begin{split}\langle
x_{1}x_{2}|x_{3}x_{4}\rangle=\langle\varphi_{1}\varphi_{2}\varphi_{3}^{\ast}\varphi_{4}^{\ast}\rangle=\delta(x_{13})\delta(x_{24})+\delta(x_{14})\delta(x_{23}).\end{split}$
(56)

The relations (54) are not canonical transformation and they are not the state
flow which was introduced in Kruthoff:2020hsi . In terms of the new canonical
variables the Hamiltonian density (34) up to the first order reads

$\begin{split}T_{00}=\mathcal{H}\equiv
H_{0}+H_{1}+\mathcal{O}(\lambda^{2})\end{split}$ (57)

with

$\begin{split}H_{0}=&\partial_{x}\varphi\partial_{x}\varphi^{\ast}\\\
H_{1}=&-2\lambda(\partial_{x}\varphi\partial_{x}\varphi^{\ast})^{2}\\\
&-\lambda\left(\varphi\partial_{x}\varphi\partial_{x}\varphi^{\ast}\partial_{x}^{2}\varphi^{\ast}+\varphi^{\ast}\partial_{x}\varphi^{\ast}\partial_{x}\varphi\partial_{x}^{2}\varphi\right).\end{split}$
(58)

As a consistency check we also work out the first-order deformation of the
number operator and momentum operator

$\begin{split}N&=-\frac{i}{2}\left(\frac{\partial\mathcal{L}}{\partial\partial_{t}\phi}\phi-\frac{\partial\mathcal{L}}{\partial\partial_{t}\phi^{\ast}}\phi^{\ast}\right)\\\
&=\phi^{\ast}\phi+\frac{\lambda}{2}\left({\phi^{\ast}}^{2}(\partial_{x}\phi)^{2}+(\partial_{x}\phi^{\ast})^{2}\phi^{2}\right)\\\
&=\varphi^{\ast}\varphi+\mathcal{O}(\lambda^{2}),\end{split}$ (59)
$\begin{split}P&=\frac{\partial\mathcal{L}}{\partial\partial_{t}\phi}\partial_{x}\phi+\frac{\partial\mathcal{L}}{\partial\partial_{t}\phi^{\ast}}\partial_{x}\phi^{\ast}\\\
&=\frac{i}{2}(\partial_{x}\phi\phi^{\ast}-\partial_{x}\phi^{\ast}\phi)(1-\lambda\,\partial_{x}\phi\partial_{x}\phi^{\ast}),\quad\\\
&=\frac{i}{2}(\partial_{x}\varphi\varphi^{\ast}-\partial_{x}\varphi^{\ast}\varphi)+\text{total
derivatives}+\mathcal{O}(\lambda^{2})\end{split}$ (60)

which are indeed undeformed as expected. We expect that the spectrum of (57)
can be derived by performing the second quantization and applying the standard
quantum perturbation theory.

###  III.3 Two-particle sector without momentum

The explicit form of the Hamiltonian (57) implies that there is only two-to-
two scattering, so without losing any generality we can focus on the two
particle sector. When the total momentum is zero, the $T\bar{T}$-deformed
spectrum satisfies a very simple equation BoseJ

$\begin{split}E_{N}(R,\lambda)=E_{N}(R+\lambda E_{N},0).\end{split}$ (61)

To illustrate the procedure of our calculation, we start from the two-particle
sector with zero momentum, $u_{1}=-u_{2}=u>0$. In this case, a general two-
particle eigenfunction is

$\begin{split}\psi=\frac{\sqrt{2}}{R}\cos\left(u(x_{1}-x_{2})\right),\end{split}$
(62)

with

$\begin{split}u_{I}=\frac{2\pi}{R}I,\quad I\in\mathbf{Z}^{+}.\end{split}$ (63)

First we compute the matrix elements of the normal ordered Hamiltonian (57) of
the field theory in the position space

$\begin{split}\langle\vec{y}|H_{0}+H_{1}|\vec{x}\rangle.\end{split}$ (64)

For example the undeformed Hamiltonian density $\mathcal{H}_{0}$ is evaluated
to be

$\begin{split}\langle\vec{y}|\mathcal{H}_{0}|\vec{x}\rangle&=\langle
0|\varphi(y_{1})\varphi(y_{2})\partial_{x}\varphi^{\ast}\partial_{x}\varphi\varphi^{\ast}(x_{1})\varphi^{\ast}(x_{2})|0\rangle\\\
&=\partial_{x}\delta(x-x_{1})\partial_{x}\delta(x-y_{2})\delta(y_{1}-x_{2})\\\
&+\partial_{x}\delta(x-x_{2})\partial_{x}\delta(x-y_{2})\delta(y_{1}-x_{1})\\\
&+\partial_{x}\delta(x-x_{1})\partial_{x}\delta(x-y_{1})\delta(y_{2}-x_{2})\\\
&+\partial_{x}\delta(x-x_{2})\partial_{x}\delta(x-y_{1})\delta(y_{2}-x_{1}).\\\
\end{split}$ (65)

Then we use the wavefunction (26) to obtain the Hamiltonian

$\begin{split}H_{0II}&\equiv\langle u_{I}|H_{0}|u_{I}\rangle=\frac{1}{2}\int
d\vec{y\,}d\vec{x}\,dx\,\psi_{I}^{\ast}(\vec{y})\psi_{I}(\vec{x})\langle\vec{y}|H_{0}|\vec{x}\rangle\\\
&=\int
d\vec{x}\,\psi^{\ast}_{I}(\vec{x})(-\partial^{2}_{x_{1}}-\partial^{2}_{x_{2}})\psi_{I}(\vec{x})\\\
&=2u_{I}^{2}\equiv\gamma I^{2},\end{split}$ (66)

where $\gamma=\frac{8\pi^{2}}{R^{2}}$. Similarly we find

$\begin{split}H_{1IJ}=-\frac{\lambda}{2}(\frac{\sqrt{2}}{R})^{2}\left(8u_{I}^{2}u_{J}^{2}R)\right)=-2\lambda\frac{\gamma^{2}I^{2}J^{2}}{R}.\end{split}$
(67)

The result $H_{1II}=-2\lambda/RH_{0II}^{2}$ coincides with the subleading term
in the $\lambda$ expansion of the exact spectrum:

$\begin{split}&E_{N}(R,\lambda)=E_{N}^{(0)}-\frac{2\lambda}{R}{E_{N}^{(0)}}^{2}+\frac{7\lambda^{2}}{R^{2}}{E_{N}^{(0)}}^{3}-\frac{30\lambda^{3}}{R^{3}}{E_{N}^{(0)}}^{4}\\\
&+\frac{143\lambda^{4}}{R^{4}}{E_{N}^{(0)}}^{5}-\frac{728\lambda^{5}}{R^{5}}{E_{N}^{(0)}}^{6}\dots\end{split}$
(68)

which is solved from the flow equation in BoseJ .

### Rayleigh-Schrödinger perturbation

The higher-order perturbative results can be obtained by using the Rayleigh-
Schrödinger perturbation:

$\begin{split}E^{(2)}_{I}=&I^{4}(-2\lambda)^{2}\frac{\gamma^{3}}{R^{2}}g_{I}(1),\\\
E^{(3)}_{I}=&I^{4}(-2\lambda)^{3}\frac{\gamma^{4}}{R^{3}}\left(g_{I}(1)^{2}-g_{I}(2)\right)\\\
E^{(4)}_{I}=&I^{4}(-2\lambda)^{4}\frac{\gamma^{5}}{R^{4}}\left(g_{I}(1)^{3}-3g_{I}(1)g_{I}(2)+g_{I}(3)\right)\\\
E^{(5)}_{I}=&I^{4}(-2\lambda)^{5}\frac{\gamma^{6}}{R^{5}}\left(g_{I}(1)^{4}-6g_{I}(1)^{2}g_{I}(2)\right.\\\
&\left.+2g_{I}(2)^{2}+4g_{I}(1)g_{I}(3)-g_{I}(4)\right)\\\ &....\end{split}$

where for convenience we have introduced the function

$\begin{split}g_{I}(a)\equiv\sum_{J\neq
I}^{\infty}\frac{I^{4(a-1)}J^{4}}{(I^{2}-J^{2})^{a}}.\end{split}$ (69)

The functions $g_{I}(1)$ and $g_{I}(2)$ are divergent, and proper
regularization has to be adopted. The first few terms of $g_{I}(a)$ are given
by

$\begin{split}&g_{I}(1)=\frac{7I^{2}}{4},\quad
g_{I}(2)=-\frac{11I^{4}}{16}+\frac{\pi^{2}I^{6}}{12},\\\
&g_{I}(3)=-\frac{I^{6}}{32}-\frac{5\pi^{2}I^{8}}{48},\\\
&g_{I}(4)=-\frac{3I^{8}}{256}+\frac{\pi^{2}I^{10}}{96}+\frac{\pi^{4}I^{12}}{720},....\end{split}$
(70)

where we have used the Dirichlet regularization to regularize $g_{I}(1)$ and
$g_{I}(2)$. For example, to compute $g(1)$ we first separate it into the
divergent piece and convergent piece

$\begin{split}g_{I}(1)=\sum_{J\neq
I}^{\infty}\left([-I^{2}-J^{2}]+\frac{I^{4}}{I^{2}-J^{2}}\right)\end{split}$
(71)

then regularize the divergent summation with the Dirichlet regularization.
Substituting the regularized functions into (III), we obtain the higher-order
quantum corrections to the spectrum:

$\begin{split}&E_{I}^{(2)}=I^{6}\lambda^{2}\frac{\gamma^{3}}{R^{2}}7,\quad
E_{I}^{(3)}=I^{8}(-\lambda)^{3}\frac{\gamma^{4}}{R^{3}}\left(30-\frac{2\pi^{2}I^{2}}{3}\right),\\\
&E^{(4)}_{I}=I^{10}(-\lambda)^{4}\frac{\gamma^{5}}{R^{4}}\left({143}-\frac{26\pi^{2}I^{2}}{3}\right),\\\
&E^{(5)}_{I}=I^{12}(-\lambda)^{5}\frac{\gamma^{6}}{R^{5}}\left(728-\frac{2\pi^{4}I^{4}-400\pi^{2}I^{2}}{5}\right),....\end{split}$
(72)

Surprisingly, these expressions are in exact match with (LABEL:Exact) if all
the transcendental terms are discarded. It was shown in Zamolodchikov:1991vx ;
Caselle:2013dra  that the appearance of the transcendental terms is the signal
of the contributions from other irrelevant operators instead of $T\bar{T}$, so
they should be discarded. If we follow this rule, we can use the Brillouin-
Wigner perturbation to obtain the exact spectrum which coincides with the one
solved from the flow equation, as we show below.

### Exact form: Brillouin-Wigner perturbation

A technical introduction to the Brillouin-Wigner perturbation can be found in
Appendix C. The exact energy satisfies

$\begin{split}E_{k}=E_{k}^{(0)}+\sum_{i=1}^{\infty}\frac{\prod_{j=0}^{i-1}H_{1m_{j}m_{j+1}}}{\prod_{j=1}^{i-1}(E_{k}-E^{(0)}_{m_{j}})},\quad
m_{0}=m_{i}=k.\end{split}$ (73)

In our case, $E_{n}^{(0)}=\gamma n^{2},\ \
H_{1nm}=-2\lambda\frac{\gamma^{2}}{R}n^{2}m^{2}$ so that

$\begin{split}E_{k}=\gamma k^{2}+\gamma
k^{4}\sum_{i=1}^{\infty}\left(-\frac{2\gamma\lambda}{R}\right)^{i}g(k,b_{k})^{i-1}\end{split}$
(74)

where $b_{k}=E_{k}/\gamma$ and

$\begin{split}g(k,b_{k})&=\sum_{m\neq
k,m=1}^{\infty}\frac{m^{4}}{b_{k}-m^{2}}=\frac{k^{4}}{k^{2}-b_{k}}+\frac{1}{2}b_{k}^{3/2}\pi\cot(\pi\sqrt{b_{k}}).\end{split}$
(75)

The equation of the energy can be recast into the following form

$\begin{split}&b_{k}=k^{2}-\frac{\beta
k^{4}}{1+\beta\left(\frac{k^{4}}{k^{2}-b_{k}}+\frac{1}{2}b_{k}^{3/2}\pi\cot(\pi\sqrt{b_{k}})\right)}\\\
\end{split}$ (76)

where $\beta=\frac{\gamma\lambda}{R}$. As in the last subsection, if we
discard the $\pi$-dependent terms this result will match the exact result
found in BoseJ . Notice that we must do this carefully. Naively, it seems
necessary to throw away the term $\pi\cot(\pi\sqrt{b_{k}})$ completely.
However, because of $b_{k}=k^{2}+O(\lambda)$, we can expand the function and
find

$\begin{split}&\pi\cot(\pi\sqrt{b_{k}})=\pi\cot(\pi(\sqrt{b_{k}}-k))\\\
&=\frac{1}{\sqrt{b_{k}}-k}+(\mbox{$\pi$-dependent terms}).\end{split}$ (77)

Then the equation of the energy becomes

$\begin{split}\frac{b_{k}}{k^{2}}=1-\frac{\beta k^{2}}{1+\beta
k^{2}\left(\frac{1}{1-b_{k}/k^{2}}+\frac{1}{2}\frac{(b_{k}/k^{2})^{3/2}}{\sqrt{b_{k}/k^{2}}-1}\right)}.\\\
\end{split}$ (78)

This is an algebraic equation, and is equivalent to

$\begin{split}\tilde{\beta}_{k}\tilde{b}_{k}^{3/2}+\sqrt{\tilde{b}_{k}}-1=0\end{split}$
(79)

where

$\begin{split}\tilde{\beta}_{k}=\beta k^{2}/2=\frac{\gamma\lambda
k^{2}}{2R},\hskip 12.91663pt\tilde{b}_{k}=\frac{E_{k}}{\gamma
k^{2}}.\end{split}$ (80)

Multiplying (79) by
$\tilde{\beta}_{k}\tilde{b}_{k}^{3/2}+\sqrt{\tilde{b}_{k}}+1$, we get

$\begin{split}\tilde{\beta}_{k}^{2}\tilde{b}_{k}^{3}+2\tilde{\beta}_{k}\tilde{b}_{k}^{2}+\tilde{b}_{k}-1=0,\end{split}$
(81)

which is the Burgers’ equation in BoseJ . A solution of (79) is simply

$\begin{split}\tilde{b}_{k}=\frac{2}{3\tilde{\beta}_{k}}\left[\cosh\left(\frac{2}{3}{\rm
arcsinh}\left(\frac{3\sqrt{3\tilde{\beta}_{k}}}{2}\right)\right)-1\right].\end{split}$
(82)

Expanding the above relation, we can reproduce the perturbative results in the
last subsection.

###  III.4 Two-particle sector with momentum

In this subsection, we consider the two-particle sector with momentum. The
general two-particle eigenfunction is

$\begin{split}\psi=\frac{1}{\sqrt{2}R}\left(e^{iu_{1}x_{1}+iu_{2}x_{2}}+e^{iu_{1}x_{2}+iu_{2}x_{1}}\right),\end{split}$
(83)

with zero-order eigenvalue

$\begin{split}E^{(0)}_{u_{1},u_{2}}=u_{1}^{2}+u_{2}^{2}.\end{split}$ (84)

When $u_{1}\neq\pm u_{2}$, the eigenvalues have a four-fold degeneracy
$E^{(0)}_{u_{1},u_{2}}=E^{(0)}_{-u_{1},u_{2}}=E^{(0)}_{u_{1},-u_{2}}=E^{(0)}_{-u_{1},-u_{2}}$.
When $u_{1}=u_{2}$, the degeneracy is two-fold:
$E^{(0)}_{u_{1},u_{2}}=E^{(0)}_{-u_{1},-u_{2}}$. The degenerate spectrum is an
obstacle to apply the flow equation (41) to derive the deformed spectrum
because the factorization formula fails. However in this case, because the
total momentum is conserved so one can focus on a sector with some fixed total
momentum $u=u_{1}+u_{2}$. Within this sector the spectrum will not be
degenerate anymore and the flow equation still works. The flow equation can be
solved by a formal series expansion in $\lambda$. The first few terms of the
deformed spectrum are given by555There is a typo in equation $(122)$ of BoseJ
: $E_{N}^{(1)}=\frac{1}{R}(-2M_{2}^{2}+2M_{1}M_{3})$.

$\begin{split}&\mathcal{E}_{u_{1},u_{2}}^{(1)}=\frac{\lambda}{R}2u_{1}u_{2}(u_{1}-u_{2})^{2},\\\
&\mathcal{E}_{u_{1},u_{2}}^{(2)}=-\frac{\lambda^{2}}{R^{2}}2u_{1}u_{2}(u_{1}-u_{2})^{2}(u_{1}^{2}-5u_{1}u_{2}+u_{2}^{2}),\\\
&\mathcal{E}_{u_{1},u_{2}}^{(3)}=\frac{\lambda^{3}}{R^{3}}2u_{1}u_{2}(u_{1}-u_{2})^{2}(u_{1}^{2}-10u_{1}u_{2}+u_{2}^{2})\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(u_{1}^{2}-3u_{1}u_{2}+u_{2}^{2}),\\\
&\dots\end{split}$ (85)

Below we will try to compute the deformed spectrum within quantum perturbation
theory. In the basis of the wavefunction (83), we find

$\displaystyle\langle u_{1},u_{2}|H_{1}|u_{3},u_{4}\rangle$ (86)
$\displaystyle=$
$\displaystyle-\frac{\lambda}{R}\delta(u_{1}+u_{2}-u_{3}-u_{4})$
$\displaystyle\times\left(8u_{1}u_{2}u_{3}u_{4}-(u_{1}+u_{2})^{2}(u_{1}u_{2}+u_{3}u_{4})\right),$

where the Dirac delta function is due to the momentum conservation. In
particular we have

$\begin{split}E_{u_{1},u_{2}}^{(1)}=\langle
u_{1},u_{2}|H_{1}|u_{1},u_{2}\rangle=\frac{\lambda}{R}2u_{1}u_{2}(u_{1}-u_{2})^{2}=\mathcal{E}_{u_{1},u_{2}}^{(1)}.\end{split}$
(87)

The second-order correction is given by a divergent summation666Without losing
generality we have assumed $u_{1}\geq u_{2}$.

$\begin{split}E_{u_{1},u_{2}}^{(2)}&=\sum_{u_{3},u_{4}}\frac{|\langle
u_{1}u_{2}|H_{1}|u_{3},u_{4}\rangle|^{2}|}{u_{1}^{2}+u_{2}^{2}-u_{3}^{2}-u_{4}^{2}}\\\
&=\frac{\lambda^{2}}{R^{2}}\sum_{u_{3}\neq
u_{1},u_{3}=-\infty}^{\infty}\frac{-8u_{1}u_{2}(u_{1}+u_{2}-u_{3})u_{3}+(u_{1}+u_{2})^{2}(u_{1}u_{2}+(u_{1}+u_{2}-u_{3})u_{3})^{2}}{2(u_{1}-u_{3})(u_{3}-u_{2})}.\end{split}$
(88)

As before to regularize it we first separate out the divergent piece by a
$1/u_{3}$ Taylor expansion

$\begin{split}\eqref{TwoMoH2}=&\sum_{u_{3}=-\infty}^{\infty}[D_{0}(u_{1},u_{2})+D_{1}(u_{1},u_{2})u_{3}+D_{2}(u_{1},u_{2})u_{3}^{2}]\\\
&-(D_{0}(u_{1},u_{2})+D_{1}(u_{1},u_{2})u_{1}+D_{2}(u_{1},u_{2})u_{1}^{2})\\\
&+\sum_{u_{3}\neq
u_{1},u_{3}=-\infty}^{\infty}C(u_{1},u_{2},u_{3}),\end{split}$ (89)

with

$\begin{split}&C(u_{1},u_{2},u_{3})=\frac{2u_{1}^{2}(u_{1}-u_{2})^{4}u_{2}^{2}}{(u_{1}-u_{3})(u_{3}-u_{2})}.\end{split}$
(90)

Then using the Dirichlet regularization

$\begin{split}\sum_{u_{3}=-\infty}^{\infty}1=\sum_{u_{3}=-\infty}^{\infty}u_{3}=\sum_{u_{3}=-\infty}^{\infty}u_{3}^{2}=0,\end{split}$
(91)

we find the final result of $E_{u_{1},u_{2}}^{(2)}$

$\begin{split}\mathcal{E}_{u_{1},u_{2}}^{(2)}=&\frac{\lambda^{2}}{R^{2}}\left(0-\left(2u_{1}(u_{1}-u_{2})^{2}u_{2}(u_{1}^{2}-6u_{1}u_{2}+u_{2}^{2})\right)\right.\\\
&\left.+2u_{1}^{2}(u_{1}-u_{2})^{2}u_{2}^{2}\right).\end{split}$ (92)

Similarly, we can read the third-order correction

$\begin{split}E_{u_{1},u_{2}}^{(3)}=\mathcal{E}_{u_{1},u_{2}}^{(3)}-\pi^{2}\frac{2\lambda^{3}}{3R^{3}}u_{1}^{3}u_{2}^{3}(u_{1}-u_{2})^{4},\end{split}$
(93)

which reduces to (LABEL:TwoSpectrum) by taking $u_{2}=-u_{1}$. The calculation
of higher-order corrections is much involved, but the feature is very similar
so we will not present the analysis here.

###  III.5 Three-particle sector

To analyze the spectrum of the three-particle sector one has to include the
second order deformation. However as we discuss before, after including the
second order deformation the theory either does not admit a perturbative
description or has ambiguities. The detail of this analysis can be found in
Appendix D. In this subsection we only compute the spectrum up to the first-
order deformation. The eigenfunction of the free three-particle theory with
$I=(I_{1},I_{2},I_{3})$ is given by

$\begin{split}\psi_{I}(x_{1},x_{2},x_{3})=\frac{1}{\sqrt{3!R^{3}}}\sum_{\sigma\in
S}\exp^{i\sum_{j=1}^{3}x_{j}u_{I\sigma_{j}}}\end{split}$ (94)

As before one can find the undeformed Hamiltonian in the position space

$\begin{split}&\langle x_{1},x_{2},x_{3}|H_{0}|y_{1},y_{2},y_{3}\rangle\\\
&=\sum_{i,j\in
S}\int\,dx\partial_{x}\delta(x-x_{i_{1}})\partial_{x}\delta(x-y_{j_{1}})\delta(x_{i_{2}}-y_{j_{2}})\delta(x_{i_{3}}-y_{j_{3}})\end{split}$
(95)

and in the momentum space

$\begin{split}\langle
u_{I}|H_{0}|u_{J}\rangle=\left(\frac{2\pi}{R}\right)^{2}\sum
I_{i_{1}}J_{j_{1}}\delta^{I_{i_{1}}}_{J_{j_{1}}}\delta^{I_{i_{2}}}_{J_{j_{2}}}\delta^{I_{i_{3}}}_{J_{j_{3}}},\end{split}$
(96)

where $S$ denotes the permutation of three particles $(1,2,3)$, and
$u_{I_{1}}=\frac{2\pi}{R}I_{1}$. Therefore the zeroth-order energy is given
by777Here we assume that $I_{1}>I_{2}>I_{3}$.

$\begin{split}E_{I}^{(0)}=\langle
u_{I}|H_{0}|u_{I}\rangle=\left(\frac{2\pi}{R}\right)^{2}(I_{1}^{2}+I_{2}^{2}+I_{3}^{2}).\end{split}$
(97)

Similarly, we can get the first-order result,

$\begin{split}&\langle x_{1},x_{2},x_{3}|H_{1}|y_{1},y_{2},y_{3}\rangle=\\\
&\sum_{i,j\in
S}\int\,dx\delta(x-x_{i_{1}})\delta(x-x_{i_{2}})\delta(x-y_{j_{1}})\delta(x-y_{j_{2}})\delta(x_{i_{3}}-y_{j_{3}})\\\
&~{}~{}~{}\left(-2\partial_{x_{i_{1}}}\partial_{x_{i_{2}}}\partial_{y_{j_{1}}}\partial_{y_{j_{2}}}-\partial_{x_{i_{1}}}\partial_{x_{i_{2}}}^{2}\partial_{y_{j_{1}}}-\partial_{x_{i_{1}}}\partial_{y_{j_{1}}}\partial_{y_{j_{2}}}^{2}\right)\end{split}$
(98)

and

$\begin{split}&\langle u_{I}|H_{1}|u_{J}\rangle\\\
&=\frac{1}{R}\left(\frac{2\pi}{R}\right)^{4}\sum_{i,j\in
S}\delta^{I_{i_{3}}}_{J_{i_{3}}}\left(-2I_{i_{1}}I_{i_{2}}J_{j_{1}}J_{j_{2}}+I_{i_{1}}I_{i_{2}}^{2}J_{j_{1}}+I_{i_{1}}J_{j_{1}}J_{j_{2}}^{2}\right)\end{split}$
(99)

The first order correction of the energy is given by

$\begin{split}E^{(1)}_{I}=&\lambda\langle u_{I}|H_{1}|u_{I}\rangle\\\
=&\lambda\frac{1}{R}\left(\frac{2\pi}{R}\right)^{4}\left(2I_{1}I_{2}(I_{1}-I_{2})^{2}\right.\\\
&\left.+2I_{2}I_{3}(I_{2}-I_{3})^{2}+2I_{3}I_{1}(I_{3}-I_{1})^{2}\right)\end{split}$
(100)

It is the same as the one in BoseJ .

The second-order correction of the energy should come from $\langle
u_{I}|H_{1}|u_{J}\rangle$ and $\langle u_{I}|H_{2}|u_{I}\rangle$

$\begin{split}E_{I}^{(2)}=\lambda^{2}\sum_{J\neq I}\frac{|\langle
u_{I}|H_{1}|u_{J}\rangle|^{2}}{E_{I}^{(0)}-E_{J}^{(0)}}+\lambda^{2}\langle
u_{I}|H_{2}|u_{I}\rangle.\end{split}$ (101)

The first part of $E_{I}^{(2)}$ is

$\begin{split}&\sum_{J\neq I}\frac{|\langle
u_{I}|H_{1}|u_{J}\rangle|^{2}}{E_{I}^{(0)}-E_{J}^{(0)}}\\\
&=\frac{1}{R^{2}}\left(\frac{2\pi}{R}\right)^{6}\sum_{J\neq
I}\frac{1}{I_{1}^{2}+I_{2}^{2}+I_{3}^{2}-J_{1}^{2}-J_{2}^{2}-J_{3}^{2}}\sum_{i_{3},j_{3}}\delta^{I_{i_{3}}}_{J_{i_{3}}}\left(-8I_{i_{1}}I_{i_{2}}J_{j_{1}}J_{j_{2}}+(I_{i_{1}}I_{i_{2}}+J_{j_{1}}J_{j_{2}})(I_{i_{1}}+I_{i_{2}})(J_{j_{1}}+J_{j_{2}})\right)^{2}\end{split}$

Notice that there is no term like
$\delta_{J_{1}}^{I_{2}}\delta_{J_{2}}^{I_{3}}$ due to
$I_{1}+I_{2}+I_{3}=J_{1}+J_{2}+J_{3}$ and $J\neq I$. After some manipulations,
we read

$\displaystyle\sum_{J\neq I}\frac{|\langle
u_{I}|H_{1}|u_{J}\rangle|^{2}}{E_{I}^{(0)}-E_{J}^{(0)}}$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}\left(\frac{2\pi}{R}\right)^{6}\left(-2I_{1}I_{2}(I_{1}-I_{2})(I_{1}^{2}-5I_{1}I_{2}+I_{2}^{2})-2I_{1}I_{3}(I_{1}-I_{3})(I_{1}^{2}-5I_{1}I_{3}+I_{3}^{2})-2I_{3}I_{2}(I_{3}-I_{2})(I_{3}^{2}-5I_{3}I_{2}+I_{2}^{2})\right)$

Matching $E_{I}^{(2)}$ with the result in BoseJ , it implies that

$\begin{split}\langle
u_{I}|H_{2}|u_{I}\rangle=&2I_{1}I_{2}I_{3}(5I_{1}^{3}-6I_{1}^{2}I_{2}-6I_{1}I_{2}^{2}+5I_{2}^{3}-6I_{1}^{2}I_{3}\\\
&+21I_{1}I_{2}I_{3}-6I_{2}^{2}I_{3}-6I_{1}I_{3}^{2}-6I_{2}I_{3}^{2}+5I_{3}^{3}).\end{split}$
(103)

##  IV Conclusion

In this work we have derived a closed form (LABEL:ttbarL) of the Lagrangian of
the $T\bar{T}$-deformed non-relativistic model by using the perturbative
computations and the method of characteristics. In addition, we computed the
deformed spectrum by using two different perturbation theories. In particular,
by using the Brillouin-Wigner perturbation, we managed to find the exact form
of the deformed energy spectrum, which can match the one obtained in BoseJ
after some appropriate regularization.

One subtle issue in our study is on the regularization. The perturbative
expansion of the deformed energy is a summation of infinite series, which is
superficially divergent. We applied the zeta-function regularization and found
that the results could match the ones in BoseJ  if we discard the
$\pi$-dependent terms. The similar $\pi$-dependent terms, which were referred
to as the terms with nonzero transcendentality, appeared also in the study on
other models Zamolodchikov:1991vx ; Caselle:2013dra . They could come from the
perturbations of higher order of $T\bar{T}$, say $T^{2}\bar{T}^{2}$. It would
be certainly interesting to understand why they appear in our perturbative
analysis.

Another interesting finding is that the $T\bar{T}$-deformation on the
Lagrangian and the one on the Hamiltonian is not equivalent if the theory has
a degenerate Legendre transformation. The deformation may remove the
degeneracy but the resulting deformed theory would not admit a perturbative
description. More importantly, we want to stress that to define the
$T\bar{T}$-deformation in the Hamiltonian formalism one has to also specify
how the constraints are deformed.

## Acknowledgements

We would like to thank Reiko Liu, Shi-Lin Zhu, Florian Loebbert and Sergey
Frolov for valuable discussions and comments. The work is in part supported by
NSFC Grants No. 11325522, No. 11735001, No. 11947301 and No. 12047502. J. T.
is also supported by the UCAS program of special research associate and by the
internal funds of the KITS.

##  Appendix A Perturbative Lagrangian

Expanding the $T\bar{T}$-deformed Lagrangian by $\lambda$,
$\mathcal{L}=\sum\lambda^{n}\mathcal{L}_{n}$, one can get a recursion relation
from the flow equation (1),

$\begin{split}\mathcal{L}_{i+1}=&\frac{1}{i+1}\sum_{k=0}^{i}\left[-\mathcal{L}_{k}\mathcal{L}_{i-k}+\mathcal{L}_{k}\vec{\beta}\cdot\frac{\partial\mathcal{L}_{i-k}}{\partial\vec{\beta}}+\mathcal{L}_{i-k}\vec{\alpha}\cdot\frac{\partial\mathcal{L}_{k}}{\partial\vec{\alpha}}-\left(\frac{\partial\mathcal{L}_{k}}{\partial\vec{x}}\cdot\vec{\alpha}\right)\left(\frac{\partial\mathcal{L}_{i-k}}{\partial\vec{y}}\cdot\vec{\beta}\right)+\left(\frac{\partial\mathcal{L}_{i-k}}{\partial\vec{\alpha}}\cdot\vec{\beta}\right)\left(\frac{\partial\mathcal{L}_{k}}{\partial\vec{\beta}}\cdot\vec{\alpha}\right)\right].\end{split}$
(104)

For one complex scalar, by the initial condition,

$\begin{split}\mathcal{L}_{0}=-V+X+Y\end{split}$ (105)

one can get the first several $\mathcal{L}_{k}$’s,

$\begin{split}\mathcal{L}_{1}=&-2iA-(V-X-Y)(V+Y)\\\
\mathcal{L}_{2}=&B-V^{3}+V^{2}X-VY^{2}-2iA(X+2Y)\\\ &+Y(X^{2}+3XY+2Y^{2})\\\
\mathcal{L}_{3}=&-4A^{2}-V^{4}+V^{3}X+B(X+2Y)-VY^{2}(3X+4Y)\\\
&-2iA(X^{2}+6XY-2(V-3Y)Y)\\\ &+Y(X^{3}+6X^{2}Y+10XY^{2}+5Y^{3})\\\
\mathcal{L}_{4}=&-V^{5}+V^{4}X+2V^{2}Y^{3}+B(X^{2}+6XY+6Y^{2})\\\
&+Y(X^{4}+10X^{3}Y+30X^{2}Y^{2}+35XY^{3}+14Y^{4})\\\ &+4A^{2}(V-3(X+2Y))\\\
&-VY(2B+Y(6X^{2}+20XY+15Y^{2}))\\\
&-2iA(2B+(X+2Y)(X^{2}+10XY+2Y(-3V+5Y)))\end{split}$ (106)

##  Appendix B Difference between two deformation formalisms

To illustrate the difference between the deformations in the Lagrangian
formalism and the Hamiltonian formalism, let us consider the well studied
$T\bar{T}$-deformed free scalar theory whose action was found in
Cavaglia:2016oda  to take Nambu-Goto form

$\begin{split}\mathcal{L}_{\lambda}=-\frac{1}{\lambda}\left(\sqrt{1+\lambda(\partial_{x}\phi)^{2}-\lambda(\partial_{t}\phi)^{2}}-1\right).\end{split}$
(107)

Taking the derivative with respect to $\partial_{t}\phi$ we can obtain the
conjugate momentum

$\begin{split}\Pi=\frac{\partial_{t}\phi}{\sqrt{1+\lambda(\partial_{x}\phi)^{2}-\lambda(\partial_{t}\phi)^{2}}}.\end{split}$
(108)

The standard Legendre transformation leads to the deformed Hamiltonian

$\begin{split}H_{\lambda}=\frac{1}{\lambda}\left(\sqrt{1+\lambda(\Pi_{\lambda}^{2}+(\partial_{x}\phi)^{2})+\lambda^{2}\Pi_{\lambda}^{2}(\partial_{x}\phi)^{2}}-1\right).\end{split}$
(109)

Now let us compare the first-order expansion of (107) and (109):

$\begin{split}\mathcal{L}_{\lambda}=&\frac{1}{2}\left((\partial_{t}\phi)^{2}-(\partial_{x}\phi)^{2}\right)+\frac{1}{8}((\partial_{t}\phi)^{2}-(\partial_{x}\phi)^{2})^{2}\lambda+\mathcal{O}(\lambda^{2})\\\
&\sim\mathcal{L}_{0}+\lambda\,\mbox{det}(T_{\mu\nu}),\\\
\mathcal{H}_{\lambda}=&\frac{1}{2}\left(\Pi^{2}+(\partial_{x}\phi)^{2}\right)-\frac{1}{8}((\Pi)^{2}-(\partial_{x}\phi)^{2})^{2}\lambda+\mathcal{O}(\lambda^{2}).\end{split}$
(110)

The first-order correction of $H_{\lambda}$ can be written as

$\begin{split}\mathcal{H}_{\lambda}=\mathcal{H}_{0}-\lambda\,\mbox{det}(T_{\mu\nu})\end{split}$
(111)

if and only if we use the zeroth-order result of $\Pi$ in (108). It implies
that if one wants to find the first-order expansion of the Hamiltonian from
the Lagrangian via Legendre transformation, he has to rewrite the time
derivative terms in terms of the conjugate momentum first, then perform the
expansion while keep the conjugate momentum. However in the non-relativistic
model we are studying, this can not be done. This fact leaves us with two
possible choices: one may take (111) as the first order correction to the
Hamiltonian or one may take the Legendre transformation of (110) as the first
order correction to the Hamiltonian. We have shown explicitly that these two
choice give rise to different results.

##  Appendix C Brillouin-Wigner Perturbation

In the textbookShankar , Rayleigh-Schrödinger perturbation theory has been
discussed. However, it is too complicated to go to a higher order. If we want
to know the correction of the energy, we have to calculate the energy and
wavefunctions simultaneously. And its physical meaning is not clear. In
contrast, the Brillouin-Wigner perturbation is easy to calculate to higher
orders. One can calculate the energy alone without knowing the wavefunctions.
It is the Feynman diagrams in quantum mechanics. One can find a good lecture
in Sun:2016 .

For a theory whose Hamiltonian is $H=H_{0}+\lambda H^{\prime}$, where
$\lambda$ is small. The energy of the $k$-th eigenstate is given by

$\begin{split}E_{n}=&E_{n}^{(0)}+\lambda
H^{\prime}_{\mathrm{nn}}+\lambda^{2}\frac{H^{\prime}_{nm}H^{\prime}_{mn}}{E_{n}-E_{m}^{(0)}}\\\
&+\lambda^{3}\frac{H^{\prime}_{nm}H^{\prime}_{mm^{\prime}}H^{\prime}_{m^{\prime}n}}{\left(E_{n}-E_{m}^{(0)}\right)\left(E_{n}-E_{m^{\prime}}^{(0)}\right)}+...\end{split}$
(112)

where

$\begin{split}H^{\prime}_{\mathrm{ij}}=\left\langle\psi_{i}^{(0)}\left|H^{\prime}\right|\psi_{j}^{(0)}\right\rangle.\end{split}$
(113)

The superscript $(0)$ denotes the zeroth correction and $\psi_{i}$ denotes the
$i$-th eigenstate. One need to sum over the repeated indices. Eq. (112) is an
equation of energy $E_{n}$ alone. One can use the iteration to get the
perturbative formula of the energy, which is just Rayleigh-Schrödinger
perturbation formula. By solving the equation, one can get the exact energy
spectrum in principle. However, the equation has infinite number of terms and
one cannot solve it in most cases. Fortunately, in the case at hand, we can
solve it exactly.

There is a physical interpretation for Brillouin-Wigner perturbation. Eq.
(112) could be understood as the summation of Feynman diagrams:
$H^{\prime}_{ij}$ denotes the vertex and $\frac{1}{E_{n}-E_{m}^{(0)}}$ denotes
the propagator. For example,

$\begin{split}&H^{\prime}_{nm}\frac{1}{E_{n}-E_{m}^{(0)}}H^{\prime}_{mm^{\prime}}\frac{1}{E_{n}-E_{m^{\prime}}^{(0)}}H^{\prime}_{m^{\prime}n}=\vbox{\includegraphics[scale={0.25}]{1.png}}\\\
\end{split}$ (114) $E_{n}=\vbox{\includegraphics[scale={0.4}]{2.png}}\\\ $
(115)

##  Appendix D Towards the second order deformation

Expanding the exact deformed Lagrangian to the second-order of $\lambda$ gives

$\begin{split}\mathcal{L}=\mathcal{L}_{0}+\lambda\mathcal{L}_{1}+\lambda^{2}\mathcal{L}_{2}\end{split}$
(116)

with (48) and

$\begin{split}&\mathcal{L}_{2}=-2\partial_{x}\phi^{\ast
2}\partial_{x}\phi^{2}+i\partial_{x}\phi\partial_{x}\phi^{\ast}\left(\partial_{t}\phi^{\ast}\partial_{x}\phi^{2}\phi^{\ast}-\partial_{t}\phi\partial_{x}\phi^{\ast
2}\phi\right)\\\
&\qquad-\frac{1}{2}\partial_{t}\phi\partial_{t}\phi^{\ast}\partial_{x}\phi\partial_{x}\phi^{\ast}\phi\phi^{\ast}+\frac{i}{2}\partial_{x}\phi^{2}\partial_{x}\phi^{\ast
2}\left(\partial_{t}\phi\phi^{\ast}-\partial_{t}\phi^{\ast}\phi\right)\\\
&\qquad+\frac{1}{4}\partial_{t}\phi\partial_{t}\phi^{\ast}(\partial_{x}\phi^{\ast
2}\phi^{2}+\partial_{x}\phi^{2}\phi^{\ast 2}).\end{split}$ (117)

This Lagrangian density is not singular so we can solve $\partial_{t}\phi$ and
$\partial_{t}\phi^{\ast}$

$\begin{split}&\partial_{t}\phi=\frac{4\Pi^{\ast}+2i(\phi+\lambda\phi^{\ast}\partial_{x}\phi^{2}+\lambda^{2}\phi\partial_{x}\phi^{2}\partial_{x}\phi^{\ast
2}-2\partial_{x}\phi^{3}\phi^{\ast}\partial_{x}\phi^{\ast})}{\lambda^{2}(\partial_{x}\phi\phi^{\ast}-\partial_{x}\phi^{\ast}\phi)},\\\
&\partial_{t}\phi^{\ast}=\frac{4\Pi-2i(\phi^{\ast}+\lambda\phi\partial_{x}\phi^{\ast
2}+\lambda^{2}\phi^{\ast}\partial_{x}\phi^{2}\partial_{x}\phi^{\ast
2}-2\partial_{x}\phi^{\ast
3}\phi\partial_{x}\phi)}{\lambda^{2}(\partial_{x}\phi\phi^{\ast}-\partial_{x}\phi^{\ast}\phi)}.\end{split}$
(118)

Therefore the Hamiltonian density is given by the Legendre transformation

$\begin{split}\mathcal{H}=&\partial_{t}\phi\Pi+\partial_{t}\phi^{\ast}\Pi^{\ast}-\mathcal{L}\\\
=&\frac{(2\Pi^{\ast}+i\phi)(2\Pi-i\phi^{\ast})}{\lambda^{2}(\partial_{x}\phi\phi^{\ast}-\partial_{x}\phi^{\ast}\phi)^{2}}+\frac{i\partial_{x}\phi^{2}\phi^{\ast}(2\Pi-i\phi^{\ast})-i\partial_{x}\phi^{\ast
2}\phi(2\Pi^{\ast}+i\phi)}{\lambda(\partial_{x}\phi\phi^{\ast}-\partial_{x}\phi^{\ast}\phi)^{2}}\\\
&+\frac{2i\Pi(\phi\partial_{x}\phi\partial_{x}\phi^{\ast}-2\partial_{x}\phi^{2}\phi^{\ast})-2i\Pi^{\ast}(\phi^{\ast}\partial_{x}\phi\partial_{x}\phi^{\ast}-2\phi\partial_{x}\phi^{\ast
2})-\partial_{x}\phi^{2}\phi^{\ast
2}+\phi\phi^{\ast}\partial_{x}\phi\partial_{x}\phi^{\ast}-\partial_{x}\phi^{\ast
2}\phi^{2}}{(\partial_{x}\phi\phi^{\ast}-\partial_{x}\phi^{\ast}\phi)^{2}}\\\
&-\frac{2\lambda\phi\phi^{\ast}\partial_{x}\phi^{3}\partial_{x}\phi^{\ast
3}}{(\partial_{x}\phi\phi^{\ast}-\partial_{x}\phi^{\ast}\phi)^{2}}+\frac{\lambda^{2}\phi\phi^{\ast}\partial_{x}\phi^{4}\partial_{x}\phi^{\ast
4}}{(\partial_{x}\phi\phi^{\ast}-\partial_{x}\phi^{\ast}\phi)^{2}}+\mathcal{O}(\lambda^{3}).\end{split}$
(119)

Because the Hamiltonian density is singular at $\lambda=0$ so the perturbation
theory is not applicable. The appearance of a non-trivial denominator is also
problematic. To proceed one may introduce proper constraints as before to
reduce the Hamiltonian system even though it is ad hoc. Based on our analysis
of the first-order deformation, it seems that the most obvious constraints are

$\begin{split}&\chi_{1}=\Pi-\frac{\partial\mathcal{L}}{\partial\partial_{t}\phi}=0,\quad\chi_{2}=\Pi^{\ast}-\frac{\partial\mathcal{L}}{\partial\partial_{t}\phi^{\ast}}=0.\end{split}$
(120)

Imposing these two constraints and using the equation of motion to get rid of
$\partial_{t}\phi$ and $\partial_{t}\phi^{\ast}$ we end up with a Hamiltonian
system:

$\begin{split}\mathcal{H}_{2}&=\partial_{x}\phi\partial_{x}\phi^{\ast}-\lambda(\partial_{x}\phi\partial_{x}\phi^{\ast})^{2}+\frac{\lambda^{2}}{4}\left(8[\partial_{x}\phi\partial_{x}\phi^{\ast}]^{3}+\partial^{2}_{x}\phi\partial^{2}_{x}\phi^{\ast}[\partial_{x}\phi\phi^{\ast}-\partial_{x}\phi^{\ast}\phi]^{2}\right),\\\
\chi_{1}&=\Pi-\frac{i\phi^{\ast}}{2}-\frac{i\lambda\phi\partial_{x}\phi^{\ast
2}}{2}+\frac{i\lambda^{2}}{4}\Big{\\{}(\partial_{x}\phi\phi^{\ast}-\partial_{x}\phi^{\ast}\phi)^{2}\partial^{2}_{x}\phi^{\ast}-2(\partial_{x}\phi\partial_{x}\phi^{\ast})^{2}\phi^{\ast}+4\partial_{x}\phi\phi(\partial_{x}\phi^{\ast})^{3}\Big{\\}}\\\
&\equiv\chi_{1}^{(0)}+\lambda\chi_{1}^{(1)}+\lambda^{2}\chi_{1}^{(2)},\\\
\chi_{2}&=\Pi^{\ast}+\frac{i\phi}{2}+\frac{i\lambda\phi^{\ast}\partial_{x}\phi^{2}}{2}-\frac{i\lambda^{2}}{4}\Big{\\{}(\partial_{x}\phi\phi^{\ast}-\partial_{x}\phi^{\ast}\phi)^{2}\partial^{2}_{x}\phi-2(\partial_{x}\phi\partial_{x}\phi^{\ast})^{2}\phi+4\partial_{x}\phi^{\ast}\phi^{\ast}(\partial_{x}\phi)^{3}\Big{\\}}\\\
&\equiv\chi_{2}^{(0)}+\lambda\chi_{2}^{(1)}+\lambda^{2}\chi_{2}^{(2)}.\\\
\end{split}$ (121)

Using the Dirac-Bergmann Algorithm one can find

$\begin{split}&[\phi(x),\phi^{\ast}(y)]_{DB}=\delta(x-y)\\\ &-\lambda
C_{1}(x,y)-\lambda^{2}\left(C_{2}(x,y)-\int
dx^{\prime}C_{1}(x,x^{\prime})C_{1}(x^{\prime},y)\right),\\\
&C_{i}(x,y)=[\Pi(y),\chi_{2}^{(i)}(x)]+[\chi_{1}^{(i)}(y),\Pi^{\ast}(x)],\quad
i=1,2.\end{split}$ (122)

Taking the ansatz of the canonical variables to be

$\begin{split}&\varphi=\phi-i\lambda\chi_{2}^{(1)}-i\lambda^{2}\chi_{2}^{(2)}+\lambda^{2}\phi_{2},\\\
&\varphi^{\ast}=\phi^{\ast}+i\lambda\chi_{1}^{(1)}+i\lambda^{2}\chi_{2}^{(2)}+\lambda^{2}\phi_{2}^{\ast},\end{split}$
(123)

and imposing the condition

$\begin{split}[\varphi(x),\varphi^{\ast}(y)]_{DB}=\delta(x-y)\end{split}$
(124)

we end up with the equation for $\phi_{2}$ and $\phi_{2}^{\ast}$

$\begin{split}&[\phi(x),\phi_{2}^{\ast}(y)]+[\phi_{2}(x),\phi^{\ast}(y)]-\frac{1}{4}\delta(x-y)(\partial_{x}\phi(x))^{2}(\partial_{x}\phi^{\ast}(x))^{2}\\\
&+\partial_{x}\partial_{y}\left(\delta(x-y)\partial_{x}\phi^{\ast}\phi^{\ast}\partial_{x}\phi\phi\right)=0.\\\
\end{split}$ (125)

Unfortunately we do not find local solutions to this equation.

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