# Bath induced phase transition in a Luttinger liquid

Saptarshi Majumdar  Université Paris Saclay, CNRS,LPTMS, 91405, Orsay, France
Laura Foini  IPhT, CNRS, CEA, Université Paris Saclay, 91191 Gif-sur-Yvette,
France  Thierry Giamarchi  Department of Quantum Matter Physics, University of
Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland  Alberto Rosso
Université Paris Saclay, CNRS,LPTMS, 91405, Orsay, France

###### Abstract

We study an XXZ spin chain, where each spin is coupled to an independent ohmic
bath of harmonic oscillators at zero temperature. Using bosonization and
numerical techniques, we show the existence of two phases separated by an
Kosterlitz-Thouless (KT) transition. At low coupling with the bath, the chain
remains in a Luttinger liquid phase with a reduced but finite spin stiffness,
while above a critical coupling the system is in a dissipative phase
characterized by a vanishing spin stiffness. We argue that the transport
properties are also inhibited: the Luttinger liquid is a perfect conductor
while the dissipative phase displays finite resistivity. Our results show that
the effect of the bath can be interpreted as annealed disorder inducing
signatures of localization.

In quantum systems, transport properties are strongly affected by the presence
of quenched disorder. The most spectacular effect is the localisation of part
of the spectrum of the hamiltonian, which is at the origin of a finite
temperature metal-insulator transition, first predicted for free fermions by
P.W.Anderson Anderson (1958). Recently, it has been argued that finite
temperature localization can also occur in presence of interactions via the
so-called many body localisation (MBL)Basko  _et al._ (2006); Ros  _et al._
(2015); Altshuler  _et al._ (1997); Nandkishore and Huse (2015); Alet and
Laflorencie (2018); Abanin  _et al._ (2019). Instead, localization is detroyed
by contact with an external bath via Variable Range Hopping (VRH)Mott (1969);
Nattermann  _et al._ (2003); Efros and Shklovskii (1975). Indeed, transport
becomes possible by the hopping of a localised electron over a distance $r$
due to the emission or absorption of phonons. Note that this process is
feasible only for acoustic phonons which are delocalized at both donor and
acceptor sites. However for systems with single degree of freedom, such as a
single spin or a single particle, baths can induce localisation. In
particular, it has been shown that subohmic baths with harmonic oscillators at
zero temperature can freeze the quantum dynamics of such systems Bray and
Moore (1982); Chakravarty (1982); Schmid (1983); Leggett  _et al._ (1987).

Figure 1: Schematic representation of the microscopic system: An one-
dimensional XXZ spin chain(blue color) with each of the spin coupled to its
individual dissipative bath(red colour). The baths are described by a
collection of simple harmonic oscillators kept at zero temperature. The
parameter $\alpha$ is a measure of the coupling strength between the bath and
the associated spin.

In our paper, we extend these works and show the existence of a genuine bath-
induced phase transition in many-body systems. To avoid processes like VRH, we
work with a bath that produces optical phonons. We focus on one dimensional
system to use an approach that allows us to probe very large systems and
account for the effect of the bath in an exact way. In practice, we choose the
system to be an XXZ spin-chain with the hamiltonian
$H_{S}=\sum_{j=1}^{N}J_{z}S_{j}^{z}S_{j+1}^{z}+J_{xy}\left(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y}\right)$
and $J_{z}/J_{xy}\in(-1,1)$. This model displays a gapless low energy spectrum
and it is in a perfectly conducting phase known as Luttinger Liquid (LL)
Giamarchi and Press (2004). Each spin $j$ of the chain is in contact with its
own independent bath of harmonic oscillators with the hamiltonian
$H_{\text{B}}=\sum_{jk}\frac{P^{2}_{jk}}{2m_{k}}+\frac{m_{k}\Omega_{k}^{2}}{2}X_{jk}^{2}$.
A different choice for local baths was studied in Lobos  _et al._ (2012). The
complete hamiltonian is given by :

$\displaystyle\begin{split}H&=H_{\text{S}}+H_{\text{B}}+H_{\text{SB}}\\\
H_{\text{SB}}&=\sum_{j=1}^{N}S_{j}^{z}\sum_{k}\lambda_{k}X_{jk}\end{split}$
(1)

Note that the coupling term $h_{j}(t)=\sum_{k}\lambda_{k}X_{jk}$ is equivalent
to a time-dependent magnetic field interacting with the spins. The time-
independent limit, $h_{j}(t)=h_{j}$, corresponds to a quenched disordered
magnetic field. Recent numerical simulations seem to suggest that quenched
disorder can induce a finite temperature MBL transition in this model Serbyn
_et al._ (2015). At zero temperature instead, a localisation transition surely
occurs and can be studied using Bosonization Giamarchi and Schulz (1987) and
powerful simulation techniques Doggen  _et al._ (2017). Here we replace the
quenched disorder by an annealed disorder produced by the bath and investigate
the possibility of a zero temperature localisation transition. To fully
characterize the bath, we need to specify the low-frequency behaviour of the
spectral function, defined as :

$J(\Omega)=\frac{\pi}{2}\sum_{k}(\lambda_{k}^{2}/m_{k}\Omega_{k})\delta(\Omega-\Omega_{k})$
(2)

In general, one has $J(\Omega)=\pi\alpha\Omega^{s}$ for
$\Omega\in(0,\Omega_{D})$. Here $\alpha$ denotes the effective coupling
strength with the bath, the cut-off $\Omega_{D}$ is the Debye frequency and
$s$ sets the nature of the bath. For our study we take $s=1$, which
corresponds to an ohmic bath. A similar model was already studied by
Bosonisation Cazalilla  _et al._ (2006) and Monte-Carlo techniques Cai  _et
al._ (2014) in a different context. However, its phase diagram remains
controversial and it is not clear how many phases appear varying the parameter
$\alpha$. Here, we introduce a novel approach, which simulates directly the
bosonised action and allows us to reach large system sizes. Our results show a
simple scenario of two phases with a KT transition between them. Increasing
$\alpha$, a dissipative phase with suppressed transport takes over the LL
phase.

Bosonised action: We map the chain with periodic condition into a 1D fermionic
system using Jordan-Wigner transformation. For $J_{z}=0$, we recover the free-
fermion problem that can be diagonised in the momentum space $q=2\pi l/(Na)$
with $a$ as lattice spacing and $l\in(-N/2,N/2)$. The Fermi momentum depends
on the total magnetisation $M$ of the spin chain, namely
$q_{F}=\pi(N-M)/(2Na)$. Two cases should be distinguished : In the zero sector
of magnetisation, $q_{F}$ is commensurate with the lattice space, while it is
incommensurate for the non-zero magnetisation sector. Here we focus on the
incommensurate case. Bosonisation allows us to include both the interaction
and the bath by linearizing the spectrum around $q_{F}$, which corresponds to
the low-energy physics of the system. Using the standard techniques (Note (1),
Sec. 1), we can map the spin-chain problem into the following field theory
action $S=S_{\text{LL}}+S_{\text{int}}$, where:

$\displaystyle S_{\text{LL}}$ $\displaystyle=\frac{1}{2\pi K}\int
dxd\tau\left[\frac{1}{u}\left(\partial_{\tau}\phi(x,\tau)\right)^{2}+u\left(\partial_{x}\phi(x,\tau)\right)^{2}\right]$
(3) $\displaystyle S_{\text{int}}$ $\displaystyle=-\frac{\alpha}{4\pi^{2}}\int
dxd\tau
d\tau^{\prime}\frac{\cos\left(2\left(\phi(x,\tau)-\phi(x,\tau^{\prime})\right)\right)}{|\tau-\tau^{\prime}|^{2}}$
(4)

Here $\phi(x,\tau)$ is a two-dimensional field living in the physical space
$x\in(0,L)$ and in imaginary time $\tau\in(0,\beta)$, where $\beta$ is the
inverse temperature. Thermodynamic quantities of the spin chain can be
expressed in terms of the correlation functions of the field $\phi$. In
particular, the propagator
$G(q,\omega_{n})=\langle\phi(q,\omega_{n})\phi(-q,-\omega_{n})\rangle$ can be
related to the susceptibility $\chi$ and spin stiffness $\rho_{s}$ by the two
equations:

$\displaystyle\chi$ $\displaystyle=\lim_{q\to 0}\lim_{\omega_{n}\to
0}\frac{q^{2}}{\pi^{2}}G(q,\omega_{n})$ (5) $\displaystyle\rho_{s}$
$\displaystyle=\lim_{\omega_{n}\to 0}\lim_{q\to
0}\frac{\omega_{n}^{2}}{\pi^{2}}G(q,\omega_{n})$ (6)

Here $\omega_{n}=2\pi n/\beta$, $n\in(-\beta/2,\beta/2)$ are the Matsubara
frequencies. When $\alpha=0$, we recover the LL action which corresponds to an
isolated XXZ spin chain and the parameters $u$ and $K$ can be directly related
to $J_{z}$ and $J_{xy}$ either by Bethe Ansatz or by bosonisation (111See SI
for additional information, Sec. 1). In this phase,
$G_{\text{LL}}(q,\omega_{n})=\pi K/(\omega_{n}^{2}/u+uq^{2})$, and hence
$\chi=K/(u\pi)$ and $\rho_{s}=uK/\pi$.

Figure 2: Calculation of different quantities for $K=0.75$ that characterizes
LL ($\alpha=1$, top row) and dissipative phase ( $\alpha=8$, bottom row). Blue
and red points correspond to $L=\beta=384$ and $L=\beta=128$, respectively.
Left : Due to symmetry, $\pi\chi=K_{r}/u_{r}$ is equal to $K/u=0.75$ for all
values of $\alpha$ and all lengthscales. Middle : For $\alpha=1$,
$\omega_{n}C(\omega_{n})$ saturates to $K_{r}/2=0.349$ as $\omega_{n}\to 0$;
whereas for $\alpha=8$, $\sqrt{\omega_{n}}C(\omega_{n})$ saturates to
$[K_{r}\pi/(8\alpha_{r}u_{r})]^{1/2}=0.392$. The other fitting constants are
$a_{1}=0.2493$ and $a_{2}=3.572$. Right : For $\alpha=1$,
$\langle\cos(\phi)\rangle$ decays as a power law, which allows us to extract
$K_{r}=0.694$, consistent with the fit of $\omega_{n}C(\omega_{n})$. For
$\alpha=8$ it saturates to a constant, as predicted by the variational ansatz
(the fit gives $c_{1}=0.62,$ $c_{2}=0.603$ and $c_{3}=-0.531$).

The bath introduces a long-range cosine interaction in $\tau$ direction only
and the strength of this potential is controlled by the parameter $\alpha$. A
perturbative RG study Cazalilla  _et al._ (2006) shows that for $K<K_{c}=0.5$,
the cosine term is relevant and the LL phase is destroyed, whereas for
$K>K_{c}$ and small $\alpha$, the system stays in LL phase but with
renormalised LL parameters $K_{r}$ and $u_{r}$. For $K\gtrsim K_{c}$, the
transition is of KT type: The critical point $\alpha_{c}(K)$ is still LL with
$K_{r}=K_{c}=0.5$. The nature of the dissipative phase is not clear : For
moderate $K$ and very large $\alpha$, the action should be gapless and
harmonic, obtained by the quadratic expansion of the cosine term. For $K\gg
K_{c}$, a large-N argument suggests the existence of a gapped disordered
phase. Monte-Carlo simulations Cai  _et al._ (2014) were performed on the 1D
hard-core bosonic chain, which can be mapped to free fermions ($K=1$).
Increasing $\alpha$, they found that $\chi$ increases and at $\alpha_{c}$, the
system undergoes a continuous second-order phase transiton with vanishing
$\rho_{s}$. Below, we propose a simple scenario able to conciliate the puzzle
of contradictory results.

Methods : To make progress, we focus on the interacting bosonised action. On
one side, we compute the correlation functions numerically by generating
equilibrated configurations from the action with the help of Langevin
dynamics. On the other hand, to understand the properties of the dissipative
phase, we improve the harmonic expansion proposed in Cazalilla  _et al._
(2006) using a variational approach where we obtain an effective quadratic
action by minimizing the variational free energy (Note (1), Sec. 2). The
propagator of this action is given by:

$\displaystyle G^{-1}_{\text{var}}(q,\omega_{n})=\frac{u_{r}q^{2}}{2\pi
K_{r}}+\frac{\alpha_{r}}{\pi^{2}}\left|\omega_{n}\right|+a_{1}\left|\omega_{n}\right|^{\frac{3}{2}}+a_{2}\omega_{n}^{2}$
(7)

The macroscopic behaviour of this phase depends only on the two parameters
$u_{r}/K_{r}$ and $\alpha_{r}$. The parameters $a_{1}$ and $a_{2}$ are
introduced to account for finite size effects. From the analysis of our
result, we will show that by varying $\alpha$, the long-distance properties
are always captured either by the LL or by the variational propagator with
renormalized parameters $u_{r},K_{r}$ and $\alpha_{r}$.

Results : In the following, we present our results for the correlation
functions of the action $S=S_{\text{int}}+S_{\text{LL}}$ with $u=1,K=0.75$ and
different $\alpha$. For our simulations, we set $\beta=L$. The first
observation is that the action of Eq.(4) is invariant under tilt
transformation(Note (1), Sec. 5). As a consequence, $\chi$ is not affected by
the presence of $S_{\text{int}}$. We measure $K_{r}/u_{r}$ both at low and
high $\alpha$ as shown in fig. 4 left. Note that the susceptibility
corresponds to $q\to 0$ limit, but due to the symmetry, $K_{r}/u_{r}$ is
invariant at all length scales and all values of $\alpha$. We conclude that
$K_{r}/u_{r}=K/u$ for all values of $\alpha$. In principle, we should now
measure the stiffness, but if we focus only on the limit $q\to 0$, it strongly
fluctuates. Instead we introduce a function $C(\omega_{n})=(1/\pi
L)\sum_{q}\left|\phi(q,\omega_{n})\right|^{2}$, which for small $\omega_{n}$
behaves as $K_{r}/2\omega_{n}$ in the LL phase and as $\sqrt{K_{r}/8\pi
u_{r}}(\alpha_{r}\omega_{n}/\pi^{2}+a_{1}\omega_{n}^{3/2}+a_{2}\omega_{n}^{2})^{-1/2}$
according to the variational prediction. From fig. 4 middle, we see that
indeed for small $\alpha$, $C(\omega_{n})$ behaves as expected for the LL
phase while for large $\alpha$, $C(\omega_{n})$ shows an agreement with the
variational approach. To confirm our prediction, we compute an independent
quantity, namely $\langle\cos(\phi)\rangle$. Due to the fluctuating nature of
the bosonised field in LL phase, this quantity goes to 0 with a finite size
behaviour $\langle\cos(\phi)\rangle_{\text{LL}}\sim L^{-K_{r}/4}$. On the
other hand, in the dissipative phase the cosine potential pins the field.
Using the variational ansatz we predict
$\langle\cos(\phi)\rangle_{\text{var}}\sim
c_{1}+c_{2}/\sqrt{\beta}+c_{3}/\beta$, where $c_{1},c_{2},c_{3}$ are fitting
cut-off dependent parameters(Note (1), Sec. 3) . Fig. 4 right confirms the
scenario of a transition between an LL to a dissipative phase described by the
variational ansatz. Moreover, the value of $K_{r}$ extracted from
$\langle\cos(\phi)\rangle$ matches nicely with the prediction of
$C(\omega_{n})$. In Fig. 3, we rationalise our results of the renormalized
parameters, obtained by varying $\alpha$. For $K=0.75$, the critical region
where the stiffness vanishes is $\alpha\in(3,4)$. Moreover just before the
transition, $K_{r}$ approaches 0.5, as predicted by KT transition. In (Note
(1), Sec. 6), we provide further results for $u=1,K=0.55$, which are also in
agreement with this picture.

Figure 3: Behaviour of different renormalised parameters as a function of
dissipative coupling $\alpha$. $K_{r}/u_{r}$ (red square points) remains
constant and equal to $K/u=0.75$ for all values of $\alpha$. $K_{r}$,
extracted from $\langle\cos(\phi)\rangle$ analysis (purple hollow circular
points) agrees with the one from the $C(\omega)$ analysis (blue solid
triangular points). It approaches $K_{c}=0.5$ as $\alpha$ reaches the critical
point. The parameter $\alpha_{r}$ (green solid circular points) starts to be
defined in the dissipative phase and increases rapidly with increase in
$\alpha$. This behaviour of the parameter allows to locate the different
phases : For $\alpha\in[0,3]$, the system is in LL phase, whereas for
$\alpha\in[4,8]$ the system is in dissipative phase. The phase transition
takes place for $\alpha\in(3,4)$.

Transport properties: With our approach, one can compute thermodynamic
quantities and does not have direct access to transport properties. However,
via Wick rotation, the conductivity can be related to the propagator :

$\sigma(\omega)=\frac{e^{2}}{\pi^{2}\hbar}\left[\omega_{n}G(q=0,\omega_{n})\right]_{i\omega_{n}\to\omega+i\epsilon}$
(8)

For Luttinger liquid, the DC conductivity
$\sigma_{\text{DC}}\equiv\text{Re}\left[\sigma(\omega\to
0)\right]=(e^{2}uK/\hbar)\delta(\omega)$, which shows the system is perfectly
conducting. For the dissipative phase, after Wick rotation,we get
$\sigma_{\text{DC}}=e^{2}/\hbar\alpha_{r}$, proving that the system has finite
conductivity. For a generic bath, $G(q=0,\omega_{n})\sim
1/(\alpha_{r}|\omega_{n}|^{s})$, and hence
$\text{Re}[\sigma(\omega)]=(e^{2}/\hbar\alpha_{r})(\epsilon/(\omega^{2}+\epsilon^{2})^{s/2})$.
Specially when the bath is subohmic ($s<1$), the DC conductivity of the system
goes to 0, which is a signature of bath induced localization in the system.

Discussion: It remains important to clarify how to conciliate our observations
of a KT transition with $K_{c}=0.5$ and the hardcore bosonsic Monte Carlo
simulations (at $K=1$ instead of $K=0.75$) that show a vanishing stiffness at
the transition Cai  _et al._ (2014). A possibility is that it is an artifact
of the commensurate-incommensurate crossover of the system as the system size
as well as the incommensurate parameter used in the Monte-carlo study is
small. Another possibility remains that our action misses some term that is
relevant for the microscopic lattice model.

Conclusion: On a more general framework, Many efforts are currently being made
to observe phase transition in open quantum systems. Often the physical bath
is replaced by quantum measurents. For a single degree of freedom, this gives
rise to the Zeno effect Misra and Sudarshan (1977); Itano  _et al._ (1990)
where the spin dyamics is completely frozen for very frequent measurements. To
observe similar phenomena in many-body systems, solving the Lindblad equation
is a very commonly used formalism Maimbourg  _et al._ (2021). However, with
this approach the numerics are often limited to very small system sizes.
Another popular method is to approximate these systems via quantum circuits
through which people have been able to show phase transitions Li  _et al._
(2018); Chan  _et al._ (2019); Skinner  _et al._ (2019); Szyniszewski  _et
al._ (2019); Turkeshi  _et al._ (2020); Jian  _et al._ (2020); Zabalo  _et
al._ (2020); Bao  _et al._ (2020); Cao  _et al._ (2019); Zhang  _et al._
(2020), but these models are very simplistic and doesn’t capture the physics
fully. Our work shows that localisation effects can be induced by optical
phonons, provided that the nature of the bath is slow enough (ohmic or
subohmic).

###### Acknowledgements.

Acknowledgments: This work is supported by ‘Investissements d’Avenir’ LabEx
PALM (ANR-10-LABX-0039-PALM) (EquiDystant project, L. Foini) and is supported
in part by the Swiss National Science Foundation under Division II. We thank
T. Maimbourg and V. Schimmenti for helpful discussions. This work was
performed using HPC/AI resources from GENCI-IDRIS (Grant 2022-AD011013581) and
GENCI-TGCC (Grant 2022-AD011013555) .

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Supplementary Material

##  Appendix A Bosonisation of the XXZ spin chain with dissipative bath

In this section, we apply Bosonisation on the one-dimensional XXZ spin chain
coupled with dissipative baths, arriving at an equivalent field theory. The
hamiltonian of the system is given by :

$\displaystyle\begin{split}H&=H_{\text{S}}+H_{\text{B}}+H_{\text{SB}}\\\
H_{\text{S}}&=\sum_{j=1}^{N}J_{z}S_{j}^{z}S_{j+1}^{z}+J_{xy}\left(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y}\right)\\\
H_{\text{B}}&=\sum_{jk}\frac{P^{2}_{jk}}{2m_{k}}+\frac{m_{k}\Omega_{k}^{2}}{2}X_{kj}^{2}\\\
H_{\text{SB}}&=\sum_{j=1}^{N}S_{j}^{z}\sum_{k}\lambda_{k}X_{kj}\end{split}$
(3)

Here we focus on the equilibrium properties at the inverse temperature
$\beta=\infty$. To tackle the dissipative problem, there are two different
approaches. One can map to an equivalent fermionic system via Jordan-Wigner
transformation and apply bosonisation to arrive at a two-dimensional field
theory. Alternatively, the quantum hamiltonian is mapped onto hard-core
bosonic system via Holstein-Primakoff transformation and then numerically
simulated via Quantum Monte-Carlo methods (Indeed Bosons don’t suffer from
Sign problem). In both cases, to integrate the bath degrees of freedom, one
has to introduce the path integral description of the system. The action
associated to the bath and the interaction between the bath and the system is
identical for bosons and fermions and is given by:

$S_{\text{B}}+S_{\text{SB}}=\int_{0}^{\beta}d\tau\sum_{j=1}^{N}\left[\left(n_{j}(\tau)-\frac{1}{2}\right)\sum_{k}\lambda_{\alpha}X_{kj}+\sum_{k}\left(m_{k}\dot{X}^{2}_{kj}+\frac{m_{k}\Omega^{2}_{k}}{2}X^{2}_{kj}\right)\right]$
(4)

Where $n_{j}=S^{z}_{j}+\frac{1}{2}$ is the density operator. Note that we have
introduced the imaginary time $\tau\in(0,\beta)$. Now, we can integrate out
the bath degrees of freedom and arrive at an effective action for the system
degrees of freedom only, where the effect of the bath is encoded in the
interacting part $S_{\text{int}}$:

$S_{\text{int}}=-\iint_{0}^{\beta}d\tau
d\tau^{\prime}\sum_{j=1}^{N}\left[n_{j}(\tau)-\frac{1}{2}\right]D(\tau-\tau^{\prime})\left[n_{j}(\tau^{\prime})-\frac{1}{2}\right]$
(5)

Here $D(\tau-\tau^{\prime})$ is the dissipative kernel which is produced from
integrating over the bath modes. In particular, its Fourier transform can be
expressed in terms of the bath spectral function $J(\Omega)$ (defined in the
main text):

$D(\omega_{n})=\frac{2}{\pi}\int_{0}^{\infty}J(\Omega)\frac{\Omega}{\omega_{n}^{2}+\Omega^{2}}$
(6)

Using the form $J(\Omega)=\pi\alpha\Omega$ ($s=1$), we get
$D(\tau-\tau^{\prime})\sim\alpha|\tau-\tau^{\prime}|^{-2}$.

The Bosonisation procedure of the XXZ spin chain is well known Giamarchi and
Press (2004). Away from the sector of zero magnetisation, one recovers the
action for the well-known Luttinger liquid model
$S_{\text{LL}}=\frac{1}{2\pi}\int_{0}^{L}dx\int_{0}^{\beta}d\tau\left[\frac{1}{uK}(\partial_{\tau}\phi(x,\tau))^{2}+\frac{u}{K}(\partial_{x}\phi(x,\tau))^{2}\right]$.
At zero magnetisation, there is an extra term in the action
$S_{\text{cos}}=-\frac{J_{z}}{2\pi^{2}}\int_{0}^{L}dx\int_{0}^{\beta}d\tau\cos(4\phi(x,\tau))$,
which is irrelevant for $K>1/2$. The constants $u$ and $K$ are called
Luttinger liquid parameters and they depend on $J_{xy}$ and $J_{z}$. These
parameters can be exactly calculated from Bethe Ansatz (eg.
$K^{-1}_{\text{Bethe ansatz}}=(2/\pi)\arccos(-J_{z}/J_{xy})$) and they match
with the Bosonisation prediction in the regime $J_{z}<<J_{xy}$
($K^{-1}_{\text{bosonisation}}=\sqrt{1+4J_{z}/\pi J_{xy}}$). However, away
from half-filling (non-zero magnetisation sector), the bosonisation prediction
between the LL parameters and the spin chain are slightly more complicated and
given by $uK=aJ_{xy}\sin(q_{F}a)$ and $u/K=uK(1+\frac{2aJ_{z}}{\pi
v_{F}}\left[1-\cos(2q_{F}a)\right])$, where $q_{F}$ is the Fermi momentum of
the spin chain and $a$ is the lattice spacing kept for dimensional matching
reasons.

To bosonise Eq. (5), we recall that the bosonized version of $S_{j}^{z}$ is
given by :

$\hat{S}^{z}=-\frac{1}{\pi}\nabla\phi+\frac{1}{\pi a}\cos(2\phi-2q_{F}x)$ (7)

Using Eq. (5), the dissipative part of the action is given by:

$\displaystyle\begin{split}S_{\text{int}}&=-\frac{1}{2\pi^{2}}\int_{0}^{L}dx\iint_{0}^{\beta}d\tau
d\tau^{\prime}\left(-\nabla\phi(x,\tau)+\frac{1}{a}\cos(2\phi(x,\tau)-2q_{F}x)\right)\\\
&\times
D(\tau-\tau^{\prime})\left(-\nabla\phi(x,\tau^{\prime})+\frac{1}{a}\cos(2\phi(x,\tau^{\prime})-2q_{F}x)\right)\end{split}$
(8)

After multiplying all the terms, one can put $a=1$, which was there for
dimensional purposes. For the expansion, we will be making a few observations
here:

  * •

At equal time ($\tau=\tau^{\prime}$), the dissipative action is
$(S_{j}^{z})^{2}$, which is identity. Hence, this term doesn’t contribute
anything to the physics and we can neglect the constant term in
$D(\tau-\tau^{\prime})$.

  * •

the terms with of the form
$\nabla\phi(\tau)\cos(2\phi(\tau^{\prime})-2q_{F}x)$ will oscillate rapidly
for non-zero magnetisation due to the $2q_{F}x$ term and hence it will
integrate to 0. Similarly, the
$\cos(2\phi(\tau)-2q_{F}x)\cos(2\phi(\tau^{\prime})-2q_{F}x)$ term can be
broken up in two terms; one of these terms will be of the form
$\cos(2(\phi(\tau)+\phi(\tau^{\prime}))-4q_{F}x)$. This term can also be
integrated to 0 due to the rapidly oscillating term $4q_{F}x$.

  * •

the $\nabla\phi(\tau)\nabla\phi(\tau^{\prime})$ term is the forward scattering
term and is irrelevant by power counting.

Hence, the action of the full system turns out to be:

$\displaystyle\begin{split}S=S_{\text{LL}}-\frac{\alpha}{4\pi^{2}}\int_{0}^{L}dx\iint_{0}^{\beta}d\tau
d\tau^{\prime}\frac{\cos(2\phi(x,\tau)-2\phi(x,\tau^{\prime}))}{|\tau-\tau^{\prime}|^{2}}\end{split}$
(9)

##  Appendix B Variational ansatz

In this section, we show a detailed calculation of our variational ansatz to
find an effective action that captures the effect of the dissipative term
Feynman (1998).

The full bosonised action of the system is given by
$S=S_{\text{LL}}+S_{\text{int}}$ :

$\displaystyle\begin{split}S_{\text{LL}}&=\frac{1}{2\pi K}\int
dxd\tau\left[\frac{1}{u}\left(\partial_{\tau}\phi(x,\tau)\right)^{2}+u\left(\partial_{x}\phi(x,\tau)\right)^{2}\right]\\\
S_{\text{int}}&=-\frac{\alpha}{4\pi^{2}}\int dxd\tau
d\tau^{\prime}\frac{\cos\left(2\left(\phi(x,\tau)-\phi(x,\tau^{\prime})\right)\right)}{|\tau-\tau^{\prime}|^{2}}\end{split}$
(10)

We would like to find an effective quadratic action of the form
$S_{\text{var}}=\frac{1}{2\beta
L}\sum_{q,\omega_{n}}\phi^{*}(q,\omega_{n})G^{-1}_{\text{var}}(q,\omega_{n})\phi(q,\omega_{n})$.
We use the variational method : we minimise the free energy
$F_{\text{var}}=-\frac{1}{\beta}\sum_{q,\omega_{n}}\log
G_{\text{var}}+\frac{1}{\beta}\langle
S-S_{\text{var}}\rangle_{S_{\text{var}}}$ (with $(S-S_{\text{var}})$ averaged
over $S_{\text{var}}$) with respect to the variational Green’s function :

$\begin{gathered}G^{-1}_{\text{var}}=\frac{1}{2\pi
K}\left(uq^{2}+\frac{\omega_{n}^{2}}{u}\right)+\frac{\alpha}{\pi^{2}}\int
d\tau
D(\tau)\left(1-\cos\omega\tau\right)e^{\left(-\frac{1}{\pi^{2}}\int_{-\infty}^{\infty}dqd\omega\
G_{\text{var}}(q,\omega)\left(1-\cos\omega\tau\right)\right)}\end{gathered}$
(11)

We try to solve this self-consistent equation making the following ansatz :
$G^{-1}_{\text{var}}(q,\omega)=\frac{1}{2\pi
K}\left(uq^{2}+\frac{\omega_{n}^{2}}{u}\right)+\frac{\alpha}{\pi^{2}}F(\omega)$,
where
$F(\omega)=a(\alpha)\left|\omega\right|^{\psi}+b(\alpha)\left|\omega\right|$.
With this assumption, for large $\tau$, the behaviour of $G(\omega)$ is
governed by the $|\omega|$ term. It can be easily shown that
$\int_{\infty}^{\infty}dqd\omega\
G_{\text{var}}(q,\omega)\left(1-\cos\omega\tau\right)\approx
C(\alpha)-\left(\frac{\tau_{c}(\alpha)}{\tau}\right)^{1/2}$, where $C(\alpha)$
and $\tau_{c}(\alpha)$ are $\alpha$ dependent constants. For a more systematic
expansion in powers of $1/\tau$ one can use the results in Santachiara  _et
al._ (2007). Putting this back into the self-consistent equation for
$F(\omega)$ we obtain :

$\begin{split}a(\alpha)\left|\omega\right|^{\psi}+b(\alpha)\left|\omega\right|&=\int
d\tau
D(\tau)\left(1-\cos\omega\tau\right)e^{-\left[C(\alpha)-\left(\frac{\tau_{c}(\alpha)}{\tau}\right)^{1/2}\right]}\\\
&\stackrel{{\scriptstyle\text{large}\ \tau}}{{\approx}}\int d\tau
D(\tau)\left(1-\cos\omega\tau\right)e^{-C(\alpha)}\left(1+\left(\frac{\tau_{c}(\alpha)}{\tau}\right)^{1/2}\right)\end{split}$
(12)

The $\omega$ dependence can be easily extracted from these equations, which
turns out to be $|\omega|$ and $|\omega|^{3/2}$. The coefficient of $|\omega|$
should be determined self-consistently and in our analysis we take it as a
fitting parameter $\alpha_{r}/\pi^{2}$. The coefficient in front of
$\omega^{2}$ will be renormalised by higher order terms from variational
analysis. Hence, the variational propagator is given at low order in $\omega$
by $G^{-1}_{\text{var}}=\frac{u}{2\pi
K}q^{2}+\frac{\alpha_{r}}{\pi^{2}}\left|\omega_{n}\right|+a_{1}\left|\omega_{n}\right|^{3/2}+a_{2}\omega_{n}^{2}$,
where $a_{1}$ and $a_{2}$ are fitting coefficients.

##  Appendix C Calculation of system size dependence of Order Parameter

In this section, we find an analytical expression for the quantity
$\langle\cos(\phi(x,\tau)\rangle$, which is equal to
$e^{-\frac{1}{2}\langle\left[\phi(\vec{r})\right]^{2}\rangle}$ for Gaussian
theories.

For our variational ansatz, we need to calculate
$\langle\cos(\phi(\vec{r}))\rangle=\exp(S_{1})=\exp\left(-\frac{1}{2\beta
L}\sum_{q,\omega_{n}}\frac{1}{\frac{u}{\pi
K}q^{2}+\frac{2\alpha}{\pi^{2}}\left|\omega_{n}\right|+2a_{1}\left|\omega_{n}\right|^{3/2}+2a_{2}\omega_{n}^{2}}\right)$.
In order to calculate the sum inside the exponential we send the limit of
integration over $q$ from $0$ to infinity, and the integral over $\omega$ from
$1/\beta$ to $1/l_{0}$, where $l_{0}$ is the microscopic cut-off. By doing so,
one can find the small $\omega$ behaviour of the sum as below :

$\begin{gathered}S_{1}=\frac{1}{4}\sqrt{\frac{K}{\pi
u}}\left[\frac{a_{1}\pi^{3}}{(2\alpha)^{\frac{3}{2}}l_{0}}-\frac{2\pi}{\sqrt{2\alpha
l_{0}}}\right]+\sqrt{\frac{\pi
K}{8u\alpha}}\frac{1}{\sqrt{\beta}}-\frac{a_{1}\pi^{3}}{4}\sqrt{\frac{K}{8u\alpha^{3}}}\frac{1}{\beta}\end{gathered}$
(13)

We put this back into the expression of $\langle\cos(\phi(\vec{r}))\rangle$
and from a large $\beta$ (zero temperature limit) expansion, we obtain the
finite size dependence of the order parameter:

$\displaystyle\begin{split}\langle\cos(\phi(\vec{r}))\rangle_{\text{var}}&=c_{1}+\frac{c_{2}}{\sqrt{\beta}}+\frac{c_{3}}{\beta}\\\
c_{1}&=\exp\left[\frac{1}{4}\sqrt{\frac{K}{\pi
u}}\left(\frac{a_{1}\pi^{3}}{(2a)^{\frac{3}{2}}l_{0}}-\frac{2\pi}{\sqrt{2\alpha
l_{0}}}\right)\right]\\\ c_{2}&=c_{1}\sqrt{\frac{\pi K}{8u\alpha}}\\\
c_{3}&=c_{1}\left(\frac{\pi
K}{16u\alpha}-\frac{a_{1}\pi^{3}}{4}\frac{K}{8u\alpha^{3}}\right)\end{split}$
(14)

##  Appendix D Numerical details

In this section, we describe the numerical procedure for this work. In a
nutshell, we and denoted $\phi_{ij}$, where $i\in[1,L]$ and $j\in[1,\beta]$
with periodic boundary conditions in both directions. Our strategy is to start
from a flat interface $\phi_{ij}=0$ at $t=0$ and then let it evolve according
to the Langevin equation Honeycutt (1992):

$\frac{d\phi_{ij}(t)}{dt}=-\frac{\delta
S[\phi_{ij}(t)]}{\delta\phi_{ij}}+\eta_{ij}(t)$ (15)

Where $\eta_{ij}(t)$ is a white noise, specified by the correlations
$\langle\eta_{ij}(t)\rangle=0$ and
$\langle\eta_{ij}(t)\eta_{i^{\prime}j^{\prime}}(t^{\prime})\rangle=2\delta_{i,i^{\prime}}\delta_{j,j^{\prime}}\delta(t-t^{\prime})$.
Note that the time $t$ that appears in Eq. (15) should not be confused with
the imaginary time $\tau$. When $t\to\infty$ the surface $\phi_{i,j}(t)$
obtained by direct integration of Eq. (15) is equilibrated with the action
$S[\phi]$. Hence, the Langevin equation, which we numerically simulate, is
given by :

$\displaystyle\begin{split}\frac{d\phi_{ij}(t)}{dt}&=\frac{u}{K\pi}\left[\phi_{i+1,j}+\phi_{i-1,j}-2\phi_{i,j}\right]+\frac{1}{uK\pi}\left[\phi_{i,j+1}+\phi_{i,j-1}-2\phi_{i,j}\right]\\\
&+\frac{\alpha}{\pi^{2}}\sum_{j^{\prime}}D(|j-j^{\prime}|)\sin\left[2(\phi_{ij^{\prime}}-\phi_{ij})\right]+\eta_{ij}(t)\end{split}$
(16)

In order to obtain a correct discretisation of the long-range kernel
$D(j-j^{\prime})$,we use the same protocol as in Zoia  _et al._ (2007). For
$\beta\to\infty$, we set :

$D(j-j^{\prime})=\int_{0}^{2\pi}\frac{d\omega}{2\pi}e^{i\omega(j-j^{\prime})}(2(1-\cos(\omega)))^{1/2}=\frac{1}{(j-j^{\prime})^{2}-\frac{1}{4}}$
(17)

At finite $\beta$, the periodic boundary condition are implemented as :

$D(j-j^{\prime})=\sum_{k=-\beta/2}^{\beta/2}\frac{1}{(|j-j^{\prime}|+k\beta)^{2}-1/4}$
(18)

To conclude, we remark that in the numerical integration the term
$\frac{\delta S[\phi_{ij}(t)]}{\delta(\phi)}$ is multiplied by $\Delta t$,
whereas $\eta_{ij}(t)$ by $\sqrt{\Delta t}$. Here we use the stochastic 2nd
order Runge-Kutta algorithm for white noise Honeycutt (1992). Using this is
preferable as this is much faster than the standard Euler’s algorithm. We
choose the value of the Langevin time-step $\Delta t=0.05$. To benchmark the
equilibration time of the surface, we used the harmonic approximation
$\sin\left[2(\phi_{ij^{\prime}}-\phi_{ij})\right]\to
2(\phi_{ij^{\prime}}-\phi_{ij})$ that can be analytically solved.

##  Appendix E Tilt Symmetry of the action

In this section, we explain why the parameter $K/u$, which identifies with the
susceptibility $\chi$, remains constant for all dissipative coupling $\alpha$.
Agoritsas  _et al._ (2010); Mézard, Marc (1990). To compute the
susceptibility, we introduce a finite magnetic field $h$ in the $z$-direction.
Then the susceptibility can be written as
$\chi=\frac{\partial^{2}}{\partial(h\beta)^{2}}(\log Z[h])$, where $Z$ is the
partition function and $\beta$ is the inverse temperature of the system. In
the bosonised language, the term $-h\sum_{j}S^{z}_{j}$ in the hamiltonian
gives rise to the term $-\frac{h}{\pi}\int(\nabla\phi(x,\tau))dxd\tau$ in the
action. Hence, the partition function of the system can be written as :

$Z[h]=\int\mathcal{D}[\phi]\exp\left(-S_{\text{LL}}-S_{\text{int}}+\frac{h}{\pi}\int\nabla\phi(x,\tau)dxd\tau\right)$
(19)

One can rewrite the terms $\frac{u}{2\pi
K}(\nabla\phi)^{2}-\frac{h}{\pi}\nabla\phi$ as $\frac{u}{2\pi
K}(\nabla\phi-\frac{hK}{u})^{2}-\frac{h^{2}K}{2\pi u}$. Introducting the tilt
$\tilde{\phi}\to\phi-\frac{hKx}{u}$, the partition function can be re-written
:

$Z[h]=\int\mathcal{D}[\tilde{\phi}]\exp\left(-S_{\text{LL}}[\tilde{\phi}]-S_{\text{int}}[\tilde{\phi}]+\frac{\beta^{2}h^{2}K}{2u\pi}\right)$
(20)

The key point is that the interacting action $S_{\text{int}}$ is invariant
under the tilt transformation
$S_{\text{int}}(\tilde{\phi}+\frac{hkx}{u})=S_{\text{int}}(\tilde{\phi})$.
From the previous equation, it can be easily seen that :

$\log Z[h]=\frac{\beta^{2}h^{2}K}{2u\pi}+\log Z[h=0]$ (21)

From this expression, the susceptibility can be easily computed, which is
finally given by:

$\chi=\frac{\partial^{2}}{\partial(\beta
h)^{2}}\frac{\beta^{2}h^{2}K}{2u\pi}=\frac{K}{u\pi}$ (22)

##  Appendix F Results for $K=0.55$

Figure 4: Calculation of different quantities for $K=0.55$ that characterizes
LL ($\alpha=0.05$, top row) and dissipative phase ( $\alpha=10$, bottom row).
Blue and red points correspond to $L=\beta=384$ and $L=\beta=128$,
respectively. Left : Due to symmetry, $\pi\chi=K_{r}/u_{r}$ is equal to
$K/u=0.55$ for all values of $\alpha$ and all lengthscales. Middle : For
$\alpha=0.05$, $\omega_{n}C(\omega_{n})$ saturates to $K_{r}/2=0.273$ as
$\omega_{n}\to 0$; whereas for $\alpha=10$, $\sqrt{\omega_{n}}C(\omega_{n})$
saturates to $[K_{r}\pi/(8\alpha_{r}u_{r})]^{1/2}=0.156$. The other fitting
constants are $a_{1}=16.61$ and $a_{2}=571.4$. Right : For $\alpha=0.05$,
$\langle\cos(\phi)\rangle$ decays as a power law, which allows us to extract
$K_{r}=0.546$, consistent with the fit of $\omega_{n}C(\omega_{n})$. For
$\alpha=10$ it saturates to a constant, as predicted by the variational ansatz
(the fit gives $c_{1}=0.788,$ $c_{2}=0.215$ and $c_{3}=0.012$). Bottom row:
Behaviour of different renormalised parameters as a function of dissipative
coupling $\alpha$. $K_{r}/u_{r}$ (red square points) remains constant and
equal to $K/u=0.55$ for all values of $\alpha$. $K_{r}$, extracted from
$\langle\cos(\phi)\rangle$ analysis (purple hollow circular points) agrees
with the one from the $C(\omega)$ analysis (blue solid triangular points). It
approaches $K_{c}=0.5$ as $\alpha$ reaches the critical point. The parameter
$\alpha_{r}$ (green solid circular points) starts to be defined in the
dissipative phase and increases rapidly with increase in $\alpha$. This
behaviour of the parameter allows to locate the different phases : For
$\alpha\in[0,0.25)$, the system is in LL phase, whereas for $\alpha\in(2,15]$
the system is in dissipative phase. The phase transition takes place for
$\alpha\in(0.25,2)$. We believe $\alpha=0.25$ to be in critical region as
$K_{r}$ extracted from $\langle\cos(\phi)\rangle$ is very close to
$K_{c}=0.5$, $\alpha_{r}$ is very small and $\omega_{n}C(\omega_{n})$
saturates for a long range of $\omega$ but then starts decreasing.


$\theta(h,{\sigma},t)=\begin{cases}\bigl{(}P_{h,{\sigma},t},\,R_{h,{\sigma},t},\,S_{h,{\sigma},t}\bigr{)},&{\sigma}>0,\\\
\bigl{(}P_{h},\,R_{h},\,S_{h}\bigr{)},&{\sigma}=0,\end{cases}$

is $\mathsf{W}^{3,0}$-continuous. Hence the following ‘‘$\theta$-evaluation’’
map

$\displaystyle\widetilde{\theta}:\mathcal{C}^{\infty}_{0}({U},\mathbb{R}^{{\mathsf{c}}}\times\mathbb{R}^{{\mathsf{n}}})\times[0;+\infty)\times[0;1]\times
U\times\mathbb{R}^{{\mathsf{b}}}\times\mathbb{R}^{{\mathsf{b}}}\times\mathbb{R}^{{\mathsf{m}}}\to\mathbb{R}^{{\mathsf{c}}}\times\mathbb{R}^{{\mathsf{c}}}\times\mathbb{R}^{{\mathsf{n}}},$
$\displaystyle\widetilde{\theta}\bigl{(}h,{\sigma},t,w,\vec{u},\vec{v},\vec{w}\bigr{)}=\begin{cases}\bigl{(}P_{h,{\sigma},t}\vec{u},\,R_{h,{\sigma},t}\vec{v},\,S_{h,{\sigma},t}\vec{w}\bigr{)},&{\sigma}>0,\\\
\bigl{(}P_{h}\vec{u},\,R_{h}\vec{v},\,S_{h}\vec{w}\bigr{)},&{\sigma}=0,\end{cases}$

is $\mathsf{W}^{r}$-continuous for all $r\geq 3$.

###### Proof.

It is a direct consequence of inequalities (6.17)-(6.19) whose right hand
sides are of the form ${\sigma}c(h,{\sigma},t)$, where $c(h,{\sigma},t)$
continuously depends on ${\sigma}$, $t$ and partial derivatives of $h$ up to
order $3$. We leave the details for the reader. ∎

Say that an $(a\times b)$-matrix $A$ has maximal rank, whenever
$\mathsf{rank}(A)=\min\\{a,b\\}$.

######  Corollary 6.5.3.

Let $a\geq 0$, $K\subset U$ be a compact subset such that
$\mathsf{rank}(P(x,0))\geq a$ for all $w\in K$. Then there exist
$\eta,\varepsilon>0$ such that $\mathsf{rank}(P_{\tilde{h},{\sigma},t}(w))\geq
a$ for all
$(\tilde{h},{\sigma},t,w)\,\in\,\mathcal{N}^{2}_{K,\eta}(h)\times[0;\varepsilon]\times[0;1]\times
K$.

A similar statement holds for matrices $S$, but one should replace
$\mathcal{N}^{2}_{K,\eta}(h)$ with $\mathcal{N}^{3}_{K,\eta}(h)$.

###### Proof.

1) Suppose that $\mathsf{rank}(P(w))\geq a$ for all $w\in K$. Since $K$ is
compact and $h$ is $\mathcal{C}^{1}$-differentiable (so $P$ is continuous),
there exists $c_{0}>0$ such that $\mathsf{rank}(P^{\prime})\geq a$ for every
matrix $P^{\prime}\in M({\mathsf{c}},{\mathsf{b}})$ satisfying
$\|P^{\prime}-P(w)\|<c_{0}$ for some $w\in K$.

Let
$\tilde{h}=(\tilde{f},\tilde{g})\in\mathcal{C}^{\infty}_{0}({U},\mathbb{R}^{{\mathsf{c}}}\times\mathbb{R}^{{\mathsf{n}}})$
and $\Bigl{(}\\!\begin{smallmatrix}\tilde{P}\ &\tilde{Q}\\\\[1.42262pt]
\tilde{R}\ &\tilde{S}\end{smallmatrix}\\!\Bigr{)}$ be its Jacobi matrix map.
Then, due to (6.17), for every $w\in K$ we have that

$\displaystyle|P_{\tilde{h},{\sigma},t}(w)-P(w)|$
$\displaystyle\leq|P_{\tilde{h},{\sigma},t}(w)-\tilde{P}(w)|+|\tilde{P}(w)-P(w)|$
(6.21) $\displaystyle\leq{\sigma}\|\tilde{f}\|_{2,K}+\|\tilde{h}-h\|_{1,K}$
$\displaystyle\leq{\sigma}\|h\|_{2,K}+{\sigma}\|\tilde{h}-h\|_{2,K}+\|\tilde{h}-h\|_{1,K}$
$\displaystyle\leq{\sigma}\|h\|_{2,K}+({\sigma}+1)\,\|\tilde{h}-h\|_{2,K}.$

Choose small $\eta,\varepsilon>0$ so that

$\varepsilon\|h\|_{2,K}+(\varepsilon+1)\eta<c_{0}.$

Then for all
$(\tilde{h},{\sigma},t,w)\,\in\,\mathcal{N}^{2}_{K,\eta}(h)\times[0;\varepsilon]\times[0;1]\times
K$ we have that $|P_{\tilde{h},{\sigma},t}(w)-P(w)|<c_{0}$, so
$\mathsf{rank}(P_{\tilde{h},{\sigma},t}(w))\geq a$.

2) Similarly, suppose that $\mathsf{rank}(S(w))\geq a$ for all $w\in K$. Then
there exists $c_{1}>0$ such that $\mathsf{rank}(S^{\prime})\geq a$ for every
matrix $S^{\prime}\in M({\mathsf{n}},{\mathsf{m}})$, satisfying
$\|S^{\prime}-S(w)\|<c_{1}$ for some $w\in K$. Assume that ${\sigma}<1$ and
let $c_{2}:=3+{\mathsf{m}}|\mu|_{1}$. Then due to (6.19) for each
$\tilde{h}=(\tilde{f},\tilde{g})\in\mathcal{C}^{\infty}_{0}({U},\mathbb{R}^{{\mathsf{c}}}\times\mathbb{R}^{{\mathsf{n}}})$
we have that

$\displaystyle|S_{\tilde{h},{\sigma},t}(w)-S(w)|$
$\displaystyle\leq|S_{\tilde{h},{\sigma},t}(w)-\tilde{S}(w)|+|\tilde{S}(w)-S(w)|$
(6.22)
$\displaystyle\leq{\sigma}\bigl{(}2|\tilde{g}|_{2,{\mathsf{m}},K}+{\sigma}|\tilde{g}|_{3,{\mathsf{m}},K}+{\mathsf{m}}|\mu|_{1}\cdot|\tilde{g}|_{2,{\mathsf{m}},K}\bigr{)}+\|\tilde{h}-h\|_{1,K}$
$\displaystyle\leq{\sigma}c_{2}\|\tilde{h}\|_{3,K}+\|\tilde{h}-h\|_{1,K}$
$\displaystyle\leq{\sigma}c_{2}\|h\|_{3,K}+({\sigma}c_{2}+1)\,\|\tilde{h}-h\|_{3,K}.$

Choose $\eta,\varepsilon>0$ so that $\varepsilon<1$ and

$\varepsilon c_{2}\|h\|_{3,K}+(\varepsilon c_{2}+1)\eta<c_{1}.$

Then, for all
$(\tilde{h},{\sigma},t,w)\in\mathcal{N}^{3}_{K,\eta}(h)\times[0;\varepsilon]\times[0;1]\times
K$ we have that $|S_{\tilde{h},{\sigma},t}(w)-S(w)|<c_{1}$, so
$\mathsf{rank}(S_{\tilde{h},{\sigma},t}(w))\geq a$. ∎

######  Corollary 6.5.4.

Assume that ${\mathsf{m}}\leq{\mathsf{n}}$, ${\mathsf{b}}\leq{\mathsf{c}}$.
Suppose a map
$h\in\mathcal{C}^{\infty}_{0}({U},\mathbb{R}^{{\mathsf{c}}}\times\mathbb{R}^{{\mathsf{n}}})$
is injective on some compact subset $K\subset U$, and

$\displaystyle\mathsf{rank}(P_{h}(x,0))={\mathsf{b}},$
$\displaystyle\mathsf{rank}(S_{h}(x,0))={\mathsf{m}},$

for every $(x,0)\in K\cap(\mathbb{R}^{{\mathsf{b}}}\times 0)$. In other words,
those matrices induce injective linear maps, which is the same here as having
maximal ranks. Then there exist $\varepsilon,\eta>0$ such that for all
$(\tilde{h},{\sigma},t)\in\mathcal{N}^{3}_{K,\eta}(h)\times[0;\varepsilon]\times[0;1]$
the map $H_{\tilde{h},{\sigma},t}$ is injective on $K$.

###### Proof.

Suppose our statement fails. Then there exist a sequence

$(h_{i},{\sigma}_{i},t_{i})\,\in\,\mathcal{C}^{\infty}_{0}({U},\mathbb{R}^{{\mathsf{c}}}\times\mathbb{R}^{{\mathsf{n}}})\times(0;+\infty)\times[0;1],\qquad
i\in\mathbb{N},$

and two sequences
$\\{(x_{i},v_{i})\\}_{i\in\mathbb{N}},\\{(y_{i},w_{i})\\}_{i\in\mathbb{N}}\subset
K$ of mutually distinct points such that

$\displaystyle\lim\limits_{i\to\infty}\|h-h_{i}\|_{3,K}=0,\qquad\lim\limits_{i\to\infty}{\sigma}_{i}=0,$
$\displaystyle
H_{h_{i},{\delta}_{i},t_{i}}(x_{i},v_{i})=H_{h_{i},{\delta}_{i},t_{i}}(y_{i},w_{i}),\
i\in\mathbb{N},$

One can assume, in addition, that

  * –

each $h_{i}$ is an embedding near $K$, since $h$ is so and embeddings near a
compact subset are open in any topology $\mathsf{W}^{r}$ for $r\geq 1$;

  * –

$\lim\limits_{i\to\infty}(x_{i},v_{i})=(x,v)$,
$\lim\limits_{i\to\infty}(y_{i},w_{i})=(y,w)$,
$\lim\limits_{i\to\infty}t_{i}=t$ for some $(x,v),(y,w)\in K$ and $t\in[0;1]$,
due to compactness of $K\times[0;1]$.

Since the $\mathcal{H}$-evaluation map is $\mathsf{W}^{3}$-continuous, see
Lemma 6.3.1, we obtain that

$h(x,v)=\lim_{i\to\infty}H_{h_{i},{\delta}_{i},t_{i}}(x_{i},v_{i})=\lim_{i\to\infty}H_{h_{i},{\delta}_{i},t_{i}}(y_{i},w_{i})=h(y,w),$

whence $(x,v)=(y,w)$ since $h$ is injective on $K$.

We clam that then $v=0$. Indeed, suppose $\|v\|>a$ for some $a>0$. Then for
sufficiently large $i$ we have that
$\delta_{i}<a<\min\\{\|v_{i}\|,\|w_{i}\|\\}$. Hence by (ii) before Lemma 6.3.1

$h_{i}(x_{i},v_{i})=H_{h_{i},{\delta}_{i},t_{i}}(x_{i},v_{i})=H_{h_{i},{\delta}_{i},t_{i}}(y_{i},w_{i})=h_{i}(y_{i},w_{i}),$

and therefore $(x_{i},v_{i})=(y_{i},w_{i})$ since each $h_{i}$ is injective.
This contradicts to the assumption that those points are distinct.

Let
$H_{h_{i},{\sigma}_{i},t_{i}}=(f_{h_{i},{\sigma}_{i},t_{i}},g_{h_{i},{\sigma}_{i},t_{i}}):U\to\mathbb{R}^{{\mathsf{c}}}\times\mathbb{R}^{{\mathsf{n}}}$
be the coordinate functions of $H_{h_{i},{\sigma}_{i},t_{i}}$. Consider two
cases.

1) Suppose that $v_{i}\not=w_{i}$ for all $i\in\mathbb{N}$. Then we can assume
that the following sequence of unit vectors
$u_{i}:=\frac{v_{i}-w_{i}}{\|v_{i}-w_{i}\|}\in\mathbb{R}^{{\mathsf{m}}}$,
$i\in\mathbb{N}$, converges to some unit vector $u$. Then, by
$\mathsf{W}^{3}$-continuity of $\theta$-evaluation map $\widetilde{\theta}$,
see Corollary 6.5.2,

$S_{h}(x,0)u=\lim\limits_{i\to\infty}\frac{g_{h,{\delta}_{i},t_{i}}(x_{i},v_{i})-g_{h,{\delta}_{i},t_{i}}(y_{i},w_{i})}{\|v_{i}-w_{i}\|}\equiv
0,$

which contradict to the assumption that $S_{h}(x,0)$ induces an injective
linear map.

2) Otherwise, we can assume that $x_{i}\not=y_{i}$ for all $i\in\mathbb{N}$.
Then, as in the previous case, we can assume that the following sequence of
unit vectors
$\tau_{i}:=\frac{x_{i}-y_{i}}{\|x_{i}-y_{i}\|}\in\mathbb{R}^{{\mathsf{b}}}$,
$i\in\mathbb{N}$, converges to some unit vector $\tau$. Then, by
$\mathsf{W}^{3}$-continuity of $\theta$-evaluation map,

$P_{h}(x,0)\tau=\lim\limits_{i\to\infty}\frac{f_{h,{\delta}_{i},t_{i}}(x_{i},v_{i})-f_{h,{\delta}_{i},t_{i}}(y_{i},w_{i})}{\|x_{i}-y_{i}\|}\equiv
0,$

which contradict to the assumption that $P_{h}(x,0)$ induces an injective
linear map. ∎

##  7\. Proof of Theorem 6.4.1

Due to Lemma 6.3.4 it suffices to show that for each $h\in\mathcal{X}$ there
exists a $\mathsf{W}^{3}$-neighborhood $\mathcal{U}$ in $\mathcal{X}$ and
$\varepsilon>0$ such that

$\mathcal{H}(\mathcal{U}\times[0;\varepsilon]\times[0;1])\subset\mathcal{X}.$

It will be convenient to say that $(\mathcal{U},\varepsilon)$ is admissible
for $h$ with respet to $\mathcal{X}$.

  1. 1)

Suppose $\mathcal{X}=\mathcal{C}^{\infty}_{-,a}({N},{F})$. As ${B}$ is
compact, there exist finitely many local trivializations of $h$

$\hat{h}_{i}=(f_{i},g_{i}):\mathbb{R}^{{\mathsf{b}}}\times\mathbb{R}^{{\mathsf{m}}}\supset{U}_{i}\to{W}_{i}\times\mathbb{R}^{{\mathsf{n}}},\qquad
i=1,\ldots,s,$

with respect to some trivialized local charts $\Phi_{i}$ and $\Psi_{i}$, and
for each $i$ a compact subset $K_{i}\subset{U}_{i}$ such that
${B}=\mathop{\cup}\limits_{i=1}^{s}\Phi_{i}(K_{i})$. Then, by Corollary 6.5.3,
there exist $\eta_{i},\varepsilon_{i}>0$ such that
$\mathsf{rank}(P_{h^{\prime},{\sigma},t}(w))\geq a$ for all
$(h^{\prime},{\sigma},t,w)\,\in\,\mathcal{N}^{2}_{K_{i},\eta_{i}}(\hat{h}_{i})\times[0;\varepsilon_{i}]\times[0;1]\times
K_{i}$. Define the following $\mathsf{W}^{2}$-open neighborhood of $h$ in
$\mathcal{C}^{\infty}_{0}({N},{F})$:

$\mathcal{U}_{i}=\bigl{\\{}\tilde{h}\in\mathcal{C}^{\infty}_{0}({N},{F})\mid\tilde{h}(\Phi_{i}(K_{i}))\subset\Psi_{i}({W}_{i}\times\mathbb{R}^{{\mathsf{n}}})\
\text{and}\
\|\Psi_{i}^{-1}\circ\tilde{h}\circ\Phi_{i}-h\|_{2,K_{i}}<\eta_{i}\bigr{\\}}.$

Then the pair
$\bigl{(}\mathop{\cap}\limits_{i=1}^{s}\mathcal{U}_{i},\min\limits_{i=1,\ldots,s}\varepsilon_{i}\bigr{)}$
is admissible for $h$ with respect to $\mathcal{C}^{\infty}_{-,a}({N},{F})$.

  2. 2)

The proof for $\mathcal{X}=\mathcal{C}^{\infty}_{|,b}({N},{F})$ is literally
the same and based on the part of Corollary 6.5.3 for matrices $S$, but in
that case $\mathcal{U}$ will be only $\mathsf{W}^{3}$-open.

  3. 3)

Suppose
$\mathcal{X}=\mathcal{C}^{\infty}_{+,a,b}({N},{F})=\mathcal{C}^{\infty}_{-,a}({N},{F})\cap\mathcal{C}^{\infty}_{|,b}({N},{F})$.
Let $(\mathcal{U}_{|},\varepsilon_{|})$ and
$(\mathcal{U}_{-},\varepsilon_{-})$ be admissible for $h$ pairs with respect
to $\mathcal{C}^{\infty}_{-,a}({N},{F})$ and
$\mathcal{C}^{\infty}_{|,b}({N},{F})$ correspondingly. Then
$(\mathcal{U}_{|}\cap\mathcal{U}_{-},\min\\{\varepsilon_{|},\varepsilon_{-}\\})$
is admissible for $h$ with respect to $\mathcal{C}^{\infty}_{+,a,b}({N},{F})$.

  4. 4)

Let $\mathcal{X}=\mathcal{E}^{\infty}_{+,{\mathsf{b}},{\mathsf{m}}}({N},{F})$
be the subset of $\mathcal{C}^{\infty}_{+,{\mathsf{b}},{\mathsf{m}}}({N},{F})$
consisting of embeddings and $h\in\mathcal{X}$. In this case we should have
that ${\mathsf{b}}\leq{\mathsf{c}}$ and ${\mathsf{m}}\leq{\mathsf{n}}$.

Let $R_{a}$ be a tubular neighborhood of ${B}$ in ${E}$ contained in ${N}$.
Then similarly to the previous cases 1) and 2) and also by Corollary 6.5.4
there exist finitely many local trivializations
$\hat{h}_{i}=(f_{i},g_{i}):\mathbb{R}^{{\mathsf{b}}}\times\mathbb{R}^{{\mathsf{m}}}\supset{U}_{i}\to{W}_{i}\times\mathbb{R}^{{\mathsf{n}}}$
of $h$, $i=1,\ldots,s,$ with respect to some trivialized local charts
$\Phi_{i}$ and $\Psi_{i}$, and for each $i$ a pair of compact subset
$K_{i},L_{i}\subset{U}_{i}$ with non-empty interiors,
$\eta_{i},\varepsilon_{i}>0$ such that

    1. (a)

$K_{i}\subset\mathrm{Int}{L_{i}}$ and
${B}\subset\mathop{\cup}\limits_{i=1}^{s}\Phi_{i}(\mathrm{Int}{K_{i}})$;

    2. (b)

$\mathsf{rank}(P_{h^{\prime},{\sigma},t}(w))={\mathsf{b}}$,
$\mathsf{rank}(S_{h^{\prime},{\sigma},t}(w))={\mathsf{m}}$, and
$H_{h^{\prime},{\sigma},t}|_{L_{i}}:L_{i}\to{W}_{i}\times\mathbb{R}^{{\mathsf{n}}}$
is injective for all
$(h^{\prime},{\sigma},t,w)\,\in\,\mathcal{N}^{3}_{L_{i},\eta_{i}}(\hat{h}_{i})\times[0;\varepsilon_{i}]\times[0;1]\times
L_{i}$;

    3. (c)

in particular, the rank of the Jacobi matrix of $H_{h^{\prime},{\sigma},t}$ at
each $w\in L_{i}$ is equal to ${\mathsf{b}}+{\mathsf{m}}=\dim({N})$, whence
$H_{h^{\prime},{\sigma},t}$ is an embedding near $L_{i}$.

For $i=1,\ldots,s$ define the following $\mathsf{W}^{3}$-open neighborhood of
$h$ in $\mathcal{E}^{\infty}_{+,{\mathsf{b}},{\mathsf{m}}}({N},{F})$:

$\mathcal{U}_{i}=\bigl{\\{}\tilde{h}\in\mathcal{E}^{\infty}_{+,{\mathsf{b}},{\mathsf{m}}}({N},{F})\mid\tilde{h}(\Phi_{i}(L_{i}))\subset\Psi_{i}({W}_{i}\times\mathbb{R}^{{\mathsf{n}}})\
\text{and}\
\|\Psi_{i}^{-1}\circ\tilde{h}\circ\Phi_{i}\|_{3,L_{i}}<\eta_{i}\bigr{\\}}.$

Put $\mathcal{U}^{\prime}=\mathop{\cap}\limits_{i=1}^{s}\mathcal{U}_{i}$ and
$\varepsilon^{\prime}=\min\limits_{i=1,\ldots,s}\varepsilon_{i}$. Then for all
$(\tilde{h},{\sigma},t)\,\in\,\mathcal{U}^{\prime}\times[0;\varepsilon^{\prime}]\times[0;1]$

    1. (d)

$H_{\tilde{h},{\sigma},t}=\tilde{h}$ on ${N}\setminus
R_{\varepsilon^{\prime}}$, and in particular, $H_{\tilde{h},{\sigma},t}$ is an
embedding on ${N}\setminus R_{\varepsilon^{\prime}}$;

    2. (e)

$H_{\tilde{h},{\sigma},t}$ is an embedding near each compact set
$\Phi_{i}(L_{i})$, $i=1,\ldots,s$, due to 0)(c).

However, this only implies that $H_{\tilde{h},{\sigma},t}$ is an immersion. To
make $H_{\tilde{h},{\sigma},t}$ embedding on all of ${N}$ we will decrease
$\mathcal{U}^{\prime}$ and $\varepsilon^{\prime}$ as follows.

By 0)(a), there exists $\varepsilon^{\prime\prime}\in(0;\varepsilon^{\prime})$
such that
$R_{\varepsilon^{\prime\prime}}\subset\mathop{\cup}\limits_{i=1}^{s}\Phi_{i}(\mathrm{Int}{K_{i}})$.
Since $h$ is injective, one can choose open neighborhoods $V,W\subset{F}$ of
disjoint compact sets $h({B})$ and $h(\overline{{N}\setminus
R_{\varepsilon^{\prime\prime}}})$ respectively such that $V\cap
W=\varnothing$. Define the following $\mathsf{W}^{0}$-open neighborhood
$\mathcal{V}$ of $h$ in
$\mathcal{E}^{\infty}_{+,{\mathsf{b}},{\mathsf{m}}}({N},{F})$:

$\mathcal{V}=\bigl{\\{}\tilde{h}\in\mathcal{E}^{\infty}_{+,{\mathsf{b}},{\mathsf{m}}}({N},{F})\mid\tilde{h}(\Phi_{i}(K_{i}))\subset
V,\ \tilde{h}(\overline{{N}\setminus\Phi_{i}(L_{i})})\subset W,\
i=1,\ldots,s\bigr{\\}}$

Note that by Lemma 6.3.1 the map
$\mathcal{H}:\mathcal{C}^{\infty}_{0}({N},{F})\times[0;+\infty)\times[0;1]\to\mathcal{C}^{\infty}_{0}({N},{F})$
is $\mathsf{W}^{0,0}$-continuous, and $\mathcal{H}(h,0,t)=h$. Therefore,
$\mathcal{H}^{-1}(\mathcal{V})$ is a $\mathsf{W}^{0}$-open neighborhood of
$h\times 0\times[0;1]$, whence there exists another neighborhood
$\mathcal{U}\subset\mathcal{U}^{\prime}\cap\mathcal{H}^{-1}(\mathcal{V})$ and
$\varepsilon<b$ such that
$\mathcal{H}(\mathcal{U}\times[0;\varepsilon]\times[0;1])\subset\mathcal{U}^{\prime}\cap\mathcal{V}$.

We claim that $\mathcal{H}(\mathcal{U}\times[0;\varepsilon]\times[0;1])$
consists of injective maps, which will finally imply that
$\mathcal{H}(\mathcal{U}\times[0;\varepsilon]\times[0;1])\subset\mathcal{E}^{\infty}_{+,{\mathsf{b}},{\mathsf{m}}}({N},{F})$,
i.e. that $(\mathcal{U},\varepsilon)$ is admissible for $h$ with respect to
$\mathcal{E}^{\infty}_{+,{\mathsf{b}},{\mathsf{m}}}({N},{F})$.

Indeed, suppose there exist
$(\tilde{h},{\sigma},t)\,\in\,\mathcal{U}\times[0;\varepsilon]\times[0;1]$ and
two distinct points $w_{1}\not=w_{2}\in{N}$ such that
$H_{\tilde{h},{\sigma},t}(w_{1})=H_{\tilde{h},{\sigma},t}(w_{2})$. If
$w_{1}\in\Phi_{i}(K_{i})$ for some $i\in\\{1,\ldots,s\\}$, then, due to
injectivity of $H_{\tilde{h},{\sigma},t}$ on $\Phi_{i}(L_{i})$, we must have
that $w_{2}\in{N}\setminus\Phi_{i}(L_{i})$. But then by 0)(d),

$H_{\tilde{h},{\sigma},t}(w_{1})=H_{\tilde{h},{\sigma},t}(w_{2})\ \in\
H_{\tilde{h},{\sigma},t}(\Phi_{i}(K_{i}))\cap
H_{\tilde{h},{\sigma},t}({N}\setminus\Phi_{i}(L_{i}))\ \subset\ V\cap W\ =\
\varnothing,$

which gives a contradiction.

Therefore
$w_{1},w_{2}\in{N}\setminus\mathop{\cup}\limits_{i=1}^{s}\Phi_{i}(\mathrm{Int}{K_{i}})\subset{N}\setminus
R_{\varepsilon^{\prime\prime}}\subset{N}\setminus R_{\varepsilon}$. Since
${\sigma}<\varepsilon$, we have that
$\tilde{h}(w_{1})=H_{\tilde{h},{\sigma},t}(w_{1})=H_{\tilde{h},{\sigma},t}(w_{2})=\tilde{h}(w_{2})$
which is also impossible, since $\tilde{h}$ is injective.

  5. 5)

If ${\mathsf{b}}=\dim({B})=\dim({C})={\mathsf{c}}$ and
${\mathsf{b}}+{\mathsf{m}}=\dim({E})=\dim({F})={\mathsf{c}}+{\mathsf{n}}$,
then
$\mathcal{E}_{0}({N},{F})\equiv\mathcal{E}^{\infty}_{+,{\mathsf{b}},{\mathsf{m}}}({N},{F})$.
Therefore 5) is a particular case of 4). ∎

##  8\. Proof of Theorem 1.1.1

Let ${B}$ be a compact manifold, ${p}:{E}\to{B}$ be a vector bundle equipped
with some orthogonal structure, and ${N}\subset{E}$ be a smooth submanifold
being also a neighborhood of ${B}$.

Evidently that for every $\mathsf{W}^{\infty}$-continuous function
${\delta}:\mathcal{C}^{\infty}_{0}({N},{E})\to(0;+\infty)$ the map (1.3) is a
${\delta}$-linearizing homotopy, whence by Corollary 6.3.2 it always extends
to a $\mathsf{W}^{\infty}$-continuous map
${H}:\mathcal{E}_{0}({N},{E})\times[0;1]\to\mathcal{C}^{\infty}_{0}({N},{E})$.
Moreover, ${H}(h,1)=\mathsf{T}_{\\!\mathsf{fib}}h={\alpha}(0,\cdot)$ on
$R_{0.2{\delta}(h)}$, see (6.7), so statement (2) of Theorem 1.1.1 also always
hold.

We will now choose ${\delta}$ so that the image of ${H}$ will be contained in
$\mathcal{E}_{0}({N},{E})$. Let $\hat{{B}}$ be a connected component of ${B}$
contained in a connected component $\hat{{E}}$ of ${E}$,
$\hat{{N}}={N}\cap\hat{{E}}$, ${\mathsf{b}}=\dim(\hat{{B}})$, and
${\mathsf{m}}=\dim(\hat{M})-{\mathsf{b}}$. Then by 5) of Theorem 6.4.1 there
exists a $\mathsf{W}^{3}$-continuous function
${\delta}^{{\mathsf{b}},{\mathsf{m}}}:\mathcal{C}^{\infty}_{0}(\hat{{N}},{E})\to(0;+\infty)$
such that the corresponding ${\delta}^{{\mathsf{b}},{\mathsf{m}}}$-linearizing
homotopy
${H}^{{\mathsf{b}},{\mathsf{m}}}:\mathcal{C}^{\infty}_{0}(\hat{{N}},{E})\times[0;1]\to\mathcal{C}^{\infty}_{0}(\hat{{N}},{E})$
leaves invariant the set
$\mathcal{E}^{\infty}_{+,{\mathsf{b}},{\mathsf{m}}}(\hat{{N}},{E})=\mathcal{E}(\hat{{N}},{E})$
of embeddings $\hat{{N}}\subset{E}$.

Let ${\delta}:\mathcal{C}^{\infty}_{0}({N},{E})\to(0;+\infty)$ be the minimum
of all functions ${\delta}^{{\mathsf{b}},{\mathsf{m}}}$, where ${\mathsf{b}}$
runs over dimensions of connected components of ${B}$, and
${\mathsf{b}}+{\mathsf{m}}$ runs over dimensions of connected components of
${E}$. Then for every $h\in\mathcal{E}_{0}({N},{E})$ and $t\in[0;1]$ the map
${H}(h,t):{N}\to{E}$ is an embedding.

Finally, due to the estimate (6.16), one can additionally decrease ${\delta}$
so that ${H}(h,t)(R_{{\delta}(h)})\subset{N}$ for all
$h\in\mathcal{E}_{0}({N},{E})$ and $t\in[0;1]$, which proves statement (1) of
Theorem 1.1.1 as well. ∎

### Acknowledgement

The authors are sincerely grateful to D. Bolotov for fruitful discussions of
vector bundle automorphisms.

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