# What is the Gravitational Field of a Mass in a Spatially Nonlocal Quantum
Superposition?

Rémi Ligez1<EMAIL_ADDRESS>R. B. MacKenzie1
<EMAIL_ADDRESS>Victor Massart1<EMAIL_ADDRESS>M.
B. Paranjape1,2<EMAIL_ADDRESS>U. A. Yajnik3<EMAIL_ADDRESS>1Groupe de physique des particules, Département de physique, Université de
Montréal, C.P. 6128, succ. centre-ville, Montréal, Québec, Canada, H3C 3J7
2Centre de recherche mathématiques, Université de Montréal, C.P. 6128, succ.
centre-ville, Montréal, Québec, Canada, H3C 3J7  3 Department of Physics,
Indian Institute of Technology Bombay, Powai, Mumbaï - 400076

###### Abstract

The study of the gravitational field produced by a spatially non-local,
superposed quantum state of a massive particle is a thrilling area of modern
physics. One question to be answered is whether the gravitational field
behaves as the classical superposition of two particles separated by a spatial
distance with half the mass located at each position or as a quantum
superposition with a far more interesting and subtle behaviour for the
gravitational field. Quantum field theory is ideally suited to probe exactly
this kind of question. We study the scattering of a massless scalar on such a
spatially nonlocal, quantum superposition of a massive particle. We compute
the differential scattering cross section corresponding to the interaction
coming from the exchange of one graviton. We find that the scattering cross
section is not at all represented by the Schrödinger-Newton picture of
potential scattering from two localized sources with half the mass at each
source. We discuss how our result would be lethal to the Schrödinger-Newton
description of gravitation interacting with quantum matter and would be
conducive to considering the gravitational field to be quantized. We comment
on the experimental feasibility of observing such effects.

###### pacs:

04.20.? q, 04.60.? m

††preprint: UdeM-GPP-TH-21-xxx††preprint: arXiv:21xx.xxx

##  I Introduction

At the core of modern physics lie the two most successful theory of physics :
General Relativity (GR) on one side and Quantum Field Theory (QFT) on the
other. Consistently combining these theories is one of the main goals of
modern theoretical physics. The most common approach is to consider the
quantum theory to be more fundamental and try to quantize gravity Kiefer .
This choice is also justified by some indirect evidence
PageGilkerIndirectEvidence . The opposite direction has also been advocated,
of “gravitizing” quantum mechanics, with some interesting arguments, see
Penrose Penrose2014gravitization ; CarlipIsQuantum .

However, we shall be interested in the standard ideas of a quantum theory of
gravity. Indeed, quantum gravity as a low energy effective field theory makes
perfect sense Donoghue-prl ; PhysRevD.50.3874 ; Donoghue:1995cz ;
Burgess:2003jk , although the directly observable predictions of such an
effective quantum field theory are still not remotely experimentally
observable. An intriguing possibility for the detection of noise due to
quantum fluctuations in the gravitational field at gravitational wave
detectors such as LIGO aLIGO  has been suggested recently Parikh . Within the
effective field theory point of view, gravitation must be treated as a fully
quantum mechanical field with all the complexities ensuing from the
possibilities of arbitrary linear superpositions of quantum states. It is in
this context that we examine the gravitational scattering of a massless
particle from the effective quantum gravitational field of a massive particle
in a spatially non-local quantum superposition.

##  II Scattering on a spatially nonlocal wave packet

The usual formalism for scattering in quantum field theory corresponds to
computing the amplitude for a given scattering process and converting the
amplitude into an experimentally observable differential scattering cross
section. The formalism is almost as old as the advent of quantum field theory,
however, generally, certain assumptions are made so that the dependence on the
initial and final wave packets disappears; see for example Peskin:1995ev .
However, here we wish to consider a process where the cross section strongly
depends on the wave packet of one of the initial states, a situation which is
not usually considered.

We consider as the scatterer a particle of mass $M$ in a linearly superposed
state described by a spatially non-local wave function, which of course
correspond to a particular momentum space wavefunction $\phi_{1}(\bm{k}_{1})$,
causing the scattering of a massless particle supposed to be in a state
$\phi_{2}(\bm{k}_{2})$. While $\phi_{2}(\bm{k}_{2})$ is assumed to be sharply
peaked at momentum $\bm{p}_{2}$, the wavefunction $\phi_{1}(\bm{k}_{1})$ is
not peaked at any $\bm{p}_{1}$, but nevertheless assumed for simplicity to be
symmetrically distributed around $\bm{p}_{1}=0$, i.e.
$\phi_{1}(-\bm{k}_{1})=\phi_{1}(\bm{k}_{1})$. This means that the calculation
is done in the convenient reference frame where $\bm{p}_{1}=0$, i.e., the
“centre of mass” frame of the wavefunction.

The formula for the differential of the cross section is given by, :

$\displaystyle d\sigma$
$\displaystyle=\left(\prod_{f}\frac{d^{3}p_{f}}{(2\pi)^{3}}\frac{1}{2E_{f}}\right)\displaystyle\int\frac{d^{3}k_{1}}{(2\pi)^{3}}\displaystyle\int\frac{d^{3}k_{2}}{(2\pi)^{3}}\frac{\left|{\cal
M}\left(k_{1},k_{2}\to\left\\{p_{f}\right\\}\right)\right|^{2}}{2E_{1}2E_{2}\left|v_{1}-v_{2}\right|}$
(1)
$\displaystyle\times\left|\phi_{1}\left(\bm{k}_{1}\right)\right|^{2}\left|\phi_{2}\left(\bm{k}_{2}\right)\right|^{2}(2\pi)^{4}\delta^{4}\left(k_{1}+k_{2}-\sum
p_{f}\right).$ (2)

For our needs, we will use the formalism of Kotkin et al kotkin  as elaborated
in Karlovets Karlovets  that employs the Wigner function formalism Wigner . We
will take the incident massless particles in on-shell momentum space wave
packets $\phi_{2}(\bm{k}_{2})$ centred and highly peaked on a momentum
$\bm{p}_{2}$. These massless particles will be incident on particles of mass
$M$ which are in on-shell, spatially nonlocal wave packets. When expressed in
terms of momentum space wave packets $\phi_{1}(\bm{k}_{1})$, these wave
packets are not highly peaked on any specific momentum as noted above. We will
be interested in the inclusive scattering cross section for the massless
particle, $p_{2}\to p_{4}$, while the non-trivial wave packet will in
principle scatter to all possible allowed final states, which will be
integrated over.

It is most convenient to consider the wave function of the scattered massive
particle, that was initially in the spatially nonlocal wave function, to be
scattered to wave functions that are eigenstates of momentum $\bm{p}_{3}$ and
to integrate over this momentum, as these do correspond to a complete set of
final states. In practice, the integration is of course not required as
energy-momentum conservation for given $k_{1},p_{2},p_{4}$ fixes the value of
$p_{3}$. Thus the scattering will give rise to final momenta $k_{1}\to p_{3}$
and $p_{2}\to p_{4}$ with integration over $\bm{k}_{1}$ smeared with wave
function $\phi_{1}(\bm{k}_{1})$ understood. Then straightforwardly simplifying
the formula in Karlovets , to the case of a particle in a momentum eigenstate
scattering on a particle in a non-trivial wave packet, we find

$d\sigma=\int\frac{d^{3}k_{1}}{(2\pi)^{3}}|T_{PW}\left(\left\\{k_{1},p_{2}\right\\}\to\left\\{p_{3},p_{4}\right\\}\right)|^{2}|\phi_{1}(\bm{k}_{1})|^{2}(2\pi)^{4}\delta^{(4)}(k_{1}+p_{2}-p_{3}-p_{4})\frac{d^{3}p_{3}}{(2\pi)^{3}}\frac{d^{3}p_{4}}{(2\pi)^{3}}$
(3)

where the plane wave amplitude
$T_{PW}\left(\left\\{k_{1},p_{2}\right\\}\to\left\\{p_{3},p_{4}\right\\}\right)$
is given by

$T_{PW}\left(\left\\{k_{1},p_{2}\right\\}\to\left\\{p_{3},p_{4}\right\\}\right)=\frac{\mathcal{M}\left(\left\\{k_{1},p_{2}\right\\}\to\left\\{p_{3},p_{4}\right\\}\right)}{\sqrt{2\epsilon_{1}2\epsilon_{2}2\epsilon_{3}2\epsilon_{4}}}$
(4)

with ${\cal
M}\left(\left\\{k_{1},p_{2}\right\\}\to\left\\{p_{3},p_{4}\right\\}\right)$
the invariant matrix element for the momentum space scattering process
Peskin:1995ev , and $\epsilon_{i}$ the energy of particle $i$. We note the
usual factor of ${\left|v_{1}-v_{2}\right|}$ in the particle flux simplifies
to unity, the velocity of the massless particle, because of the assumed
symmetry of the wave function $\phi_{1}(\bm{k}_{1})$.

The normalized wave function for the particle of type 1, which is taken to be
spatially nonlocal, in momentum space has the form

$|\phi_{1}(\bm{k}_{1})|^{2}=4\left(\pi\sigma^{2}\right)^{3/2}e^{-\sigma^{2}|\bm{k}_{1}|^{2}}\frac{2+e^{2i\bm{r}\cdot\bm{k}_{1}}+e^{-2i\bm{r}\cdot\bm{k}_{1}}}{1+e^{-|\bm{r}|^{2}/(\sigma^{2})}}$
(5)

where $\sigma$ is the width of a Gaussian wave packet that is superposed at
spatial position $\bm{r}$ and $\bm{-r}$.

##  III The 1-graviton exchange scattering amplitude

The amplitude is easily computed using the linearized gravitational theory and
subsequent graviton propagator and matter vertices, following Donoghue
PhysRevD.50.3874 ; Donoghue:1995cz  (see also DeWittI ; DeWittII ; DeWittIII ;
veltman1976quantum ; Burgess:2003jk ; Scadron ), as prescribed by the Feynman
diagram Fig. (1)

$p_{3}$$\phi_{1},k_{1}$$p_{4}$$p_{2}$$q$ Figure 1: 1-graviton exchange
scattering

Applying momentum conservation, we get for the amplitude

$\displaystyle\mathcal{M}$
$\displaystyle=\frac{(-i)^{2}\kappa^{2}(k_{1}^{\mu}p_{3}^{\nu}+k_{1}^{\nu}p_{3}^{\mu}-(p_{2}\cdot
p_{4})\eta^{\mu\nu})(\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha}-\eta_{\mu\nu}\eta_{\alpha\beta})(p_{2}^{\alpha}p_{4}^{\beta}+p_{2}^{\beta}p_{4}^{\alpha}-(p_{2}\cdot
p_{4})\eta^{\alpha\beta})}{2q^{2}}$
$\displaystyle=\frac{\kappa^{2}\left[2(k_{1}\cdot p_{2})(k_{1}\cdot
p_{4})+(k_{1}\cdot p_{2})(p_{2}\cdot p_{4})-(k_{1}\cdot p_{4})(p_{2}\cdot
p_{4})-(p_{2}\cdot p_{4})^{2}\right]}{p_{2}\cdot p_{4}}$ (6)
$\displaystyle=\frac{\kappa^{2}}{1-\bm{\hat{}}p_{2}\cdot\bm{\hat{}}p_{4}}\Big{[}2M^{2}-2M\bm{k}_{1}\cdot(\bm{\hat{}}p_{2}+\bm{\hat{}}p_{4})+M(\omega_{2}-\omega_{4})(1-\bm{\hat{}}p_{2}\cdot\bm{\hat{}}p_{4})+2|\bm{k}_{1}|^{2}$
$\displaystyle+2(\bm{k}_{1}\cdot\bm{\hat{}}p_{2})(\bm{k}_{1}\cdot\bm{\hat{}}p_{4})-\bm{k}_{1}\cdot(\omega_{2}\bm{\hat{}}p_{2}-\omega_{4}\bm{\hat{}}p_{4})(1-\bm{\hat{}}p_{2}\cdot\bm{\hat{}}p_{4})-\omega_{2}\omega_{4}(1-\bm{\hat{}}p_{2}\cdot\bm{\hat{}}p_{4})^{2}\Big{]}$
(7)

where $\kappa$ is the gravitational coupling constant, $\omega_{2}$ and
$\omega_{4}$ are short hand notation for $p_{2}^{0}$ and $p_{4}^{0}$ and the
momentum transfer is $q=p_{3}-k_{1}=p_{2}-p_{4}$ with $q^{2}=-2p_{2}\cdot
p_{4}=-2\omega_{2}\omega_{4}(1-\bm{\hat{p}_{2}\cdot\bm{\hat{}}p_{4}})$ as
$p_{2}$ and $p_{4}$ are massless and on shell.

##  IV The scattering cross section

The amplitude must now be folded in with the wave function, the energy
denominators and factors of $2\pi$ as in Eqn. (3) and then integrated over
$d^{3}p_{3}$ which removes the spatial delta functions and integrated over
$d\omega_{4}$ which removes the temporal delta function yielding the
differential scattering cross section

$\frac{d\sigma}{d\Omega_{4}}=\int\frac{d^{3}k_{1}}{2^{9}\pi^{5}}|\phi_{1}(\bm{k}_{1})|^{2}\frac{\left|\mathcal{M}\right|^{2}}{{\epsilon_{1}\omega_{2}\epsilon_{3}\omega_{4}}}\frac{\omega_{4}^{2}}{\left|f^{\prime}_{\delta}(\omega_{4})\right|}.$
(8)

The energy conserving delta function is given by
$\delta(f_{\delta}(\omega_{4}))=\delta(\epsilon_{1}+\omega_{2}-\epsilon_{3}-\omega_{4})$
where the complications arise because
$\epsilon_{3}=\sqrt{M^{2}+({\bm{k}_{1}+(\bm{p}_{2}}-\omega_{4}(\bm{\hat{p}_{4}})^{2}}$.
$\epsilon_{1}=\sqrt{M^{2}+\bm{k}_{1}^{2}}$ is the energy of the Fourier
component corresponding to momentum $\bm{k}_{1}$ of the spatially nonlocal
particle wave function and $\omega_{2}$ is the energy of the incoming massless
particle. The full expression for the cross section is a complicated,
unenlightening jumble, however its multipole expansion does shed some light on
the gravitational interaction.

##  V Multipole expansion of the scattering cross section

We can write the cross section Eqn.(8) as

$\frac{d\sigma}{d\Omega_{4}}=\int\frac{d^{3}k_{1}}{2^{9}\pi^{5}}|\phi_{1}(\bm{k}_{1})|^{2}g(\bm{k}_{1})\left|\mathcal{M}\right|^{2}$
(9)

and then $g(\bm{k}_{1})\left|\mathcal{M}\right|^{2}$ admits an expansion in
powers of $\bm{k}_{1}$ as

$g(\bm{k}_{1})\left|\mathcal{M}\right|^{2}=\left.g(\bm{k}_{1})\left|\mathcal{M}\right|^{2}\right|_{\bm{k}_{1}=0}+{{1}\over
2}\left.\partial_{\bm{k}_{1}^{i}}\partial_{\bm{k}_{1}^{j}}\left(g(\bm{k}_{1})\left|\mathcal{M}\right|^{2}\right)\right|_{\bm{k}_{1}=0}{\bm{k}_{1}^{i}}{\bm{k}_{1}^{j}}+\cdots$
(10)

where the terms odd in $\bm{k}_{1}$ are absent due to symmetry. Then the
scattering cross section admits the multipole expansion

$\displaystyle\frac{d\sigma}{d\Omega_{4}}=$
$\displaystyle\alpha\int|\phi_{1}(\bm{k}_{1})|^{2}\frac{d^{3}k_{1}}{(2\pi)^{3}}+\beta^{ij}\int
k_{1i}k_{1j}|\phi_{1}(\bm{k}_{1})|^{2}\frac{d^{3}k_{1}}{(2\pi)^{3}}$ (11)
$\displaystyle=\alpha+\frac{\beta^{ij}\delta_{ij}}{3}\int|\bm{k}_{1}|^{2}|\phi_{1}(\bm{k}_{1})|^{2}\frac{d^{3}k_{1}}{(2\pi)^{3}}+\beta^{ij}\int\left(k_{1i}k_{1j}-\frac{|\bm{k}_{1}|^{2}}{3}\delta_{ij}\right)|\phi_{1}(\bm{k}_{1})|^{2}\frac{d^{3}k_{1}}{(2\pi)^{3}}+\cdots$
(12)
$\displaystyle=\alpha+\frac{\beta^{ij}\delta_{ij}}{3}\left(\frac{3}{2\sigma^{2}}-\frac{|\bm{r}|^{2}}{\sigma^{4}}\frac{1}{1+e^{|\bm{r}|^{2}/\sigma^{2}}}\right)+\beta^{ij}\left(\frac{|\bm{r}|^{2}}{3}\delta_{ij}-r_{i}r_{j}\right)\frac{1}{\sigma^{4}\left(1+e^{|\bm{r}|^{2}/\sigma^{2}}\right)}$
(13)

where evidently
$\alpha=\left.\frac{1}{2^{6}\pi^{2}}g(\bm{k}_{1})\left|\mathcal{M}\right|^{2}\right|_{\bm{k}_{1}=0}$
and
$\beta^{ij}=\frac{1}{2^{7}\pi^{2}}\left.\partial_{\bm{k}_{1}^{i}}\partial_{\bm{k}_{1}^{j}}\left(g(\bm{k}_{1})\left|\mathcal{M}\right|^{2}\right)\right|_{\bm{k}_{1}=0}$
and where we have used the integral

$\int
k_{1i}k_{1j}|\phi_{1}(\bm{k}_{1})|^{2}\frac{d^{3}k_{1}}{(2\pi)^{3}}=\frac{\delta_{ij}}{2\sigma^{2}}-\frac{r_{i}r_{j}}{\sigma^{4}}\frac{1}{1+e^{|\bm{r}|^{2}/\sigma^{2}}}.$
(14)

The leading terms in the expansion about $\bm{k}_{1}=0$ are found after a
somewhat tedious calculation. For the second derivative we will use:

$\displaystyle\left.\partial_{\bm{k}_{1}^{i}}\partial_{\bm{k}_{1}^{j}}\left(g(\bm{k}_{1})\left|\mathcal{M}\right|^{2}\right)\right|_{\bm{k}_{1}=0}$
$\displaystyle=$
$\displaystyle\mathcal{M}^{2}\partial_{\bm{k}_{1}^{i}}\partial_{\bm{k}_{1}^{j}}g+2\mathcal{M}(\partial_{\bm{k}_{1}^{i}}g\partial_{\bm{k}_{1}^{j}}\mathcal{M}$
$\displaystyle+$
$\displaystyle\partial_{\bm{k}_{1}^{j}}g\partial_{\bm{k}_{1}^{i}}\mathcal{M})+2g(\partial_{\bm{k}_{1}^{i}}\mathcal{M}\partial_{\bm{k}_{1}^{j}}\mathcal{M}+\left.\mathcal{M}\partial_{\bm{k}_{1}^{i}}\partial_{\bm{k}_{1}^{j}}\mathcal{M})\right|_{\bm{k}_{1}=0}$
(15)

We will need the expression for $\omega_{4}$ obtained from the energy
conservation delta function $\omega_{4}=\epsilon_{1}+\omega_{2}-\epsilon_{3}$,
and its derivatives. We find that the leading terms in the expansion in powers
of $1/M$ of $\omega_{4}$ and its derivatives are given by

$\displaystyle\omega_{4}$
$\displaystyle=\omega_{2}-\frac{1}{M}\left({\omega_{2}^{2}}+\bm{k}_{1}\cdot\bm{p}_{2}-\omega_{2}(\bm{k}_{1}+\bm{p}_{2})\cdot\bm{\bm{\bm{\bm{\hat{p}_{4}}}}}\right)+$
$\displaystyle+\frac{1}{M^{2}}\left(\omega_{2}^{3}-2\omega_{2}^{2}(\bm{k}_{1}+\bm{p}_{2})\cdot\bm{\bm{\bm{\bm{\hat{p}_{4}}}}}\right.$
$\displaystyle\left.+\omega_{2}(\bm{k}_{1}\cdot\bm{\bm{\bm{\bm{\hat{p}_{4}}}}}+\bm{p}_{2}\cdot\bm{\hat{p}_{4}})^{2}+(\bm{k}_{1}\cdot\bm{p}_{2})(\omega_{2}-\bm{k}_{1}\cdot\bm{\bm{\bm{\bm{\hat{p}_{4}}}}}-\bm{p}_{2}\cdot\bm{\bm{\bm{\bm{\hat{p}_{4}}}}})\right)$
(16) $\displaystyle\left.\omega_{4}\right|_{\bm{k}_{1}=0}$
$\displaystyle=\omega_{2}$ (17)
$\displaystyle\partial_{i}\left.\omega_{4}\right|_{\bm{k}_{1}=0}$
$\displaystyle=\frac{\omega_{2}}{M}({\hat{p}_{4i}}-{\hat{p}_{2i}})$ (18)
$\displaystyle\partial_{i}\partial_{j}\left.\omega_{4}\right|_{\bm{k}_{1}=0}$
$\displaystyle=\frac{\omega_{2}}{M^{2}}(2{{\hat{p}_{4i}}{{\hat{p}_{4j}}}}-{\hat{p}_{2i}}{\hat{p}_{4j}}-{\hat{p}_{2j}}{\hat{p}_{4i}}).$
(19)

Then we find, using the above, relatively easily, to leading order in powers
of $M$

$\displaystyle\left.\mathcal{M}\right|_{\bm{k}_{1}=0}$
$\displaystyle=\kappa^{2}\frac{2M^{2}}{1-\bm{\hat{p}_{2}\cdot\bm{\hat{}}p_{4}}}$
(20) $\displaystyle\left.\partial_{i}\mathcal{M}\right|_{\bm{k}_{1}=0}$
$\displaystyle=\frac{\kappa^{2}}{1-\bm{{\hat{p}_{2}\cdot\bm{\hat{}}p_{4}}}}\left(-2M({\hat{p}_{2i}+\hat{p}_{4i}})\right)$
(21)
$\displaystyle\left.\partial_{i}\partial_{j}\mathcal{M}\right|_{\bm{k}_{1}=0}$
$\displaystyle=\frac{\kappa^{2}}{1-{\bm{\hat{p}_{2}\cdot\bm{\hat{}}p_{4}}}}\left(4\delta_{ij}+2({\hat{p}_{2i}\hat{p}_{4j}}+{\hat{p}_{2j}\hat{p}_{4i}})\right).$
(22)

A lengthier and more tedious calculation, which includes the calculation of
$1/|f^{\prime}_{\delta}|$, gives $g$ and its derivatives, again expanded in
powers of $1/M$,

$\displaystyle g(\omega_{4},\bm{k}_{1})$
$\displaystyle=\frac{\omega_{4}}{M^{2}\omega_{2}}\Big{(}1-\frac{\omega_{2}(1-\bm{\hat{p}_{2}\cdot\bm{\hat{}}p_{4}})-\bm{k}_{1}\cdot\bm{\hat{p}_{4}}}{M}+\frac{\omega_{2}^{2}-|\bm{k}_{1}|^{2}-2\omega_{2}(\bm{k}_{1}+\bm{p}_{2})\cdot\bm{{\bm{\bm{\bm{\hat{p}_{4}}}}}}+(\bm{k}_{1}\cdot\bm{\bm{\bm{\bm{\hat{p}_{4}}}}+\bm{p}_{2}\cdot\bm{\hat{}}p_{4}})^{2}}{M^{2}}\Big{)}$
$\displaystyle\left.g\right|_{\bm{k}_{1}=0}$ $\displaystyle=\frac{1}{M^{2}}$
(23) $\displaystyle\left.\partial_{i}g\right|_{\bm{k}_{1}=0}$
$\displaystyle=\frac{1}{M^{3}}(2{\hat{p}_{4i}-\hat{p}_{2i}})$ (24)
$\displaystyle\left.\partial_{i}\partial_{j}g\right|_{\bm{k}_{1}=0}$
$\displaystyle=\frac{2}{M^{4}}\left(3{\hat{p}_{4i}\hat{p}_{4j}-\hat{p}_{2i}\hat{p}_{4j}-\hat{p}_{2j}\hat{p}_{4i}}-\delta_{ij}\right)$
(25)

Then putting all this together, we find

$\alpha=\frac{\kappa^{4}M^{2}}{16\pi^{2}(1-\bm{\hat{p}_{2}\cdot\bm{\hat{}}p_{4}})^{2}}$
(26)
$\beta^{ij}=\Big{(}\frac{\kappa^{4}}{16\pi^{2}(1-\bm{\hat{p}_{2}\cdot\bm{\hat{}}p_{4}})^{2}}\Big{)}(\delta_{ij}+3{\hat{p}_{2i}\hat{p}_{2j}})$
(27)

The term $\alpha$ gives exactly the limiting small momentum transfer
gravitational deflection of a massless particles from a massive particle,
Scadron .

##  VI Discussion and Conclusions

The term proportional to $\alpha$, the lowest order monopole term, correspond
to the scattering cross section of a single point like mass $M$, the analog of
the Rutherford/Thompson cross section of a massless particle scattering from a
point like Newtonian potential. We see that the higher order terms coming from
the
$\beta^{ij}\delta_{ij}=\Big{(}\frac{\kappa^{4}}{16\pi^{2}(1-{\bm{\hat{\bm{p}}_{2}}}\cdot\bm{\bm{\bm{\bm{\hat{p}_{4}}}}})^{2}}\Big{)}6$,
contribute to the monopole. This means that the scattering cross section is
able to probe the non point like nature of the monopole part of the scattering
particle’s wave function. This addition to the monopole contribution is given
by

$\Big{(}\frac{\kappa^{4}}{16\pi^{2}(1-\bm{\hat{p}_{2}}\cdot\bm{\bm{\bm{\bm{\hat{p}_{4}}}})}^{2}}\Big{)}2\left(\frac{3}{2\sigma^{2}}-\frac{|\bm{r}|^{2}}{\sigma^{4}}\frac{1}{1+e^{|\bm{r}|^{2}/\sigma^{2}}}\right).$
(28)

However the actual quadrupole-type contribution is interestingly nothing like
what would be expected if the gravitational field behaved according to the
Schrödinger-Newton prescription, Diosi ; Penrose . The Schrödinger-Newton
prescription would have the corresponding gravitational field as if one half
the mass were concentrated at each of the two spatially nonlocal points,
Diosi2 , a configuration which has a quadrupole moment

$M\left(\frac{|\bm{r}|^{2}}{3}\delta_{ij}-r_{i}r_{j}\right).$ (29)

Such a gravitational field yields a contribution to the (gravitational)
scattering cross section

$\frac{\kappa^{4}M^{2}\omega_{2}^{2}}{16\pi^{2}(1-\bm{\hat{p}_{2}\cdot\bm{\hat{}}p_{4}})^{2}}(\hat{p}_{4i}-\hat{p}_{2i})(\hat{p}_{4j}-\hat{p}_{2j})\left(\frac{|\bm{r}|^{2}}{3}\delta_{ij}-r_{i}r_{j}\right)$
(30)

obtained from a presumed Newtonian potential scattering from two point sources
of mass $M/2$ located at the two peaks of the spatially nonlocal wave
function. This result is not at all what we find from, Eqn.(13)

$\frac{\kappa^{4}}{16\pi^{2}\sigma^{4}\left(1+e^{|\bm{r}|^{2}/\sigma^{2}}\right)}\frac{3{\hat{p}_{2i}\hat{p}_{2j}}}{{(1-\bm{\hat{p}_{2}\cdot\bm{\hat{}}p_{4}})^{2}}}\left(\frac{|\bm{r}|^{2}}{3}\delta_{ij}-r_{i}r_{j}\right),$
(31)

where $\kappa=\sqrt{8\pi G}$ which gives a coefficient $4G^{2}$. Indeed the
quadrupole term from the calculation from one graviton exchange scattering is
exponentially small as $|\bm{r}|\gg\sigma$ compared to the result expected
from scattering from a potential where the mass is split between two postions
as prescribed by the Newton-Schrödinger formalism. This is lethal to the
Schrödinger-Newton formalism, the calculation corresponding to one graviton
exchange is clearly more justifiable. Additionally, our result shows that the
wave function is only probed by the incoming massless particle’s direction
relative to the direction of separation $\bm{r}$ something that is hopefully
experimentally verifiable.

Useful further calculations would be to compute the gravitational contribution
to the self energy of a massive particle in a spatially nonlocal wave
function. One would look for the amplitude and behaviour of the self energy as
a function of the spatial separation $r$ of the nonlocal wave function. A
behaviour as $1/r$ of the self energy would correspond to the Newtonian
potential and the corresponding $1/r^{2}$ law of attraction of the two non-
local lumps. A calculation of the amplitude would indicate how the two
nonlocal lumps behave gravitationally with respect to each other.

It would be very interesting to measure any of these phenomena although
probably not technically feasible presently. However, it is not out of the
realm of possibility to have a spatially non-local superposition about a
$10^{9}$ atoms. A Bose condensate would be proposed, and it can be imagined
that such condensates could be launched in a atom interferometer. Then the
gravitational interactions of the two quantum superposed masses might be
measurable K .

## Acknowledgments

We thank the Indian Institute of Technology Bombay for its hospitality where
this work was started. We thank the Ministère des relations internationales et
de la francophonie du gouvernement du Québec and the Programme de Coopération
Québec-Maharashtra and NSERC of Canada for financial support. The work of VM
was further supported by the Department of Physics and the Faculty of Graduate
Studies of the Université de Montréal.

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# Emergent U(1) symmetry in non-particle-conserving one-dimensional models

Zakaria Jouini  Institute of Physics, Ecole Polytechnique Fédérale de Lausanne
(EPFL), CH-1015 Lausanne, Switzerland  Natalia Chepiga  Kavli Institute of
Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The
Netherlands  Loïc Herviou  Institute of Physics, Ecole Polytechnique Fédérale
de Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Frédéric Mila  Institute of
Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland

(September 3, 2024)

###### Abstract

The properties of stable Luttinger liquid phases in models with a non-
conserved number of particles are investigated. We study the Luttinger liquid
phases in one-dimensional models of hard-core boson and spinless fermion
chains where particles can be created and annihilated three by three on
adjacent sites. We provide an intuitive and systematic method based on the
flow equation approach, which accounts for additional terms in the
correlations generated by the $\mathbb{Z}_{3}$-symmetric interactions. We find
that despite the emergence of U(1) symmetry under renormalization, the
observables are still affected by its breaking in the bare Hamiltonian. In
particular, the standard bosonization mapping becomes insufficient to capture
the full behavior of correlation functions.

##  I Introduction

The observation of density-wave order in recent experiments on one-dimensional
systems of Rydberg atomsKeesling  _et al._ (2019); Bernien  _et al._ (2017)
has brought back unsolved questions about the nature of the commensurate
melting of period-$p$ phasesOstlund (1981); Huse and Fisher (1982); Huse  _et
al._ (1983); Howes (1983); den Nijs (1988); Huse and Fisher (1984). For $p>2$,
a floating phase, characterized by incommensurate and algebraic correlations,
separates the $\mathbb{Z}_{p}$-ordered and the disordered phasesHaldane  _et
al._ (1983); Ostlund (1981); Huse and Fisher (1984); Katsura  _et al._ (2015).
For $p=3$ and $p=4$, the extension of the floating phase is still debatedSelke
and Yeomans (1982); Duxbury  _et al._ (1984); Huse and Fisher (1984); Giudici
_et al._ (2019); Chepiga and Mila (2019); Whitsitt  _et al._ (2018); Chepiga
and Mila (2021a, b); Rader and Läuchli ; Nyckees  _et al._ (2021); Nyckees and
Mila (2022). One open question is the existence of a direct and continuous
transition in the chiral universality class that would occur before the
floating phase developsHuse and Fisher (1982, 1984). The quantum version of
this problem can be formulated in terms of hard-core bosons associated with
domain walls in the commensurate structure of the density wave Schulz  _et
al._ (1982). The resulting Hamiltonian exhibits a $\mathbb{Z}_{p}$ symmetry as
it contains terms that create and annihilate $p$ adjacent particles. When
these perturbations to the free-fermion fixed point are irrelevant, the U(1)
charge conservation is restored at the scaling limit in an extended
incommensurate Luttinger liquid phaseFendley  _et al._ (2004); Chepiga and
Mila (2021b), equivalent to the floating phase of the 2D classical problemHuse
and Fisher (1984).

The properties of the Luttinger liquid phase are described by a bosonic
conformal field theoryHaldane (1981); Giamarchi (2003). The correspondence
between the operators at the lattice scale and the bosonic fields in the
continuum limit can be in principle constructed from selection rules dictated
by the symmetry of the lattice Hamiltonian. When the latter exhibits U(1)
symmetry, this correspondence is generally given by the standard bosonization
mappingGiamarchi (2003). For models that do not conserve the number of
particles, the latter becomes insufficient to capture the long-distance
behavior of the correlations. A similar problem was discussed recently in the
case of nonsymmorphic 1D modelsYang  _et al._ (2022a, b). We provide in this
paper a way to derive a bosonic representation of lattice operators in
$\mathbb{Z}_{p}$-symmetric 1D models. Using the flow equation approachWegner
(1994), introduced by Wegner in 1994, a continuous unitary transformation is
designed to restore the U(1) symmetry perturbatively in an effective
Hamiltonian. The action of these transformations on the lattice operator
generates an expansion in terms of the bosonic fields of the Luttinger liquid
theory. The bosonization mappings obtained from this procedure are sufficient
to capture the long-distance behavior of the correlations. We illustrate this
method on two chains of spinless fermions and hard-core bosons where particles
are created and annihilated three by three on adjacent sites. Our findings are
assessed by density matrix renormalization group (DMRG)White (1992);
Schollwöck (2005); Östlund and Rommer (1995); Schollwöck (2011) simulations.

The paper is organized as follows. In Section II, we discuss the phase
diagrams of the hard-core boson and spinless fermion models. Section III
provides a brief introduction to the flow equation approach. In Section IV,
the flow equation approach is applied to the fermionic model to derive
perturbatively a U(1)-symmetric effective Hamiltonian. A modified bosonic
representation of the single-fermion operator is then derived and used to
calculate correlation functions inside the Luttinger liquid phase. In Section
V, the same procedure is applied to the continuum limit of the hard-core boson
model, using the generator that diagonalizes the dual sine-Gordon model. The
results are summarized in Section VI.

##  II models and phase diagrams

We consider one-dimensional models of hard-core bosons and spinless fermions
where particles are created and annihilated three by three on adjacent sites.
The two models share the feature of a Luttinger liquid phase that remains
stable when the $\mathbb{Z}_{3}$-symmetric interaction is turned on. In this
phase, the low-energy properties of the system are described by the Luttinger
liquid Hamiltonian

$H_{\text{LL}}=\frac{1}{2\pi}\int
dx\,vK[\partial_{x}\theta(x)]^{2}+\frac{v}{K}[\partial_{x}\phi(x)]^{2},$ (1)

where $K$ is the Luttinger parameter and $v$ is the velocity. The fields
$\phi$ and $\theta$ are bosonic in nature and satisfy the commutation relation
$[\phi(x),\theta(y)]=i\pi\,\text{sign}(y-x)/2$. The correlations decay
algebraically with an exponent controlled by $K$ and oscillate with an
incommensurate wave vector proportional to the Fermi wave vector $k_{F}$. We
present in this section the arguments for the stability of the Luttinger
liquid phase and discuss the nature of the transitions out of it.

###  II.1 Hard-core bosonic model

The Hamiltonian of the hard-core boson model is given by

$H=\sum_{i}-t(b_{i+1}^{\dagger}b_{i}+\text{h.c.})-\mu
n_{i}+\lambda(b_{i}^{\dagger}b_{i+1}^{\dagger}b_{i+2}^{\dagger}+\text{h.c.}),\\\
$ (2)

where $b^{\dagger}_{i}$ and $b_{i}$ are respectively the creation and
annihilation operators of hard-core bosons at site $i$, and
$n_{i}=b^{\dagger}_{i}b_{i}$ is the density operator. The hard-core constraint
amounts to a restriction of the occupation number to $n_{i}=0$ and $n_{i}=1$.
Accordingly, the operators satisfy the commutation relation
$[b_{i},b^{\dagger}_{j}]=(1-2n_{i})\delta_{i,j}$. The hard-core boson model
was recently introduced as a dual description of the transition to phases with
a density-wave order of period 3 in chains of Rydberg atoms Whitsitt  _et al._
(2018); Fendley  _et al._ (2004). Its phase diagram is studied extensively in
Ref. Chepiga and Mila, 2021b. We recall here the main results. Without loss of
generality, the hopping amplitude is set to $t=1$. At $\lambda=0$, Eq. (2) is
a free fermion Hamiltonian that can be mapped in the continuum limit to the
Luttinger liquid Hamiltonian (1) with a Luttinger exponent $K=1$ and a
velocity $v=2\sin(k_{F})$. When $\lambda\neq 0$, the U(1) symmetry is reduced
to a $\mathbb{Z}_{3}$ symmetry. The stability of the Luttinger liquid phase
follows from the scaling analysis of the Hamiltonian in the continuum limit,
obtained by applying the bosonization mappingGiamarchi (2003). Since the
bosonic representation of the hard-core boson operators takes the form $b\sim
e^{i\theta}$, the Hamiltonian reduces, at half filling ($\mu=0$) and up to the
most relevant term, to

$H\sim H_{\text{LL}}+\frac{g}{\pi\alpha^{2}}\int dx\cos(3\theta(x)),$ (3)

where $g$ is a dimensionless coupling and $\alpha$ is a real-space cutoff. A
standard renormalization group (RG) analysisGiamarchi (2003) yields the RG
equations

$\begin{split}\frac{dK(l)}{dl}&=\frac{9}{4}g^{2}(l),\\\
\frac{dg(l)}{dl}&=\Big{(}2-\frac{9}{4K(l)}\Big{)}g(l).\end{split}$ (4)

When $K<9/8$, the coupling constant decays exponentially under renormalization
such that the Luttinger liquid Hamiltonian is recovered with effective
parameters $K^{*}$ and $v^{*}$ that depend on the bare coupling constants in
Eq. (3). The U(1) symmetry is thus restored at the scaling limit. It manifests
itself as a symmetry under translation of the dual field $\theta$. At $K=9/8$,
the system undergoes a Kosterlitz-Thouless (KT) transitionKosterlitz and
Thouless (1973) into a $\mathbb{Z}_{3}$-ordered phase by pinning the dual
field in the minima of the cosine, i.e., at $\theta_{n}=2\pi n/3$. Upon
varying the chemical potential $\mu$ inside the Luttinger liquid phase, a
commensurate-incommensurate transition into a disordered phase occurs. This
transition is in the Pokrovsky-Talapov (PT) universality classPokrovsky and
Talapov (1979), characterized by a dynamical exponent $z=2$ and an
incommensurate correlations wave vector that approaches its commensurate value
with a singularity proportional to $|\mu-\mu_{c}|^{1/2}$. The KT and PT lines
are expected to meet at a Lifshitz point that would appear before the three-
state Potts pointHuse and Fisher (1984). Between the two points, the
commensurate melting is direct and takes place in the chiral universality
class. These predictions are confirmed numerically in Ref. Chepiga and Mila,
2021b.

###  II.2 Fermionic model

The Hamiltonian of the fermionic model is given by

$H=\sum_{i}-t(c_{i+1}^{\dagger}c_{i}+\text{h.c.})-\mu
n_{i}+\lambda(c_{i}^{\dagger}c_{i+1}^{\dagger}c_{i+2}^{\dagger}+\text{h.c.}),\\\
$ (5)

where $c^{\dagger}_{i}$ and $c_{i}$ are the creation and annihilation operator
of spinless fermions. It differs from the hard-core boson model (2) by a
string operator when the Jordan-Wigner transformation is applied.

FIG. 1: (a)-(b) Scaling of the incommensurate wave vector as a function of the
chemical potential close to the PT transition for (a) $\lambda=0.2$ and (b)
$\lambda=2$. (c) Luttinger parameter $K$ as a function of $\lambda$ at
$\mu=0$, extracted numerically from the decay of the two-fermion correlations.
This procedure is only valid when $K$ is larger than $\sqrt{3}$ (see Section
IV.3.2). The system is simulated with $N=600$ sites.

Similar to the bosonic case, the stability of the Luttinger liquid phase in
Eq. (5) can be investigated using the bosonization mappingGiamarchi (2003). In
terms of the fields $\phi$ and $\theta$, the Hamiltonian reads

$H\sim H_{\text{LL}}+\frac{g}{\pi\alpha^{2}}\int
dx\,\cos(3\phi(x)-3k_{F}x)\cos(3\theta(x)),$ (6)

where terms that oscillate at $k_{F}$ are neglected as they vanish under the
integration for all fillings of the band inside the Luttinger liquid phase.
Eq. (6) becomes non-oscillating at $k_{F}=2\pi/3$. The perturbation to the
Luttinger liquid Hamiltonian has a scaling dimension $\Delta=9(K+1/K)/4$ and
becomes relevant when $\Delta<2$. This inequality has however no solution and
the coupling constant $g$ decays exponentially under renormalization for all
values of $K$. It should be noted that terms such as $\cos(6\theta)$ and
$\cos(6\phi)$ are expected to arise along the RG flow and can lead to a KT
transition when $K>9/2$ or $K<2/9$. DMRG computations of $K$ suggest however
that it increases slowly enough from $K=1$ for the generated terms to remain
irrelevant even at large values of $\lambda$. For instance, the numerical
results at $\mu=0$ indicate an extension of the Luttinger liquid phase at
least up to $\lambda\sim 50$ (Fig.1). Nevertheless, a KT transition at larger
values of $\lambda$ cannot be excluded. As the chemical potential is varied in
the Luttinger liquid phase, a PT transition line can be identified numerically
using the scaling behavior of the wave vector (Fig.1).

##  III Flow equation approach

The general idea behind the flow equation approach introduced by
Wegner(Wegner, 1994) is to apply continuous unitary transformations to the
Hamiltonian in order to bring it into a more band-diagonal form. The approach
consists in a renormalization scheme where states with large energy
differences are first decoupled while smaller energy differences are later
suppressed along the flow.

The formalism of the method is based on the parametrization of a set of
unitarily equivalent Hamiltonians $H(l)=U(l)H(0)U^{\dagger}(l)$. By taking the
derivative with respect to $l$, the problem is recast into the differential
equation

$\frac{dH(l)}{dl}=[\eta(l),H(l)],$ (7)

where

$\eta(l)=\frac{dU(l)}{dl}U^{\dagger}(l)$ (8)

is the anti-hermitian generator of the flow. The latter can be chosen
appropriately to diagonalize the bare Hamiltonian $H(0)$. The canonical choice
proposed by WegnerWegner (1994) is given by
$\eta(l)=[H_{\text{d}}(l),H_{\text{od}}(l)]$, where $H_{\text{d}}(l)$ and
$H_{\text{od}}(l)$ are the diagonal and the off-diagonal parts of the flowing
Hamiltonian $H(l)$. From this definition of the generator, it can be
shownKehrein (2006) that

$\frac{d\text{Tr}(H^{2}_{\text{od}}(l))}{dl}=-2\text{Tr}(\eta^{\dagger}(l)\eta(l))\leq
0,$ (9)

which indicates that the flow gradually brings the Hamiltonian into a more
band-diagonal form. A fixed point of the flow is reached when $\eta(l)$
vanishes. The diagonal and off-diagonal parts of the flowing Hamiltonian then
commute and the Hamiltonian becomes block-diagonal with respect to the
symmetry of the non-interacting part. Thus, the flow equation approach
provides a systematic way to design a unitary transformation that recovers the
U(1) symmetry in models that do not conserve the number of particles. Its
drawback is the proliferation of terms along the flow. Truncations schemes are
hence needed to keep the calculations tractable.

Once the diagonal Hamiltonian is obtained from the flow equation procedure,
the change of basis associated with $\eta(l)$ can be applied to the operators.
Given an operator $O$, its transformation along the flow is dictated by the
flow equation

$\frac{dO(l)}{dl}=[\eta(l),O(l)].$ (10)

The expectation value in the ground state of the bare Hamiltonian can be
evaluated using the relation

$\bra{\psi_{\text{gs}}}O\ket{\psi_{\text{gs}}}=\bra{\psi_{\text{gs}}(\infty)}O(\infty)\ket{\psi_{\text{gs}}(\infty)},$
(11)

where
$\ket{\psi_{\text{gs}}(\infty)}=U^{\dagger}(\infty)\ket{\psi_{\text{gs}}}$ is
the ground state of the diagonal Hamiltonian $H(\infty)$.

In the following sections, we apply the flow equation procedure to the models
in Eqs. (2),(5). The transformed bosonic representations of hard-core bosonic
and fermionic operators is derived from the generator of the flow that
restores the U(1) symmetry in the Hamiltonians.

##  IV Flow equation approach to
the fermionic model

We derive in this section a U(1)-symmetric effective Hamiltonian that
describes the low-energy properties of the fermionic Hamiltonian (5).

###  IV.1 Flow of the Hamiltonian

We proceed by writing the Hamiltonian in the Fourier basis in order to
separate it into a diagonal part $H_{0}$ and an interaction part $\lambda
H_{3}$. We have

$H=\sum_{k}\xi_{k}c_{k}^{\dagger}c_{k}+\frac{\lambda}{3!\sqrt{N}}\sum_{k,q}B_{k,q}(c_{k}^{\dagger}c_{q}^{\dagger}c_{-k-q}^{\dagger}-\text{h.c.}),\\\
$ (12)

where

$\begin{split}\xi_{k}&=-2\cos(k)-\mu\\\
B_{k,q}&=2i[\sin(2k+q)-\sin(2q+k)-\sin(k-q)].\end{split}$ (13)

Along the flow, other interaction terms that are not initially present are
generated. They are incorporated in the following ansatz for the flowing
Hamiltonian $H(l)$:

$H(l)=H_{0}(l)+\lambda H_{3}(l)+\lambda^{2}H_{U}(l),$ (14)

where

$H_{U}(l)=\frac{1}{N}\sum_{k,q,p}U_{k,q,p}(l)c^{\dagger}_{k+p}c^{\dagger}_{q-p}c_{q}c_{k}.$
(15)

The normal-ordering of terms in Eq.(14) with respect to the Fermi sea of the
diagonal Hamiltonian $H_{0}(l)$ is implicit. It is systematically carried out
during the calculation to truncate the generated terms in a controlled manner.
We note that three-body interaction and six-fermion terms also arise along the
flow. They are neglected in the ansatz (14) due to their large scaling
dimension. By taking the generator as $\eta(l):=[H_{0}(l),\lambda H_{3}(l)]$,
which reads

$\eta(l)=\frac{\lambda}{3!\sqrt{N}}\sum_{k,q}B_{k,q}(l)\alpha_{k,q}(l)(c^{\dagger}_{k}c^{\dagger}_{q}c^{\dagger}_{-k-q}+\text{h.c.}),$
(16)

with $\alpha_{k,q}=\xi_{k}+\xi_{q}+\xi_{-k-q}$, we obtain a set of flow
equations:

$\begin{split}\frac{dB_{k,q}(l)}{dl}&=-\alpha^{2}_{k,q}(l)B_{k,q}(l),\\\
\frac{dU_{k,q,p}(l)}{dl}&=-\frac{1}{2}[\alpha_{k+p,q-p}(l)+\alpha_{k,q}(l)]B_{k,q}(l)B_{k+p,q-p}(l)\\\
&\hskip 56.9055pt\times[2\Theta(k_{F}-|k+q|)-1]\\\
\frac{d\xi_{k}(l)}{dl}&=-\frac{\lambda^{2}}{N}\Big{[}\sum_{|q|<k_{F}}\alpha_{k,q}(l)B^{2}_{k,q}(l)\Theta(k_{F}-|k+q|)\\\
&+\sum_{|q|>k_{F}}\alpha_{k,q}(l)B^{2}_{k,q}(l)[1-\Theta(k_{F}-|k+q|)]\Big{]}\end{split}$
(17)

with the initial conditions $B_{k,q}(0)=B_{k,q}$, $\xi_{k}(0)=\xi_{k}$, and
$U_{k,q,p}(0)=0$. We have omitted for brevity the $l$-dependence in the bare
Hamiltonian, i.e. $H\equiv H(0)$. The derivation of the flow equations is
detailed in Appendix A. To the leading order in $\lambda$, it is sufficient to
take the bare value $\alpha_{k,q}$ in the flow equation of $B_{k,q}(l)$. The
solution shows that the three-site term decays to zero along the flow as
$B_{k,q}(l)=B_{k,q}e^{-\alpha^{2}_{k,q}l}$. On the other hand, the generated
terms take finite values at $l=\infty$ given by

$\begin{split}U_{k,q,p}(\infty)&=-\frac{1}{2}\frac{\alpha_{k+p,q-p}+\alpha_{k,q}}{\alpha^{2}_{k,q}+\alpha^{2}_{k+p,q-p}}B_{k,q}B_{k+p,q-p}\\\
&\hskip 56.9055pt\times[2\Theta(k_{F}-|k+q|)-1]\\\
\xi_{k}(\infty)&=\xi_{k}-\frac{\lambda^{2}}{2N}\Big{[}\sum_{|q|<k_{F}}\frac{B^{2}_{k,q}}{\alpha_{k,q}}\Theta(k_{F}-|k+q|)\\\
&\hskip
28.45274pt+\sum_{|q|>k_{F}}\frac{B_{k,q}^{2}}{\alpha_{k,q}}[1-\Theta(k_{F}-|k+q|)]\Big{]}\end{split}$
(18)

By construction, the two-body interaction matrix $U_{k,q,p}$ is symmetrized
with respect to the permutations of fermion operators in Eq. (15). The
divergence as $\alpha_{k,q}$ goes to zero in the renormalized quantities (18)
is an artifact of the limit $l\to\infty$. In fact, the contribution of the
flow to the dispersion vanishes when $\alpha_{k,q}=0$. Finally, we note that
Wegner’s prescription is relaxed by taking in the definition (16) of $\eta(l)$
only the three-site term and not all the interacting part of $H(l)$. As a
result, the approach becomes perturbative in $\lambda$. Terms that are
generated along the flow have a second-order dependence in $\lambda$ and can
be eliminated by including them in a redefinition of the generator. This
induces higher-order corrections to the flow equations (17). Thus, the flow
equation procedure enables to push the $\lambda$-dependence of terms that
break the U(1) symmetry to higher orders, thereby restoring the symmetry
perturbatively.

###  IV.2 Bosonization of the effective Hamiltonian

We investigate in this section the low-energy properties of the Hamiltonian
(5) starting from the effective Hamiltonian
$H_{\text{eff}}=H_{0}(\infty)+\lambda^{2}H_{U}(\infty)$. After taking the
long-range limit in the two-body interaction term, only particle-hole
excitations within a momentum range $\Lambda$ around the Fermi points $\pm
k_{F}$ are retained. The renormalized dispersion relation can then be
linearized around the Fermi point and the fermion operators separated into
right and left modes:

$c_{k}=\Theta(\Lambda-|k-k_{F}|)c_{R,k}+\Theta(\Lambda-|k+k_{F}|)c_{L,k},$
(19)

where $\Theta$ is the Heaviside function. After constructing fermionic fields
from these modes, i.e.,

$c(x)=\frac{1}{\sqrt{N}}\sum_{k}e^{ikx}c_{k}=:c_{R}(x)+c_{L}(x),$ (20)

we obtain in the continuum limit a Hamiltonian that describes the low-lying
states of $H_{\text{eff}}$. It is given by

$\begin{split}H_{\text{eff}}=&-i\tilde{v}_{F}\int
dx\,\big{[}c^{\dagger}_{R}(x)\partial_{x}c_{R}(x)-c^{\dagger}_{L}(x)\partial_{x}c_{L}(x)\big{]}\\\
&+4\lambda^{2}g_{2}\int dx\,\rho_{R}(x)\rho_{L}(x),\end{split}$ (21)

where $\tilde{v}_{F}=\partial\xi_{k}(\infty)/\partial k|_{k=k_{F}}$ is the
renormalized Fermi velocity, $\rho_{R}(x)$ and $\rho_{L}(x)$ are respectively
the density operators of the right and left branches, and
$g_{2}=U_{k_{F},-k_{F},0}$ is the forward scattering matrix element. Due to
the symmetry of the two-body interaction matrix, $g_{4}$ scattering processes
that couple fermions at the same branch vanish. Finally, the factor 4 accounts
for the two possible $g_{2}$ processes and the two backscattering processes,
i.e., $g_{1}=-U_{k_{F},-k_{F},2k_{F}}$, which for spinless fermions coincide
with $g_{2}$ processes. We note that since $g_{2}>0$, the interaction is
attractive, which indicates that the three-site term in the bare Hamiltonian
favors the occupation of three adjacent sites.

We now apply the bosonization mapping between the fermionic fields
$c_{R},c_{L}$ and the bosonic fields $\phi,\theta$. It is given by Giamarchi
(2003)

$c_{r}(x)=\frac{F_{r}}{\sqrt{2\pi\alpha}}e^{irk_{F}x}e^{-i[r\phi(x)-\theta(x)]},$
(22)

where $r=1$ for $r=R$ and $r=-1$ for $r=L$. Here, $\alpha\sim 1/\Lambda$ is a
short-distance cut-off and $F_{r}$ are unitary operators called Klein factors.
They follow the commutation relations
$\\{F_{r},F^{\dagger}_{r^{\prime}}\\}=2\delta_{r,r^{\prime}}$ and ensure the
anti-commutation of fermions from different species. Using Eq. (22), the
Hamiltonian (21) can be reduced to the Luttinger liquid Hamiltonian (1) with a
renormalized velocity $u$ and a Luttinger parameter, given to the second order
of $\lambda$ by

$K=1+\frac{4}{\pi}\sin(k_{F})\sin^{2}\Big{(}\frac{k_{F}}{2}\Big{)}\lambda^{2}.$
(23)

The derivation of Eq. (23) is detailed in Appendix C. Here, $k_{F}$ is defined
by the filling of the renormalized band $\xi_{k}(\infty)$ and differs from its
value at the non-interacting limit $\lambda=0$. The relation between $k_{F}$
in Eq. (23) and the bare chemical potential can be obtained by setting
$\xi_{k_{F}}(\infty)=0$. This leads to the auto-coherent equation

$\mu=-2\cos(k_{F})-\lambda^{2}I(k_{F},\mu),$ (24)

where $I$ is the sum over the $q$ modes in Eq. (18).

FIG. 2: (a) Example of the local density profile at $\lambda=0.1$ and
$\mu=1.2$. (b) Analytical calculation and DMRG simulations of the Luttinger
parameter $K$ at $\lambda=0.1$ as a function of the bare chemical potential
$\mu$, obtained by solving Eq. (24).

The behavior of the Luttinger parameter is compared to DMRG results, along a
small $\lambda$ cut, where the perturbative calculations still hold (Fig.2).
$K$ is obtained numerically by fitting the profile of the local density, which
for open boundary conditions exhibits Friedel oscillations. According to
conformal field theory, we have White  _et al._ (2002); Fabrizio and Gogolin
(1995)

$\expectationvalue{n_{j}}\propto\frac{\cos(2k_{F}j+\beta)}{[(N/\pi)\sin(\pi
j/N)]^{K}},$ (25)

with a phase shift $\beta$. The DMRG results for $K$ follow the form described
by Eq. (23). The absence of a reflection symmetry with respect to $\mu$ is due
to the breaking of particle-hole symmetry by the three-site interaction in Eq.
(5). The maximal deviation between the analytical and the numerical results is
of order $10^{-3}$. Its origin is twofold. First, it should be noted that by
considering only scattering processes with momentum transfer $p\sim 0,2k_{F}$
in the effective Hamiltonian (21), we have neglected terms that oscillate with
a wave vector $2k_{F}$. These terms contribute to the renormalization of $K$
at the $4^{\text{th}}$ order of $\lambda$, mainly at half-filling. Secondly,
three-body and six-fermion interaction terms are discarded in the early stages
of the calculations. Including them in the generator of the flow also provides
corrections of order $4$ in $\lambda$ to the two-body interaction matrix.

When the edges of the band $\xi_{k}(\infty)$ are crossed, the system undergoes
a PT transition as the density of fermions respectively vanishes or saturates.
For small $\lambda$, the shape of the transition line can be extracted from
the solution of Eq. (24) at $k_{F}=0,\pi$ (Fig.3). As the transition is
approached from the Luttinger liquid phase, the two-body interaction matrix
vanishes, i.e., $U_{k_{F},-k_{F},p}=0$ for all $p$ and the Luttinger parameter
tends, up to the second order in $\lambda$, towards its non-interacting value
$K=1$. This result indicates that although the particles are not conserved in
the bare Hamiltonian, they still behave as free fermions in the low density
limit.

FIG. 3: (a)-(b) Theoretical prediction for the Pokrovsky-Talapov transition
line at (a) $k_{F}=0$ and (b) $k_{F}=\pi$, as a function of $\lambda$.

###  IV.3 Transformed Fermionic operator

In this section, a modified bosonic representation of the single fermion
operator is derived from the flow to the effective Hamiltonian (21).
Consequences of these modifications on the correlation functions of bare
operators are discussed.

####  IV.3.1 Flow of single fermion operators

In order to evaluate observables in the ground state of $H_{\text{eff}}$, the
operators need also to be transformed. In particular, the form of the single
fermion operator under the transformation $U(\infty)$ that block-diagonalizes
the Hamiltonian can be obtained by solving the flow equation

$\frac{dc_{k}(l)}{dl}=[\eta(l),c_{k}(l)].$ (26)

Similar to the flow of the Hamiltonian, a closed form of the solution to Eq.
(26) is not tractable and truncations should be carried out. We take here the
ansatz

$\begin{split}c_{k}(l)=c_{k}&+\frac{\lambda}{\sqrt{N}}\sum_{q}\gamma_{k,q}(l)c^{\dagger}_{q}c^{\dagger}_{-k-q}\\\
&-\frac{2\lambda}{3\sqrt{N}}\sum_{q,p}\gamma_{q,p}(l)\big{[}c^{\dagger}_{q}c^{\dagger}_{p}c^{\dagger}_{-p-q}-c_{q}c_{p}c_{-p-q}\big{]}c_{k}.\end{split}$
(27)

It will later be argued that higher-order terms in $\lambda$ do not bring any
qualitative change to the behavior of the correlation functions. The flow
equation reads

$\frac{d\gamma_{k,q}(l)}{dl}=-\frac{1}{2}B_{k,q}(l)\alpha_{k,q}(l),$ (28)

with the initial condition $\gamma_{k,q}(0)=0$. The solution is given by
$\gamma_{k,q}(\infty)=-B_{k,q}/2\alpha_{k,q}$. Since the U(1) symmetry is
recovered in the rotated basis, the standard bosonization mapping (22) can be
applied. We start by constructing a fermionic field $\tilde{c}(x)$ from the
transformed modes $c_{k}(\infty)$ in the same fashion as in Eq. (20). Given
that the mode $k$ decouples from the other modes in the last term of Eq. (27),
the Fourier transform yields a local operator that will at most lead to a
renormalization of the numerical prefactors in correlation function. Hence, we
only consider

$\begin{split}\tilde{c}(x)\sim\,c(x)+\frac{\lambda}{N}\int
dydz\,\Gamma(z,z-y)c^{\dagger}(x+y)c^{\dagger}(x+z),\\\ \end{split}$ (29)

where

$\Gamma(x,y)=\frac{1}{N}\sum_{k,q}e^{-ikx}e^{-iqy}\gamma_{k,q}(\infty).$ (30)

The most relevant operators in the bosonic representation of Eq. (29) are
extracted by carrying out operator product expansions (OPE) in the product of
vertex operators. We have

$\begin{split}c^{\dagger}(x+y)c^{\dagger}(x+z)&\sim\frac{1}{\pi\alpha}\sin(k_{F}(z-y))e^{-2i\theta(x)}.\end{split}$
(31)

By inserting this expression back into Eq. (29) and carrying out the double
integration, we obtain

$\begin{split}\tilde{c}(x)\sim\,&\frac{e^{ik_{F}x}}{\sqrt{2\pi\alpha}}e^{-i[\phi(x)-\theta(x)]}+\frac{e^{-ik_{F}x}}{\sqrt{2\pi\alpha}}e^{i[\phi(x)+\theta(x)]}\\\
&+\frac{4\lambda}{\pi\alpha}\sin(k_{F})e^{-2i\theta(x)}.\\\ \end{split}$ (32)

From the structure of the generator $\eta$, the form of higher-order terms in
the transformed operator can be guessed. We will only consider operators of
order $\lambda^{2}$ as these can contribute to the second-order expansion of
the correlations by combining with the $\text{zero}^{\text{th}}$-order part of
Eq. (27). Given an operator $O(c,c^{\dagger})$ generated at the second order
in $\lambda$ , it transforms under U(1) rotations as
$O(e^{i\alpha}c,e^{-i\alpha}c^{\dagger})=e^{in\alpha}O(c,c^{\dagger})$, where
$n$ can only take the values $1,7,-5$. Since the U(1) rotations act on the
bosonic fields as $\phi\to\phi$ and $\theta\to\theta+\alpha$, this indicates
that operators with a scaling dimension smaller than those in Eq. (32) cannot
be generated. The terms that transform as a single-fermion operator will
merely add a $\lambda^{2}$-dependence to the prefactor of the corresponding
vertex operators. Thus, the transformed fermion operator truncated to the
second order in $\lambda$ takes the form

$\begin{split}\tilde{c}(x)\sim\,&C_{1}e^{ik_{F}x}e^{-i[\phi(x)-\theta(x)]}\\\
&+C_{1}e^{-ik_{F}x}e^{i[\phi(x)+\theta(x)]}+C_{2}e^{-2i\theta(x)},\\\
\end{split}$ (33)

with $C_{1}\sim 1+O(\lambda^{2})$ and $C_{2}\sim O(\lambda)$.

####  IV.3.2 Fermionic correlation functions

Using the effective bosonic representation of the transformed fermion operator
(33), we can now compute correlation functions in the Luttinger liquid phase.
Consider the point-split product of bare-fermion operator

$F_{p}(x)=\lim_{\Delta\to 0}\prod_{n=0}^{p-1}c(x+n\Delta).$ (34)

The correlations in the ground state of the bare Hamiltonian can be evaluated
in the Luttinger liquid ground state of $H_{\text{eff}}$ using the relation

$\expectationvalue{F_{p}(x)^{\dagger}F_{p}(y)}_{\text{GS}}=\expectationvalue{\tilde{F}^{\dagger}_{p}(x)\tilde{F}_{p}(y)}_{\text{LL}},$
(35)

where $\tilde{F}_{p}(x)=U(\infty)F_{p}(x)U^{\dagger}(\infty)$ is obtained by
replacing $c$ with the transformed field $\tilde{c}$ in Eq. (34). The
correlations of $p$ fermions are deduced from the well-known result
(Giamarchi, 2003) for the correlation function of vertex operators in the
Luttinger liquid Hamiltonian (1):

$\expectationvalue{e^{i[n\phi(x)+m\theta(x)]}e^{-i[n\phi(y)+m\theta(y)]}}_{\text{LL}}\sim\frac{1}{|x-y|^{\frac{n^{2}K}{2}+\frac{m^{2}}{2K}}}.$
(36)

Accordingly, Eqs. (33) and (35) yield the one-fermion correlations

$\begin{split}\expectationvalue{F_{1}^{\dagger}(x)F_{1}(y)}_{\text{GS}}\sim&\,\,2(C_{1})^{2}\frac{\cos(k_{F}r)}{r^{\frac{1}{2K}+\frac{K}{2}}}+(C_{2})^{2}\frac{1}{r^{\frac{2}{K}}},\end{split}$
(37)

where $r=|x-y|$. Similarly, the two-fermion correlations can be obtained from
the bosonic representation of $F_{2}$. We have

$\begin{split}\tilde{F}_{2}(x)\sim\,\,&(C_{1})^{2}e^{2i\theta(x)}+C_{1}C_{2}e^{ik_{F}x}e^{-i[\phi(x)+\theta(x)]}\\\
&+C_{1}C_{2}e^{-ik_{F}x}e^{i[\phi(x)-\theta(x)]},\end{split}$ (38)

which leads to

$\begin{split}\expectationvalue{F_{2}^{\dagger}(x)F_{2}(y)}_{\text{GS}}\sim&\,\,(C_{1})^{4}\frac{1}{r^{\frac{2}{K}}}+2(C_{1}C_{2})^{2}\frac{\cos(k_{F}r)}{r^{\frac{1}{2K}+\frac{K}{2}}}.\end{split}$
(39)

FIG. 4: One fermion and two-fermion correlation functions in the ground state
of the bare Hamiltonian (5) at $\mu=1.86$ and $\lambda=3$. The dashed line are
provided as a guide to the eye. The DMRG calculations are performed on a
system of size $N=2100$. The Luttinger parameter $K$ is extracted from Friedel
oscillations as discussed in section IV.2 and the coefficients in Eqs. (37)
and (39) are obtained from a least square fit.

We note that a single-fermion operator appears in the bosonic representation
of $F_{2}$. It is an example of a term that is not initially present in the
standard bosonization mapping but is generated by the
$\mathbb{Z}_{3}$-symmetric interaction along the flow. Its consequence is a
crossover between two power-laws in the two-fermion correlations. At short
distance, the correlations decay with the standard exponent $2/K$ of the two-
fermion operator. At large distance, a single-fermion part with an exponent
$\frac{K}{2}+\frac{1}{2K}$, oscillating with a wave vector $k_{F}$, takes over
the correlations when $K<\sqrt{3}$. The crossover takes place at length scale
$l\sim 1/\lambda^{4K/(3-K^{2})}$. Numerically, it can be observed near the PT
transition line, where the Luttinger parameter tends to $K=1$ while $\lambda$
remains small enough for $l$ to remain smaller than the system size. The
numerical results for the correlations (Fig.4), fitted with Eqs. (37),(39) by
the least-square method, are in concordance with the analytical calculations.
Finally, it should be noted that a two-fermion operator is also generated in
the bosonic representation of the single fermion operator (see Eq. (33)). Its
effect on the correlation remains however small compared to the single-fermion
part.

##  V Flow equation approach to
the hard-core bosonic model

We turn now to the hard-core boson model (2). The commutation relation of the
hard-core boson operator renders the calculation of the flow on the lattice
difficult. It is more convenient to consider the Hamiltonian in the continuum
limit, where it reduces to the dual sine-Gordon model:

$H=\frac{v_{F}}{2\pi}\int
dx\,[\partial_{x}\theta(x)]^{2}+[\partial_{x}\phi(x)]^{2}+\frac{g}{\pi\alpha^{2}}\int
dx\cos(\beta\theta(x)),$ (40)

with $\beta=3/\sqrt{K}$. Eq. (40) is obtained from the bosonization mapping
(3) by absorbing the Luttinger parameter $K$ into a redefinition of the
bosonic fields, i.e., $\phi\to\phi\sqrt{K}$ and $\theta\to\theta/\sqrt{K}$.
The duality transformation $\phi\leftrightarrow\theta$ and $K\to 1/K$ recovers
the sine-Gordon Hamiltonian, which is studied with the flow equation approach
in Refs.Kehrein, 1999, 2001. We review here the main result of this work and
establish the duality correspondence with Eq. (40) to obtain a low-energy
effective Hamiltonian.

###  V.1 Flow of the sine-Gordon Hamiltonian

We introduce the vertex operators
$V_{r}(\beta,x)=\,:e^{i\beta[r\phi(x)-\theta(x)]}:$, where $r=R,L$ denotes the
left and right species. The relation between our notations and those used in
the reference can be found in Appendix B. The columns refer to the normal
ordering with respect to the ground state of the non-interacting part in Eq.
(40). After performing the duality transformation, the interaction part of the
Hamiltonian (40) reads

$H_{3}=\frac{g}{2\pi\alpha^{2}}\Big{(}\frac{2\pi\alpha}{L}\Big{)}^{\frac{\tilde{\beta}^{2}}{4}}\int
dx\big{[}V_{R}(\tilde{\beta}/2,x)V_{L}(-\tilde{\beta}/2,x)+\text{h.c.}\big{]},$
(41)

with $L$ the total length of the chain and $\tilde{\beta}(K)=\beta(1/K)$.
Combined with the non-interacting part, Eq. (41) is the starting point of
calculations of the flow equations carried out in Ref.Kehrein, 1999, 2001. The
Hamiltonian is diagonalized by a generator
$\eta(l)=\eta^{(1)}(l)+\eta^{(2)}(l)$, where

$\begin{split}\eta^{(1)}(l)&=-2iv_{F}\int dxdy\frac{\partial u(y,l)}{\partial
y}\\\ &\hskip
42.67912pt\times\big{[}V_{R}(\tilde{\beta}/2,x)V_{L}(-\tilde{\beta}/2,x-y)+\text{h.c.}\big{]},\\\
\end{split}$ (42)

and $u(x,l)$ is obtained from the Fourier transform of
$u(k,l)=\frac{g(l)}{4\pi^{2}\alpha^{2}}\Big{(}\frac{2\pi\alpha}{L}\Big{)}^{\frac{\tilde{\beta}^{2}}{4}}e^{-4v_{F}^{2}k^{2}l}$.
Here, $g(l)$ is a running coupling that flows to zero in the weak-coupling
regime, i.e., inside the Luttinger liquid phase. It is initially given by the
bare coupling constant in Eq. (40). $\eta^{(2)}(l)$ generates the flow of the
parameter $\beta$. Its expression can be found in Ref. Kehrein, 2001. At the
end of the flow, an effective Hamiltonian
$H(\infty)=H_{0}+H_{\text{d}}(\infty)$ is obtained, where $H_{0}$ denotes the
non-interacting part in the bare Hamiltonian and

$\begin{split}H_{\text{d}}(\infty)=\sum_{k>0}\omega_{k}(\infty)\Big{[}\tilde{P}_{R}(-k)\tilde{P}^{\dagger}_{R}(-k)+\tilde{P}^{\dagger}_{R}(k)\tilde{P}_{R}(k)\\\
+\tilde{P}_{L}(k)\tilde{P}^{\dagger}_{L}(k)+\tilde{P}^{\dagger}_{L}(-k)\tilde{P}_{L}(-k)\Big{]},\end{split}$
(43)

with
$\omega_{k}(\infty)=-v_{F}g^{2}\frac{\cos(\pi\tilde{\beta}^{2}/4)}{2\Gamma^{2}(\tilde{\beta}^{2}/4)}k|\alpha
k|^{(\tilde{\beta}^{2}-8)/2}$. $\tilde{P}_{R}(k)$ and $\tilde{P}_{L}(k)$ are
soliton and antisoliton creation and annihilation operator defined as the
Fourier transform of vertex operators (see Appendix B). The effective
Hamiltonian of the dual sine-Gordon Hamiltonian (40) is deduced from the dual
transformation of Eq. (43). The latter acts on the soliton and antisoliton
operators as $\tilde{P}_{R}(k)\to P^{\dagger}_{R}(-k)$ and
$\tilde{P}_{L}(k)\to P_{L}(k)$, where $P_{r}(k)$ is obtained from
$\tilde{P}_{r}(k)$ by replacing $\tilde{\beta}$ with $\beta$. Moreover, the
flow equations for $g$ and $K$ in Eq. (40) can be deduced (see Appendix A):

$\begin{split}\frac{dK(l_{\text{RG}})}{dl_{\text{RG}}}&=\frac{9}{4\Gamma(\frac{9}{4K(l_{RG})}-1)}g^{2}(l_{\text{RG}}),\\\
\frac{dg(l_{\text{RG}})}{dl_{\text{RG}}}&=\Big{(}2-\frac{9}{4K(l_{RG})}\Big{)}g(l_{\text{RG}}),\end{split}$
(44)

where $l_{\text{RG}}$ is the parameter of the RG flow. It is related to the
parameter of the flow equation approach $l$ by
$l_{\text{RG}}=\frac{1}{2}\ln(\frac{32l}{\alpha^{2}})$. The RG equations,
derived in Eq. (4), are recovered by an expansion of the Gamma function around
the critical value $K_{c}=9/8$. Finally, we note that since the soliton and
antisoliton operators transform under U(1) rotations as $P_{r}(k)\to
e^{i\alpha}P_{r}(k)$, the U(1) symmetry is restored in the effective
Hamiltonian.

###  V.2 Transformed hard-core boson operator

From the Jordan-Wigner transformation and the bosonization mapping (22), a
bosonic representation of the hard-core boson can be derived Giamarchi (2003).
It is given by

$b(x)=\frac{e^{i\theta(x)}}{\sqrt{2\pi\alpha}}\big{[}1+\cos(2\phi(x)-2k_{F}x)\big{]}.$
(45)

We calculate in this section the transformation of Eq. (45) along the flow
that diagonalizes the dual sine-Gordon model.

####  V.2.1 Flow of the hard-core boson operator

The flow of the hard-core boson operator is evaluated using the generator of
the dual sine-Gordon Hamiltonian. The latter is given by the dual of Eq. (42),
i.e.,

$\begin{split}\eta_{\text{dual}}^{(1)}(l)&=-2iv_{F}\int dxdy\frac{\partial
u(y,l)}{\partial y}\\\ &\hskip
42.67912pt\times\big{[}V_{R}(\beta/2,x)V_{L}(\beta/2,x-y)+\text{h.c.}\big{]}.\\\
\end{split}$ (46)

We take the following ansatz for the flowing hard-core boson operator,
truncated to the most relevant terms:

$\begin{split}b(x,l)&=[C_{1}(l)+C_{2}(l)\cos(2\phi(x)-2k_{F}x)]e^{i\theta(x)}\\\
&\hskip
14.22636pt+[C_{3}(l)+C_{4}(l)\cos(2\phi(x)-2k_{F}x)]e^{-2i\theta(x)},\end{split}$
(47)

with $C_{1}(0)=C_{2}(0)=1/\sqrt{2\pi\alpha}$, and $C_{3}(0)=C_{4}(0)=0$. Terms
with larger scaling dimensions can be neglected in describing the behavior of
the correlation functions. Since the bosonic fields in Eq. (47) are those of
the original Hamiltonian (3), the flow of the rescaled operator need to be
considered. We calculate here the flow of $e^{i\theta(x)}$, from which the
flow equation for $C_{2}(l)$ can be deduced. We have

$e^{i\theta(x)/\sqrt{K}}=\Big{(}\frac{2\pi\alpha}{L}\Big{)}^{\beta^{2}/36}V_{R}(-\beta/6,x)V_{L}(-\beta/6,x).$
(48)

The details of the calculation can be found in Appendix D. We summarize here
the main steps of the derivation. First, the commutator of Eq. (48) with the
hermitian conjugate part of the generator (46) leads to less relevant terms
that are truncated in the ansatz (47). The most relevant operator in the
remaining part of $[\eta_{\text{dual}}^{(1)}(l),e^{i\theta(x)/\sqrt{K}}]$ is
extracted by an OPE of the vertex operators. The parameters of the vertex
operators in Eqs. (46) and (48) combine to produce the operator
$e^{-2i\theta(x)}$ in the ansatz (47). Namely,

$e^{-2i\theta(x)/\sqrt{K}}=\Big{(}\frac{2\pi\alpha}{L}\Big{)}^{\beta^{2}/9}V_{R}(\beta/3,x)V_{L}(\beta/3,x).$
(49)

The flow equation for $C_{3}(l)$ is then given by

$\frac{dC_{3}(l)}{dl}=-C_{1}(l)\frac{4v_{F}g(l)}{\Gamma(\beta^{2}/12)^{2}}\frac{2\pi}{L}\sum_{k>0}k|\alpha
k|^{\beta^{2}/6-2}e^{-4v_{F}^{2}k^{2}l}.$ (50)

To the leading order in the bare coupling constant $g$, we can replace
$C_{1}(l)$ by its bare value. Moreover, the running coupling constant can be
replaced by its approximate solution in the weak-coupling regime Sabio and
Kehrein (2010). From the RG equations (4), we have

$g(l_{\text{RG}})\sim
ge^{(2-\beta^{2}/4)l_{\text{RG}}}=g\Big{(}\frac{32l}{\alpha^{2}}\Big{)}^{1-\beta^{2}/8}.$
(51)

Finally, the solution of Eq. (50) at the end of the flow reads

$C_{3}(\infty)=-\frac{4v_{F}g}{\sqrt{2\pi\alpha}}D_{\beta}\int_{0}^{\infty}dk\,k^{\beta^{2}/6-1}f(2v_{F}k),$
(52)

with $D_{\beta}=\frac{(32)^{1-\beta^{2}/8}}{\Gamma(\beta^{2}/12)}$ and
$f(k)=k^{\beta^{2}/4-4}\Gamma(2-\beta^{2}/8,k^{2})$.

####  V.2.2 Bosonic correlation functions

We consider the correlation functions of the point-split product of $p$ hard-
core bosonic operators

$B_{p}(x)=\lim_{\Delta\to 0}\prod_{n=0}^{p-1}b(x+n\Delta).$ (53)

As in section IV.3, the correlation functions in the ground state of the bare
Hamiltonian (2) can be evaluated from the bosonic representation of the
transformed operator $\tilde{B}_{p}(x)=U(\infty)B_{p}(x)U^{\dagger}(\infty)$.
It is derived from $B_{p}(x)$ by substituting the hard-core boson operators
with their transformed counterpart $b(x,\infty)$. The one-boson and two-boson
operators are given by

$\begin{split}\tilde{B}_{1}(x)&\sim
C_{1}e^{i\theta(x)}+C_{3}e^{-2i\theta(x)},\\\
\tilde{B}_{2}(x)&\sim(C_{1})^{2}e^{2i\theta(x)}+2C_{1}C_{3}e^{-i\theta(x)},\end{split}$
(54)

where the coefficients are taken at the end of the flow, i.e. at $l=\infty$.
The oscillating terms in Eq. (47) are neglected since they have larger scaling
dimensions. Using the result (36) for the correlation functions in the
Luttinger liquid Hamiltonian, we obtain

$\displaystyle\expectationvalue{B^{\dagger}_{1}(x)B_{1}(y)}_{\text{GS}}$
$\displaystyle\sim(C_{1})^{2}\frac{1}{r^{\frac{1}{2K}}}+(C_{3})^{2}\frac{1}{r^{\frac{2}{K}}},$
(55) $\displaystyle\expectationvalue{B^{\dagger}_{2}(x)B_{2}(y)}_{\text{GS}}$
$\displaystyle\sim(C_{1})^{4}\frac{1}{r^{\frac{2}{K}}}+4(C_{1}C_{3})^{2}\frac{1}{r^{\frac{1}{2K}}},$
(56)

with $r=|x-y|$. For $p=3$, a combination of $e^{i\theta(x)}$ in the product
(53) can lead to a vanishing exponent. In this case, a higher-order expansion
in the point-splitting parameter $\Delta$ need to be carried out. Moreover,
the Klein factors become necessary as their anti-commutation prevents
artificial cancellations. They are reintroduced in the ansatz by replacing
$\cos(2\phi)$ with $F_{R}e^{2i\phi}+F_{L}e^{-2i\phi}$. The details of the
derivation are presented in Appendix D. Finally, the correlation function
takes the form

$\begin{split}\expectationvalue{B^{\dagger}_{3}(x)B_{3}(y)}_{\text{GS}}\sim&D_{0}\frac{1}{r^{\frac{9}{2K}}}+D_{1}\frac{1}{r^{2}}+D_{2}\frac{\cos(2k_{F}r)}{r^{2K}}\\\
&+D_{3}\frac{\cos(2k_{F}r)}{r^{2K+2}}+D_{4}\frac{1}{r^{4}},\end{split}$ (57)

where $D_{1}$, $D_{2}$, $D_{3}$ and $D_{4}$ depend on the coefficient of the
transformed operator $b(x,\infty)$.

FIG. 5: (a) One-boson and two-boson correlation functions at $\mu=1$,
$\lambda=0.06$ and (b) Connected three-boson correlation functions at
$\mu=1.9$, $\lambda=0.2$, in the ground state of the bare Hamiltonian. The
dashed line are provided as a guide to the eye. The DMRG calculations are
performed on a system of size $N=1200$. The Luttinger parameter $K$ and the
Fermi wave vector $k_{F}$ are extracted from Friedel oscillations as discussed
in section IV.2 and the coefficients in the correlations are obtained from a
least square fit.

As shown in Fig.5, the correlations decay algebraically at short distance with
the exponent $p^{2}/2K$, associated with the operator $e^{ip\theta(x)}$ in the
standard bosonic representation of the product of $p$ hard-core bosons. At
large distances, the two-boson and three-boson correlations undergo a
crossover to power-laws with smaller exponents, induced by the terms generated
along the flow. In particular, the two-boson correlations exhibit a crossover
to the one-boson correlations. Similarly, the three-boson correlations
acquire, among other terms, an oscillating part with a wave vector $2k_{F}$
that decays with an exponent $2K$. We also note that the presence of gradients
of bosonic fields in $\tilde{B}_{3}(x)$ leads to a non-vanishing expectation
of the three-boson operator inside the Luttinger liquid phase. Since these
terms are generated to the first order of the coupling $g\propto\lambda$, the
expectation decays linearly with the interaction strength and vanish at the
non-interacting point $\lambda=0$. This result is confirmed by the numerical
simulations (Fig.6).

FIG. 6: DMRG calculations of the expectation value of the three-boson operator
$B_{3}$, in the ground state of the bare Hamiltonian (2), as a function of the
three-site interaction coupling $\lambda$ and at $\mu=1$.

##  VI Conclusion

Using the flow equation approach, we have provided an analytical derivation of
modified bosonic representations of hard-core boson and spinless fermion
operators that take account of the $\mathbb{Z}_{3}$ symmetry of the
Hamiltonian. As a result, the correlation functions of $p$-particles are
dominated in the long distances by the power-law decay of operators that are
not initially present in the bosonic representation of the operators. These
calculations can be straightforwardly generalized to
$\mathbb{Z}_{n}$-symmetric models. Since the generator of the flow exhibits
the symmetry of the bare Hamiltonian by construction, terms that transform
covariantly under $\mathbb{Z}_{n}$-rotations will be generated in the bosonic
representation of single-particle operators. For instance, in the hard-core
boson model with creation/annihilation operators on four adjacent sites, which
is of interest in the open problem of commensurate melting of density-waves,
the bosonic representation of a product of $p$ bosons is expected to contain
terms that are associated with $p-4m$ particles, where $m\in\mathbb{Z}$. Some
of these terms have smaller scaling dimension than the bare operator and will
dominate the correlations at long distances.

More generally, this paper demonstrates that even if an emergent U(1) symmetry
is present, the long distance correlation functions can have a behaviour that
differs from the naive expectation and that the flow equation approach is a
very useful tool to identify the correct bosonic representation of lattice
operators. It would be interesting to investigate the extension of this method
to models with an emergent non-abelian symmetry.

## Acknowledgments

The work has been supported by the Swiss National Science Foundation (FM, ZJ)
grant 212082 and by the Delft Technology Fellowship (NC). Numerical
simulations have been performed on the Dutch national e-infrastructure with
the support of the SURF Cooperative and the support of the Scientific IT and
the Application Support Center of EPFL. We thank Melvyn Nabavi for proof
reading the manuscript.

##  APPENDIX A DERIVATION OF THE FLOW EQUATIONS

###  A.1 Flow equations for the fermionic Hamiltonian

An efficient way of computing the flow equations is to use Wick’s theorem for
operator products to collect the contributions to the different generated
terms in $[\eta,H_{3}]$. For convenience, we write the interaction term as

$H_{3}=\frac{1}{3!\sqrt{N}}\sum_{\vec{k}}\tilde{B}_{k_{1},k_{2},k_{3}}\Big{(}c^{\dagger}_{k_{1}}c^{\dagger}_{k_{2}}c^{\dagger}_{k_{3}}+c_{k_{1}}c_{k_{2}}c_{k_{3}}\Big{)},$
(58)

where $\vec{k}=(k_{1},k_{2},k_{3})$ and

$\tilde{B}_{k_{1},k_{2},k_{3}}=\delta_{k_{1}+k_{2}+k_{3},0}\sum_{\sigma\in
S_{3}}\epsilon(\sigma)e^{-ik_{\sigma(2)}}e^{-2ik_{\sigma(3)}},$ (59)

is purely imaginary. $S_{3}$ is the symmetric group of order 3, and
$\epsilon(\sigma)$ is the sign of the permutation. The contribution from
$[\eta,H_{3}]$ to the flow of $U_{k,q,p}$ and $\xi_{k}$ stems from the
following term:

$\begin{split}[\eta,H_{3}]\to&\frac{1}{(3!)^{2}N}\sum_{\vec{k}}\tilde{B}_{k_{1},k_{2},k_{3}}\tilde{B}_{q_{1},q_{2},q_{3}}\tilde{\alpha}_{k_{1},k_{2},k_{3}}\Big{(}[c^{\dagger}_{k_{1}}c^{\dagger}_{k_{2}}c^{\dagger}_{k_{3}},c_{q_{1}}c_{q_{2}}c_{q_{3}}]+(\vec{k}\leftrightarrow\vec{q})\Big{)},\end{split}$
(60)

where $\tilde{\alpha}_{k_{1},k_{2},k_{3}}=\sum_{i=1}^{3}\xi_{k_{i}}$. To
proceed with the calculation, it is useful to apply Wick’s theorem in order to
normal order the generated terms with respect to the Fermi sea in the ground
state of the free Hamiltonian $H_{0}$. The contractions of two-fermion
operators are given by

$\wick{\c{1}c^{\dagger}_{k}\c{1}c_{q}}=\Theta(k_{F}-|k|)\delta_{k,q},\,\,\wick{\c{1}c_{k}\c{1}c^{\dagger}_{q}}=[1-\Theta(k_{F}-|k|)]\delta_{k,q}.$
(61)

To compute the flow equation of the two-body interactions, we need to collect
all the normal-ordered terms that result from single contractions in Eq.(60).
These are given by

$\begin{split}:c^{\dagger}_{k_{1}}c^{\dagger}_{k_{2}}c^{\dagger}_{k_{3}}::c_{q_{1}}c_{q_{2}}c_{q_{3}}:\,\,&\to\Theta(k_{F}-|k_{1}|)\delta_{k_{1},q_{1}}:c^{\dagger}_{k_{2}}c^{\dagger}_{k_{3}}c_{q_{2}}c_{q_{3}}:+\text{other
single contractions}\\\
&=\Big{(}\frac{1}{2!}\Big{)}^{2}\sum_{\sigma,\sigma^{\prime}\in
S_{3}}\epsilon(\sigma)\epsilon(\sigma^{\prime})\Theta(k_{F}-|k_{\sigma(1)}|)\delta_{k_{\sigma(1)},q_{\sigma^{\prime}(1)}}:c^{\dagger}_{k_{\sigma(2)}}c^{\dagger}_{k_{\sigma(3)}}c_{q_{\sigma^{\prime}(2)}}c_{q_{\sigma^{\prime}(3)}}:,\\\
\end{split}$ (62)

where the columns denote the normal-ordering with respect to the Fermi sea and
the factor $(1/2!)^{2}$ avoids over-counting contractions. Similarly,

$:c_{q_{1}}c_{q_{2}}c_{q_{3}}::c^{\dagger}_{k_{1}}c^{\dagger}_{k_{2}}c^{\dagger}_{k_{3}}:\,\,\to\Big{(}\frac{1}{2!}\Big{)}^{2}\sum_{\sigma,\sigma^{\prime}\in
S_{3}}\epsilon(\sigma)\epsilon(\sigma^{\prime})[1-\Theta(k_{F}-|k_{\sigma(1)}|)]\delta_{k_{\sigma(1)},q_{\sigma^{\prime}(1)}}:c^{\dagger}_{k_{\sigma(2)}}c^{\dagger}_{k_{\sigma(3)}}c_{q_{\sigma^{\prime}(2)}}c_{q_{\sigma^{\prime}(3)}}:.$
(63)

Therefore, the contribution to the two-body interaction term reads

$\begin{split}[\eta,\lambda
H_{3}]\to&\Big{(}\frac{1}{2!}\Big{)}^{2}\frac{\lambda^{2}}{(3!)^{2}N}\sum_{\vec{k},\vec{q}}\tilde{B}_{k_{1},k_{2},k_{3}}\tilde{B}_{q1,q_{2},q_{3}}(\tilde{\alpha}_{k_{1},k_{2},k_{3}}+\tilde{\alpha}_{q_{1},q_{2},q_{3}})\\\
&\hskip 142.26378pt\times\sum_{\sigma,\sigma^{\prime}\in
S_{3}}\epsilon(\sigma)\epsilon(\sigma^{\prime})[2\Theta(k_{F}-|k_{\sigma(1)}|)-1]\delta_{k_{\sigma(1)},q_{\sigma^{\prime}(1)}}:c^{\dagger}_{k_{\sigma(2)}}c^{\dagger}_{k_{\sigma(3)}}c_{q_{\sigma^{\prime}(2)}}c_{q_{\sigma^{\prime}(3)}}:\\\
=&\Big{(}\frac{1}{2!}\Big{)}^{2}\frac{1}{(3!)^{2}N}\sum_{\vec{k},\vec{q}}\tilde{B}_{k_{\sigma^{-1}(1)},k_{\sigma^{-1}(2)},k_{\sigma^{-1}(3)}}\tilde{B}_{q_{\sigma^{\prime-1}(1)},q_{\sigma^{\prime-1}(2)},q_{\sigma^{\prime-1}(3)}}(\tilde{\alpha}_{k_{\sigma^{-1}(1)},k_{\sigma^{-1}(2)},k_{\sigma^{-1}(3)}}+\tilde{\alpha}_{q_{\sigma^{\prime-1}(1)},q_{\sigma^{\prime-1}(2)},q_{\sigma^{\prime-1}(3)}})\\\
&\hskip 142.26378pt\times\sum_{\sigma,\sigma^{\prime}\in
S_{3}}\epsilon(\sigma)\epsilon(\sigma^{\prime})[2\Theta(k_{F}-|k_{1}|)-1]\delta_{k_{1},q_{1}}:c^{\dagger}_{k_{2}}c^{\dagger}_{k_{3}}c_{q_{2}}c_{q_{3}}:\\\
=&\Big{(}\frac{3!}{2!}\Big{)}^{2}\frac{1}{(3!)^{2}N}\sum_{\vec{k},\vec{q}}\tilde{B}_{k_{1},k_{2},k_{3}}\tilde{B}_{k_{1},q_{2},q_{3}}(\tilde{\alpha}_{k_{1},k_{2},k_{3}}+\tilde{\alpha}_{k_{1},q_{2},q_{3}})[2\Theta(k_{F}-|k_{1}|)-1]:c^{\dagger}_{k_{2}}c^{\dagger}_{k_{3}}c_{q_{2}}c_{q_{3}}:.\\\
\end{split}$ (64)

In the second line, the change of indices $k_{\sigma^{\prime}(i)}\to k_{i}$
for $i=1,2,3$ is made in order to shift the permutations dependence to the
indices of the interaction matrix. Finally, the last line is obtained from the
anti-symmetry of the interaction matrix under permutation, i.e.,
$B_{k_{\sigma(1)},k_{\sigma(2)},k_{\sigma(3)}}=\epsilon(\sigma)B_{k_{1},k_{2},k_{3}}$.
A similar result can be obtained for terms generated from double contractions,
which contribute to the dispersion $\xi_{k}$. After writing Eq. (64) in terms
of $B_{k,q}=\tilde{B}_{k,q,-k-q}$ and
$\alpha_{k,q}=\tilde{\alpha}_{k,q,-k-q}$, the contribution to the two-body
term reads

$[\eta,\lambda
H_{3}]\to\frac{\lambda^{2}}{4N}\sum_{k,q,p}B_{k+p,q-p}B_{q,k}(\alpha_{k+p,q-p}+\alpha_{k,q})[2\Theta(k_{F}-|k+q|)-1]:c^{\dagger}_{k+p}c^{\dagger}_{q-p}c_{q}c_{k}:.\\\
$ (65)

Thus, the flow equation of the two-body interaction is given by

$\frac{dU_{k,q,p}(l)}{dl}=-\frac{1}{4}B_{k+p,q-p}(l)B_{k,q}(l)(\alpha_{k+p,q-p}(l)+\alpha_{k,q}(l))[2\Theta(k_{F}-|k+q|)-1].$
(66)

###  A.2 Flow equations for the hard-core bosonic Hamiltonian

The flow equations of the sine-Gordon model are derived in Ref. Kehrein, 2001.
We have

$\begin{split}\frac{d\tilde{\beta^{2}}(l_{\text{RG}})}{dl_{\text{RG}}}&=-\frac{\tilde{\beta}^{4}(l_{\text{RG}})}{4\Gamma(\frac{\tilde{\beta}^{2}(l_{\text{RG}})}{4}-1)}g^{2}(l_{\text{RG}}),\\\
\frac{dg(l_{\text{RG}})}{dl_{\text{RG}}}&=\Big{(}2-\frac{\tilde{\beta}^{2}(l_{\text{RG}})}{4}\Big{)}g(l_{\text{RG}}).\end{split}$
(67)

Eqs. (44) are deduced from the relation $\tilde{\beta}=3\sqrt{K}$ and the
duality transformation $K\to 1/K$.

##  APPENDIX B BOSONIZATION DICTIONARY

###  B.1 Bosonic fields

The bosonic field $\phi$, $\theta$ are constructed from the density modes
$\rho^{\dagger}_{r}(p)$, where $r=R,L$ denotes their species. We have

$\begin{split}\phi(x)=-\frac{i\pi}{L}\sum_{p\neq
0}\frac{e^{-\alpha|p|/2-ipx}}{p}\big{[}\rho_{R}^{\dagger}(p)+\rho_{L}^{\dagger}(p)\big{]},\\\
\theta(x)=\frac{i\pi}{L}\sum_{p\neq
0}\frac{e^{-\alpha|p|/2-ipx}}{p}\big{[}\rho^{\dagger}_{R}(p)-\rho^{\dagger}_{L}(p)\big{]},\end{split}$
(68)

where

$[\rho_{r}^{\dagger}(p),\rho_{r^{\prime}}^{\dagger}(-q)]=-\delta_{r,r^{\prime}}\delta_{p,q}\frac{rpL}{2\pi}.$
(69)

The relation between the density modes and the normal modes of the sine-Gordon
model is given by

$\begin{split}\rho^{\dagger}_{R}(p)=\sqrt{|p|}\sigma_{1}(p),\\\
\rho^{\dagger}_{L}(p)=\sqrt{|p|}\sigma_{2}(p).\end{split}$ (70)

The relation between the fields in Eq. (68) and the fields $\tilde{\phi}$ and
$\tilde{\theta}$ introduced in the reference is given by

$\begin{split}\phi(x)=\sqrt{\pi}\tilde{\phi}(x),\\\
\theta(x)=-\sqrt{\pi}\tilde{\theta}(x).\end{split}$ (71)

###  B.2 Vertex operators

We define the vertex operators as

$V_{r}(\beta,x)=\,:e^{i\beta[r\phi(x)-\theta(x)]}:.$ (72)

Note that this definition coincides with the vertex operators
$\tilde{V}_{r}(\beta,x)=\,:e^{i\sqrt{\pi}\beta[r\tilde{\phi}(x)-\tilde{\theta}(x)]}:$
introduced in the reference. In terms of the density modes, Eq. (72) reads

$V_{r}(\beta,x)=\,:\exp(r\beta\frac{2\pi}{L}\sum_{p\neq
0}\frac{e^{-\alpha|p|/2-ipx}}{p}\rho^{\dagger}_{r}(p)):.$ (73)

The normal ordering with respect to the vacuum defined by

$\begin{split}\rho^{\dagger}_{R}(p<0)\ket{0}=0,\\\
\rho^{\dagger}_{L}(p>0)\ket{0}=0,\end{split}$ (74)

yields

$V_{r}(\beta,x)=\Big{(}\frac{L}{2\pi\alpha}\Big{)}^{\beta^{2}/2}e^{i\beta[r\phi(x)-\theta(x)]}.$
(75)

The bosonization mapping is then given by

$\psi_{r}(x)=\frac{e^{irk_{F}x}}{\sqrt{L}}V_{r}(-1,x).$ (76)

####  B.2.1 Operator product expansion

The operator product expansion of a product of vertex operators is given by

$\begin{split}V_{R}(\beta,x)V_{R}(-\gamma,y)\sim\Big{(}\frac{L/2\pi}{i(y-x)+\alpha}\Big{)}^{\beta\gamma}V_{R}(\beta-\gamma,x),\\\
V_{L}(\beta,x)V_{L}(-\gamma,y)\sim\Big{(}\frac{L/2\pi}{i(x-y)+\alpha}\Big{)}^{\beta\gamma}V_{L}(\beta-\gamma,x).\end{split}$
(77)

####  B.2.2 Exchange relations

The order of vertex operators can be exchanged using the following relations:

$\begin{split}V_{R}(-\gamma,y)V_{R}(\beta,x)&\sim
V_{R}(\beta,x)V_{R}(-\gamma,y)\times\frac{[i(y-x)+\alpha]^{\beta\gamma}}{[i(x-y)+\alpha]^{\beta\gamma}},\\\
V_{L}(-\gamma,y)V_{L}(\beta,x)&\sim
V_{L}(\beta,x)V_{L}(-\gamma,y)\times\frac{[i(x-y)+\alpha]^{\beta\gamma}}{[i(y-x)+\alpha]^{\beta\gamma}},\end{split}$
(78)

and $[V_{R}(\gamma,x),V_{L}(\delta,y)]=0$ for all $\gamma,\delta$.

####  B.2.3 Solition and antisoliton operators

The soliton and antisoliton operators in the effective Hamiltonian (43) are
defined as the Fourier transform of the vertex operators:

$\tilde{P}_{r}(k)=\Bigg{[}\frac{\Gamma(\tilde{\beta}^{2}/4)}{2\pi
L}\Big{(}\frac{L|k|}{2\pi}\Big{)}^{1-\frac{\tilde{\beta}^{2}}{4}}\Bigg{]}^{1/2}\int
dx\,e^{-ikx}V_{r}(-\tilde{\beta}/2,x),\,\,\,\text{for $r=R,L$}.$ (79)

##  APPENDIX C DERIVATION OF THE LUTTINGER LIQUID HAMILTONIAN

We derive here the Luttinger parameter and the velocity inside the Luttinger
liquid phase of the model (5). We consider the particle-hole excitations close
to the Fermi point $\pm k_{F}$ in the two-body interactions of the effective
Hamiltonian $H_{\text{eff}}=H_{0}(\infty)+\lambda^{2}H_{U}(\infty)$. They
consist in two $g_{2}$ processes and two $g_{1}$ processes. Since the fermions
are spinless, these two processes are indistinguishable and the two-body
interaction term reduces to

$H_{U}(\infty)=\frac{4g_{2}}{N}\sum_{p}\rho_{R}(p)\rho_{L}(-p),$ (80)

where

$g_{2}=U_{k_{F},-k_{F},0}=-\frac{16\sin^{2}(k_{F})\sin^{4}(k_{F}/2)}{2+4\cos(k_{F})+3\mu},$
(81)

and

$\rho_{r}(p)=\sum_{k}c^{\dagger}_{r,k+p}c_{r,k}$ (82)

are the Fourier components of the density operators at the right ($r=R$) and
left branches ($r=L$). Using the expressions of the density operators in terms
of the fields $\phi$ and $\theta$:

$\begin{split}\rho_{R}(x)&=-\frac{1}{2\pi}\big{[}\partial_{x}\phi(x)-\partial_{x}\theta(x)\big{]},\\\
\rho_{L}(x)&=-\frac{1}{2\pi}\big{[}\partial_{x}\phi(x)+\partial_{x}\theta(x)\big{]},\end{split}$
(83)

the two-body interaction becomes

$H_{U}(\infty)=\frac{4g_{2}}{(2\pi)^{2}}\int
dx\,[\partial_{x}\phi(x)]^{2}-[\partial_{x}\theta(x)]^{2}.$ (84)

Eq. (84) is then combined with the non-interacting Luttinger liquid
Hamiltonian such that the Hamiltonian remains quadratic. We obtain

$H_{U}(\infty)=\frac{u}{2\pi}\int
dx\,K[\partial_{x}\phi(x)]^{2}+\frac{1}{K}[\partial_{x}\theta(x)]^{2},$ (85)

where

$\begin{split}K&=\Big{[}\frac{1-2g_{2}\lambda^{2}/\pi\tilde{v}_{F}}{1+2g_{2}\lambda^{2}/\pi\tilde{v}_{F}}\Big{]}^{1/2},\\\
u&=\tilde{v}_{F}\Big{[}1-\Big{(}\frac{2g_{2}\lambda^{2}}{\pi\tilde{v}_{F}}\Big{)}^{2}\Big{]}^{1/2},\end{split}$
(86)

and $\tilde{v}_{F}=\partial\xi_{k}/\partial k|_{k=k_{F}}$ is the renormalized
velocity. Eq. (23) is obtained from an expansion of Eq. (86) to the second
order of $\lambda$.

##  APPENDIX D SOME DETAILS OF THE CALCULATIONS

###  D.1 Derivation of the flow of the hard-core boson operator

We give here the detailed calculation of the flow equation for $C_{3}$ in Eq.
(47). We have

$[\eta_{\text{dual}}^{(1)}(l),e^{i\theta(x)/\sqrt{K}}]=-2iv_{F}\Big{(}\frac{2\pi\alpha}{L}\Big{)}^{\beta^{2}/36}\int
dydz\frac{\partial u(z,l)}{\partial
y}\big{[}V_{R}(\beta/2,y)V_{L}(\beta/2,y-z)+\text{h.c.},V_{R}(-\beta/6,x)V_{L}(-\beta/6,x)\big{]}.$
(87)

Since the terms obtained from the hermitian conjugation in Eq. (87) are less
relevant terms, we only consider the commutator

$\begin{split}\big{[}V_{R}(\beta/2,y)V_{L}(\beta/2,y-z),V_{R}(-\beta/6,x)V_{L}(-\beta/6,x)\big{]}&=V_{R}(\beta/2,y)V_{R}(-\beta/6,x)V_{L}(\beta/2,y-z)V_{L}(-\beta/6,x)\\\
&\times\Big{\\{}1-\Big{[}\frac{i(x-y)+\alpha}{i(y-x)+\alpha}\Big{]}^{\beta^{2}/12}\Big{[}\frac{i(y-z-x)+\alpha}{i(x-y+z)+\alpha}\Big{]}^{\beta^{2}/12}\Big{\\}}\\\
&\sim\Big{(}\frac{L}{2\pi}\Big{)}^{\beta^{2}/6}V_{R}(\beta/3,x)V_{L}(\beta/3,x)\\\
&\times\Big{\\{}[i(x-y)+\alpha]^{-\beta^{2}/12}[i(y-z-x)+\alpha]^{-\beta^{2}/12}-[i(y-x)+\alpha]^{-\beta^{2}/12}[i(x-y+z)+\alpha]^{-\beta^{2}/12}\Big{\\}},\end{split}$
(88)

where an OPE of the vertex operator is carried out in last line. By inserting
this expression in Eq. (87), we obtain, after a few steps of calculations,

$\begin{split}&[\eta^{(1)}_{\text{dual}}(l),e^{i\theta(x)/\sqrt{K}}]\to-2iv_{F}\alpha^{\beta^{2}/6}\frac{g(l)}{4\pi^{2}\alpha^{2}}\frac{2\pi}{L}\sum_{k}(-ik)e^{-4v_{F}^{2}k^{2}l}e^{-2i\theta(x)/\sqrt{K}}\\\
&\hskip 28.45274pt\times\int
dydz\,e^{-ikz}\Big{\\{}[i(x-y)+\alpha]^{-\beta^{2}/12}[i(y-z-x)+\alpha]^{-\beta^{2}/12}-[i(y-x)+\alpha]^{-\beta^{2}/12}[i(x-y+z)+\alpha]^{-\beta^{2}/12}\Big{\\}}\\\
&\hskip
82.51282pt=-\frac{v_{F}g(l)}{\pi^{2}}\frac{2\pi}{L}\sum_{k}ke^{-4v_{F}^{2}k^{2}l}f_{\beta}(k)e^{-2i\theta(x)/\sqrt{K}},\end{split}$
(89)

with

$\begin{split}f_{\beta}(k)&=\int dydz\,e^{-ik\alpha
z}[1-iy]^{-\beta^{2}/12}[1+i(y-z)]^{-\beta^{2}/12}\\\ &=|\alpha
k|^{\beta^{2}/6-2}\frac{4\pi^{2}}{\Gamma(\beta^{2}/12)^{2}}\Theta(k).\end{split}$
(90)

Therefore, the flow equation for $C_{3}(l)$ is given by

$\frac{dC_{3}(l)}{dl}=-C_{1}(l)\frac{4v_{F}g(l)}{\Gamma(\beta^{2}/12)^{2}}\frac{2\pi}{L}\sum_{k>0}k|\alpha
k|^{\beta^{2}/6-2}e^{-4v_{F}^{2}k^{2}l}.$ (91)

###  D.2 Derivation of $\tilde{B}_{3}(x)$

We give here the detailed calculation of the three-boson correlations in Eq.
(57). We start by reordering the vertex operators in the product
$\tilde{B}_{3}(x)$. This leads to

$\begin{split}\tilde{B_{3}}(x)&=\lim_{\Delta\to
0}\sum_{n,m,l=0,1}\Big{[}A_{n}+B_{n}\cos(2\tilde{\phi}(x))\Big{]}\Big{[}A_{m}+B_{m}(-1)^{p_{n}}\cos(2\tilde{\phi}(x+\Delta))\Big{]}\\\
&\hskip
113.81102pt\times\Big{[}A_{l}+B_{l}(-1)^{p_{n}+p_{m}}\cos(2\tilde{\phi}(x+2\Delta))\Big{]}e^{i[p_{n}\theta(x)+p_{m}\theta(x+\Delta)+p_{l}\theta(x+2\Delta)]},\\\
\end{split}$ (92)

where $\tilde{\phi}(x)=\phi(x)-k_{F}x$, $p_{n}=1-3n$, $A_{0}=C_{0}$,
$B_{0}=C_{1}$, $A_{1}=C_{3}$ and $B_{1}=C_{4}$. To avoid artificial
cancellations, it is necessary to reintroduce the Klein factors in the
definition of the ansatz. This amounts to the following substitution:

$\begin{split}B_{n}\cos(2\phi(x))&\to\frac{1}{2}B_{n}\big{[}F_{R}e^{2i\phi(x)}+F_{L}e^{-2i\phi(x)}\big{]},\\\
A_{n}&\to\frac{1}{2}A_{n}(F_{R}+F_{L}).\end{split}$ (93)

By carrying out an expansion in the splitting parameter $\Delta$, we obtain
for a term at position $x+(n-1)\Delta$ in the product of Eq. (92):

$\begin{split}&\frac{1}{2}\big{[}F_{R}e^{2i\phi(x+(n-1)\Delta)}+F_{L}e^{-2i\phi(x+(n-1)\Delta)}\big{]}\\\
&\hskip
56.9055pt\sim\frac{1}{2}F_{R}\Big{\\{}1+2i[(n-1)\Delta\partial_{x}\phi+\frac{(n-1)^{2}\Delta^{2}}{2}(\partial_{x}\phi)^{2}]-2(n-1)^{2}\Delta^{2}\partial_{x}^{2}\phi\Big{\\}}e^{2i\phi}\\\
&\hskip
71.13188pt+\frac{1}{2}F_{L}\Big{\\{}1-2i[(n-1)\Delta\partial_{x}\phi+\frac{(n-1)^{2}\Delta^{2}}{2}(\partial_{x}\phi)^{2}]-2(n-1)^{2}\Delta^{2}\partial_{x}^{2}\phi\Big{\\}}e^{-2i\phi},\end{split}$
(94)

where the position of the fields is at $x$ and is omitted for brevity.
Similarly, we have

$\begin{split}e^{i[p_{n}\theta(x)+p_{m}\theta(x+\Delta)+p_{l}\theta(x+2\Delta)]}\sim
e^{i[p_{n}+p_{m}+p_{l}]\theta}e^{i[(p_{m}+2p_{l})\Delta\partial_{x}\theta+(\frac{1}{2}p_{m}+2p_{l})\Delta^{2}\partial_{x}^{2}\theta]}\\\
\sim
e^{i[p_{n}+p_{m}+p_{l}]\theta}\big{[}1+i[(p_{m}+2p_{l})\Delta\partial_{x}\theta+(\frac{1}{2}p_{m}+2p_{l})\Delta^{2}\partial_{x}^{2}\theta]-\frac{1}{2}(p_{m}+2p_{l})^{2}\Delta^{2}(\partial_{x}\theta)^{2}\Big{]}.\end{split}$
(95)

After collecting all the terms that have a scaling dimension smaller than the
bare operator $e^{3i\theta(x)}$, we obtain

$\begin{split}\tilde{B}_{3}(x)\sim&(C_{0})^{3}e^{3i\theta(x)}+[8(C_{1})^{2}C_{4}-(C_{2})^{2}C_{4}]\cos(2\tilde{\phi})-12\Delta
C_{1}C_{2}C_{4}\partial_{x}\theta\\\
&+\Delta[32C_{1}C_{2}C_{3}-8(C_{1})^{2}C_{4}+(C_{2})^{2}C_{4}]\partial_{x}\phi\sin(2\tilde{\phi})-72\Delta^{2}(C_{1})^{2}C_{3}(\partial_{x}\theta)^{2}\\\
&+i\Delta^{2}[156(C_{1})^{2}C_{3}-2(C_{2})^{2}C_{3}+12C_{1}C_{2}C_{4}]\partial^{2}_{x}\theta+16\Delta^{2}(C_{2})^{2}C_{3}(\partial_{x}\phi)^{2}.\end{split}$
(96)

It should be noted that for simplicity of the calculation, the normal-ordering
of the operators in Eq. (92) is not taken before carrying out the Taylor
expansion. The latter modifies the prefactors of the generated terms in Eq.
(96).

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