$\displaystyle\epsilon_{+}(K;q)$ $\displaystyle=-2t_{0}(\cos q_{x}+\cos
k_{0}\cos q_{y})+2t_{b}\sin 2k_{0}\cos 2q_{y}$
$\displaystyle\epsilon_{-}(K;q)$ $\displaystyle=t_{0}\sin k_{0}(2\sin
q_{y}-\sin 2q_{y})$ (207)

In the long-wavelength limit, $q\ll 1$, Eq.199 can be simplified as

$\displaystyle E_{q}$ $\displaystyle=\sqrt{2n_{0}U[t_{0}q_{x}^{2}+(t_{0}\cos
k_{0}-4t_{b}\sin 2k_{0})q_{y}^{2}+O(q^{4})]}$ $\displaystyle+t_{0}\sin
k_{0}q_{y}^{3}+O(q^{5})$ (208)

where the Doppler shift term does not vanish, but goes as $q_{y}^{3}$ instead
of $q_{y}$ in the continuum. Again, there is no chance to drive a QPT in this
special case.

(c) Injecting $H_{b3}$.

The single particle spectrum takes form

$\displaystyle\epsilon_{k}$
$\displaystyle=-t_{0}e^{ik_{x}}-t_{0}e^{ik_{y}}-it_{b}e^{i(k_{x}+k_{y})}+h.c.$
$\displaystyle=-2t_{0}(\cos k_{x}+\cos k_{y})+2t_{b}\sin(k_{x}+k_{y})$ (209)

which develops a single minimum at $k=K=(k_{0},k_{0}),\quad$ with (assuming
$t_{0}>0$)

$\displaystyle\epsilon_{K}$
$\displaystyle=-\frac{1}{2\sqrt{2}t_{b}}(3t_{0}+\sqrt{t_{0}^{2}+8t_{b}^{2}})\sqrt{4t_{b}^{2}-t_{0}^{2}+t_{0}\sqrt{t_{0}^{2}+8t_{b}^{2}}},\quad$
$\displaystyle k_{0}$
$\displaystyle=\arcsin\Big{(}\frac{t_{0}-\sqrt{t_{0}^{2}+8t_{b}^{2}}}{4t_{b}}\Big{)}$
(210)

where $k_{0}$ satisfies:

$t_{0}\sin k_{0}+t_{b}\cos 2k_{0}=0$ (211)

which can be used to simplify many expressions and $|k_{0}|\leq\frac{\pi}{4}$
holds for any ratio of $t_{b}/t_{0}$.

One can calculate

$\displaystyle\epsilon_{+}(K;q)$ $\displaystyle=-2t_{0}\cos k_{0}(\cos
q_{x}+\cos q_{y})+2t_{b}\sin 2k_{0}\cos(q_{x}+q_{y})$
$\displaystyle\epsilon_{-}(K;q)$ $\displaystyle=2t_{0}\sin k_{0}[\sin
q_{x}+\sin q_{y}-\sin(q_{x}+q_{y})]$ (212)

In the long wave-length limit $q\ll 1$ ( BZ size ), Eq.199 can be simplified
as

$\displaystyle E_{q}$ $\displaystyle=\sqrt{2n_{0}U[t_{0}\cos
k_{0}(q_{x}^{2}+q_{y}^{2})-t_{b}\sin 2k_{0}(q_{x}+q_{y})^{2}+O(q^{4})]}$
$\displaystyle-\frac{1}{3}t_{0}\sin
k_{0}[q_{x}^{3}+q_{y}^{3}-(q_{x}+q_{y})^{3}]+O(q^{5})$ (213)

By using the constraint Eq.211, one can show the quantity inside the square
root is positive, while that outside vanishes as $t_{b}$ as $t_{b}\to 0$.

One can see the Doppler shift term does not vanish, but goes as $q^{3}$
instead of linear in the continuum. The first term remains linear. So the
Doppler shift term is always sub-leading to the first term in the long
wavelength limit. There is no chance to drive a QPT in this special case
either.

###  C.2  The calculations in the new basis with $H_{b1}$.

It is also tempting to study the BEC of the bosons in the new basis Eq.121
where the BEC condensation momentum gets back to $0$. Rewriting
$\tilde{b}_{k}=\sqrt{N_{0}}\delta_{k,0}+\tilde{\psi}_{k}$ and applying the
Bogoliubov theory leads to:

$\displaystyle
H=E_{0}+\sqrt{N_{0}}[-2(t_{0}+\sqrt{t_{0}^{2}+t_{b1}^{2}})-\mu+Un_{0}](\tilde{\psi}_{0}+\tilde{\psi}_{0}^{\dagger})$
$\displaystyle+\sum_{k}[(\tilde{\epsilon}_{k}-\mu)\tilde{\psi}_{k}^{\dagger}\tilde{\psi}_{k}+Un_{0}(2\tilde{\psi}_{k}^{\dagger}\tilde{\psi}_{k}+\frac{1}{2}\tilde{\psi}_{k}\tilde{\psi}_{-k}+\frac{1}{2}\tilde{\psi}_{k}^{\dagger}\tilde{\psi}_{-k}^{\dagger})]+\cdots$
(214)

Setting the linear term vanishing leads to
$\mu=n_{0}U-2(t_{0}+\sqrt{t_{0}^{2}+t_{b1}^{2}})$. The quadratic Hamiltonian
can be diagonalized by a Bogoliubov transformation:

$\displaystyle\tilde{E}_{k}=\sqrt{[\tilde{\epsilon}_{k}+2T][\tilde{\epsilon}_{k}+2T+2n_{0}U]}$
(215)

where $T=t_{0}+\sqrt{t_{0}^{2}+t_{b1}^{2}}$ was defined below Eq.203. It has
no Doppler shift term. This basis explained the exact vanishing of the Doppler
shift term in the $H_{b1}$ case, but not the other two cases with $H_{b2}$ and
$H_{b3}$.

Note that the above Bogoliubov calculation of the SF phase automatically break
the C- symmetry at $z=1$, so no Higgs mode can be found. This is expected,
because the above calculations only apply to away from integer fillings where
there is no C- symmetry anyway. But as mentioned below Eq.204, it may apply to
the $z=2$ case where either $p$ or $h$ condenses, after considering the
difference between the density of bosons $n_{0}$ and that $\rho_{0}$ of either
$p$ or $h$, the interaction between the bosons $U$ and that of $u$ between $p$
or $h$. This is the main difference between $b_{i}$ and the order parameter
$\psi$ in Eq.3 and Eq.81.

##  Appendix D  The Galileo invariance, enlargement of a Fermi surface and its
explicit breaking in artificially generated 2d/3d SOC case

It was known that the Electro-Magnetic (EM) fields satisfy Maxwell equations
which is intrinsically Lorentz invariant. The Klein-Gordan (KG) equations for
spin-0 bosons $\phi$ and Dirac equations for spin-1/2 fermions $\psi$ are also
Lorentz invariant. So the KG equation and Dirac equation with a rest-mass $m$
couple to external EM fields in the minimal coupling scheme are both gauge
invariant and Lorentz invariant. The intrinsic velocity is the speed of light
$c_{l}$. Under the Lorentz transformation

$\displaystyle x^{\prime}=\Lambda
x,~{}~{}\Lambda=e^{i\omega_{\alpha\beta}L_{\alpha\beta}}$ (216)

where $L_{\alpha\beta}=-L_{\beta\alpha}$ stands for the three rotation
$J_{i},i=1,2,3$ and the three boosts $Ks_{i},i=1,2,3$ ( Eq.179 ), then KG
spin-0 bosons and Dirac spin-1/2 fermions or spin-1 gauge field $A_{\mu}$ give
its scalar, spinor and vector representation of the Lorentz group
respectively:

$\displaystyle\phi^{\prime}(x^{\prime})$ $\displaystyle=\phi(x),$
$\displaystyle\psi^{\prime}(x^{\prime})$
$\displaystyle=\Lambda_{1/2}\psi(x),~{}~{}\Lambda_{1/2}=e^{i\omega_{\mu\nu}S_{\mu\nu}}$
$\displaystyle A^{\prime}_{\mu}(x^{\prime})$ $\displaystyle=\Lambda
A_{\mu}(x),$ (217)

where
$S_{\mu\nu}=\frac{i}{4}[\gamma_{\mu},\gamma_{\nu}],\Lambda^{-1}_{1/2}\gamma_{\mu}\Lambda_{1/2}=\Lambda_{1/2}\gamma_{\mu}$
and $\gamma_{\mu}$ are the 4 Dirac $\gamma$ matrix.

When contrasted to Eq.87 for the non-relativistic case, one can see the
chemical potential is kept to be zero $\mu=0$ in Eq.217. This is because any
$\mu\neq 0$ breaks Lorentz invariance. So the chemical potential shift in
Eq.87 is a unique feature for the Galileo boost. Indeed, it is this shift
which is responsible for the step (II) shift in the QPT boundary from the Mott
to the BSF in Fig.8.

Taking the non-relativistic limit $v/c_{l}\ll 1$ where $v=k/m$ limit, the KG
or the Dirac equation in an EM field just reduces to the S-equation with a
mass $m$ in the EM field ( or the spin-1 gauge field $A_{\mu}$ in Eq.217 )
which is GT ( see Eq.229 ). For the Dirac equation, there is an additional
Zeeman field term $-\mu_{B}\vec{\sigma}\cdot\vec{B}$ where
$\mu_{B}=\frac{e\hbar}{2mc}$ is the Bohr magneton and the $\vec{\sigma}$ is
the Pauli matrix. This Zeeman term is obviously GI. When pushing to higher
orders, the Dirac equation in a scalar potential $A_{0}$ reduces to S-equation
with a mass $m$ in the scalar field ( See Eq.111 ) plus several new terms such
as the, $p^{4}$ term, the spin-orbit coupling (SOC) term and Darwin term which
break the GI at the order of $(v/c)^{2}$. The gauge invariance is always kept
in such a limit ( Appendix A ). Indeed, as shown below, the SOC breaks the
Galileo invariance.

In this appendix, we first discuss the GT in the Schrodinger equation in the
first quantization which clarify a lot of confusions in the existing
literatures or even textbooks, then show that the Fermi Surface of non-
interacting fermions ( in the second quantization ) breaks Lorentz invariance,
but owns Galileo invariance. Then we show that in the second quantization
language, any Luttinger liquid in 1d or the spin-orbit coupling SOC ; QCD
break the Galileo invariance.

1\. Galileo transformation on the single particle Schrodinger equation: first
quantization

It was known that the non-relativistic Schrodinger equation ( 1st quantization
form )

$i\hbar\frac{\partial\psi}{\partial
t}=[-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}+V(x,t)]\psi$
(218)

where the single-body potential is usually time independent when the sample is
static $V(x,t)=V(x)$.

Eq.218 transforms under the Galileo transformation
$x^{\prime}=x-vt,t^{\prime}=t$ as:

$i\hbar\frac{\partial\psi^{\prime}}{\partial
t^{\prime}}=[-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{\prime
2}}+V^{\prime}(x^{\prime},t^{\prime})]\psi^{\prime}$ (219)

where the wavefunction and the potential when the sample is static are related
to those when the sample is moving by:

$\displaystyle\psi(x,t)$
$\displaystyle=e^{i(k_{0}x^{\prime}-E_{0}t^{\prime}/\hbar)}\psi^{\prime}(x^{\prime},t^{\prime})$
$\displaystyle V^{\prime}(x^{\prime},t^{\prime})$
$\displaystyle=V(x^{\prime}+vt^{\prime},t^{\prime})$ (220)

where $\psi(x,t)=\psi(x^{\prime}-vt^{\prime},t^{\prime})$ and
$k_{0}=-\frac{mv}{\hbar},E_{0}=\frac{\hbar^{2}k^{2}_{0}}{2m}=\frac{1}{2}mv^{2}$.
Of course, if $V=0$, it is Galileo invariant (GI). For the hydrogen problem,
the central potential is the Coulomb interaction
$V(\vec{x}-\vec{R})=-\frac{Ze^{2}}{|\vec{x}-\vec{R}|}$, because the atomic
center is also moving under the GT, then
$V(\vec{x}-\vec{R})=V(\vec{x}^{\prime}-\vec{R}^{\prime})$. So it remains GI.
Its many-body generalization in a periodic lattice potential was shown in
Eq.111 and Eq.112.

In general, due to $V^{\prime}(x^{\prime},t^{\prime})\neq
V(x^{\prime},t^{\prime})$ Schrodinger equation is not Galileo invariant,
because it is time-independent when the sample is static, but becomes time-
dependent when the sample is moving, so can not be GI. For example, if one
takes the harmonic potential $V_{h}(x)=\frac{1}{2}m\omega^{2}x^{2}$ and the
center of the potential is fixed at $x=0$, so not moving under the GT. Then
the potential becomes time dependent
$V^{\prime}(x^{\prime},t^{\prime})=V(x^{\prime}+vt^{\prime},t^{\prime})=\frac{1}{2}m\omega^{2}(x^{\prime}+vt^{\prime})^{2}$
when the sample is moving, therefore breaks the GI. For a typical optical
lattice potential $V_{o}(x)=\cos^{2}\pi
x/a=\cos^{2}\pi(x^{\prime}+vt^{\prime})/a$, then it becomes time-dependent
which is periodic not only in the space with $x^{\prime}\to x^{\prime}+a$, but
also in the time $t^{\prime}\to t^{\prime}+na/v$, or the joint space-time
symmetry.

2\. A moving FS in second quantization: Galileo invariance and enlargement of
the FS in a moving sample

Using the prescription presented in the main text, we immediately find the FS
observed when the sample is moving with the velocity $v\hat{y}$ in the second
quantization form:

$H_{M}=\psi^{\dagger}(\vec{k})[\frac{\hbar^{2}k^{2}}{2m}-\mu-
ck_{y}]\psi(\vec{k}),~{}~{}~{}\mu=\frac{\hbar^{2}k^{2}_{0F}}{2m}$ (221)

where $k_{0F}$ is the Fermi momentum when the sample is static. In a grand
canonical ensemble, one need also introduce the chemical potential $\mu$ which
also plays the role of the energy $E$ in the canonical ensemble.

Eq.221 can be written as:

$H_{M}=\psi^{\dagger}(\vec{k})[\frac{\hbar^{2}}{2m}[(\vec{k}-\vec{k}_{0})^{2}-(k^{2}_{0F}+k^{2}_{0})]\psi(\vec{k})$
(222)

where $\vec{k}_{0}=mc/\hbar\hat{y}$ is the FS center shift due to the boost.
In the fermion case, one can perform the similar set of transformation as
listed in Eq.87, one can show Eq.222 is also Galileo invariant, but no TPT. Of
course, the GI is always emergent, because there always exist higher order
derivative terms such as $k^{4},k^{6},.....$ which break the GI explicitly.

For 1d case, one can identify the two Fermi points when the sample is moving:

$\displaystyle k^{+}_{F}=k_{0}+\sqrt{k^{2}_{0}+k^{2}_{0F}}>0$ $\displaystyle
k^{-}_{F}=k_{0}-\sqrt{k^{2}_{0}+k^{2}_{0F}}<0$ (223)

which shows the Fermi momentum is shifted from $\pm k_{0F}$. Because, no
change of topology of FS, so no topological phase transition (TPT) tuned by
the boost. However, the FS gets enlarged
$k^{+}_{F}-k^{-}_{F}=2\sqrt{k^{2}_{0}+k^{2}_{0F}}>2k_{0F}$. This should not be
too surprising: because there is a relative motion between the sample and the
reservoir in a grand canonical ensemble, then the extra fermions just comes
from the reservoir. This interesting phenomenon can be easily extended to 2d
or 3d.

In fact, Eq.222 can also be viewed as the kinetic term in the interacting
boson case in Eq.190. In the boson case, one focus on the BEC momentum, so the
shift can also be transformed away by introducing the new field $\tilde{\psi}$
and then the new effective chemical potential $\tilde{\mu}$ in Eq.87. So the
boson case in Eq.190 also has the Galileo invariance. However, in contrast to
the fermion case, the effective chemical potential $\tilde{\mu}$ can tune the
QPT from the Mott to SF with $z=2$ as shown in Sec.VI. Both the boson and
fermion case are also due to the Galileo invariance of the non-relativistic
Schrodinger equation at $V=0$ presented in the first quantization language in
the last subsection.

3\. 2d/3d artificial Spin-orbit coupling and split FS with opposite helicities
in second quantization: breaking the Galileo invariance by an effective Zeeman
field.

As stressed in the conclusion section, here we only focus on the artificially
generated SOC here SOC , so one can simply apply the GT. One can add a 3d Weyl
or 2d Rashba like SOC weyl  term $\lambda\vec{k}\cdot\vec{S}$ to Eq.221:

$H_{M,SOC}=\psi^{\dagger}(\vec{k})[\frac{\hbar^{2}k^{2}}{2m}-\mu-
ck_{y}+\lambda\vec{k}\cdot\vec{S}]\psi(\vec{k}),~{}~{}~{}\mu=\frac{\hbar^{2}k^{2}_{0F}}{2m}$
(224)

In the helicity basisweyl ; soc1 ; soc2 ; soc3 , the SOC alone SOC  without
the boost can also be written as a linear $k$ term with $\pm$ sign ( see
Eq.228 ). So it is interesting to look at the combined effects of SOC and the
boost.

Eq.224 can be written as

$H_{M,SOC}=\psi^{\dagger}(\vec{k})[\frac{\hbar^{2}}{2m}[(\vec{k}-\vec{k}_{0})^{2}-(k^{2}_{0F}+k^{2}_{0})]+\lambda\vec{k}\cdot\vec{S}]\psi(\vec{k})$
(225)

where $\vec{k}_{0}=mc/\hbar^{2}\hat{x}$ is the FS center shift due to the
boost.

After performing the similar set of transformation as in Eq.87, Eq.225
becomes:

$H_{M,SOC}=\psi^{\dagger}(\vec{q})[\frac{\hbar^{2}}{2m}[q^{2}-(k^{2}_{0F}+k^{2}_{0})]+\lambda\vec{q}\cdot\vec{S}+\lambda\vec{k}_{0}\cdot\vec{S}]\psi(\vec{q})$
(226)

where the extra term $\lambda\vec{k}_{0}\cdot\vec{S}$ breaks the Galileo
invariance. It plays the role of a Zeeman field which is due to the
combination of both the SOC $\lambda$ and the boost $\vec{k}_{0}$. So we
conclude the SOC breaks the Galileo invariance.

It is easy to find its two eigen-energies at 2d:

$E(q_{x},q_{y})=\frac{\hbar^{2}}{2m}[q^{2}-(k^{2}_{0F}+k^{2}_{0})\pm
2k_{R}\sqrt{k^{2}_{0}+2k_{0}q_{x}+q^{2}}]$ (227)

where $k_{R}=m\lambda$ corresponds the recoil momentum soc1 . Setting
$k_{0}=0$ recovers the SOC when the sample is static
$\frac{\hbar^{2}}{2m}[q^{2}\pm 2k_{R}q-k^{2}_{0F}]$ which leads to one large (
and one small) FS with negative ( positive ) helicity and the Fermi momentum

$k_{F\pm}=\sqrt{k^{2}_{0F}+k^{2}_{R}}\pm k_{R}$ (228)

which is due to the SOC, indeed can be contrasted with Eq.223 which is due to
the boost. Of course, any boost breaks the joint spin-momentum rotation
symmetry, so the helicity is not a good quantum number anymore.

As shown in the last subsection, there is no TPT tuned by the boost in the
absence of SOC. However, as $k_{0}$ increases in Eq.226, there is a
competition between the enlargement of the FS and the presence of SOC which
plays the role of the Zeeman field $\lambda\vec{k}_{0}\cdot\vec{S}$. It would
be interesting to see if such a competition will lead to any TPT. For the
dramatic effects of Weyl type SOC in interacting fermionic systems in a
continuum, see weyl ; soc1 ; soc2 . Similarly, one can show that the
conventional $\lambda\vec{S}\cdot\vec{L}$ in materials SOC  also breaks the
Galileo invariance. But as alerted in the conclusion and the appendix A, one
need to consider the relativistic effects at the order of $(v/c)^{2}$. For the
conventional SOC, see the review SLrev1 .

##  Appendix E  Emergent Galileo invariance of many-body systems in an
external magnetic field

The GT in Eq.111 discussed in Sec.VII in the main text only contains the
lattice potential, no external gauge field. Here we discuss the role of the
external gauge field which shows quite different behaviours than just the
lattice scalar potential in Eq.111. Following the strategies used in the last
section, we will first present the GI in the first quantization in a canonical
ensemble, then analyze its form in the second quantization in either a
canonical or a grant canonical ensemble.

###  E.1  Galileo transformation on many body Schrodinger equation: the first
quantization

In a quantum Hall system subject to a uniform magnetic field, assuming
electron spins are all polarized due to the large Zeeman splitting, so one
only need to consider the orbital effects of the magnetic field for these
spinless fermions, the lattice potential $V_{1}(x)$ can be ignored in such an
effective continuum system, then Eq.111 changes to:

$\displaystyle i\hbar\frac{\partial\Psi(x_{1},x_{2},\cdots x_{N},t)}{\partial
t}=[\sum_{i}\frac{1}{2m}[-i\hbar\frac{\partial}{\partial
x_{i}}-\frac{e}{c_{l}}\vec{A}(x_{i})]^{2}$
$\displaystyle+\sum_{i}eA_{0}(x_{i})+\sum_{i<j}V(x_{i},x_{j})]\Psi(x_{1},x_{2},\cdots
x_{N},t)$ (229)

where $\vec{B}=\nabla\times\vec{A}=B\hat{z}$. Because of the possible
involvement of the time component $A_{0}$ of the gauge field, one may also
need to put it into the Hamiltonian in any case. As shown below, even
$A_{0}=0$ at beginning, it may be generated by a GT. The crucial difference
than Eq.111 is the appearance of the speed of light $c_{l}$ in the minimal
coupling to $\vec{A}$ which need to be considered carefully when performing
the GT. So Eq.111 and Eq.229 are two different class of quantum MBS in Fig.1.
Setting $A_{0}=0,~{}~{}~{}\vec{A}=0$ recovers the Helium 4 case which is
charge neutral and clearly Galileo invariance.

Similarly,
$V(x^{\prime}_{1}-x^{\prime}_{2})=V(x_{1}-x_{2})=e^{2}/|x_{1}-x_{2}|$ is
translational and Galileo invariant retardation . Note that Eq.229 also
applies to the charge neutral interacting bosons in a rotating traprotation
where the artificial gauge field
$\vec{A}=\frac{1}{2}\vec{\omega}\times\vec{r}$ comes from the rotation close
to the trapping frequency $\omega=\omega_{t}$. In this artificial gauge field
case, then the speed of light $c_{l}$ does not appear either.

The gauge potential $\vec{A}(x)$ usually breaks the translational invariance,
however, the magnetic field does not. For example, in the Landau gauge
$A_{x}=0,A_{y}=Bx$ or $A_{x}=-By,A_{y}=0$, the gauge potential breaks
translational invariance along a given direction $x$ or $y$. In the symmetric
gauge $\vec{A}=\frac{1}{2}\vec{B}\times\vec{r}$, namely
$A_{x}=-\frac{1}{2}By,A_{y}=\frac{1}{2}Bx$, it breaks translational invariance
completely, but keeps rotational symmetry around an arbitrarily chosen origin
chosen at $z=0$.

However, in a moving frame, the space-time, the many-body wavefunction and the
gauge potential in the (symmetric gauge) in the lab frame are related to those
in the moving frame

$\displaystyle x^{\prime}$ $\displaystyle=x+vt,~{}~{}~{}t^{\prime}=t$
$\displaystyle\partial_{t}$
$\displaystyle=\partial_{t^{\prime}}+v\partial_{x^{\prime}},~{}~{}~{}\partial_{x}=\partial_{x^{\prime}}$
(230)

The many-body wavefunction changes as the same as in Eq.112

$\displaystyle\Psi(x_{i},t)=e^{i(k_{0}\sum_{i}x^{\prime}_{i}-NE_{0}t^{\prime}/\hbar)}\Psi^{\prime}(x^{\prime}_{i},t^{\prime})$
(231)

where
$k_{0}=-\frac{mv}{\hbar},E_{0}=\frac{\hbar^{2}k^{2}_{0}}{2m}=\frac{1}{2}mv^{2}$.
While the gauge field changes as

$\displaystyle A_{0}$
$\displaystyle=0,A_{x^{\prime}}(y^{\prime})=-\frac{1}{2}By^{\prime},A_{y^{\prime}}(x^{\prime},t^{\prime})=\frac{1}{2}Bx^{\prime}-\frac{1}{2}Bvt^{\prime}$
$\displaystyle A^{\prime}_{0}$
$\displaystyle=A_{0}+\frac{v}{c_{l}}A_{x^{\prime}}=-\frac{v}{2c}By^{\prime},A^{\prime}_{x^{\prime}}=A_{x^{\prime}},A^{\prime}_{y^{\prime}}=A_{y^{\prime}}$
(232)

where $[A_{0}]=[A_{x}]$, $c_{l}$ is the intrinsic velocity of the EM field. In
a contrast, the periodic lattice ( Coulomb ) potential listed below Eq.112 is
GI.

Then one can check the changes in electric and magnetic field ( be careful of
$A^{0}=-A_{0}$ in the Lorentz signature )

$\displaystyle\vec{E}^{\prime}$
$\displaystyle=-\frac{1}{c_{l}}\frac{\partial\vec{A}}{\partial t}-\nabla
A_{0}=\frac{v}{c_{l}}B\hat{y}^{\prime}\ll B,$ $\displaystyle\vec{B}^{\prime}$
$\displaystyle=\nabla\times\vec{A}=B\hat{z}^{\prime}$ (233)

which turns out to be the LT of $E$ and $B$ to the order of $v/c_{l}$, but
dropping $(v/c_{l})^{2}$ and higher orders. When taking the non-relativistic
limit $v/c_{l}\ll 1$, naively, one may simply drop this electric field
generated by the GT. However, the above manipulations demonstrate that one
must keep this small electric field to the order of $v/c_{l}$ to keep the
Galileo invariance Lforce  ( Appendix A ) This should also be expected because
the speed of light $c_{l}$ does appear in the minimal coupling in the many-
body Schrodinger equation Eq.229. Of course, it also appears in the magnetic
length $l_{0}=\sqrt{\frac{\hbar c_{l}}{eB}}$. This is dramatically different
than the charge neutral systems such as the Boson Hubbard model and the ionic
lattice model Eq.111 discussed in the main text where there is simply no
external magnetic field, so one can safely take $c_{l}\rightarrow\infty$
limit.

In the moving frame, one can perform the $U(1)$ gauge transformation to get
rid of the time component $A_{0}$ in Eq.232,

$\displaystyle\Psi\to\Psi e^{-i\frac{1}{2}Bvt^{\prime}y^{\prime}}$ (234)

where $\chi=-\frac{1}{2}Bvt^{\prime}y^{\prime}$. Then the gauge field in
Eq.232 becomes:

$\displaystyle A^{\prime}_{0}$ $\displaystyle\to
A^{\prime}_{0}-\frac{1}{c_{l}}\partial_{0}\chi=0,$ $\displaystyle
A^{\prime}_{x^{\prime}}$ $\displaystyle\to
A^{\prime}_{x^{\prime}}+\partial_{x^{\prime}}\chi=-\frac{1}{2}By^{\prime},$
$\displaystyle A^{\prime}_{y^{\prime}}$ $\displaystyle\to
A^{\prime}_{y^{\prime}}+\partial_{y^{\prime}}\chi=\frac{1}{2}Bx^{\prime}-Bvt^{\prime}$
(235)

One can check that $\vec{E}^{\prime}$ and $\vec{B}^{\prime}$ stay the same. In
real time, one need to consider $A^{0}=-A_{0}$ to get the sign right.

Similarly, one can perform the $U(1)$ gauge transformation to recover the
spatial component of the gauge field,

$\displaystyle\Psi\to\Psi e^{i\frac{1}{2}Bvt^{\prime}y^{\prime}}$ (236)

where $\chi=\frac{1}{2}Bvt^{\prime}y^{\prime}$. Then the gauge field in Eq.232
becomes:

$\displaystyle A^{\prime}_{0}$ $\displaystyle\to
A^{\prime}_{0}-\frac{1}{c_{l}}\partial_{0}\chi=-\frac{v}{c}By^{\prime},$
$\displaystyle A^{\prime}_{x^{\prime}}$ $\displaystyle\to
A^{\prime}_{x^{\prime}}+\partial_{x^{\prime}}\chi=-\frac{1}{2}By^{\prime},$
$\displaystyle A^{\prime}_{y^{\prime}}$ $\displaystyle\to
A^{\prime}_{y^{\prime}}+\partial_{y^{\prime}}\chi=\frac{1}{2}Bx^{\prime}$
(237)

So in the combination of the GT Eq.230,231,232 and the gauge transformation
Eq.236, one can recover the spatial component $\vec{A}$, but still generates a
time component $A^{\prime}_{0}=\frac{v}{c}By^{\prime}$ which leads to the
electric field $\vec{E}^{\prime}=\frac{v}{c_{l}}B\hat{y}^{\prime}\ll B$. One
can also see the differences between the GT and gauge transformation, the
former changes $E$ or $B$, but the latter must keep both the same.

One can repeat the calculations for the two Landau gauges $A_{x}=0,A_{y}=Bx$
or $A_{x}=-By,A_{y}=0$.

The Laughlin ground state wavefunction in the symmetric gauge indeed breaks
the translational invariance, but keeps rotational symmetry. In the lab frame,
it takes the form:

$\Psi(z_{1},z_{2},\cdots,z_{N},t)=\Pi_{i<j}(z_{i}-z_{j})^{m}e^{-\sum_{i}\frac{|z_{i}|^{2}}{4l^{2}_{0}}}e^{-iE_{G}t/\hbar}$
(238)

where $E_{G}$ is the ground state energy, $m$ is odd ( even) for fermion (
boson ) respectively, $l_{0}=\sqrt{\frac{\hbar c_{l}}{eB}}$ is the magnetic
length. In the moving frame, the Laughlin wavefunctions change as Eq.230 in
its $N$ coordinate and Eq.231 under the GT. The Jastraw factor keeps the
Galileo invariance, but the Gaussian factor does not. It was known that the
Jastraw factor may act as the conformal blocks of the chiral edge theory, so
it maybe interesting to explore how does it relate to the edge reconstruction
to be discussed in the appendix H-3 and Fig.17. Similar GT maybe applied to
Jain’s CF wavefunctions also jain . The bulk-edge correspondence can be stated
as that the ground state wavefunction in the bulk can be expressed as the
conformal blocks of the CFT in the edge. More generally, the bulk ground state
wavefunction at a given time slice ( namely 2d spacial dimension ) can be
viewed as the partition function of the CFT.

###  E.2  Galileo transformation on many-body QFT: the second quantization

Eq.229 can be written in the second quantization language:

$\displaystyle{\cal H}$ $\displaystyle=\int
d^{2}x\psi^{\dagger}(\vec{x})[\frac{1}{2m}[-i\hbar\frac{\partial}{\partial
x}-\frac{e}{c_{l}}\vec{A}(x)]^{2}-eA_{0}(x)-\mu]\psi(\vec{x})$
$\displaystyle+\int
d^{2}x_{1}d^{2}x_{2}\psi^{\dagger}(\vec{x}_{1})\psi(\vec{x}_{1})V_{2}(x_{1}-x_{2})\psi^{\dagger}(\vec{x}_{2})\psi(\vec{x}_{2})$
(239)

where the single-body gauge potential $\vec{A}(\vec{x}),A_{0}(\vec{x})$ and
the two-body interaction $V_{2}(x_{1}-x_{2})$ are automatically incorporated
into the kinetic term and the interaction term respectively. By adding the
chemical potential $\mu$, we also change the canonical ensemble with a fixed
number of particles $N$ in the first quantization to the grand canonical
ensemble in the second quantization.

Following the step leading from Eq.2 to Eq.3, one can obtain the effective
action in the moving frame with the velocity $v\hat{x}$ ( for notational
simplicity, we drop the $\prime$ in the moving frame ):

$\displaystyle{\cal H}=\int
d^{2}x\psi^{\dagger}(\vec{x})[\frac{1}{2m}[-i\hbar\frac{\partial}{\partial
x}-\frac{e}{c_{l}}\vec{A}(x)]^{2}$ $\displaystyle-
iv\partial_{x}-eA_{0}(x)-\mu]\psi(\vec{x})$ $\displaystyle+\int
d^{2}x_{1}d^{2}x_{2}\psi^{\dagger}(\vec{x}_{1})\psi(\vec{x}_{1})V_{2}(x_{1}-x_{2})\psi^{\dagger}(\vec{x}_{2})\psi(\vec{x}_{2})$
(240)

It is easy to show that Eq.240 takes the same form as Eq.239 after formally
making the same GT as in Eq.87

$\tilde{\psi}=\psi e^{-ik_{0}x},~{}~{}~{}~{}\tilde{\mu}=\mu+\frac{mv^{2}}{2}$
(241)

in a grand canonical ensemble. While the gauge field ( in the symmetric gauge
) changes as Eq.232. The wavefunction exponential factor in Eq.231 in the
first quantization is equivalent to the shift of the momentum and the chemical
potential as listed in Eq.241 in the second quantization. Indeed, $-\mu\int
d^{2}x\psi^{\dagger}(\vec{x})\psi(\vec{x})=-\mu N$ where $N$ is the number of
particles in Eq.231 in the canonical ensemble.

For a canonical ensemble with a fixed number of particles $N$ which is more
convenient for the FQH ( See Eq.244 ) to be discussed in the following
appendix:

$\tilde{\psi}=\psi e^{-i(k_{0}x-E_{0}t/\hbar)}$ (242)

where
$k_{0}=-\frac{mv}{\hbar},E_{0}=\frac{\hbar^{2}k^{2}_{0}}{2m}=\frac{1}{2}mv^{2}$

For a small number of $N$, this is the end of story. Unfortunately, it is
still difficult to see what are the effects of such a GT in the present second
quantization scheme in the thermodynamic limit $N\to\infty$. This should not
be too surprising, because as advocated by P. W. Anderson, “ More is different
”, there are many emergent quantum or topological phenomena such as quantum
magnetism, superconductivity, superfluidity, fractional Quantum Hall effects
etc which can nowhere be seen in such a formal model Eq.229. This kind of
formal model looks complete, exact to some level, but not effective to see any
emergent phenomena. So only when proceeding further and deeper to get an
effective model to see the signature of emergent quantum or topological
phenomenon, one may start to see the real effects of such a GT on such
emergent phenomenon. In the present case, the emergent phenomenon is the
fractional Quantum Hall effects which can be best seen by introducing the
Chern-Simon gauge fields to be discussed in the following appendix.

##  Appendix F  Emergent Galileo invariance in the Chern-Simon effective
action of FQH in the bulk: comments on HLR theory with $z=2$ and Son’s Dirac
theory with $z=1$ at $\nu=1/2$ filling.

The new quantum phases and novel quantum phase transitions discovered in the
main text still fall into Ginsburg-Landau symmetry breaking picture. It was
known that topological transitions without accompanying symmetry breaking are
beyond Ginsburg-Landau scheme. The Fractional Quantum Hall systems (FQH) CB ;
hlr ; blqh  mentioned in the introduction have Galileo invariance. It dictates
that the mass appearing in the Landau level spacing ( or cyclotron frequency
$\omega_{c}=eB/mc_{l}$ ) is simply the bare mass $m$ which is not renormalized
by any inter-particle interaction $V(x_{i}-x_{j})$ in Eq.229 hlr ; CB . This
result may be considered as the counter-part in a strongly interacting bosonic
system $\rho_{s}(T=0)=\rho$, namely, it is 100 % superfluid at $T=0$ despite
many atoms are kicked out of the condensate at zero momentum to high momentum
states due to the atom-atom interaction $V(x_{i}-x_{j})$.

As alerted in the last appendix, it is impossible to see how the FQH emerge
from the microscopic Hamiltonian Eq.229 in its first quantization or Eq.239 in
its second quantization. One may need to construct effective actions to study
this emergent quantum phenomena. Both bosonic CB ; blqh ; BLQHwave  and
fermion CS effective action fradkin ; hlr  have been constructed to study the
FQHE by performing a singular gauge transformation to attach even or odd
number of fluxs to electrons. It is important to check if the GI has been kept
in such a singular gauge transformation which despite was claimed to be exact
formally, due to its singularity, could be very dangerous in practice. One may
perform a singular gauge transformation by attaching $\tilde{\phi}=2$ flux
quanta to the electron operator $\psi$:

$\psi_{CF}(\vec{x})=\psi(\vec{x})e^{i\tilde{\phi}\int
d^{2}\vec{x}^{\prime}arg(\vec{x}-\vec{x}^{\prime})\rho(\vec{x}^{\prime})}$
(243)

where
$\rho(\vec{x})=\psi^{\dagger}(\vec{x})\psi(\vec{x})=\psi^{\dagger}_{CF}(\vec{x})\psi_{CF}(\vec{x})$
is the electron or composite fermion (CF) density.

This flux attachment to the electron operator $\psi$ can be formally achieved
by introducing the Chern-Simon (CS) gauge field $a_{\mu}$ coupled to the CF.
The Lagrangian corresponding to Eq.240 in the canonical ensemble becomes CB ;
hlr ; onehalf0 ; blqh

$\displaystyle{\cal
L}_{CF}[\psi]=\psi^{\dagger}_{CF}(-i\partial_{t}-\frac{e}{\hbar}A_{0}-a_{0})\psi_{CF}$
$\displaystyle-\frac{1}{2m}|(\partial_{i}-i\frac{e}{\hbar
c_{l}}A_{i}-ia_{i})\psi_{CF}|^{2}+\frac{i}{4\pi\tilde{\phi}}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}$
$\displaystyle+\int
d^{2}x_{1}d^{2}x_{2}[\psi^{\dagger}(\vec{x}_{1})\psi(\vec{x}_{1})-\bar{n}]V_{2}(x_{1}-x_{2})[\psi^{\dagger}(\vec{x}_{2})\psi(\vec{x}_{2})-\bar{n}]$
(244)

where $(A_{0},A_{i})$ is the external magnetic field and $(a_{0},a_{i})$ are
the Chern-Simon gauge field, $\bar{n}$ is the average density of the
electrons. One may also note that the former is a real EM field, so the speed
of light $c_{l}$ must be kept, while the latter has nothing to do with the
light speed $c_{l}$.

In the kinetic term, it may be convenient to absorbing the external gauge
field into the CS field by defining

$a_{\mu}+A_{\mu}=(a_{0}+\frac{e}{\hbar}A_{0},a_{i}+\frac{e}{\hbar
c_{l}}A_{i})$ (245)

which is a suitable combination of the dynamic CS field $a_{\mu}$ having no
intrinsic velocity with the external gauge field $A_{\mu}$ having the speed of
light as the intrinsic velocity. This right combination make the two different
type of gauge fields transform the same under the GT.

Considering the functional integral measure $\int{\cal D}\psi_{CF}{\cal
D}\psi^{\dagger}_{CF}=\int{\cal D}\psi{\cal D}\psi^{\dagger}$, then it is
straightforward to show Eq.244 is GI under the space-time Eq.247, the CF
operator Eq.242, the external gauge field Eq.232 and the CS gauge field
Eq.248. In fact, all the three terms in Eq.244 are separately GI under the GT.

At the half filling case $\nu=1/2$, the average value of the CS field
$a_{\mu}$ and the external gauge field $A_{\mu}$ just cancels each other
$\langle a_{\mu}+A_{\mu}\rangle=0$ in Eq.245. Eq.244 becomes the starting
point of the HLR theory hlr . Unfortunately, Eq.244 does not respect the
particle-hole (PH) symmetry at $\nu=1/2$ expected in the $m\to 0$ limit hlr ;
MS ; onehalfexp ; onehalf0 ; onehalf1 ; onehalf2 ; onehalf3 ; MSrev . When an
effective action does not get the right symmetry inherited from the
microscopic Hamiltonian, it will problems in many physical quantities onehalf0
; MS ; MSrev . Some ad hoc ways could be developed to fix some of these
problems onehalf2 ; onehalf3 , but to fix all these problems at a fundamental
level must come up with a theory which respects the fundamental PH symmetry.
The root of such fundamental problems may just come from the singular gauge
transformation Eq.243. It is non-perturbative and formally exact, but mix up
all the energy scales of the original Hamiltonian Eq.240. This maybe
problematic because the low energy sectors in the transformed Hamiltonian
Eq.244 may actually correspond to some high energy levels in the original
Hamiltonian Eq.240, so screw up the PH symmetry at the LLL. However, the CS
action can be used to re-derive all the global topological properties of the
wavefunctions, anyon excitations and the fractional statistics from the second
quantization, this is because these topological properties are robust against
these local energy level re-distributions ( This is also the underlying
mechanism of topological quantum computing which is robust against local
perturbations ). Unfortunately, it may not be used to calculate correctly all
the local properties such as the energy gaps, correlation functions etc. This
is because these local properties are usually sensitive to these local energy
level re-distributions, so they may only be accurately calculated in the first
quantization using the wavefunctions jain ; MSfield . As mentioned in the
conclusion, these difficulties maybe directly related to the fact that in
contrasts to the Maxwell theory, it is still unknown how to regularize a CS
theory on a lattice. In this regard, the status of FQH maybe similar to string
theory which is a first quantization theory, while the string field theory
which is a second quantization theory is just too difficult to formulate so
far.

This fact motivated Son to develop an effective 2-component Dirac theory
onehalf1 ; onehalf2  with a finite chemical potential also coupled to the
external gauge field $A_{\mu}$ and the CS field $a_{\mu}$, but with no CS term
for $a_{\mu}$. This Dirac fermion theory has the exact PH symmetry in the zero
Dirac mass limit, therefore can be used to address several difficult problems
onehalf0  suffered in the original HLR theory hlr ; onehalf3 . However, it
breaks the GI of the microscopic Hamiltonian of interaction electrons in an
external magnetic field Eq.239. As shown here, the HLR theory does have the GI
inherited from the microscopic Hamiltonian. Therefore, in terms of the PH
symmetry, the Son’s Dirac fermion theory ( can be called $z=1$ ) has a clear
advantage over the HLR theory. However, in terms of the GI, the HLR theory (
can be called $z=2$ ) seems has an edge over the Son’s Dirac fermion theory.
It has been quite difficult to distinguish the two competing theory in any
experiments onehalfexp ; onehalf3 . On the face value, the lack of GI in Son’s
Dirac theory may be its deficiency. In reality, it may not. As concluded in
the caption of Fig.11, the PH symmetry is a must for any sensible low energy
theory in the LLL, but the space-time symmetry like the GI may not.

In fact, Dirac fermion interacting with Coulomb interaction coupled the CS
gauge theory has been developed to investigate the QH to QH transition field1
; field2 ; Ginvye  theory. It is well known a lattice system is a common one
leading to Dirac or Weyl fermions. Son’s Dirac fermion may emerge from a low
energy effective theory under the LLL projection. If so the LLL projection may
give a completely new mechanism leading to Dirac fermion. In addition to the
Kohn theorem on the inter-Landau level spacing dictated by the GI, the GI does
not have any other implications in the LLL physics which has the exact PH
symmetry at $\nu=1/2$. So an effective theory at the LLL may not even need to
care about the GI anymore. Unfortunately, despite some numerical evidences to
support the nature of Dirac fermion with the $\pi$ Berry phase, such a
microscopic derivation leading to a possible Dirac fermion with a finite
potential is still lacking. In the following appendix H, we show that the bulk
gapped CS theory has the GI, the low energy excitations along the edge derived
from it inherits the GI from the bulk, in contrast to a recent claim made in
QHedge1 .

##  Appendix G  Observing the FQH chiral edge state with $z=1$ in a moving
Hall bar.

As alerted at the beginning of Sec.II, the main difference between the space-
time transformation such as the GT and an internal transformation such as the
gauge transformation is that in the former, the parameters in the Hamiltonian
may need also make the corresponding changes as shown in the context of VII
and VIII in the context of boson Hubbard model, while in the latter, the
parameters do not change. This is the main difference between the two
dramatically different transformations. This subtly is used in this section to
show the FQH Chiral edge has emergent GI. Following the strategies used in the
main text, we will explore its various experimental detectable consequences
from both effective actions and the microscopic calculations.

###  G.1  Effective chiral edge description

In the following, we take the real time formalism, so the bulk CS term
describing the Abelian FQH state at $\nu=1/m$ is

${\cal
L}_{CS}=\frac{m}{4\pi}\epsilon^{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}$
(246)

where $\nu=1/m$ is the filling factor.

It is important to observe that the bulk CS term has no intrinsic velocity
relating the space $x$ and the time $t$. Dimensional analysis
$[a_{0}\partial_{\alpha}]=[a_{\alpha}\partial_{0}]$ shows that
$[a_{0}]=[ca_{\alpha}]$ where $c$ is any velocity.

It can be shown that the bulk CS term is invariant under the GT with any boost
velocity $c$ Eq.247 and Eq.248 which are the real time version of Eq.53 and
Eq.54 adapted to the boost along x-direction:

$\displaystyle t^{\prime}$ $\displaystyle=t,~{}~{}~{}~{}x^{\prime}=x+ct,$
$\displaystyle\partial_{t}$
$\displaystyle=\partial_{t^{\prime}}+c\partial_{x^{\prime}},~{}~{}~{}\partial_{x}=\partial_{x^{\prime}}$
(247)

The CS field transforms as :

$a^{\prime}_{0}=a_{0}+ca_{x},~{}~{}~{}~{}a^{\prime}_{x}=a_{x},~{}~{}~{}~{}a^{\prime}_{y}=a_{y}$
(248)

That is expected, as shown in the last appendix G, the bulk FQH systems is GI.

When deriving the chiral edge effective action, we take a similar approach to
study a moving SF as done in SSS . Assuming the chiral edge state moves along
the x-edge with a drift velocity $v$ ( See Eq.258 from the microscopic
evaluations ), then we take the co-moving $S^{\prime}$ frame moving together
with the edge mode, so that $c=v$ which is the intrinsic velocity of the edge
mode. In this co-moving frame, we also choose the temporal gauge

$a^{\prime}_{0}=0$ (249)

which vanishes in the co- moving frame only.

As shown in wen ; wenedge , this gauge imposes the constraint
$f^{\prime}_{ij}=0$ which can be solved by
$a^{\prime}_{i}=\partial^{\prime}_{i}\phi$. Substituting the solution into
Eq.246 leads to the chiral edge action in the co-moving frame:

$S_{Co-}=\frac{m}{4\pi}\int
dx^{\prime}dt^{\prime}\partial^{\prime}_{t}\phi\partial^{\prime}_{x}\phi$
(250)

Now one need to get back to the lab frame by performing the GT Eq.247 and
Eq.248. Then Eq.248 becomes

$a^{\prime}_{0}=a_{0}+va_{x}=0,~{}~{}~{}~{}a^{\prime}_{x}=a_{x},~{}~{}~{}~{}a^{\prime}_{y}=a_{y}$
(251)

Then Eq.250 transfers back to that in the lab frame;

$S_{L}=\frac{m}{4\pi}\int
dxdt(\partial_{t}-v\partial_{x})\phi\partial_{x}\phi$ (252)

which is the chiral edge state of the Abelian QH state at the filling
$\nu=1/m$ along the edge along x-direction. Then one only need to change the
sign of $v$ of the other edge.

Eq.252 was first derived by Wen wen ; wenedge . What is new here is we derive
it by GT which relates Eq.250 in the co-moving frame to Eq.252 in the lab
frame. Our derivation establishes a deep connection between the GT and the
chiral edge state and also brings additional insights on the connections
between the bulk theory Eq.246 which is GI to the chiral edge action Eq.252
which is not GI.

It can be pushed further to get some new results. First, one can perform the
GT Eq.247 with any boost velocity $c$ along the x-direction on the effective
action Eq.252 again to reach the action in $S^{\prime}$ frame ( still drop the
$\prime$ for simplicity ):

$S_{B,l}=\frac{m}{4\pi}\int
dxdt(\partial_{t}+c\partial_{x}-v\partial_{x})\phi\partial_{x}\phi$ (253)

Setting $c=v$ recovers that in the co-moving frame Eq.250. When $c<v$, the
edge moves along the same direction as in the lab frame. When $c=v$, it
becomes zero in the co-moving frame. Then one need to go beyond the linear
order to see $\omega\sim k^{2}_{x}$ with the edge dynamic exponent $z=2$. When
$c>v$, it reverses the direction. So despite the edge action Eq.253 is GI, the
behaviour of the chiral edge modes does depend on which frame it is observed.
The edge velocity in the other side will be $-(v+c)$ whose magnitude increase
monotonically ( Fig.17 ). In fact, the longitudinal boost can be simply
interpreted as a longitudinal Doppler shift term.

Figure 17:  How are the Abelian chiral edge modes viewed in different moving
frames. $c$ is the longitudinal boost, $v$ is the intrinsic drift velocity.
(a) the Lab frame (b) in a frame moving slower than the drift velocity (c) co-
moving frame with the edge mode. The dot means zero velocity in the top edge.
Then one need to go beyond the linear order to see $\omega\sim k^{2}_{x}$ with
the edge dynamic exponent $z=2$. These higher order terms do break the
emergent GI of the chiral edge action. (d) in a frame moving faster than the
drift velocity. The top edge reverses its direction, then the two edge modes
at the top and the bottom move along the same direction. The bulk stays the
same FQH state. The bulk- edge correspondence is enriched in the moving frame.

Second, one can also study the edge modes when boosting along the normal
direction to the x-edge chen . Then one may need to study from the bulk Chern-
Simon action Eq.246 and re-derive the effective action under such a transverse
boost along y-direction. Extending Eq.247 to a transverse boost leads to:

$\displaystyle t^{\prime}$
$\displaystyle=t,~{}~{}~{}~{}x^{\prime}=x+vt,y^{\prime}=y+ct$
$\displaystyle\partial_{t}$
$\displaystyle=\partial_{t^{\prime}}+v\partial_{x^{\prime}}+c\partial_{y^{\prime}},~{}~{}~{}\partial_{x}=\partial_{x^{\prime}},\partial_{y}=\partial_{y^{\prime}}$
(254)

The CS field transforms as :

$a^{\prime}_{0}=a_{0}+va_{x}+ca_{y},~{}~{}~{}~{}a^{\prime}_{x}=a_{x},~{}~{}~{}~{}a^{\prime}_{y}=a_{y}$
(255)

Eq.249 and Eq.250 remain the same. While Eq.251 changes to

$a^{\prime}_{0}=a_{0}+va_{x}+ca_{y}=0,~{}~{}~{}~{}a^{\prime}_{x}=a_{x},~{}~{}~{}~{}a^{\prime}_{y}=a_{y}$
(256)

Eq.253 changes to

$S_{B,t}=\frac{m}{4\pi}\int
dxdt(\partial_{t}-v\partial_{x}+c\partial_{y})\phi\partial_{x}\phi|_{y=ct}$
(257)

which can be transformed back to Eq.252 after making the substitution Eq.261.
Unfortunately, the physical interpretation of this effective action under the
transverse boot is not clear from itself. However, it becomes clear from the
microscopic calculations as to be shown in the following. In fact, the
transverse boost can be simply interpreted as a transverse Doppler shift term.

###  G.2  The microscopic calculations.

In fact, the chiral edge state moves along the x-edge with the drift velocity
given by:

$v_{x}=c_{l}\frac{E_{y}}{B_{z}}$ (258)

where one notes the three perpendicular directions: the x-edge, the electric
field $E_{y}$ normal to the edge and the external magnetic field perpendicular
to the sample.

Now we will try to understand the physical meanings of the velocity $v-c$ in
the effective action Eq.253. According to Eq.233,
$E^{\prime}_{y}=E_{y}-\frac{c}{c_{l}}B_{z}$, so the drift velocity in Eq.258
changes under the GT as

$v_{x}=c_{l}\frac{E_{y}}{B_{z}}\to v^{\prime}_{x}=v_{x}-c$ (259)

So $v-c$ is nothing but the edge velocity in the $S^{\prime}$ frame, so one
can rewrite Eq.253 as:

$S_{b,l}=\frac{m}{4\pi}\int
dxdt(\partial_{t}-v^{\prime}\partial_{x})\phi\partial_{x}\phi$ (260)

which takes the identical form as Eq.252 in the lab frame when one replace the
edge velocity $v$ in the lab frame by that $v^{\prime}$ in the $S^{\prime}$
frame. In the co-moving frame $v^{\prime}=0$, then one need to go beyond the
linear order to see $\omega\sim k^{2}_{x}$ with the edge dynamic exponent
$z=2$. In fact, these higher order terms always exist, they do break the
emergent GI of the chiral edge action. Eq.259 can be compared to the shift of
the chemical potential in Eq.87 for the $z=2$ SF-Mott transition.

One can also study microscopically the edge states when boosting along the
normal direction to the x-edge chen . According to Eq.233,
$E^{\prime}_{x}=E_{x}+\frac{c}{c_{l}}B_{z},E^{\prime}_{y}=E_{y}$, so the drift
velocity in Eq.258 changes under the GT as

$\vec{v}^{\prime}=c_{l}\frac{(E_{x}\hat{x}+E_{y}\hat{y})\times\hat{z}}{B}=c_{l}\frac{E_{y}}{B_{z}}\hat{x}-c\hat{y}=\vec{v}-\vec{c}$
(261)

where $\vec{c}=c\hat{y}$ stands for the transverse boost. It shows it is
nothing but the GT of the edge velocity in the $S^{\prime}$ frame.

Now the physical interpretation of the effective action Eq.257 under the
transverse boot is quite straightforward from the microscopic perspective: the
drift velocity along the x-edge remains the same as Eq.258. Its y-component is
nothing but the boost velocity. So the two perpendicular motions just decouple
from each other.

It is constructive to compare the Abelian Chiral edge states with that of a
Quantum Anomalous Hall ( QAH ) under a transverse boost QAHinj . The QAH can
be considered as an integer Quantum Hall, but due to the spin-orbital coupling
which breaks the GI explicitly. Under a transverse boost, the chiral edge
state of the Chern insulator transforms non-trivially due to the SOC: For
example, its edge velocity becomes $\sqrt{v^{2}-c^{2}}$ which indicates that
the two perpendicular motions: the spinor edge wave propagation along the
x-edge and the boost along the y- axis are coupled to each other through the
SOC. Indeed, its 2-component spinor $\chi_{\pm}$ also depends on the
transverse boost sensitively. There is a chiral edge state when $c<v$, but not
anymore when $c>v$. This is because the SOC in the QAH violates the GI
explicitly. While the spin is completely polarized in the FQH, it has a GI.

It is also interesting to extend the chiral edge theory of the Abelian FQH to
non-Abelian such as the Read-Moore state or Read-Rezayi state which have
multi-channelschen  with different edge drift velocities. Gravitational
anomaly tenfold  has been studied in the bulk, but not in the edge yet. Here,
we only study the flat bulk and flat boundary, so may not need to worry about
the Gravitational anomaly yet. In the curved space with a generic matric, it
could be a different class of problems chen .

###  G.3  Implications in the bulk through the bulk-edge correspondence

There are recent works called holography mapping or dual between topological
order at dimension $d$ and the symmetry at one dimensional lower at $d-1$, or
reversely, the symmetry is a shadow of topological order of one dimension
higher wengapless ; holgraphicorder . It could be also called topological
orderd+1/symmetryd duality in analogy to $AdS_{d+1}/CFT_{d}$ duality. It aims
to establish the connection between the classification of symmetry or symmetry
breaking at dimension $d$ and the classification of topological orders at one
higher dimension $d+1$. This holographic point of view maybe taken just as an
extension of bulk-edge correspondence in FQH or TQPT: For example, the $1+1$d
Wen’s chiral edge theory ( a $c=1$ CFT with Virasora algebra symmetry ) is the
edge theory of the Abelian Chern-Simon theory which describes the topological
orders of Abelian FQH states. Similarly, the $1+1$d chiral WZW CFT ( with
$SU(2)_{k}$ KM symmetry ) is the edge theory of the Non-Abelian Chern-Simon
theory which describes the topological orders of Non-Abelian FQH such as
Moore-Read state or Read-Razayi state, etc wen . So in the context of FQH, it
is just the CFT/TQFT correspondence.

It is well known that the edge determines the bulk, but not the other way
around. The same bulk may correspond to different boundaries. Namely, the bulk
is the center of the boundary. So this holographic point view from projecting
topological orders to symmetries at one dimensional lower, or the symmetry can
be viewed as the image of topological orders at one higher dimension is
surjective instead of injective. This fact may makes its application subtle.
As shown in Fig. 17, the edge undergos edge reconstruction transition in
different inertial frames. How do the bulk topological orders change in
different inertial frames ? It is possible that the edge reconstruction in a
moving frame may just have changed the edge without affecting the bulk
qualitatively. Namely, as long as the bulk gap remains open, the bulk stay in
the same topological phase. The bulk is always more robust than its
corresponding edge. However, if the bulk gap also vanish, the system may get
to some gapless topological ordered phases. It remains interesting to find how
the bulk gap changes in a moving frame.

###  G.4  Discussions

A recent work QHedge1  pointed out a striking result: the fact that Wen’s
chiral edge theory is not invariant under the GT along the straight edge is an
intrinsic deficiency of theory which need to be fixed in a refined theory. Our
results show that there is probably no deficiencies in the chiral edge theory
bert1982 ; wen ; stone , in contrast to the claim made in QHedge1 . The Wen’s
Chiral edge theory is 253 is emergent GI already. Only at the edge re-
construction QCP in Fig.17c, one need to consider the higher order terms in
the edge mode dispersion which break the emergent GI explicitly.

Another important example is that it was known that the Coulomb interaction is
responsible for the fact that QPT from one FQH phase to another FQH phase has
the dynamic exponent $z=1$ which seems in-compatible with the Galileo
invariance. The $z=1$ may imply that both the quasi-particle and quasi-hole
excitation gaps vanish at the QCP, but may have different gaps away from it.
Indeed, the bare mass $m$ can be sent to $0$ in the chiral edge states and
near the $z=1$ bulk QCP. It hints new emergent space-time structure near the
TPT in the bulk and also in the chiral edge theory which may be interpreted as
a TPT in the real space.

We studied two class of problems. The first class is the SF-Mott transition in
a lattice where the lattice potential plays an important role. One must pay
special attention on how the underlying lattice transforms under the GT. Due
to the $U(1)$ symmetry breaking in the SF side, the number is not conserved,
one must use the second quantization approach to construct the effective
actions to study the symmetry breaking QPT from the Mott to SF phase. As shown
in Sec.VII-VIII, the first quantization wavefunction approach is not
convenient to explore the effects of the exponential factor
$e^{i(k_{0}\sum_{i}x^{\prime}_{i}-NE_{0}t^{\prime}/\hbar)}$ near such a QPT in
different inertial frames. As shown in the main text, it becomes quite
involved to explore how this QPT changes in a moving frame from the effective
actions in the second quantization.

The second class is the FQH in an external magnetic field where the external
magnetic field also plays an important role. One must pay special attention on
how the external magnetic field transforms under the GT. FQH has no symmetry
breaking, the particle number $N$ is conserved. The first quantization in
terms of the wavefunction at a given number of particles $N$ works best for
the bulk gapped topological phases. However, one may still need to construct
effective actions to describe the gapless edge excitations. In this case, the
exponential factor $e^{i(k_{0}\sum_{i}x^{\prime}_{i}-NE_{0}t^{\prime}/\hbar)}$
in Eq.231 has a clear physical meaning: it just stands for the global center
of mass (COM) motion of the gapped edge excitations, topological ordered
phase: $k_{0}/m$ is nothing but the COM motion, $E_{0}$ is the shift of the
ground state energy density. This is indeed the case by our specific
calculations from Wen’s chiral edge theory: its edge velocity along a straight
edge just transforms as a velocity under the GT. In a sharp contrast, despite
it is quite powerful to study the global bulk topological bulk properties, the
second quantization in terms of topological quantum field theory is not
successful yet to explore any local bulk properties. However, when one study
the $z=1$ bulk TQP from one FQH to any FQH by tuning the filling or the
external magnetic field, the wavefunction approach is the first quantization
may not be useful anymore, one is forced to use effective action approach in
the second quantization field1 ; field2 . Then the exponential factor could
play quite non-trivial important roles near such a TPT.

The third class of problem which we only mention in the conclusion section is
the combination of the two classes: in the presence of both lattice and the
external magnetic field, namely, the Abelian flux problem slowgold . It
remains outstanding to explore the emergent space-time in this class.

##  Appendix H  The evolution of the Hamiltonian From High energy physics to
Condensed matter physics

Standard model: $SU(3)\times SU(2)\times U(1)$, strong interaction and
electro-weak interaction, gluons/quarks, $W^{\pm},Z_{0}$ and photons. Lorentz
invariance at $(T=0,\mu=0)$ and CPT symmetry. Einstein’s special relativity
dictates that Different inertial frames are related by the Lorentz
transformation (LT). The standard model must be Lorentz invariance (
Covariance ) at $(T=0,\mu=0)$ and CPT symmetry. There is No quantum phase
transition at $T=0$. Different inertial frames are related by LT. The ground
state or vacuum stay the same in different inertial frames The speed of light
$c_{l}$ is the only relevant velocity scale. In particle/nuclear physics
experiments such as Fermi lab, J-lab or RHIC, what one can tune is: increase
the scattering energy, tune the scattering angle, what is the temperature ?
One may just take it to be zero.

Of course, any $T>0$ breaks the LI. Any $\mu>0$ also breaks the LI. In the
RHIC experiments at $(T>0,\mu>0)$ there is a quark Confinement and de-
confinement ( first-order ) transition at a finite $T_{dec}\sim 144$ Mev $\sim
10^{12}$ K to the Quark-gluon plasma phase in the QCD. When the quarks start
to detach from the nucleons, one must introduce a finite chemical potential
$\mu>0$.

In the Electro-weak interaction, there is a symmetry breaking $SU(2)\times
U(1)\to U(1)_{EM}$ which leads to the Higgs particle, the massive
$W^{\pm},Z_{0}$ and the massless photons $A_{\mu}$. So it also has a finite
temperature phase transition $T_{EW}\sim 10^{16}$ K above which the symmetry
breaking is restored.

How does $T$ depend on different inertial frames was found in Eq.162. How do
$T_{dec},T_{EW}$ depend on different inertial frames should follow the same
equation. Unfortunately, It is extremely difficult to go to a different
inertial frame at a speed close to $c_{l}$ to measure any such effects.

How do the particle physics evolve into condensed matter ? As the energy gets
lower and lower, the ions start to detach from electrons, then one must
introduce a finite chemical potential $\mu>0$. It breaks the LI. Then the ions
may start to form a lattice to break translational invariance: Then What one
can tune in a solid is much more than in a particle accelerator: Temperature
T, magnetic field B, electric field E, Pressure ( strain ) P, electron
density, n, by gate voltage….. Then there are quantum phase transitions at
$T=0$ tuned by these parameters, of course, also corresponding finite
temperature phase transitions. This is the essence of P. W. Anderson: More is
different !

There are two kinds of condensed matter systems:

1\. Continuous systems where the underlying lattice is not important.

Superfluid Helium 4: one tune the pressure P to drive a 1st order transition
from SF to a solid transition at $T=0$ and $p=p_{c}$ Translational symmetry
breaking, the relevant velocity is the sound velocity $v_{s}\ll c_{l}$. LT
reduce to the Galileo transformation (GT) in the $c_{l}\to\infty$ limit.

Fractional quantum Hall systems: tune B or n to change the filling, there are
TPT from one quantum Hall Plateau to another, TPT without symmetry breaking,
the relevant velocity is the Fermi velocity $v_{F}\ll c_{l}$.

As said in Appendix F,G,H, the LT reduce to a generalized Galileo
transformation upto the linear order in $v_{F}/c_{l}\ll 1$

2\. Lattice systems which are much more involved

A lot of QPT in lattice systems at $T=0$: SF-Mott transition, AFM to VB
transition,…… Intrinsic velocity is determined by the Wannier functions in a
lattice structure $v_{L}\ll c_{l}$.

In the $c_{l}\to\infty$ limit, the LT reduce to GT. How does the QPT at $T=0$
or the thermal phase transition at $T>0$ change under the GT or any low
velocity limit of LT ? A missing component of P. W. Anderson: More is
different ! This question was addressed in the present work in the context of
the SF-Mott transition. In contrast to the particle scattering experiments, it
is not too difficult to go to a different inertial frame or in a moving sample
at a low speed $c\ll c_{l}$ to measure any such effects. So the moving
velocity $c$ become a new tuning parameter to drive new QPT.

##  Appendix I  Combining statistical mechanics with Quantum mechanics,
special relativity, quantum field theory and general relativity

In standard undergraduate courses, we learned Quantum mechanics, statistical
mechanics, special relativity, then then the combination of Quantum mechanics
and statistical mechanics leads to Quantum statistical mechanics which
resolved many paradoxes suffered in classical statistical mechanics such as
the third law of thermodynamics, the specific heat of a solid $C_{v}\sim
T^{3}$ which leads to the support for a quantum theory. Then the combination
of Quantum mechanics and the special relativity leads to the Dirac equation
whose low velocity limits leads to the electron spin, SOC, etc. But so far, no
combination of statistical mechanics with the special relativity yet.

This difficulty can be circumvented by promoting Dirac equation to
relativistic quantum field theory. There are two ways to do the quantization:
Canonical quantization and path integral, the latter seems more powerful in
the relativistic quantum field theory. Then when getting to a finite
temperature, one can start to write a partition function of the QFT by going
to imaginary time, this way directly and naturally incorporate the special
relativity, QFT and statistical mechanics. It also establishes the deep
connections between the imaginary time and the inverse temperature.

$\tau=it=\hbar/k_{B}T$ (262)

which is, in fact, compatible with the Heisenberg uncertainty relation $\Delta
E\cdot\Delta t\sim\hbar$ with $\Delta E\sim k_{B}T$ in an equilibrium system
with a temperature $T$. This connection seems can only be established by the
path integral instead of canonical quantization. This fact shows that the path
integral approach is beyond the canonical quantization when combing quantum
field theory with the statistical mechanics. It has been used to establish the
LT law of the temperature in Eq.162.

In classical general relativity or the Unruh radiation discussed in Sec.IX-C,
the temperature can be identified in the metric part containing the time which
is nothing but that of a black hole solution of the Einstein equation. It
leads to the black-hole Thermodynamics. The Ads/CFT automatically leads to the
identification of the $T$ in the boundary CFT with that of the bulk black hole
in the AdS geometry. The Hawking-page transition in the bulk corresponds to
the confinement to de-confinement transition in the quark-gluon plasma phase
in the boundary QCD. The SYK models show that the quantum information
scrambling rate with the Lyapunov exponent $\lambda_{L}=2\pi/\beta$ in the SYK
model in the boundary saturates that of the bulk black hole in the JT gravity.
So Ads/CFT, with the relativistic CFT on the boundary in general, the SYK
which is non-relativistic in particular, establish the one to one connection
between the quantum statistical mechanics in a temperature $T$ on the boundary
with that in the bulk in the AdS geometry.

##  Appendix J  Relation and difference between a closed system with a
reservoir and an open system with a bath

It is important to distinguish the Reservoir from the environment ( or called
bath sometimes ).

In the former which is called a closed system, the system is an equilibrium
and unitary system which exchange energies and particle numbers with the
reservoir ( Fig.2,15 ). One only need to introduce a chemical potential $\mu$
to take care of the effects of the reservoir. Its density matrix evolution is
just the Heisenberg equation of motion. The systems’s energy is always
conserved. This belongs to the grand-canonical ensemble in the text-book
statistical mechanics.

Taking the simplest example, the free fermi gas at a temperature $T$. How do
the non-interacting fermions reach a temperature $T$, if they do not interact
with each other ? This is due to the Reservoir which gives the temperature $T$
in either canonical or grand canonical ensemble. All the electrons interact
with the reservoir, but not with each other, to acquire the same temperature
$T$. Will the reservoir mediate the mutual interactions between or among
fermions ? No, the R is so large than it will not do it. Just like two persons
standing on the earth will not interact with each other due to both kicking
the ground.

While in the latter which is called an open system, it is an non-equilibrium
and non-unitary system. Its density matrix evolution has a Linblard term There
are all kinds of interactions between the system and the bath degree of
freedoms: for example, in a cavity system, the cavity photons may leak into
the photon bath outside the cavity. or qubits couple to their environments.
This process leads to the de-coherence which may be described by a Master
equation.

Taking a simple example: how a moving condensate such as an exciton superfluid
in semi-conductor bilayer emit photons ehbl1 ; ehbl2 ; ehbl3 . Then the
photons act as the environment. Obviously, when the condensate is moving, the
pattern of emitted photons also change. Then one can just detect the emitted
photons instead of performing the scattering experiments in Fig.15. Another
example is the qubit-bath interaction discussed in mark .

Due to the existence of the reservoir, one must distinguish the case A with
the case B. Einstein’s special theory of relativity is about two point
particles, but a many-body system is a trinity with the reservoir playing an
important role. So our theory only applies to the case A, but not the case B.
Unfortunately, the experimentalists may do case B easily, much more difficult
to do the case A.

It may be instructive to do a more fundamental re-derivation on the case A
where there is a relative motion between the sample and the reservoir in both
canonical and grand-canonic ensemble. What we did here is just perform a
Lorentz or Galileo transformation on the system without touching the
reservoir, then this relative motion indeed shows many interesting effects,
especially near a quantum phase transition in the system. We think this maybe
the most direct way to look at the effects of this relative motion between the
system and the reservoir. The existence of a lattice in the system makes a
huge deal here. Indeed, it would be nice to have an independent way to do it
in a future publication.

##  Appendix K  Brief comments on the case B: an inertial moving frame

In this work, we addressed the case A in Fig.2 and Fig.15 where there is a
relative motion between the system and the reservoir. Then one only need to
perform the LT on the degree of freedoms in the system. In the case B in Fig.2
and Fig.15, there is no relative motion between the sample and the reservoir,
only the observer is moving, so one must perform LT on the degree of freedoms
in both the system and the reservoir. One knows how to perform a LT or GT to
the system, but not the combined system of the system plus the reservoir yet.
So case (B) remains unknown, but interesting to explore.

However if the system has a Galileo invariance such as superfluid Helium 4 in
the normal phase or the FQH, then we may assume that there is no particle
transfer between the system and the reservoir in the case (B). The chemical
potential change in Eq.87 need to be offset by the corresponding change in the
reservoir, so Eq.87 may be replaced by ( for a moving sample, see the comment
in evenso  )

$\tilde{\psi}=\psi
e^{-ik_{0}y},~{}~{}~{}~{}\tilde{\mu}=\mu+\frac{mc^{2}}{2}=\mu+\frac{c^{2}}{4v^{2}_{y}}\rightarrow\mu$
(263)

If we apply this rule to Eq.115, then it seems except the simple plane-wave
factor in the boson or fermion field which leads to the shift of the BEC
momentum or the shift of the FS center, there is no other effects in a
formally GI many-body lattice system, namely no wedding cake rearrangements in
Fig.10 or no the FS enlargement in the Appendix E-2. As alerted in the last
appendix, it may be instructive to do a more fundamental derivation on the
case B also. The experiment in this case may be more easily achieved than the
case A by just moving the detector.

There are previous result that Eq.162 should be replaced by

$\displaystyle\frac{T}{T_{0}}=1/\sqrt{1-(\frac{v}{c_{l}})^{2}}>1$ (264)

This can not be right. When $v/c_{l}\to 1^{-}$, $T\to\infty$ approaches to
infinity. This can not be right, because the QFT breaks down at such a high
temperature.

There are also previous result that Eq.162 should be

$\displaystyle\frac{T}{T_{0}}=1$ (265)

This can not be right either, because it is GT result holding only in the
$c_{l}\to\infty$ limit.

In fact, Eq.162 was first achieved by the late giants such as Planck,
Einstein, Pauli and Laue Planck ; Einstein ; Pauli ; Laue  from completely
different approach: namely by incorporating the special relativity into the
Thermodynamics. Unfortunately, at late stages of his life, Einstein became
confused on what is the correct way to do such an incorporation, so not sure
about the final answer. Based on various different phenomenological
assumptions on the LT transformation properties of different thermodynamic
quantities, all the three results Eq.162, 264, 265 can be reachedthermo1 .
These confusions or disputes can only be resolved from the fundamental ( or
microscopic ) statistical mechanics. The thermodynamic is just an effective (
or macroscopic ) description of the fundamental ( or microscopic ) statistical
mechanics. The quantum statistical mechanics naturally leads to all the zeroth
to the third law of thermodynamics. Especially, the third law is completely
due to the quantum nature of matter at $T=0$.

As explained in appendix J, the confusions suffered by Einstein in combining
the special relativity with the Thermodynamics can be circumvented by putting
QFT at a finite $(T,\mu)$. of course, these modern developments happened after
the late greats, of Planck, Einstein, Pauli and Laue. This modern approach
naturally lead to the correct answer Eq.162. The fundamental quantity is the
partition function ${\cal Z}$ in Eq.164 in Sec.IX-B. Indeed, the GKP-W
relation in the Ads/CFT is also through the equality of the partition function
on both sides. The Lorentz invariance of the ${\cal N}=4$ SUSY Yang-Millis
$SU(N)$ gauge field on the boundary just corresponds to the diffeomorphism on
the $AdS^{5}\times S^{5}$ in the bulk. The duality between the SYK and the JT
gravity is also through the equality of the partition function on both sides
in the low energy limit, both of which are described by the Schwarzian.

It is important to point out that at $\mu=0$, both case A and B are the same
in the relativistic QFT case. Because the PH symmetry dictates that there are
nether particles nor holes in the vacuum state despite it is full of quantum
fluctuations. However, any $\mu>0$ represents the existence of a reservoir
which exchange particles with the system, then A and B become different.
Eq.162 only holds for the case A. For a relativistic thermal QFT at a finite
$(T,\mu)$ such as the quark-gluon phase in RHIC, one may also assume $(T,\mu)$
do not change under the LT in case B. If so, Eq. 265 may hold in case B. But
it is not known if this assumption is valid. We leave this for a future work.
Note that Eq.162, 264, 265 become the same when taking $c_{l}=\infty$ with the
difference appearing only in $(v/c)^{2}$.

It is tempting to think A and B are same for the particle-hole symmetric $z=1$
in Fig.9. Indeed, fixed $n$ and fixed $\mu$ are in general different in the
original BH model Eq.1 and Fig.9: at fixed $\mu$, then $n$ changes, while at
fixed $n$, then $\mu$ changes. Only at the PH symmetry point fixing $\mu$ and
fixing $n$ become the same. So there is a tantalizing possibility that case A
and B are the same at this enlarged PH symmetry line. However, this is in-
correct due to that the ” vacuum ” is the Mott state with $\langle n\rangle=1$
instead of a real vacuum. The effective action to describe the P-H excitations
above the Mott phase is a ”pseudo-Lorentz invariant ” instead of ”Lorentz
invariant ”, so our results still only apply to case A. Intuitively, if it had
also been alos applied to case B, then one would have reached a paradox: an
insulating Mott phase in the lab frame turns into a dissipationless SF in a
moving frame.

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  * (54) This strategy maybe contrasted to the composite fermion approach hlr ; onehalf0 : the even number of flux attachments transfer the fermions to composite fermions is a non-perturbation treatments, then the residual Chern-Simon gauge field fluctuations will be treated perturbatively.
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  * (60) This intrinsic velocity $v_{c}=t_{0}/t_{b1}$ is completely different from the $v_{y,z=1}$ listed in Eq.139. The former is solely detrmined by the Wannier functions, while the latter also depends on the bare boost valocity $v$ through Eq.132. This dramatic qualtitative difference has also been stressed in the introduction.
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  * (64) One should not confuse this PH current or P ( or H ) current with the moving velocity of the atoms loaded on the lattice in teh moving sample. For example, well inside the Mott state, every lattice site captures one atom, so the atom moves together with the lattice. So the atom in the Mott phase also moves with the same velocity $\vec{v}$. However, due to the lack of PH excitations in the $z=1$ case and the particle or hole excitations in the $z=2$ case, the Mott phase carries no net ( or mean field ) PH current or P ( or H ) current respectively. See Sec.VIII for the differences between the P/H excitations and the original bosons.
  * (65) The Mott state in the moving sample turns into the SF phase should not be too counter-intuitive. As shown explicitly in Eq.27 for $z=1$ where the $C$ symmetry was spontaneously broken by the BSF phase and Eq.87 for $z=2$ where the $C$ symmetry was explicitly broken by the Hamiltonian, the moving just increases the effective chemical potential $\tilde{\mu}$ and leads the Mott to the BSF transition with $z=2$. This may be contrasted to the original Mott to the SF transition tuned by $\mu$ in Fig.9. See also the conclusion Sec.XI-1.
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  * (83) The Weyl SOC in cold atom systems is the coupling between a momentum shift induced by external lasers and the pseudo-spin which is two hyperfine structures, so it is just a quantum simulated SOC which is not a relativistic effect. However, just like in the atoms explained in the Appendix A, the Rashba SOC in non-centro symmetric semi-conductor systems is a relativistic effect modified by the crystal field. If so, one may need to consider the time contraction effects upto the $(v/c_{l})^{2}$ order of the LT.
  * (84) Even for the impurities randomly distributed over $\vec{R}_{i}$, the potential due to the impurities $V_{1}(\vec{x})=\sum_{\vec{R}_{i}}v(\vec{x}-\vec{R}_{i})$ completely breaks the translational invariance, but is still Galileo invariant. The moving impurity case is outlined in SSS . This is not suprising, becasue the translation and GT belong to two different sectors of the Galileo group $R^{3}\rtimes SO(3)$. See also Appendix A.
  * (85) In principle, there could also be long-range regarded interaction between electrons mediated by the spacial component of the dynamic gauge field $\vec{A}$ in the Coulomb gauge. These retarded interactions do violate the Galileo invariance, but they are much smaller than the instantaneous Coulomb interaction mediated by the dynamic time component $A_{0}$, so can be neglected up to the order of $(v_{F}/c_{l})^{2}$ where $v_{F}$ is the intrinsic Fermi velocity of the electrons. So the Galileo invariant Eq.111 remains a very good approximation up to the order of $(v_{F}/c_{l})^{2}$.
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  * (88) The same transformation from the field operator to the tight-binding limit was used in lightatom1 ; lightatom2  to derive the coupling of photon scattering to the valence bond order parameter. This fact provides the mechanism for the possible light or atom scattering detection of VBS or VBS supersolid orders.
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  * (92) For the fermionic Hubbard model with spin-1/2, there is a C- symmetry at half-filling $n=1$. The PH transformation is simply $f_{i\sigma}\to f^{\dagger}_{i\sigma}$. For the bosonic Hubbard model, there is also a C- symmetry at integer-filling $n=1$. Unfortunately, it is still not known what is the PH transformation in terms of the boson operator $b_{i}$. This fact may also be related to new space-time emerge from such a $z=1$ QCP, see Sec.VIII and Fig.11 for the detailed discussions on this important point.
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