# Full-ECE: A Metric For Token-level Calibration on Large
Language Models

Han Liu∗
Tsinghua University
Baichuan Inc.
&Yupeng Zhang
Baichuan Inc.
&Bingning Wang†
Baichuan Inc.
Weipeng Chen
Baichuan Inc.
& Xiaolin Hu
Tsinghua University
&
Equal contributionCorresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Deep Neural Networks (DNNs) excel in various domains but face challenges in
providing accurate uncertainty estimates, which are crucial for high-stakes
applications. Large Language Models (LLMs) have recently emerged as powerful
tools, demonstrating exceptional performance in language tasks. However,
traditional calibration metrics such as Expected Calibration Error (ECE) and
classwise-ECE (cw-ECE) are inadequate for LLMs due to their vast vocabularies,
data complexity, and distributional focus. To address this, we propose a novel
calibration concept called full calibration and introduce its corresponding
metric, Full-ECE. Full-ECE evaluates the entire predicted probability
distribution, offering a more accurate and robust measure of calibration for
LLMs.

##  1 Introduction

Deep Neural Networks(DNNs) have achieved remarkable success across various
domains, demonstrating superior performance in many tasks. Despite these
impressive achievements, a critical challenge remains: the calibration of
these models’ predictions. Calibration refers to the model’s ability to
provide uncertainty estimates that accurately reflect the true likelihood of
its predictions being correct. In many high-stakes applications such as
healthcare Leibig et al. (2017); Dolezal et al. (2022), self-driving
Michelmore et al. (2020), and protein engineering Greenman et al. (2023), it
is not sufficient for a model to be highly accurate; it must also provide
reliable estimates of uncertainty to ensure safety and robustness.

Large Language Models Achiam et al. (2023); Touvron et al. (2023); Brown et
al. (2020) have recently emerged as powerful tools in the DNN area,
demonstrating exceptional performance in a wide array of language tasks. It is
essential to ensure that these models are well-calibrated at the token level.
Token-level calibration refers to the alignment of the predicted probability
distribution for each token with the true distribution observed in the corpus.
This ensures that the probabilities assigned to each token reflect their
actual occurrence likelihoods. Token-level calibration differs from
traditional classification task calibration in several aspects:

  1. 1.

Vocabulary Size: Token-level calibration deals with a vast number of classes,
typically in the hundreds of thousands according to the vocabulary size,
vastly surpassing traditional classification tasks.

  2. 2.

Data Complexity and Imbalance: LLMs are trained on datasets containing tens of
billions of tokens, introducing significant complexity and diversity compared
to standard datasets. Moreover, token-level calibration grapples with highly
imbalanced token distributions, posing additional challenges.

  3. 3.

Distribution Focus: Traditional calibration methods focus on the top-1
prediction. In contrast, token-level calibration for LLMs must account for the
entire probability distribution across all tokens, as inference involves
sampling from this distribution, making every token’s probability significant.

These distinctions underscore the inadequacy of traditional calibration
metrics like Expected Calibration Error (ECE) Naeini et al. (2015) and
classwise-ECE (cw-ECE) Kull et al. (2019) when applied to LLMs. ECE primarily
considers the probability of the most likely token, which, although suitable
for traditional classification tasks, fails to capture the nuances of the
token distribution in LLMs. In contrast, cw-ECE evaluates calibration across
each class but struggles with the extreme class imbalance and the sheer number
of classes in LLMs. As the vocabulary size increases, the frequency of many
tokens diminishes, making it challenging to obtain reliable calibration
metrics based on these rare tokens.

To address these challenges, we propose a novel calibration concept called
full calibration and introduce its corresponding metric, Full-ECE. Full-ECE
evaluates the calibration of the entire predicted probability distribution,
encompassing all tokens rather than focusing on the top-1 prediction. This
comprehensive approach ensures that the calibration metric is robust to
imbalances and vast class count of LLMs, providing a more accurate measure of
LLMs’ calibration.

Figure 1: (a), (b), and (c) demonstrate how ECE, cw-ECE, and Full-ECE
aggregate statistics per bin for a 4-class classification task. Red numbers
represent maximum probability values, and blue dashed boxes indicate ground
truth label categories. ECE considers only these red maximum values, cw-ECE
computes ECE for each class individually, while Full-ECE enhances cw-ECE by
combining bins across different classes for statistical analysis.

##  2 Background

Calibration refers to the process of adjusting the predictive probabilities of
a model so that they reflect the true likelihood of outcomes. For example, if
a model predicts an event with a probability of 70%, that event should occur
approximately 70% of the time in the long run. The existing concepts of
calibration can mainly be divided into two types: confidence calibration and
classwise calibration.

To introduce these two types of calibration, we first need to explain some
fundamental concepts. Let $P_{data}$ denote the probability distribution of
data. A dataset consists of finite $(x,y)$ pairs where $x\sim P_{data}$ is a
data instance and $y\in\mathcal{Y}=\\{1,2,...,K\\}$ is the ground-truth class
label of $x$. A model takes in the instance $x$ and output its prediction
vector $p\in\mathbb{R}^{K}$. Let $p^{(k)}$ denote the $k$-th dimension of $p$.

###  2.1 Confidence Calibration

In confidence calibration, the index corresponding to the maximum value in the
model’s predicted probability distribution
$\hat{y}=\text{argmax}_{k\in\mathcal{Y}}\\{p^{(k)}\\}$ is taken as the model’s
prediction, and the associated probability value
$\hat{p}=\text{max}_{k\in\mathcal{Y}}\\{p^{(k)}\\}$ is considered as the
model’s confidence. A model is confidence-calibrated if for any confidence
$c\in[0,1]$,

$\displaystyle\mathbb{P}(\hat{y}=y|\hat{p}=c)=c.$ (1)

A widely used metric for evaluating confidence calibration is the ECE Naeini
et al. (2015), which is defined as the expected absolute difference between
the model’s confidence and its accuracy:

$\displaystyle{\text{ECE}}=\mathbb{E}_{\hat{p}}|\mathbb{P}(\hat{y}=y|\hat{p})-\hat{p}|.$
(2)

Since we only have finite samples, ECE cannot be directly calculated using the
definition provided above. In practical calculations, we replace the above
definition with a discretized version of ECE in which the interval $[0,1]$ is
divided into $M$ equispaced bins. Let $B_{m}$ denote the indices of samples
with confidences belonging to the $m$-th bin (i.e.
$(\frac{m-1}{M},\frac{m}{M}]$). The accuracy of this bin is
$A_{m}=\frac{1}{|B_{m}|}\sum_{i\in B_{m}}\mathds{1}(\hat{y}_{i}=y_{i})$. The
average confidence of this bin is $C_{m}=\frac{1}{|B_{m}|}\sum_{i\in
B_{m}}\hat{p}_{i}$. The discretized version of ECE is defined as

$\displaystyle{\text{ECE}}=\sum_{m=1}^{M}\frac{|B_{m}|}{N}|A_{m}-C_{m}|,$ (3)

where $N$ is the number of samples in the dataset.

###  2.2 Classwise Calibration

Classwise calibration is concerned with calibrating the predicted
probabilities for each individual class in a multi-class classification
problem. It ensures that the predicted probabilities across different classes
are accurate and properly calibrated. A model is classwise-calibrated if for
any class $k$ and any probability $q$,

$\displaystyle\mathbb{P}(y=k|p^{(k)}=q)=q.$ (4)

Similar to ECE for confidence calibration, the cw-ECE Kull et al. (2019) for
classwise calibration is defined as

$\displaystyle{\text{cw-
ECE}}=\frac{1}{K}\sum_{k=1}^{K}{\text{class-}}k{\text{-ECE}},$ (5)

where ${\text{class-}}k{\text{-ECE}}$ is computed on the $k$-th class:

$\displaystyle{\text{class-}}k{\text{-ECE}}=\mathbb{E}_{p^{(k)}}|\mathbb{P}(y=k|p^{(k)})-p^{(k)}|.$
(6)

The discretized version of cw-ECE is defined as

$\displaystyle{\text{cw-
ECE}}=\sum_{k=1}^{K}\sum_{m=1}^{M}\frac{|B_{m,k}|}{NK}|A_{m,k}-C_{m,k}|,$ (7)

where $B_{m,k}$ denotes the set of indices of samples whose predicted
probabilities of the $k$-th class lie in the $m$-th bin,
$A_{m,k}=\frac{1}{|B_{m,k}|}\sum_{i\in B_{m,k}}\mathds{1}(y_{i}=k)$ and
$C_{m,k}=\frac{1}{|B_{m,k}|}\sum_{i\in B_{m,k}}p^{(k)}_{i}$.

##  3 Full Calibration: Calibrating the Whole Distribution Predicted by LLMs

As mentioned earlier, ECE considers only the calibration of the most probable
class in the probability distribution, while classwise-ECE (cw-ECE) takes into
account the calibration of all classes in the probability distribution.
However, as shown in Equations (5) and (7), for each class $k$, cw-ECE
requires calculating its corresponding ECE metric. ECE is a statistical metric
that requires a sufficient number of samples for each class to be calculated
accurately. In token-level calibration, it is impossible to ensure that each
token appears frequently enough in the test set. We analyzed the distribution
of token occurrences in a test set containing 5000 sentences and found that
30% of the tokens do not appear in the dataset at all, and 41% of the tokens
appear only 1-10 times. This indicates that 71% of the tokens appear fewer
than 10 times, making cw-ECE evaluation inaccurate.

To address the issues of existing calibration metrics, ECE and cw-ECE, in
token-level calibration, we propose a new and more suitable calibration
concept: full calibration, along with its corresponding calibration metric,
Full-ECE.

Unlike confidence calibration, where the index corresponding to the maximum
value in the model’s predicted probability distribution is taken as the
model’s prediction, full calibration views the model’s output as a process of
sampling from the predicted probability distribution. The sampled index
$y^{*}$ is taken as the output token, and its corresponding probability value
$p^{*}$ is considered as its confidence. A model is full-calibrated if for any
$q\in[0,1]$, the following holds:

$\displaystyle\mathbb{P}(y^{*}=y|p^{*}=q)=q.$ (8)

Naturally, similar to the definition of ECE, we can define the metric
corresponding to full calibration as Full-ECE:

$\displaystyle{\text{Full-
ECE}}=\mathbb{E}_{p^{*}}|\mathbb{P}(y^{*}=y|p^{*})-p^{*}|.$ (9)

According to Bayes’ theorem and the law of total probability,

$\displaystyle\mathbb{P}(y^{*}=y|p^{*})$
$\displaystyle=\frac{\mathbb{P}(y^{*}=y,p^{*})}{\mathbb{P}(p^{*})}$ (10)
$\displaystyle=\frac{\sum_{k=1}^{K}\mathbb{P}(y^{*}=y=k,p^{*})}{\sum_{k=1}^{K}\mathbb{P}(y^{*}=k,p^{*})}.$

Similar to the previously defined discrete ECE and cw-ECE, we divide the
probability interval [0,1] of each class $k$ into $M$ equally spaced bins.
Each bin $B_{m,k}$ represents the set of indices of samples whose predicted
probabilities of the $k$-th class lie in the $m$-th bin. Let $|B^{*}_{m}|$
represent the sum of the quantities in the $m$-th bin $B_{m,k}$ for all
classes $k$, i.e. $|B^{*}_{m}|=\sum_{k=1}^{K}|B_{m,k}|$. Base on (9) and (10),
Full-ECE can be discretized and represented as

Full-ECE
$\displaystyle=\sum_{m=1}^{M}\frac{|B^{*}_{m}|}{N}|A^{*}_{m}-C^{*}_{m}|,$ (11)

where

$\displaystyle A^{*}_{m}=\frac{\sum_{k=1}^{K}\sum_{i\in
B_{m,k}}\mathds{1}(y_{i}=k)}{|B^{*}_{m}|},$ (12)

and

$\displaystyle C^{*}_{m}=\frac{\sum_{k=1}^{K}\sum_{i\in
B_{m,k}}p_{i}^{(k)}}{|B^{*}_{m}|}.$ (13)

Figures 1 depicts the aggregation methods of ECE, cw-ECE, and Full-ECE for
each bin. ECE focuses solely on the highest probability value within each
distribution, whereas cw-ECE considers the distribution across all classes
$k$, while Full-ECE combines the statistics of different classes within the
same bin. This approach addresses the issue in token-level calibration where
cw-ECE faces the challenge of having too few samples for many classes.

##  4 The Robustness of Full-ECE

| 2B params | 7B params
---|---|---
RSD of cw-ECE | 40.93% | 44.90%
RSD of Full-ECE | 7.56% | 8.36%
Table 1: The RSD of Full-ECE and cw-ECE for different models as $M$ varies.

Both Full-ECE and cw-ECE are metrics designed to evaluate the calibration of
the entire predicted distribution. From their definitions in the discrete
case, it can be observed that both of them require the probability interval
[0,1] to be divided into $M$ equal-length bins. A robust metric should exhibit
lower relative variation as the value of $M$ changes. We evaluated the
relative standard deviation (RSD) of Full-ECE and cw-ECE for different values
of $M$ in the set $\\{5,10,20,50,100,200,500\\}$. Our experiments were
conducted on two models: a 2-billion parameter GPT model and a 7-billion
parameter GPT model trained on 1 trillion tokens. The lower the RSD, the more
stable the metric is. The experimental results, shown in Table 1, demonstrate
that for both models, the RSD of Full-ECE as $M$ varies is significantly lower
than that of cw-ECE. Therefore, Full-ECE is more stable across different
values of $M$ compared to cw-ECE, highlighting its advantage in evaluating
token-level calibration in large language models.

##  5 Continuous Improvement of Full-ECE during LLM Training

Figure 2: Full-ECE with varying training data

Previously, we demonstrated across different dimensions that Full-ECE is a
more suitable metric than ECE and cw-ECE for evaluating uncertainty in LLMs.
Another criterion for effective model evaluation is reliability and
discriminability, where metrics should consistently improve with model
capability. We tested the Full-ECE metric ($M=10$) at different training
stages of the Baichuan-2 7BYang et al. (2023) model and observed a consistent
downward trend (shown in Figure 2), indicating a continuous improvement in
token-level calibration throughout training.

##  6 Conclusion

We introduce the concept of full calibration and propose its corresponding
metric, Full-ECE, to address the challenges in calibrating LLMs. Unlike
traditional metrics such as ECE and cw-ECE, Full-ECE evaluates the entire
predicted probability distribution across all tokens, From both experimental
and theoretical perspectives, we have demonstrated that Full-ECE provides a
more robust calibration measure for LLMs operating with vast vocabularies and
complex data distributions.

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# Dual approach to soft-core anyonic Lieb-Liniger fluids

Gerard Valentí-Rojas<EMAIL_ADDRESS>Naquidis
Center, Institut d’Optique Graduate School, 91127, Palaiseau, France  SUPA,
Institute of Photonics and Quantum Sciences, Heriot-Watt University,
Edinburgh, EH14 4AS, United Kingdom  Patrik Öhberg  SUPA, Institute of
Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh, EH14 4AS,
United Kingdom

###### Abstract

The identity of quantum matter can be effectively altered by means of gauge
fields. In two spatial dimensions this is illustrated by the Chern-Simons
flux-attachment mechanism, but such a mechanism is not possible in lower
dimensions. Here, we study a one-dimensional interacting Bose gas in the
presence of a gauge field. This model can be explicitly mapped into an
interacting anyonic system by a large gauge transformation, indicating a
statistical transmutation analogous to that of Chern-Simons. The Bogoliubov
spectrum in the weakly-interacting limit reveals the presence of a roton
minimum arising from the statistical interaction. At a mean-field level chiral
solitons are recovered. Should these be understood as quantum bound states, it
is natural to interpret them as corresponding to localised anyonic
quasiparticles. Hydrodynamic arguments highlight the presence of dispersive
chiral shock waves in the propagation of a wavepacket due to a Riemann-Hopf
nonlinearity. Numerical calculations show the presence of both chiral soliton
trains and shock waves.

Introduction.—  Quantum particles have an intrinsic identity. Their properties
can be summarised in a handful of magnitudes accounting for the most relevant
degrees of freedom available. In a condensed matter context, these are mass
($m$), charge ($q$), spin ($s$), statistics ($\eta$) and lifetime ($\tau$),
among few others. Out of these properties, spin and statistics encode the
quantum identity of matter as they have no classical analogue. Quasiparticles
appear as localised excitations in interacting many-body systems and possess
similar qualities $(m^{*},q^{*},s^{*},\eta^{*},\tau^{*})$ with the important
difference that they are emergent, as opposed to the prior, which are
fundamental.

For a long time, quasiparticles have been found to inherit part of their
identity from the underlying fields due to adiabatic continuity [1]. Anyons
[2, 3, 4, 5], however, constitute an important counterpoint to this framework,
as quasiparticles with widely different properties from the original
microscopic interacting system. The exotic nature of anyons, as compared to
conventional quantum particles, has opened entirely new research avenues and
already constitutes a key element for near-future technologies [6, 7, 8, 9].

While systems giving rise to anyonic and fractionalised excitations have been
thoroughly explored [10, 11, 12, 13], a general physical understanding for how
conventional quantum systems can yield exotic matter featuring anyons is
lacking. One such mechanism in two spatial dimensions is flux attachment [4,
5], a by-product of a many-body Aharonov-Bohm effect in which electrically-
charged particles capture magnetic flux quanta and become Chern-Simons gauge-
dressed entities with transmuted statistics [14]. In one spatial dimension
anyonic statistics is still allowed, but the mechanism of flux attachment is
no longer possible as there is no notion of magnetic flux. The physical
intuition for anyons in this context is much less clear. This is manifest in
the lack of consensus for what an anyon actually is in setups such as a one-
dimensional quantum fluid 111An exception to this is the case of Majorana zero
modes [58, 59], for which much more consensus exists.. More concretely, given
an initial one-dimensional continuous system whose components have a given set
of conventional $(m,q,s,\eta,\tau)$, how does one obtain a fluid with anyonic
quasiparticles and what do they look like? Is there a notion of a statistical
gauge field in this context in analogy to well-known Fractional Quantum Hall
(FQH) fluids [16, 17, 18]?

Here, we explore an explicit example of a “conventional” quantum system with
$(\eta)$ that becomes “exotic” with $(\eta^{*})$ by coupling it to a certain
gauge field. More concretely, we discuss the connection between a bosonic
fluid in one spatial dimension and its composite dual model, the soft-core
anyonic Lieb-Liniger model [19, 20, 21]. We comment on phenomenological
features of the model interpreted in the bare (bosonic) basis. This serves as
a concrete application of the composite particle duality [22], which argues
that the notion of statistical gauge fields is valid in any dimension. We
discuss the formation of exotic solitons interpreted as bound states, shock
waves at long times, and asymmetric dynamics.

Dual Quantum Fluids.—  Our starting point is a locally-interacting bosonic
fluid in one spatial dimension. An implementation and manipulation of such a
model can be achieved, for instance, with ultracold gases in an optical
waveguide [23]. According to the composite particle duality [22], the coupling
of such a conventional state of quantum matter to an appropriate statistical
gauge field allows the system to be exactly transformed into an anyonic
system. More concretely, the bosonic quantum fluid with Lagrangian density

$\mathcal{L}_{\text{B}}=i\hbar\,\hat{\Psi}^{\dagger}D_{t}\hat{\Psi}-\frac{\hbar^{2}}{2m}\big{(}D_{x}\hat{\Psi}\big{)}^{\dagger}\big{(}D_{x}\hat{\Psi}\big{)}-\frac{g}{2}\,\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi}$
(1)

and conventional equal-time commutation relations

$\displaystyle\big{[}\hat{\Psi}(x)\,,\hat{\Psi}(x^{\prime})\big{]}=\big{[}\hat{\Psi}^{\dagger}(x)\,,\hat{\Psi}^{\dagger}(x^{\prime})\big{]}=0\;,$
(2)
$\displaystyle\big{[}\hat{\Psi}\,(x)\,,\hat{\Psi}^{\dagger}(x^{\prime})\big{]}=\delta\,(x-x^{\prime})\;,$
(3)

is equivalent to a soft-core anyonic Lieb-Liniger gas

$\tilde{\mathcal{L}}_{\text{C}}=i\hbar\,\hat{\Psi}_{\text{C}}^{\dagger}\,\partial_{t}\hat{\Psi}_{\text{C}}-\frac{\hbar^{2}}{2m}\,\partial_{x}\hat{\Psi}_{\text{C}}^{\dagger}\,\partial_{x}\hat{\Psi}_{\text{C}}-\frac{g}{2}\,\hat{\Psi}_{\text{C}}^{\dagger}\hat{\Psi}_{\text{C}}^{\dagger}\hat{\Psi}_{\text{C}}\hat{\Psi}_{\text{C}}\;,$
(4)

with the composite field obeying the anyonic algebra

$\displaystyle\hat{\Psi}_{\text{C}}^{\dagger}(x)\,\hat{\Psi}_{\text{C}}^{\dagger}(x^{\prime})-e^{\,\frac{i}{\hbar}\gamma\,\text{sgn}\,(x-x^{\prime})}\,\hat{\Psi}_{\text{C}}^{\dagger}(x^{\prime})\,\hat{\Psi}_{\text{C}}^{\dagger}(x)=0\,,$
(5)
$\displaystyle\hat{\Psi}_{\text{C}}(x)\,\hat{\Psi}_{\text{C}}^{\dagger}(x^{\prime})-e^{\,-\frac{i}{\hbar}\gamma\,\text{sgn}\,(x-x^{\prime})}\,\hat{\Psi}_{\text{C}}^{\dagger}(x^{\prime})\,\hat{\Psi}_{\text{C}}(x)=\delta\,(x-x^{\prime})\,,$
(6)

upon redefining the bosonic field as a composite via the large gauge
transformation

$\hat{\Psi}\,(t,x)=e^{\,\frac{i}{\hbar}\hat{\Phi}(t,x)}\,\hat{\Psi}_{\text{C}}\,(t,x)\;,$
(7)

where $\hat{\Phi}$ is a disorder field or a generalised Jordan-Wigner string

$\hat{\Phi}\,(t,x)=\gamma\int_{-\infty}^{\infty}dx^{\prime}\;\Theta\,(x-x^{\prime})\,\hat{n}\,(t,x^{\prime})\;,$
(8)

and where the Heaviside step function $\Theta(x)$ plays the role of a kink.
The bosonic matter field in Eq. (1) is minimally coupled to a statistical
gauge field $\hat{a}_{\mu}$ via the gauge-covariant derivative
$D_{\mu}=\partial_{\mu}-i\hbar^{-1}\hat{a}_{\mu}\,$, where the form of the
gauge field is set by the conserved current constraint

$\hat{J}^{\,\mu}=\gamma^{-1}\epsilon^{\,\mu\nu}\hat{a}_{\nu}\;\;\;\;\;\;\;\text{and}\;\;\;\;\;\;\;\partial_{\mu}\,\hat{J}^{\,\mu}=0\;,$
(9)

and where the covariant Noether current density is defined in the conventional
way

$\hat{J}^{\,\mu}\,(t,x)=\bigg{(}\,\hat{\Psi}^{\dagger}\hat{\Psi}\,,\;\frac{\hbar}{2mi}\,\Big{[}\hat{\Psi}^{\dagger}D_{x}\,\hat{\Psi}-(D_{x}\,\hat{\Psi})^{\dagger}\,\hat{\Psi}\Big{]}\,\,\bigg{)}\;.$
(10)

Since both systems are equivalent, it is a matter of taste whether to work
with the original or transformed formulation. We refer to this as working on
either the bare (B) or the composite (C) basis. While the composite basis
might prove useful for theoretical calculations [19, 20, 21], ultimately the
eventual experimental system is bosonic, so the latter basis is preferable to
understand the system from an implementation standpoint.

Elementary Excitations.—  The Hamiltonian formulation in the bare basis can be
written as $H=H_{\text{LL}}+H_{\text{G}}\,$, where

$H_{\text{LL}}=\int
dx\;\bigg{(}\frac{\hbar^{2}}{2m}\partial_{x}\hat{\Psi}^{\dagger}\partial_{x}\hat{\Psi}+\frac{g}{2}\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi}\bigg{)}$
(11)

refers to the usual Lieb-Liniger model, and

$H_{\text{G}}=H_{\chi}+H_{\text{3b}}=2\gamma\int
dx\;\bigg{(}\colon\\!\hat{n}\,\hat{j}_{x}\colon\\!-\frac{3\gamma}{4m}\hat{\Psi}^{\dagger}\hat{n}^{2}\hat{\Psi}\bigg{)}$
(12)

results from the coupling to the statistical gauge field. Here,
$(\colon\\!\bullet\colon\\!)$ denotes bosonic normal ordering and
$\hat{j}_{x}$ is the ungauged current density. We consider the weakly-
repulsive Bogoliubov regime and calculate the excitation spectrum in the
conventional way [24, 25]. The key novel term is the density-current chiral
interaction

$\displaystyle
H_{\chi}=\frac{2\gamma}{\text{Vol}}\sum_{k_{1},k_{2},q}\bigg{(}\frac{\hbar
k_{1}}{m}\bigg{)}\;\hat{b}^{\dagger}_{k_{2}+q}\hat{b}^{\dagger}_{k_{1}-\frac{q}{2}}\hat{b}_{k_{1}+\frac{q}{2}}\hat{b}_{k_{2}}\;,$
(13)

which explicitly breaks parity ($\mathcal{P}$) and time-reversal
($\mathcal{T}$) symmetries. The other local interaction terms in
$H_{\text{3b}}$, namely $\sim\gamma^{2}\colon\hat{n}^{2}\colon$ and
$\sim\gamma^{2}\colon\hat{n}^{3}\colon\,$ merely renormalise the conventional
contact interparticle interactions in the Bogoliubov limit to $g\rightarrow
g_{\text{eff}}\,$. Like in conventional Bogoliubov theory for the dilute Bose
gas, we expand in low momenta and consider only zero total momentum state
$\Delta k=k_{\text{in}}-k_{\text{out}}=0$. We find that $H_{\chi}$ merely
modifies the standard contact interaction by

$g\longrightarrow\tilde{g}(k)=g_{\text{eff}}\,\bigg{(}1+\gamma\frac{4\hbar
k}{mg_{\text{eff}}}\bigg{)}\;.$ (14)

Hence the Bogoliubov spectrum reads as

$\mathcal{E}_{k}=\sqrt{\bigg{(}\frac{\hbar^{2}k^{2}}{2m}\bigg{)}^{2}+\tilde{g}(k)\,\frac{n_{0}\,\hbar^{2}k^{2}}{m}\,\;\;}\;\;,$
(15)

where $n_{0}=N/\text{Vol}$ is the number density. This dispersion relation can
be seen in Figure 1. The spectrum has a characteristic phonon-maxon-roton
shape, unlike the usual weakly-interacting Bose gas.

Figure 1: Bogoliubov spectrum in dimensionless units for $\gamma=0$ (dashed)
and for increasing values of $0<\gamma\leq 1/4$ (gray). For non-vanishing
values of the statistical interaction, an asymmetry in the spectrum develops
and a roton minimum appears. As $\gamma/g_{\text{eff}}$ is increased, the
roton is softened until it reaches zero energy. For $\gamma>1/4$, the spectrum
becomes complex-valued (not shown).

We appreciate three characteristic features:

  * •

There are gapless phonons.

  * •

There is a roton minimum for $\text{sgn}(\gamma k)<0$.

  * •

It is asymmetric in $k$.

## On phonons.

Some insight can be gained from comparing this spectrum with that of the
Zhang-Hansson-Kivelson (ZHK) model [26, 27, 28, 29] for a charged superfluid
coupled to a Chern-Simons gauge field in two spatial dimensions. We observe a
qualitative difference with respect to phonons. Here, phonons do not acquire a
topological mass gap, so there is no Anderson-Higgs mechanism. In one
dimension there is no sense of a magnetic field, so no notion of cyclotron
motion. Thus, the cyclotron frequency, that determines the size of the
magnetophonon gap in two spatial dimensions [30, 28], vanishes in the one-
dimensional limit. This implies that the system is compressible, unlike a FQH
fluid.

## On asymmetry.

The spectrum shows some similarities with that of a spin-orbit-coupled
condensate [31, 32]. Given the spectral asymmetry, two critical velocities
corresponding to $\pm k$ flow shall be defined with respect to the ($+$)
phonon and ($-$) roton. Landau’s criterion for superfluidity has to be
modified accordingly due to the breaking of parity. This is a recurrent
feature in spin-orbit-coupled condensates [32, 33]. The speed of sound is also
affected as a consequence. The particular case of a superfluid coupled to a
certain class of density-dependent gauge fields has been studied at a
classical hydrodynamic level in Ref. [34]. Buggy finds an extended analytical
expression for the speed of sound in the condensate that is dependent on the
statistical parameter. The propagation of phonons in such systems is, in
general, not only asymmetric but also highly non-trival.

## On rotons.

As the ratio $\gamma/g_{\text{eff}}$ becomes of order one, a roton minimum
develops. This feature is also found in the ZHK model. As the statistical
interaction increases the roton is softened down to $\mathcal{E}_{k}=0$ at
finite $k<0$. For sufficiently large values of the ratio, the spectrum becomes
complex-valued, signalling an instability. The roton satisfies the usual
dispersion

$\mathcal{E}_{\text{rot}}(k)=\frac{\hbar^{2}(k-k_{r})^{2}}{2m^{*}}+\Delta$
(16)

with gap $\Delta$ and effective mass $m^{*}$. At $k=k_{r}$ the roton has zero
net group velocity at non-vanishing momentum. The gap of the roton minimum
depends on the relative strength of the statistical and interparticle
interactions. In this sense, the statistical gauge field yields an effect
similar to conventional long-range interactions, such as in dipolar gases [35,
36]. We find that for the ratio $\gamma/g_{\text{eff}}\rightarrow 1/4$ the gap
vanishes $\Delta\rightarrow 0$, signalling a practically null energy cost for
the creation of this collective mode at finite wave-number.

## On solitons.

Solving the classical equations of motion for the matter field in terms of a
generalised chiral Gross-Pitaevskii equation, reveals unusual soliton
solutions [37, 38, 39, 40, 41, 42, 43], which are stable for one chirality,
but unstable for the other. A bright chiral soliton has been experimentally
implemented and observed by Frölian et al. [44]. The chiral aspect of the
soliton appears as an instability to backscattering, something which both
experiments [44] and numerical simulations in the current work confirm. Such a
solution should be robust upon additional $\mathcal{N}-$body interactions [45,
46]. This soliton is the one-dimensional analogue of an anyonic ZHK vortex. It
is plausible to conclude that it is a gauge-charge composite, namely a linear
anyon.

Figure 2: Schematic of a dispersive shock wave formation by a Riemann-Hopf
nonlinearity. A wavepacket propagates to the right, with initial stages of the
dynamics being approximately linear. Due to a Riemann-Hopf nonlinearity, a
density overhang is created in the direction of motion. At breaking time
$t=t_{c}$, the Riemann-Hopf term creates a mathematical gradient singularity
known as a gradient catastrophe. This manifests in the formation of a shock
front. The nonlinearity is regularised by higher order dispersive terms
creating highly oscillatory behaviour which can yield solitons in its wake.

## On Shock Waves.

Regarding the dynamics of the system, even the simplest semi-classical
hydrodynamic analysis reveals that the current nonlinearity will give rise to
shock waves. In other words, for moderate values of the parameter $\gamma$ a
propagating wavepacket will eventually develop a density overhang and
experience a wave-breaking effect of a form similar to that described in
Figure 2. The reason being the presence of a dominant Riemann-Hopf
nonlinearity in addition to higher order terms in the conservation of current

$0=\partial_{t}n+\partial_{x}\,\big{(}n\,\tilde{v}_{s}\big{)}=\partial_{t}n+\partial_{x}\,\big{(}n\,v_{s}\big{)}-\frac{2\gamma}{m}\,n\,\partial_{x}n\;,$
(17)

where the last term creates a Riemann wave. The gauged superfluid velocity is
defined in the conventional way

$\tilde{v}_{s}=\frac{\hbar}{m}\,\bigg{(}\partial_{x}\theta-\frac{1}{\hbar}a_{x}\bigg{)}=v_{s}-\frac{\gamma}{m}n\;.$
(18)

The gradient catastrophe will be driven by the statistical parameter $\gamma$,
which governs the strength of the nonlinearity. The catastrophe will be
regularised by higher order dominant terms in the corresponding Euler
equation. We expect this regularisation to be dispersive [47] and the shock
wave to be chiral. The existence of shock waves at long times is generic,
while its characterisation requires detailed numerical and analytical
machinery beyond the scope of this work. However, preliminary confirmation of
this intuition can be gained by generating a momentum kick $k_{0}$ in the
wavepacket near a wall [48]. When the kick and the gauge coupling have the
same sign, i.e. $\text{sgn}(\gamma k_{0})>0$, we observe an asymmetric
expansion and the creation of a robust density wave modulation after the
contact with a wall (see Figure 3 right). For the case of opposite signs, we
observe the creation of a chiral soliton train (see Figure 3 left). For the
latter case, solitons get destroyed one by one upon backscattering with the
wall.

Figure 3: Anomalous reflection dynamics for density-current nonlinear
wavepacket boosted with a $\lvert k_{0}\rvert=3$ momentum kick and $\gamma=2$.
(Left) Creation of a $k_{0}<0$ chiral soliton train with $N=2$ solitons. The
“small” soliton propagates at higher group velocity than the “big” one. When
in contact with the wall and backscattered, the solutions are unstable. For
times $t\sim 7-8$, the backscattered wavepacket and the “big” soliton
interact. The soliton solution is robust against this perturbation. At late
times, the “big” soliton also bounces off the wall and gets destroyed. (Right)
For $k_{0}>0$ we observe an asymmetric expansion followed by shock wave after
reflection on a wall. The highly oscillatory behaviour is characteristic of
dispersive contributions regularising a gradient catastrophe. The shock wave
is robust at late times and results in a regular density modulation. All
parameters are in dimensionless units.

We would also like to draw attention to the fact that the last term in Eq.
(17) can be seen as a source term to the conventional current conservation.
Provided that this contribution is generated by a gauge field, this source
term can be interpreted as an unusual chiral anomaly in a non-Galilean fluid.

Discussion and Conclusions.—  A paradigmatic example of conventional quantum
matter, such as an interacting Bose gas can be exactly mapped into an anyonic
quantum fluid upon coupling to a certain gauge field. This situation is well-
understood in two spatial dimensions for the case of FQH fluids, where the
Chern-Simons mechanism of flux attachment transmutes the quantum identity of
the many-body system. This might even be experimentally tunable [49].

We have studied a spatially one-dimensional interacting bosonic system.
Strictly speaking, there is no Chern-Simons theory in one spatial dimension,
no mechanism of flux attachment, no cyclotron motion, and no sense of
topological quantisation. However, we advocate for the existence of a remnant,
a gauge field that plays an analogous role in altering the commutation
relations of the underlying matter fields, i.e. statistically transmuting
their quantum identity. This lower dimensional analogue can be understood as
arising from a chiral axion term [22], which is also the natural dimensional
reduction of the Abelian Chern-Simons term.

Because of the low dimensionality of the system, the gauge field can be fully
eliminated leading to a residual statistical interaction in the form of a
local $\mathcal{P}$ and $\mathcal{T}$ breaking interparticle interaction term.
Its effect is significant, as it gives rise to important elementary
excitations not present in the usual non-gauged bosonic model, namely chiral
solitons and rotons. These are reminiscent of the Chern-Simons magnetorotons
and anyonic ZHK vortices. An immediate natural question is whether these
chiral solitons have fractional charge and statistics. We argue that such
classical solutions can be understood as quantum bound states, and correspond
to a gauge-matter composite that can be identified with one-dimensional
anyons. Of course, a more convincing answer that addresses this issue should
come from a fully quantum treatment of these solutions, which we did not
pursue in this work. Other important aspects of the current model are (i) the
absence of an Anderson-Higgs mechanism, and thus, the presence of massless
Goldstone modes; and (ii) the creation of chiral dispersive shock waves, and
the possibility of generating chiral solitons trains or chiral density waves
in the propagation of the fluid at moderate values of the gauge coupling. All
of the above is within experimental reach with current state-of-the-art
techniques in ultracold atoms.

Further Considerations.—  It is understood from exact results in the
conventional Lieb-Liniger model [50, 51, 52] that exact results beyond a
Bogoliubov analysis reveal the existence of a “replica” excitation branch in
addition to the one already discussed. These two branches are in one-to-one
correspondence with fermionic particle-hole branches for excitations. In other
words, a more profound analysis reveals particle-like (Lieb I mode) and hole-
like (Lieb II mode) excitations. The Lieb I mode corresponds to the usual
Bogoliubov phononic branch, while the Lieb II mode can be identified with the
mean-field Gross-Pitevskii dark soliton branch [53, 54]. The exact Bose-Fermi
correspondence [55] between the Lieb-Liniger and Cheon-Shinegara fluids [56]
might be the way out as it provides a duality between two models with
manifestly different quantum identities.

The exact correspondence between the statistically-gauged interacting Bose gas
and the soft-core anyonic Lieb-Liniger fluid extends the above notions. Guided
by intuition, we are tempted to conjecture a soliton-particle excitation
correspondence between bare-composite bases. This is suggestive of an order-
disorder duality in which one can either choose to see a system of bosons with
solitonic excitations or, treating those solitons as composite entities 222We
note here that the idea of interpreting certain chiral solitons as composite
particles has been discussed in other contexts [60]. More generally, this goes
in the same spirit as the sine-Gordon/massive-Thirring correspondence.,
reinterpret the theory as a gas of interacting effective anyonic particles.
Alas, more detailed analysis should be conducted.

Acknowledgments.—  We warmly thank A. Celi and L. Tarruell for useful
discussions. G.V-R. acknowledges financial support from EPSRC CM-CDT Grant No.
EP/L015110/1.

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