#  Topological phase transitions, invariants and enriched bulk-edge
correspondences in fermionic gapless systems with extended Fermi surface

Fadi Sun1,2 and Jinwu Ye1,2,3 1 The School of Science, Great Bay University,
Dongguan 523000, Guangdong, China
2 Institute for Theoretical Sciences, Westlake University, Hangzhou 310024,
Zhejiang, China
3 Department of Physics and Astronomy, Mississippi State University, MS 39762,
USA

###### Abstract

Topological phases and topological phase transitions (TPT) are among the most
fantastic phenomena in Nature. Here we show that injecting a current may lead
to new topological phases, especially new gapless topological metallic phases
with extended Fermi surfaces (FSs) through novel class of TPTs in the bulk or
the boundary. Specifically, we study the quantum anomalous Hall (QAH) system
in a square lattice under various forms of injecting currents. In addition to
the previously known Chern insulator ( which will be called even Chern
insulator here ), band insulator and band metal (BM), we find three new
topological phases we name as: the gapped odd Chern insulator (Odd CI), the
gapless odd Chern metal (Odd CM) and even Chern metal (Even CM). The Chern
number may not be effective anymore in characterizing the topological gapless
phases with extended FS. It is the Hall conductance which acts as the new
topological invariant in such gapless systems. Its jump is a universal integer
or non-integer across the even CM/BM or odd CM/BM TPT respectively where there
is also a corresponding TPT in the Longitudinal (L-) edge modes. The Odd/even
CM to BM transition is a novel class of TPT without any non-analyticity in the
ground state energy density. This presents the first example of a TPT which is
not a quantum phase transition (QPT). The original bulk-edge correspondence is
enriched into bulk/Longitudinal (L-)/Transverse (T-) edge correspondence. The
L- edge reconstruction may happen earlier, later or at the same time as the
bulk TPT respectively in the even CI/odd CI/odd CM sequence with the edge
dynamic exponent $z_{L}=3$, in the even CI/even CM/odd CM sequence with
$z_{L}=2$ or a direct even CI/odd CM with a flat edge. The disappearance of
the T- edge always happen at the same time as the bulk TPT with a universal
edge critical behaviour. We classify all the possible bulk and edge TPTs and
also evaluate the thermodynamic quantities such as the density of states,
specific heat, compressibility and the Wilson ratio in all the phases and also
their quantum scaling forms near all these TPTs. The methods may be applied to
explore new topological phases of other Hamiltonians in any lattice in any
dimension in any forms of injecting current. Doing various experiments by
injecting different sorts of currents may become an effective way to drive
various topological phases to new topological gapped or gapless metallic
phases through novel bulk or edge TPTs.

##  I Introduction

The Anomalous Hall Effect(AHE) due to the Berry phase of itinerant electrons
in a quantum Ferromagnet in real space AHE1  or in momentum space AHE2 ;
Haldane2004  has been under intense investigations since more than 20 years
ago. It involves the spin-orbit coupling which couples the orbital motion of
electrons to the spin polarizationAHE3 . The AHE in a metallic Ferromagnet is
in general, not quantized, so can take any value. However, it can be quantized
in an insulator, called Quantum Anomalous Hall (QAH) effect. The original QAH
was proposed haldane  even earlier in a honeycomb lattice model soon after the
discovery of integer and fractional quantum Hall effects (FQHE). Since the
first experimental realization of the quantum anomalous Hall (QAH) effect in
Cr doped Bi(Sb)2Te3 thin films QAHthe ; QAHexp , it has also been observed in
many other materials such as both Cr doped and V doped (Bi,Sb)2Te3 films
QAHexp2 . More recently, it was also discovered in the twisted bilayer
graphene TBGAHE . It was realized with cold atoms with the fermions 40K in
haldaneexp . The bosonic version of QAH was also implemented with spinor
bosons 87Rb in 2dsocbec . The connections between the non-interacting
fermionic QAH and the interacting bosonic version of QAH and various quantum
or topological phase transitions in interacting bosonic systems were explored
in QAHboson .

It turns out that the FQHE and QAH are just two early members of the vast
number of topological quantum matter kane ; zhang ; tenfold ; wenrev . The
topological quantum matter is one of the main themes in condensed matter
physics, lattice gauge theories and topological quantum field theories kane ;
zhang ; tenfold ; wenrev . It not only contain new physical concepts, rich and
profound phenomena with deep mathematical structures, but may also have some
potential industry applications such as dissipationess transmissions, quantum
communications and topological quantum computing.

There have been flurries of classifications of both gapped and gapless
topological phases kane ; zhang ; tenfold ; wenrev . In the known non-
interacting gapless case, the gapless source comes from the Dirac fermions,
Weyl fermions and various line/ring nodes in the bulk. We call this class of
bulk gapless system with Fermi points or lines, linear dispersion $\omega\sim
k$ and vanishing density of states (DOS) $D(\omega)\sim\omega$ as having the
dynamic exponent $z=1$. There are still little works on how to characterizing
a bulk topological gapless system with extended Fermi surface (FS), neither
much work on topological phase transitions (TPT) between various topological
phases. Furthermore, how to define topological invariants to characterize a
bulk gapless system with extended FS and what are the associated edge modes
remain outstanding problems. Because these extended FS originates from point-
like Fermi points with quadratic band touching $\omega\sim k^{2}$ and constant
DOS $D(\omega)\sim const.$ at a TPT, we call such class of gapless systems as
having the bulk dynamic exponent $z=2$ which are the main focus of the present
work.

There are two complementary approaches to address these outstanding problems.
One way is to use SPT or SET to classify by various mathematical tools such as
Co-homology, K-theory (for non-interacting fermions), Co-bordisms and tensor
categories or its higher order versions tenfold ; wenrev . Another is to start
from a concrete parent Hamiltonian hosting various topological phases, then
explore its topological phases and TPTs under various deformations breaking
different symmetries haldane ; dimer1 ; dimer2 ; dimer3 ; Kit1 ; Kit2 ;
tenfold . Here we take the second approach, but limit to non-interacting
fermionic systemshaldane . We focus on the simplest and widely experimentally
realized topological phase: the quantum anomalous Hall (QAH) phase haldane ;
QAHthe ; QAHexp ; QAHexp2 ; haldaneexp ; QAHboson  and also establish its
connection to the un-quantized AHE. Specifically, we start from a QAH
Hamiltonian which hosts some known gapped topological phases such as Chern
insulator and band insulator and see how a Parity (P-) breaking injecting
current Fig.1a,b or a P- persevering chemical potential or energy dispersion
drives the parent QAH Hamiltonian into new topological gapped or gapless
phases through new classes of topological phase transitions (TPT). We also
define the new ”topological bulk invariants” of these $z=2$ topological
gapless phase, investigate the associated new edge state structure, explore
enriched bulk-edge correspondence and new longitudinal/transverse edge-edge
correspondence. During this establishment, we propose a classification of the
QAH insulators and AHE metals in Fig.2.

Figure 1:  New topological phases and bulk or edge TPTs are generated by (a)
Injecting a longitudinal current into the QAH sample in a strip geometry. (b)
Injecting a transverse current into the QAH sample in a strip geometry.

This work contains 3 parts. In part I, under an injecting current (Fig.1a,b),
we evaluate the Hall conductivity at both zero and finite temperatures. Then
we study the non-analytical behaviours of the ground state energy and map out
the global phase diagram in the Zeeman field/injecting current plane Fig.3.
The 2 old phase turn into 4 phases: the old (even) Chern insulator with the
Chern number ${\rm Ch}_{-}=\pm 1$ and quantized Hall conductivity
$\sigma_{xy}=\pm 1\times e^{2}/h$ when $c<v$, a new $z=2$ Odd Chern metal
(OCM) phase with ${\rm Ch}_{-}=\pm 1$, extended Fermi surfaces (FS) and un-
quantized universal Hall conductivity $|\sigma_{xy}|=v/c<1$ (in the unit of
$e^{2}/h$) when $c>v$. The emergence of the particle and hole FS in the Odd
Chern metal phase is responsible for the reduced Hall conductivity which is
still independent of $h/t$ and many other microscopic details of the
Hamiltonian. The old band insulator with Ch${}_{-}=0,\sigma_{xy}=0$ and a new
band metal phase with Ch${}_{-}=0,\sigma_{xy}=0$ with extended FSs. The TPT
from the Chern insulator to the band insulator is a 3rd order with the dynamic
exponent $z=1$, while both the Chern insulator to Odd Chern metal, and the
band insulator to band metal are second order ones with $z=2$. Strikingly, the
TPT from the Odd Chern metal to the band metal is novel: it has no non-
analyticity to infinite order in the ground state energy, but the Chern number
of the band jumps $\Delta{\rm Ch}_{-}=\pm 1$ and the Hall conductivity has a
universal non-integer jump $\Delta\sigma_{xy}=v/c$. This presents the first
example of a TPT from both bulk and associated edge properties, but not a QPT
in conventional wisdom sachdev ; tqpt ; weyl . We also evaluate various
thermodynamic quantities such as the density of states, specific heat,
compressibility and Wilson ratio at a finite $T$ in all the 4 phases and also
their quantum scaling forms near all these bulk TPTs.

Figure 2:  The six-fold hexagon phase diagram of the 6 quantum/topological
phases: 3 insulators: BI: band insulator, Even CI: even Chern insulator, Odd
CI: odd Chern insulator; 3 metals: BM: band metal, Even CM: even Chern metal,
Odd CM: odd Chern metal are sitting in the 6 vertices of the hexagon. $z=1,2$
are the dynamic exponent of the bulk QPT, $z_{L}=2,3$ are the dynamic exponent
of the longitudinal edge TPT. The bulk TPT from the even or odd CM to the BM
is infinite order with the unit jump in the Hall conductance
$\Delta\sigma_{H}=\pm 1$ or non-integer $\Delta\sigma_{H}=\pm v/c<1$
respectively. It is $\Delta\sigma_{H}$ which plays the role of topological
invariants in the 3 metal phases. It maybe the first example of a TPT which is
not a QPT. The TPT/QPT from the even CI to the odd CM usually goes in two
steps: even CI/odd CI/odd CM (the longitudinal-edge TPT happens earlier than
the bulk) or even CI/even CM/odd CM (the L- edge TPT happens later than the
bulk) with the odd CI or the even CM as the intermediate phase respectivelySSS
, but it could also go directly (the L-edge TPT happens at the same time as
the bulk QPT) with $z=2$ and completely flat L- edge shown in Fig.3. The
direct TPT/QPT may be viewed as a fine tuning or a multi-critical point. The
Transverse (T-) edge mode disappears always at the same time as the bulk TPT
with its velocity vanishing as $\sqrt{v^{2}-c^{2}}$. All the metallic phases
have no T-edge mode. Due to the absence of $C-$ symmetry, the even CM to the
even CM transition is absent. The Universal Hall conductance jump from the odd
CM to its T-reversal odd CM partner is twice of that from the odd CM to the
BM. Some BM may also have non-vanishing Chern number and associated floating
edge modes above the bulk modes. But practically, it is the same phase as the
BM except it has a larger AHE. If no lines connecting two phases, then it
means there is no direct transition between the two except through some fine
tuned multi-critical point.

Then we study the edge states in a strip geometry when the injecting current
is either parallel (longitudinal) or transverse to the edge, from both the
lattice model and from the effective low energy theory. During the bulk TPT
from the even Chern insulator to the Odd Chern metal, the edge in a parallel
injection also undergoes very unusual edge TPT: Inside the Chern insulator
$c<v$ before the TPT, the two edge modes on the two opposite sides of the
sample (Fig.1c) flow along the opposite directions $v_{L}>0,v_{R}<0$, but one
side flow slower than the other side $|v_{R}|<|v_{L}|$. Only the edge mode
contributes to $\sigma_{H}=\pm 1$. At the TPT $c=v$, one edge mode becomes
completely flat with zero velocity $v_{R}=0$ which can be viewed as a fine
tuning to a multi-critical point (Fig.2). Inside the odd Chern metal after the
TPT $c>v$, the two opposite sides of the sample start to flow along the same
direction $v_{L}>0,v_{R}>0$, but one side flows much slower than the other
side $|v_{R}|\ll|v_{L}|$. Both bulk and edge contribute to the transport.
However, in a transverse boost, the edge mode velocity behaves as
$\sqrt{v^{2}-c^{2}}$ before the TPT $c<v$, but were squeezed out after the TPT
$c>v$, so no T-edge modes in the Odd Chern metal anymore. Only the bulk
contributes to the transport. So the $z=2$ Odd Chern metal has the exotic edge
modes with $v_{L}v_{R}>0$ in a parallel injection (Fig.1a), but not in a
transverse injection (Fig.1b). We may call this new phenomenon a new
longitudinal ( L-) /transverse( T-) edge correspondence under the current
injection. As alerted in the last paragraph, the novel bulk TPT from Odd Chern
metal to band metal can also be viewed from the edge in a parallel boost: the
former has one edge mode with $v_{L}v_{R}>0$, the latter none. Obviously, the
Odd Chern metal with $z=2$ is clearly different from the previously known
topological semi-metals such as Dirac metal or Weyl semi-metal with $z=1$
tenfold .

It maybe necessary to stress that one need to distinguish two mathematical
quantities which have different physical meanings inside different phases: the
Chern number $Ch_{-}$ versus the Hall conductivity $\sigma_{xy}$. They are the
same in the Chern insulator, but different in the Odd Chern metal: the Chern
number is defined for a band only, therefore independent of the filling of the
band bosonicQAH . Even so, it still has a clear physical meaning when the
boost is parallel to the edge in a strip geometry even in a Odd Chern metal:
it stands for the contribution from the edge states in both Chern insulator
and Odd Chern metal. Inside the Chern insulator $c<v$, this is the only
contribution to the $\sigma_{xy}$, but inside the Odd Chern metal $c>v$, due
to the gapless extended FS in the bulk, the bulk states also contribute, so
one have the decomposition $\sigma_{xy}=1+(v/c-1)=v/c<1$ where the first term
comes from the edge state which is quantized, the second term is from the bulk
FS which is un-quantized. So when the boost is parallel to the edge, there is
an enriched bulk-edge correspondence inside the topological Odd Chern metal:
the Chern number defined for the bulk band gives the quantized edge
contribution, independent of the fillings of the band. But the Hall
conductivity $\sigma_{xy}$ receives the total contribution from the edge + the
bulk, depends on the fillings, so not quantized. However, when the boost is
perpendicular to the edge, the Chern number defined for a band only has
mathematical sense, but no physical meaning in the Odd Chern metal phase. In
this case, there is no edge state anymore, so no contributions from the edge,
$\sigma_{xy}=0+v/c$ completely comes from the bulk.

In Part II which is on the gauge-invariant current injection case, one need to
consider the combined effects of the two currents, the first is a NN $n=1$
Non-Abelian gauge-invariant current term, the second is a NNN $n=2$ (higher
order) current term. The first term can be treated exactly in a transformed
basis by combining it with the NN hopping and SOC term in the original QAH
Hamiltonian. It leads to a very counter-intuitive effect: an band insulator
near the TPT from the Chern insulator to the band insulator in the lab frame
turns into a Chern insulator, but not the other way around. In the transformed
basis, if treating the NNN $n=2$ current term as an independent one just like
a NNN injecting current, the results achieved on NN $n=1$ injecting current in
the Part-I can be applied here with some notable differences: (1) The Global
phase diagram changes to Fig.24 where one can see that a new topologically
gapped phase we named odd Chern insulator phase intervening between the the
even CI and the odd CM in Fig.3. It has the same bulk properties as the even
Chern insulator, but with different edge properties. Its L- edge modes satisfy
the exotic relation $v_{L}v_{R}>0$ similar to the odd CM, its T- edge modes
satisfy the conventional relation $v_{L}v_{R}<0$ just as in the even Chern
insulator. So the direct TPT from the Even CI to the odd CM in Fig.3 splits
into two in Fig.24: In the L- edge, the edge mode undergoes its own edge TPT
from the even Chern insulator to the odd Chern insulator with an L- edge
dynamic exponent $z_{L}=3$ before the bulk TPT from the odd CI to the odd CM.
(2) The TPT from the odd CI to the odd CM does not happen at a constant $c=v$,
but depends on the Zeeman field $h/t$ in a lobe shape. The Odd Chern metal’s
Hall conductivity is not just given by $v/c$, but also depends on the Zeeman
field $h/t$. The band metal’s Hall conductivity is not zero anymore, but also
depends on the Zeeman field $h/t$. (3) Remarkably, the Hall conductivity jump
from the odd Chern metal to the band metal remains the same universal number
as the $n=1$ case. So does the jump from the odd Chern metal to its T-reversal
odd Chern number partner which is twice as that from the OCM to the BM. This
could be a universal salient feature of the odd Chern metal (Fig.2) ! This
fact establish the Hall conductivity jump as the new ”topological invariants”
characterizing the gapless topological phases with extended FS. (4) Due to the
$n=2$ NNN feature, the Doppler shift in the 4 nodes become the same sign. This
is contrast to the $n=1$ NN injecting current case where the 4 nodes have 2+
and 2- Doppler shifts.

As a byproduct, taking some results from moving , for the QAH or AHE due to
the artificially generated SOC which is a non-relativistic effect, we find
that a Galileo boost $v$ on a lattice leads to the $n=1$ NN gauge-invariant
current and $n=2$ NNN current. So the results achieved in Sec.V and Sec.VI can
also be applied to a moving sample. So if an insulator is band or Chern type
may depend on if the observer is moving relative to the lattice. Then after
absorbing the $n=1$ NN gauge-invariant current into the QAH Hamiltonian by a
unitary transformation, the NNN current term in the transformed basis changes
sign after some critical boost velocity $v_{c}\sim 1cm/s$ solely determined by
the Wannier functions. So does the Doppler shift near the four nodes. All
these new features are subject to the scattering measurements in the moving
frame in Fig.34.

In terms of the SPT language, despite the original QAH Hamiltonian breaks the
time-reversal symmetry explicitly, it still has a charge (C-) conjugation
symmetry and also a parity (P-) symmetry. An injecting current or a moving
sample breaks P-symmetry, but keeps the C-symmetry. Because it is a non-
interacting system, the C-symmetry is never broken during the evolution, so it
is the C-symmetry protected topological phases and TPTs driven by the $n=1$
and $n=2$ current.

In part III, we study the P-preserving deformation such as an energy
dispersion. It leads to a even Chern metal phase which has the same bulk
properties as the odd Chern metal. But the Universal Hall conductance jump
from the even CM to the band metal is an integer number. It also has a
dramatically different L-edge mode properties than the odd CM: the L-edge mode
satisfies the conventional $v_{L}v_{R}<0$ instead of the exotic
$v_{L}v_{R}>0$. A real material contains both P-breaking and P-persevering
components and is examined in Sec. VIII. We find that as the parameter
changes, the generic AHE will be either in even-like Chern metal or odd-like
Chern metal: there is a edge reconstruction between the two with a L- edge
exponent $z_{l}=2$. We propose a complete classification of AHE metals leading
to un-quantized QAH effect as the Band metal (BM), odd Chern metal and even
Chern metal, while the gapped phases leading to quantized QAH effect as BI, CI
and odd CI (Fig.2). The BM is nothing but the previously well studied one
contributing to the un-quantized AHE AHE2 ; Haldane2004 . While the itinerant
metal contributing the AHE due to the Berry phase acquired by electrons moving
in the non-coplanar spin texture in the real space in a Ferromagnet does not
fall into this non-interacting classification AHE1 .

Experimentalists got used to apply magnetic field, electric field, or strain,
pressure, neutron scattering, muon spin rotation, etc. Here, we show that
injecting various forms of currents may be an effective way to bring out a lot
of information on the topological phases, also drive them to new phases
through novel TPTs. Alternatively, for SOC which is a non-relativistic effect,
putting the sample in a strip shape to move in a trail, then perform various
scattering experiments such as neutron, X-ray scattering or ARPES may also be
helpful for artificially generated QAH systems.

The rest of the paper is organized as follows: we will first study the QAH
under a P-breaking injecting current ( Fig.1a,b ). We will study the bulk
properties of both systems via lattice theory and continuum effective theory
in the thermodynamic limit, then investigate the corresponding edge properties
in a strip geometry in both longitudinal and transverse edge via also both
lattice theory and the continuum effective theory. Then in the first two
appendices, we investigate the QAH under a P-preserving chemical potential or
energy dispersion by the similar approaches. The P-breaking and P-preserving
Hamiltonian are two different kinds of deformations leading to different bulk
phases and topological phase transitions, also different edge properties. A
real material contains both and will be examined in Sec.VIII. In Sec.IX, we
summarize ”Topological invariants” and the enriched bulk/L-edge/T-edge
correspondence in gapless fermionic systems with extended Fermi surface. The
experimental detections are analyzed in Sec.X.

##  II The bulk properties: the microscopic lattice theory

The quantum anomalous Hall model on a square lattice takes the form

$\displaystyle
H_{\text{QAH}}=-\\!\sum_{i}[c_{i}^{\dagger}(t\sigma_{z}-it_{s}\sigma_{x})c_{i+x}\\!+\\!c_{i}^{\dagger}(t\sigma_{z}-it_{s}\sigma_{y})c_{i+y}$
$\displaystyle\quad+h.c.]-h\sum_{i}(n_{i\uparrow}-n_{i\downarrow})+U\sum_{i}n^{2}_{i}-\mu\sum
n_{i}\>.$ (1)

Without loss of generality, we assume that $t>0$ and $t_{s}>0$. In this work,
we focus on the non-interacting limit $U=0$, but the chemical potential can be
zero or non-zero.

Figure 3: The global Phase diagram of the Lattice Hamiltonian Eq.2 with a
fixed $t_{s}/t=1$. The phase diagram does depend on the ratio $t_{s}/t$. For
simplicity, we take $t_{s}/t=1$ in this figure only. Then the black phase
boundary is a straight line. But $t_{s}/t$ is taken as any value in the
continuum effective actions. The topological metal phases are named as
A1,A2,B1,B2,C1,C2. Green means the Particle (P-) FS, the yellow Hole (H-) FS.
The inserts plot the Fermi surface of different metal phases within the 1st
Brillouin Zone, where we use the notation A1/B1 for the phase boundary between
A1 and B1, etc. The first box stands for the band insulator where there is no
FS, the second is the metal-insulator transition where there are two Fermi
points (one P-, one H-) emerging with the quadratic band touchings (so the
dynamic exponent $z=2$), they evolve into two (one P-, one H-) Fermi pockets
in the band metal. Then they touch quadratically at $(0,0)$ in the A1-CM/BM
QCP with not any non-analyticity. A1 corresponds to Fig.9d, A1/B1 to Fig.13d.
The P- FS expands to A1/C1, the bottom part of the FS becomes straight in
A1/B1/C1. B1 corresponds to Fig.13e. The collision of the two P- FS from B1
leads to B1/C1. B1/B2 differs from B1 by the conic touching of P- and H- Fermi
pocket at $(0,\pi)$ and $(\pi,0)$ with not any non-analyticity. Then the split
between the P- and H- FS from C1/C2 leads to C1. See also Fig.12, 13 and
Fig.16, 17 near $h/t=\pm 4$ and $h/t=0$ respectively in the continuum
calculation. See also Fig.23 for higher order $n=2$ boost.

Under an injecting current (for its motivation from a Galileo transformation,
see appendix E), one obtains the injected Hamiltonian:

$\displaystyle
H_{\text{inj}}=H_{QAH}+\sum_{i}(it_{b,x}c_{i}^{\dagger}c_{i+x}+it_{b,y}c_{i}^{\dagger}c_{i+y}+h.c.)$
(2)

The non-injected Hamiltonian Eq.1 has the Charge conjugation C-symmetry and
the parity P-symmetry $k_{x}\to-k_{x},k_{y}\to-
k_{y};\sigma_{x}\to-\sigma_{x},\sigma_{y}\to-\sigma_{y}$. but no Time Reversal
T-symmetry. The injected Hamiltonian Eq.2 breaks the P-symmetry, still
respects the C-symmetry: $\sigma_{x}\mathcal{K}$,($\mathcal{K}$ denotes the
complex conjugate),

$\displaystyle(\sigma_{x}\mathcal{K})H(\mathbf{k})(\sigma_{x}\mathcal{K})=-H(-\mathbf{k})$
(3)

The C-symmetry guarantees a relation between upper band and lower band
$E_{+}(\mathbf{k})=-E_{-}(-\mathbf{k})$, and the Berry curvature
$\Omega_{+}(\mathbf{k})=-\Omega_{-}(-\mathbf{k})$. We also have
$\Omega_{\pm}(\mathbf{k})=\Omega_{\pm}(-\mathbf{k})$. Note that the QAH
belongs to the Class A, but still with the particle number conserved. Because
this C-symmetry does not conserve the particle number, so it can not be
understood as the existence of an anticomutating symmetry operators, otherwise
the system would belong to the Class D instead of the Class A tenfold .

For simplicity, we study the injection along the $y$ direction, thus
$t_{b,x}=0$ and $t_{b,y}=t_{b}$. It can be easily generalized to any injection
direction. In momentum space, the Hamiltonian becomes

$\displaystyle H_{\text{inj}}=\sum_{k}c_{k}^{\dagger}\\{$
$\displaystyle\\!-[h+2t(\cos k_{x}+\cos k_{y})]\sigma_{z}+2t_{s}\sin
k_{x}\sigma_{x}$ $\displaystyle\\!+2t_{s}\sin k_{y}\sigma_{y}-2t_{b}\sin
k_{y}\sigma_{0}\\}c_{k}$ (4)

The Diagonalization of Eq.(4) leads to the two bands

$\displaystyle E_{\pm}(\mathbf{k})=-2t_{b}\sin k_{y}$
$\displaystyle\pm\sqrt{[h+2t(\cos k_{x}\\!+\\!\cos
k_{y})]^{2}\\!+\\!4t_{s}^{2}(\sin^{2}k_{x}\\!+\\!\sin^{2}k_{y})}$ (5)

Since $E_{+}(\mathbf{k})\geq E_{-}(\mathbf{k})$ always holds for a fixed
$\mathbf{k}$, we will call the $E_{+}$ the upper band and the $E_{-}$ the
lower band. When $t_{b}$ is sufficiently small, it is in a insulating phase;
When $t_{b}/t_{s}$ is sufficiently large, it is in a metallic phase, with hole
surface is given by $E_{-}(\mathbf{k})=0$ and electronic surface given by
$E_{+}(\mathbf{k})=0$.

The critical $t_{b}$ are determined by the minimization problem
$\min_{k}E_{+}(\mathbf{k};t_{b})=0$. In the full range of $t_{b}/t_{s}$, the
Fermi surfaces (FS ) can be rather complicated, see Fig.3.

When $4t>h>0$, it is in the Chern insulator phase. At the critical
$t_{b}/t_{s}=1$ and the Fermi points are located at
$(\pi,\arccos(\frac{2t-h}{2t}))$, it moves into the A1 Odd Chern metal phase,
then as $t_{b}$ increases further to $t_{b}=\sqrt{t_{s}^{2}+h(h+4t)/4}$, the
previous Fermi points grow-up into a Fermi surface with the emergence of the
other Fermi point at
$(0,\arccos[\frac{-t(2t+h)}{2(t^{2}+t_{b}^{2}-t_{s}^{2})}])$ ( See A1/B1 ), it
moves into the B1 Odd Chern metal phase. When $t>t_{b}$ increases further,
these two Fermi surfaces can collide at B1/C1 to move into the C1 phase.

When $h>4t$, it is in the band insulator phase. At the critical
$t_{b}=\sqrt{t_{s}^{2}+h(h-4t)/4}$, the Fermi points are located at
$(0,\arccos(\frac{2t}{h-2t}))$, it moves into the band metal phase.

The TPTs in Fig.3 can be classified into 4 classes: (1) The linear band
touching due to the Dirac points are 3rd TPT with $z=1$. In the $t_{b}=0$, it
is a Dirac point with emergent Lorentz invariance. The $t_{b}\neq 0$ drives it
into a boosted Dirac point. (2) The emergency of the P- or H- Fermi point in
insulator/metal, A1/B1 are quadratic band touching 2nd order TPT with the
dynamic exponent $z=2$. (3) Band metal/A1, B1/B2,C1/C2, even the M point B1/B2
(C1/C2) can be understood as the the conic band touching between the P- and
the H- FS. They do not have any non-analyticity in the ground state energy.
(4) The P-/P- FS ( equivalently H-/H- FS ) collision in A1/C1, B1/C1 are TPT
with universal sub-leading scalings weyl ; class3 .

Table 1:  The classification of bulk QPT or TPT. 4 classes TPTs | CI-BI | CI-OCM | OCM-BM | P-FS/P-FS collision
---|---|---|---|---
Dynamic exponent | $z=1$ | $z=2$ | No | saddle point cone
Order | 3rd | 2nd | Infinite order | 3rd, 5th..
Scaling | Yes | Yes | No | sub-leading
$R_{W}=T\kappa_{u}/C_{v}$ | $\frac{2\ln 2}{9\zeta(3)}$ | $\frac{3}{\pi^{2}}$ | $\frac{3}{\pi^{2}}$ | sub-leading

where we only list one representative of CI-BI, CI-CI in the second column,
CI-OCM, BI-BM, OCM-OCM in the 3rd column, OCM-BM, OCM-OCM in the 4th column,
H-FS/H-FS in the 5th column. The Odd CM can also be replaced by the Even CM
except the ECM-ECM transition does not exist as demonstrated in Sec.VII and
VIII ( See Fig.29 and Fig.37 ). The Wilson ratio (WR) was evaluated in the
lattice in Sec.II-C-3 and in the continuum effective theories in Sec.III-A-3c
and III-B-3c. The WR for all the gapped phases ( CI and BI )
$R_{W}=(T/\tilde{\Delta})^{2}$ ( See Eq.19 ) is not listed in the Table. In
fact, $R_{W}=\frac{3}{\pi^{2}}$ also holds inside the all the gapless phases (
CM and BM ), because the OCM to the BM transition has no analyticity anyway.
We did not list the edge reconstruction transition from the CI to odd CI in
Fig.23 and even CM to odd CM in Fig.29 with the longitudinal dynamic exponent
$z_{L}=3$ and $z_{L}=2$ respectively. For the bulk or edge properties of these
phases, see Table II.

In fact, as elucidated in Fig.3, if one look at $B1\to B1/C1\to C1$, the B1/C1
QCP is reached by the collision of the two P- FS (or equivalently, the two
H-FS) from B1, so this class of TPT can be similarly investigated by the
method developed in weyl , universal subleading scalings can be derived. One
can also similarly study the TPT from $A1\to A1/C1\to C1$. In the following,
we are mainly interested in the experimentally most relevant case
$t_{b}/t_{s}$ is not too large, so focus on A1,A2,B1,B2 phases and class-1 and
class-2 TPTs, but do not discuss C1 and C2 phase and the class-3 TPT in any
details. The A1,A2,B1,B2 phases can be distinguished by $\sigma_{xy}$ and
Fermi surfaces (FS) topology: The A1 phase has $0<\sigma_{xy}<1$ and the FS is
just one part, the B1 phase has $0<\sigma_{xy}<1$ and the FS consists of two
disconnected parts. A2(B2) have the same FSs as A1(B1), but with opposite sign
of $\sigma_{xy}$.

###  II.1  The quantum Hall response at zero and finite temperature:
Topological phase transitions (TPT).

In order to calculate Berry connections $\mathbf{A}$ and Berry curvatures
$\Omega$, it is convenient to rewrite the boosted QAH Hamiltonian Eq.(2) in
terms of $(d_{0},\vec{d})$ vectors:

$\displaystyle
H(k)=d_{0}(k)\sigma_{0}+d_{x}(k)\sigma_{x}+d_{y}(k)\sigma_{y}+d_{z}(k)\sigma_{z},$
$\displaystyle d_{0}(k)=-2t_{b}\sin k_{y},\>\mathbf{d}=(d_{x},d_{y},d_{z})$
$\displaystyle d_{x}(k)=2t_{s}\sin k_{x},\>d_{y}(k)=2t_{s}\sin k_{y},\>$
$\displaystyle d_{z}(k)=-[h+2t(\cos k_{x}+\cos k_{y})]$ (6)

Then the Berry Connections and Berry curvatures can be evaluated as:

$\displaystyle A_{\pm,i}(\mathbf{k})$
$\displaystyle\\!=\\!i\langle\pm,\mathbf{k}|\partial_{k_{i}}|\pm,\mathbf{k}\rangle=\frac{(d_{y}\partial_{k_{i}}d_{x}-d_{x}\partial_{k_{i}}d_{y})}{2|\mathbf{d}|(|\mathbf{d}|\pm
d_{z})}$ $\displaystyle\Omega_{\pm,xy}(\mathbf{k})$
$\displaystyle\\!=\\!\partial_{k_{x}}\\!A_{\pm,y}\\!-\\!\partial_{k_{y}}\\!A_{\pm,x}\\!=\\!\mp\frac{1}{2|\mathbf{d}|^{3}}\mathbf{d}\\!\cdot\\!\partial_{x}\mathbf{d}\\!\times\\!\partial_{y}\mathbf{d}$
(7)

Since the $t_{b}$ term is proportional to the unit matrix $\sigma^{0}$, so it
does not affect the eigenvectors, then the Berry connections and Berry
curvatures are exactly the same as $t_{b}=0$ case. Note that $\sigma_{xy}$ is
related to $\Omega_{\pm,xy}$ and $\sigma_{yx}=-\sigma_{xy}$ is related to
$\Omega_{\pm,yx}$. For later calculations on Hall response, we will only
consider $\sigma_{xy}$ and drop subscript $xy$ in $\Omega$.

####  II.1.1 Zero temperature Hall conductance and TPT

As long as $h\neq 0,\pm 4t$, the upper band and the lower band are well
separated, thus one can calculate the Chern number of the lower band via
integrating the Berry curvature $\Omega_{-}(\mathbf{k})$ over the entire
Brillouin zone (BZ) which is a torus $\mathbb{T}^{2}$:

$\displaystyle{\rm
Ch}_{-}\\!=\\!\frac{1}{2\pi}\int_{\mathbb{T}^{2}}d^{2}\mathbf{k}\>\Omega_{-}(\mathbf{k})=\begin{cases}0,&|h/t|>4;\\\
+1,&4>h/|t|>0;\\\ -1,&0>h/|t|>-4;\\\ \end{cases}$ (8)

In the insulating phase, only the lower band is full occupied, thus the Chern
number is the zero temperature Hall conductance in unit $e^{2}/h$, that is
$\sigma_{H}={\rm Ch}_{-}$. In the metallic phase, both band are partially
filled, thus the zero temperature Hall conductance reduces to $\sigma_{H}={\rm
Ch}_{-}\times|t_{s}/t_{b}|$.

In any case, the zero temperature Hall conductance can be expressed as

$\displaystyle\sigma_{H}$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{T}^{2}}d^{2}\mathbf{k}\>\sum_{s=\pm}\Omega_{s}(\mathbf{k})\Theta(-E_{s}(\mathbf{k}))$
$\displaystyle={\rm Ch}_{-}\times\min(1,|t_{s}/t_{b}|)\>.$ (9)

As shown in Fig.4, the Hall conductivity show plateau structure during the
scanning of $h$: it is zero for band insulator and band metal, show a plateau
with value $-t_{s}/t_{b}$ for A2 phase and B2 phase, show a plateau with value
$t_{s}/t_{b}$ for A1 phase and B1 phase, becomes zero again for band insulator
and band metal.

We conclude that the zero temperature Hall conductance in the metallic phase
is reduced by a factor $|t_{s}/t_{b}|<1$ relative to its quantized value in
the Chern insulator phase ( Fig.4 ). In both the insulting phase and metallic
phase, they are not that sensitive to the microscope details. As to be shown
in the following sections, they can all be reproduced via the analytical
evaluations of relevant integrals in the continuum theory.

####  II.1.2 Finite temperature Hall conductance

At finite temperature, one only need to replace the step function in Eq.9 by
the Fermi distribution function

$\displaystyle\sigma_{H}(T)$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{T}^{2}}d^{2}\mathbf{k}\,\sum_{s=\pm}\Omega_{s}(\mathbf{k})f(E_{s}(\mathbf{k}))$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{T}^{2}}d^{2}\mathbf{k}\,\Omega_{-}(\mathbf{k})[f(E_{-}(\mathbf{k}))-f(E_{+}(\mathbf{k}))]$
$\displaystyle={\rm
Ch}_{-}+\frac{1}{\pi}\int_{\mathbb{T}^{2}}d^{2}\mathbf{k}\Omega_{+}(\mathbf{k})f(E_{+}(\mathbf{k}))$
(10)

where $f(E)=1/[\exp(E/T)+1]$. The finite $T$ Hall conductances as varying
parameters of $h/t$ or $t_{b}/t_{s}$ are plotted in Fig.5

Figure 4: The zero temperature $T=0$ Hall conductance $\sigma_{H}$ as a
function of $h/t$ for fixed $t_{b}/t_{s}=0,1.2$ in (a) Across the 3rd order
TPT with $z=1$ from BI/CI, CI/CI, CI/BI, the $\sigma_{H}(T=0)$ has an integer
jump $\Delta\sigma_{H}=-1,2,1$ respectively, (b) As shown in Fig.9, there is
no non-analytical behaviours ( infinite order ) for the BM/A2, B1/B2, A1/BM
TPTs, but $\sigma_{H}(T=0)$ still has a universal non-integer jump
$\Delta\sigma_{H}=-t_{s}/t_{b}=-1/1.2=-5/6,2\times 5/6,5/6$ respectively,
independent of any other microscopic details. When across the 2nd order TPT
with $z=2$ from the BI/BM, A2/B2, and A1/B1, $\sigma_{H}(T=0)$ has no changes.
In a generic case, the BM should contribute to a un-quantized AHE ( See Fig.24
and Table 1 ), but it vanishes in this particular $n=1$ case due to some fine
tuning. As a function of $t_{b}/t_{s}$ for fixed $h/t=1,5$ in (c) a 2nd order
QPT with $z=2$ and (d) $\sigma_{H}=0$ in both the BI and BM. We expect the
class-3 TPT of B1/C1 and A1/C1 weyl  has no changes in $\sigma_{H}(T=0)$
either. See also Fig.14 and Fig.18 near $h/t=\pm 4$ and $h/t=0$ respectively
in the continuum calculations.  Figure 5: The finite temperature $T>0$ Hall
conductance $\sigma_{H}$ as a function of $h$ for fixed $t_{b}/t_{s}=0,1.2$ in
(a) and (b) and as function of $t_{b}/t_{s}$ for fixed $h/t=1,5$ in (c) and
(d), where we also choose $t=t_{s}=1$, thus $T=0.1$ means $k_{B}T/t=0.1$.

###  II.2  The ground-state energy and Quantum phase transitions (QPT)

We are interested in the non-analytical behaviours in the ground-state energy
density on the lattice which can be numerically calculated via

$\displaystyle
E_{\text{GS}}^{\text{(lat)}}\\!\\!=\\!\\!\int_{\mathbb{T}^{2}}\\!\\!\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}[E_{+}(\mathbf{k})\Theta(-E_{+}(\mathbf{k}))+E_{-}(\mathbf{k})\Theta(-E_{-}(\mathbf{k}))]$
(11)

The quantum phase transitions can be driven either by tuning by the Zeeman
field $h/t$ or the boost $t_{b}/t$.

####  II.2.1 QPTs driven by the boost $t_{b}/t$

In the insulating phase, the lower band is full occupied, the ground-state
energy density is

$\displaystyle
E_{\text{GS}}^{\text{(lat)}}\\!=\\!\\!\int_{\mathbb{T}^{2}}\\!\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}E_{-}(\mathbf{k};t_{b})\\!=\\!\\!\int_{\mathbb{T}^{2}}\\!\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}E_{-}(\mathbf{k};t_{b}\\!=\\!0)$
(12)

where the last equality is due to that the $t_{b}$-dependent part in Eq.5 is
odd in $\vec{k}$, so vanishes after integration over entire Brillouin zone
(BZ). Thus the $E_{\text{GS}}^{\text{(lat)}}$ is independent of $t_{b}$ in the
insulating phase. However, in the metallic phase, due to the partial filling
of the upper and lower bands, the $E_{\text{GS}}^{\text{(lat)}}$ does depend
on $t_{b}$. Thus near the insulating-metallic transition driven by $t_{b}$ at
some critical value $t_{b,c}$, the ground-state energy density
$E_{\text{GS,I}}^{\text{(lat)}}=const$ in the insulating side
$(t_{b}<t_{b,c})$,
$E_{\text{GS,M}}^{\text{(lat)}}\propto(t_{b}-t_{b,c})^{\alpha}$ in the
metallic side $(t_{b}>t_{b,c})$. The index $\alpha>0$ indicates the order of
the phase transition is $\lceil\alpha\rceil$. For example, if $\alpha\leq 1$,
then is the first order transition with a cusp at $t_{b}=t_{b,c}$; if
$1<\alpha\leq 2$, then is the second order QPT shown in Fig.6,7.

Figure 6: The ground-state energy density $E_{\text{GS}}^{\text{(lat)}}$ as
the function of $t_{b}/t$, with fixed $t=t_{s}=1$ and $h=0.5$ in Fig.3. From
(a),(b),(c) are $E_{\text{GS}}^{\text{(lat)}}$’s zeroth-/first-/second-order
derivative with respect to $t_{b}$. The vertical dashed lines correspond to
the critical $t_{b}$, which separate the Chern insulator, A1,B1 and C1. An
obvious second order QPT discontinuity is shown at $t_{b}=1.0,1.25$ both with
$z=2$, the cusp at $t_{b}=1.44$ shows a third order one in the class-3 weyl
respectively. See also table I.  Figure 7: The same as Fig.6, but with $h=4.5$
in Fig.3. The only vertical dashed line corresponds to the critical $t_{b}$
which separates the band insulator from the band metal. An obvious second
order QPT discontinuity with $z=2$ is shown at $t_{b}=1.25$.

Because any phase transitions are independent of how they are approached or
scanned, so scanning $t_{b}/t$ or $h/t$ should research consistent results.
That is indeed the case as shown in the following section.

####  II.2.2 QPTs driven by $h/t$

We first study the TPT between band insulator and Chern Insulator. Because
$E_{GS}$ is $t_{b}$ independent in the insulating phases, so the TPTs are the
same as the no-boost $t_{b}=0$ case. In Fig.8 we numerically evaluate the
ground-state energy Eq.(12) as a function of scanning $-6<h/t<6$ with fixed
$t_{b}=0$ which is identical to the $t_{b}/t<1$ case. An obvious third order
discontinuity is shown between the band insulator and the Chern insulator.

Now we study TPTs between metals where $E_{GS}$ becomes $t_{b}$ independent.
Scanning $-6<h/t<6$ with fixed $t_{b}>t_{s}$, we meet consecutively band
insulator, band metal, A2 Odd Chern metal, B2 Odd Chern metal, B1, A1, band
metal, band insulator. In the Fig.9, a clear second order discontinuity
appears between band insulator/band metal A2/B2 phase, and A1/B1 phase.
However, no any order discontinuity is found The band metal/A2, B2/B1, A1/band
metal transitions, so they could be just infinite-order TPT. But they can
still be distinguished by the Hall conductance shown in Fig.4.

Figure 8: The ground-state energy density $E_{\text{GS}}$ as function of $h$,
with fixed $t=t_{s}=1$, $t_{b}=0$. From left to right are $E_{\text{GS}}$’s
0th-/1st-/2nd-/3rd-order derivative with respect to $h$. The vertical dashed
lines correspond to the critical $h$, which separate band insulator, Chern
insulator (-1), Chern insulator (+1), band insulator. An obvious 3rd order QPT
discontinuity with $z=1$ is shown at $h/t=-4,0,+4$.  Figure 9: The ground-
state energy density $E_{\text{GS}}$ as function of $h$, with fixed
$t=t_{s}=1$, $t_{b}=1.2$. From left to right are $E_{\text{GS}}$’s
0th-/1st-/2nd-/3rd-order derivative with respect to $h$. The vertical dashed
lines correspond to the critical $h$, which separate band insulator, tribal
metal, A2, B2, B1, A1, band metal, band insulator. A clear 2nd order QPT
discontinuity with $z=2$ is obtained between band insulator/band metal A2/B2,
A1/B1 phase. However, no non-analytical behaviours are found for the BM/A2,
B2/B1, A1/BM transitions, even in the third-order derivatives and beyond. The
underlying physical mechanisms for these infinite order TPT are explored from
the continuum effective theory in Sec.III-A and III-B.

###  II.3  Thermodynamic Quantities

We will first discuss the density of states ( DOS ), then use it to compute
several experimentally measurable quantities.

####  II.3.1 The density of states (DOS)

From the Hamiltonian (6), one can find the Matsubara Green’s function

$\displaystyle G(\mathbf{k},i\omega_{n})$
$\displaystyle=[i\omega_{n}-H(\mathbf{k})]^{-1}$
$\displaystyle=\frac{P_{+}(\mathbf{k})}{i\omega_{n}-E_{+}(\mathbf{k})}+\frac{P_{-}(\mathbf{k})}{i\omega_{n}-E_{-}(\mathbf{k})}$
(13)

where
$P_{\pm}=\frac{1}{2}[1\pm\mathbf{d}\cdot\boldsymbol{\sigma}/|\mathbf{d}|]$ are
the projection operators onto the $s=\pm$ upper/lower bands and
$P_{+}+P_{-}=1$.

The total DOS $D(\omega)=D_{+}(\omega)+D_{-}(\omega)$ is

$\displaystyle D(\omega)$
$\displaystyle=-\frac{1}{\pi}\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\mathfrak{Im}\operatorname{Tr}[G_{R}(\mathbf{k},\omega)]$
$\displaystyle=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}[\delta(\omega-
E_{+}(\mathbf{k}))+\delta(\omega-E_{-}(\mathbf{k}))]$ (14)

It should be convenient to introduce the DOS for each band

$\displaystyle
D_{\pm}(\omega)=\int\frac{d^{2}\mathbf{k}}{4\pi^{2}}\delta(\omega-
E_{\pm}(\mathbf{k})),$ (15)

The DOS on a lattice contain some van-Hove singularities when $\omega$ far
away from $0$, which makes numerical calculation on DOS time consuming. The
lattice DOS is plotted in Fig.10. For $\omega\sim 0$, in a metallic phase
where $t_{b}/t>1$, the DOS is nearly, but not exactly a constant.

Figure 10:  At fixed $t=t_{s}=1$ and $h=1$, the density of states (DOS) of
Eq.(4) at different $t_{b}/t_{s}$ values: (a) $t_{b}/t_{s}=0.5$ in Chern
insulator, (b) $t_{b}/t_{s}=1.0$ on the QCP between Chern Insulator and A1 Odd
Chern metal with $z=2$, (c) $t_{b}/t_{s}=1.16$ inside the A1 Odd Chern metal.
The cusps are Van-Hove singularities. Note that (c) inset is showing the total
density of states is nearly, but not exactly flat near $\omega=0$. Compare to
Fig.15 and Fig.19 near $h/t=\pm 4$ and $h/t=0$ respectively in the continuum
calculation.

####  II.3.2 The specific heat and compressibility

Here, we will make use of the DOS to evaluate the two conserved quantities,
then compute the Wilson ratio. The Helmholtz free energy density

$\displaystyle
F(T)=-T\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\ln[2(1+\cosh(E_{+}(\mathbf{k})/T))]$
$\displaystyle\\!=\\!E_{\text{GS}}\\!+\\!\\!\int\\!\\!\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}(|E_{+}\\!(\mathbf{k})|\\!-\\!T\ln[2(1\\!+\\!\cosh\\!\frac{E_{+}\\!(\mathbf{k})}{T})])$
(16)

where the zero temperature part of $F(T)$ is nothing but the ground-state
energy density $E_{\text{GS}}$ in Eq.11.

The specific heat (at a constant volume)
$C_{v}(T)=-T\frac{\partial^{2}F}{\partial T^{2}}$:

$\displaystyle
C_{v}(T)=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\frac{[E_{+}(\mathbf{k})]^{2}}{T^{2}[1+\cosh(E_{+}(\mathbf{k})/T)]}$
$\displaystyle=\\!\\!\int\\!\\!\frac{d\omega\>\omega^{2}D_{+}(\omega)}{T^{2}[1+\cosh(\omega/T)]}=\\!\\!\int\\!\\!\frac{d\omega\>\omega^{2}D(\omega)}{2T^{2}[1+\cosh(\omega/T)]}$
(17)

The (isothermal) uniform compressibility
$\kappa_{u}(T)=\sum_{s=\pm}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{\partial
f(E_{s,k})}{\partial E_{s,k}}$ is

$\displaystyle\kappa_{u}(T)=\int\frac{d^{2}k}{(2\pi)^{2}}\frac{[E_{+}(\mathbf{k})]^{2}}{T^{2}[1+\cosh(E_{+}(\mathbf{k})/T)]}$
$\displaystyle=\\!\int\\!\frac{d\omega\>D(\omega)}{2T[1+\cosh(\omega/T)]}=\\!\int\\!\frac{d\omega\>D_{+}(\omega)}{T[1+\cosh(\omega/T)]}$
(18)

Below we discuss their low temperature behaviours.

In the gapped phases, i.e. Chern insulator or band insulator, we denote the
gap as

$\displaystyle\tilde{\Delta}=\min_{k}E_{+}(k)>0,$ (19)

then $D(\omega)=0$ for entire $-\tilde{\Delta}<\omega<\tilde{\Delta}$ and
$D(\tilde{\Delta})=D_{2}>0$ at the gap edge ( Fig.10a ), thus
$C_{v}(T)=2D_{2}\tilde{\Delta}^{2}T^{-1}e^{-\tilde{\Delta}/T}$. Similarly,
$\kappa_{u}(T)=2D_{2}Te^{-\tilde{\Delta}/T}$. Both are exponentially
suppressed in $T$.

In the gapless phases, i.e. A or B Odd Chern metal phase or band metal phase,
we have $\tilde{\Delta}=\min_{k}E_{+}(k)<0$ and $D(0)=D_{0}>0$ ( Fig.10c ),
thus $C_{v}(T)=\frac{\pi^{2}}{3}D_{0}T$, linear in $T$. Similarly,
$\kappa_{u}(T)=D_{0}+\cdots$ where the sub-leading $T$ dependence $\cdots$ can
be best evaluated in the continuum theory to be evaluated in Sec.III.

For the 3rd order QCPs with $h=0,\pm 4t$ and $t_{b}/t<1$, we have
$\tilde{\Delta}=\min_{k}E_{+}(k)=0$ and $D(\omega)=D_{1}\omega$ for small
$\omega$ just like a Dirac fermion, thus $C_{v}(T)=9\zeta(3)D_{1}T^{2}$ where
$\zeta(3)\approx 1.2021$ is the Riemann Zeta function:

$\displaystyle\zeta(s)=\sum^{\infty}_{n=1}1/n^{s}=\frac{1}{\Gamma(s)}\int^{\infty}_{0}\frac{x^{s-1}}{e^{x}-1}dx$
(20)

Similarly, $\kappa_{u}(T)=2\ln 2D_{1}T$ where $\ln(2)\approx 0.6931$.

For the other QCPs which are 2nd order, i.e. $\tilde{\Delta}=0$ and
$t_{b}/t_{s}>1$, or at the boundary between $A1$ and $B1$, we always have non-
zero $D(0)=D_{0}>0$( Fig.10b ) , thus $C_{v}(T)=\frac{\pi^{2}}{3}D_{0}T$.
Similarly, $\kappa_{u}(T)=D_{0}+\cdots$. where again the sub-leading $T$
dependence $\cdots$ can also be best evaluated in the continuum theory to be
evaluated in Sec.III.

####  II.3.3 The Wilson ratio

The Wilson ratio is defined as the ratio of the two conserved quantities
$R_{W}=T\kappa_{u}/C_{v}$ which has the following low temperature behaviours.

In the gapped phases, i.e. Chern insulator phase or band insulator phase,
$R_{W}=(T/\tilde{\Delta})^{2}$ where $\tilde{\Delta}$ is given in Eq.19.

In the gapless phases, i.e. A or B Odd Chern metal phase or band metal phase,
$R_{W}=\frac{3}{\pi^{2}}\approx 0.3040$.

For the 3rd order QCP with $z=1$ near $h=0,\pm 4t$ and $t_{b}/t_{s}<1$,
$R_{W}=\frac{2\ln 2}{9\zeta(3)}\approx 0.1281$.

For the other QCP which are 2nd order with $z=2$,
$R_{W}=\frac{3}{\pi^{2}}\approx 0.3040$.

These results will also be confirmed by the analytic calculations from the
continuum theory in Sec.III-A-3(c) near $h/t=\pm 4$ and III-B-3(c) near
$h/t=0$ respectively. They are also listed in the last line of the Table-I.

So we conclude the Wilson ratio can be used to distinguish all the gapped,
gapless and QCPs. However, one need $Ch_{-}$ or $\sigma_{H}$, especially the
longitudinal/transverse edge to be discussed in Sec.V and VI to distinguish
the topology.

###  II.4  QPT versus TPT

Quantum phase transition (QPT) is characterized by the change of the ground
state energy as shown in Sec.II-B. One diagnose the QPT by the non-analytical
behaviours of ground state energy density: namely, by taking consecutive
derivatives on the ground state energy density at $T=0$ with respect to the
tuning parameter such as the injection/boost or the Zeeman field until hitting
the singularity sachdev ; tqpt ; weyl . The number of derivatives needed to
reach the singularity gives the 1st, 2nd, or higher order QPT. The DOS and the
dynamic exponent $z$ can also be extracted. Then at a finite $T$ near the QPT,
various physical quantities such as the specific heat, compressibility and
Wilson ratio satisfy the corresponding scaling functions or sub-leading
scalings weyl .

While Topological phase transition (TPT) is characterized by the change of
topological invariants such as the Chern number $Ch_{-}$ and quantum Hall
conductance as shown in Sec.II-A. For the non-interacting Fermi system tqpt ;
weyl ; topoSF , there is also the corresponding changes in the Fermi surface
topology as shown in Fig.3.

QPT, especially in interacting bosonic or quantum spin system may not
necessarily be a TPT. For example, the QPT from the BI-BM in Fig.12 is a pure
2nd order QPT with $z=2$ which is not a TPT. Of course, the well known SF-Mott
QPT, AFM to VBS, etc are not TPT sachdev . However, the QPT from CI to OCM is
also a 2nd order one with $z=2$, but despite there is no change in the Chern
number $Ch_{-}$, there is also a corresponding changes in both longitudinal
and transverse edge modes. So it is also a TPT. However, in general, a TPT
must be also a QPT. For example, the TPT from the CI to BI is a TPT where the
Chern number $Ch_{-}$ and the Hall conductance $\sigma_{H}$ changes by $\pm
1$. Of course, there is also an corresponding changes in the edge modes due to
the conventional bulk-edge correspondence. At the same time, it is also a 3rd
order QPT with $z=1$. Of course, the well-known TPT from FQH to insulator
transition moving  is also a QPT with $z=1$. However, for the very first time,
we discover an counter-example to this general believe: the TPT from the OCM
to the BM is not a QPT ! This maybe the very first example of a TPT which is
NOT a QPT. Similar classifications also apply to Fig.16. See also Table I and
Table II. See also Sec.VIII-C for more concrete discussions.

##  III  The bulk effective theory in the continuum limit

In the momentum space, Eq.2 becomes:

$\displaystyle H(\mathbf{k})$ $\displaystyle=-[h+2t(\cos k_{x}+\cos
k_{y})]\sigma_{z}+2t_{s}\sin k_{x}\sigma_{x}$ $\displaystyle+2t_{s}\sin
k_{y}\sigma_{y}-2t_{b}\sin k_{y}\sigma_{0}$ (21)

When $h\sim 4t$, there are low-energy excitations near
$\mathbf{K}_{3}=(\pi,\pi)$, it reduces to

$\displaystyle H_{3}(\mathbf{K}_{3}+\mathbf{k})$
$\displaystyle=-[h-4t+t(k_{x}^{2}+k_{y}^{2})]\sigma_{z}-2t_{s}k_{x}\sigma_{x}$
$\displaystyle-2t_{s}k_{y}\sigma_{y}+2t_{b}k_{y}\sigma_{0}$ (22)

When $h\sim 0$, there are low-energy excitations near both
$\mathbf{K}_{1}=(\pi,0)$ and $\mathbf{K}_{2}=(0,\pi)$, it reduces to

$\displaystyle H_{1}(\mathbf{K}_{1}+\mathbf{k})$
$\displaystyle=-[h+t(k_{x}^{2}-k_{y}^{2})]\sigma_{z}-2t_{s}k_{x}\sigma_{x}$
$\displaystyle+2t_{s}k_{y}\sigma_{y}-2t_{b}k_{y}\sigma_{0}$ $\displaystyle
H_{2}(\mathbf{K}_{2}+\mathbf{k})$
$\displaystyle=-[h-t(k_{x}^{2}-k_{y}^{2})]\sigma_{z}+2t_{s}k_{x}\sigma_{x}$
$\displaystyle-2t_{s}k_{y}\sigma_{y}+2t_{b}k_{y}\sigma_{0}$ (23)

When $h\sim-4t$, there are low-energy excitations near $\mathbf{K}_{0}=(0,0)$,
it reduces to

$\displaystyle H_{0}(\mathbf{k})$
$\displaystyle=-[h+4t-t(k_{x}^{2}+k_{y}^{2})]\sigma_{z}+2t_{s}k_{x}\sigma_{x}+2t_{s}k_{x}\sigma_{x}$
$\displaystyle-2t_{b}k_{y}\sigma_{0}$ (24)

The low-energy physics near all the 4 Dirac points can be written in a generic
form

$\displaystyle H(\mathbf{k})=$
$\displaystyle(\Delta+\alpha_{x}k_{x}^{2}+\alpha_{y}k_{y}^{2})\sigma_{z}\\!+v_{x}k_{x}\sigma_{x}\\!+v_{y}k_{y}\sigma_{y}\\!-ck_{y}\sigma_{0}\>$
(25)

where the velocities $v_{x,y}$ and “mass” $\alpha_{x,y}$ must be non-zero. In
this work, all the above equations correspond to the isotropic case
$|v_{x}|=|v_{y}|$ and $|\alpha_{x}|=|\alpha_{y}|$. As stressed below Eq.2,
both the C-symmetry (charge symmetry) and P-symmetry (parity symmetry) exist
at $c=0$, and P-symmetry is broken at $c\neq 0$, but the C-symmetry still
holds at $c\neq 0$.

Diagonalization of the effective Hamiltonian leads to the energy dispersion

$\displaystyle\epsilon_{\pm}(\mathbf{k})\\!=\\!\pm\sqrt{(\Delta\\!+\\!\alpha_{x}k_{x}^{2}\\!+\\!\alpha_{y}k_{y}^{2})^{2}\\!+\\!v_{x}^{2}k_{x}^{2}\\!+\\!v_{y}^{2}k_{y}^{2}}\\!-\\!ck_{y}$
(26)

The half-filling condition and C-symmetry ensure the Fermi energy is always
zero. By examining the minima of upper band
$\min_{\mathbf{k}}\epsilon_{+}(\mathbf{k})$, there exists a critical velocity
$c_{0}$ and $c<c_{0}$ the $\epsilon_{+}$ is empty and the system is in an
insulating phase; and $c>c_{0}$ the $\epsilon_{+}$ is partially filled and the
system is in a metallic phase; When $\alpha_{y}\Delta>0$, the critical
velocity is $c_{0}=\sqrt{\smash[b]{v_{y}^{2}+4\alpha_{y}\Delta}}$; when
$\alpha_{y}\Delta<0$, the critical velocity is $c_{0}=|v_{y}|$. See Fig.11 for
a geometric interpretation of the critical velocity in the two cases.

Figure 11: Geometric determination of the critical velocity (a)
$\alpha\Delta>0$ and (b) $\alpha\Delta<0$. The condition
$0=\min_{\mathbf{k}}\epsilon_{+}(\mathbf{k})$ is equivalent to
$0=\min_{k}[(\Delta+\alpha_{y}k_{y}^{2})^{2}-(c^{2}-v^{2})k_{y}^{2}]$. The
symmetry axis of the parabola is $k_{y}^{2}=-\Delta/\alpha$. In both cases,
the two Fermi points are located at momentum $\pm\sqrt{|\Delta/\alpha|}$ shown
in Fig.13. It may be contrasted the Geometric determination of the critical
velocities inside a superfluid with or without a roton minimum moving .

When re-write the continuum Hamiltonian Eq.25 in the form of Eq.6:

$\displaystyle
H(k)=d_{0}(k)\sigma_{0}+d_{x}(k)\sigma_{x}+d_{y}(k)\sigma_{y}+d_{z}(k)\sigma_{z},$
$\displaystyle d_{0}(k)=-ck_{y},\>\mathbf{d}=(d_{x},d_{y},d_{z})$
$\displaystyle d_{x}(k)=v_{x}k_{x},\>d_{y}(k)=v_{y}k_{y},\>$ $\displaystyle
d_{z}(k)=\Delta+\alpha_{x}k_{x}^{2}+\alpha_{y}k_{y}^{2}$ (27)

and then the Berry Connections and Berry curvatures stay the same form as in
Eq.7:

$\displaystyle A_{\pm,i}(\mathbf{k})$
$\displaystyle=\frac{(d_{y}\partial_{k_{i}}d_{x}-d_{x}\partial_{k_{i}}d_{y})}{2|\mathbf{d}|(|\mathbf{d}|\pm
d_{z})}$ $\displaystyle\Omega_{\pm,xy}(\mathbf{k})$
$\displaystyle=\mp\frac{1}{2|\mathbf{d}|^{3}}\mathbf{d}\cdot\partial_{x}\mathbf{d}\times\partial_{y}\mathbf{d}$
(28)

which takes the identical form as Eq.7. But it is integrated over $R^{2}$
here, while it is over a Torus $\mathbb{T}^{2}$ there.

It can be written in an explicit form

$\displaystyle A_{\pm}(\mathbf{k})$
$\displaystyle=\frac{v_{x}v_{y}(k_{y},-k_{x})}{2\sqrt{A^{2}+\bar{k}^{2}}[\sqrt{A^{2}+\bar{k}^{2}}\pm
A]}$ $\displaystyle\Omega_{\pm,xy}(\mathbf{k})$
$\displaystyle=\mp\frac{v_{x}v_{y}(\Delta-\alpha_{x}k_{x}^{2}-\alpha_{y}k_{y}^{2})}{2[A^{2}+\bar{k}^{2}]^{3/2}}$
(29)

where
$A=\Delta+\alpha_{x}k_{x}^{2}+\alpha_{y}k_{y}^{2},\bar{k}^{2}=v_{x}^{2}k_{x}^{2}+v_{y}^{2}k_{y}^{2}$.

Since the $c$ boost term does not affect the eigenvectors, the Berry
connections and Berry curvatures are exactly the same as $c=0$ case.

Because the upper band and lower band are always separated either directly or
indirectly, thus a Chern number of lower band, independent of its filling, can
always be evaluated via the integral ${\rm
Ch}_{-}=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\Omega_{-}(\mathbf{k})d^{2}\mathbf{k}$,
which gives

$\displaystyle\mathrm{Ch}_{-}\\!=\\!\operatorname{sgn}(v_{x}v_{y})[2\operatorname{sgn}(\Delta)\\!-\\!\operatorname{sgn}(\alpha_{x})\\!-\\!\operatorname{sgn}(\alpha_{y})]/4,$
(30)

where $\operatorname{sgn}()$ denotes the sign function footnote00 .

The band Chern number, which is independent of the filling, is closely related
to the zero temperature Hall conductance which does depend on the filling. The
linear response theory gives the intrinsic Hall conductance Qi2006  in the
unit $e^{2}/h$ as

$\displaystyle\sigma_{H}=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\sum_{s=\pm}\Omega_{s}(\mathbf{k})\Theta(-\epsilon_{s}(\mathbf{k}))d^{2}\mathbf{k}$
(31)

where the unit step function $\Theta(\epsilon)=1$ when $\epsilon>0$ and $=0$,
when $\epsilon\leq 0$. it just replaces the $\mathbb{T}^{2}$ in a lattice Eq.9
by $R^{2}$ in the continuum.

Due to the C-symmetry, the intrinsic Hall conductance can be separated into
two parts

$\displaystyle\sigma_{H}$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\Omega_{-}(\mathbf{k})d^{2}\mathbf{k}+\frac{1}{\pi}\int_{\mathbb{R}^{2}}\Omega_{+}(\mathbf{k})\Theta(-\epsilon_{+}(\mathbf{k}))d^{2}\mathbf{k}$
$\displaystyle={\rm Ch}_{-}+\nu_{b}$ (32)

whose interpretation in terms of the edge states in a strip geometry will be
given in Sec.IV-A-1.

When $\max\epsilon_{-}(\mathbf{k})<0<\min\epsilon_{+}(\mathbf{k})$,
$\nu_{b}=0$ and $\sigma_{H}={\rm Ch}_{-}$. However, when
$\min\epsilon_{+}(\mathbf{k})<0<\max\epsilon_{-}(\mathbf{k})$, the evaluation
of the second integral in Eq.(32) gives

$\displaystyle\nu_{b}=\operatorname{sgn}({\rm
Ch}_{-})\frac{\operatorname{sgn}(\alpha_{y}\Delta)-1}{2}\frac{c-|v_{y}|}{c}$
(33)

which indicates that $\nu_{b}$ and ${\rm Ch}_{-}$ always have opposite sign.
So $|\sigma_{H}|$ is always smaller than $|{\rm Ch}_{-}|$.

Using Eq.(30), one can reproduce the correct Chern number calculated from the
lattice Hamiltonian Eq.6.

I) When $h\sim 4t$, $\Delta=4t-h$,
$\alpha_{x}=\alpha_{y}=-t$,$v_{x}=v_{y}=-2t_{s}$, thus ${\rm
Ch}_{-}=[\operatorname{sgn}(4t-h)+\operatorname{sgn}(t)]/2$;

If $t>0$, ${\rm Ch}_{-}=+1$ when $h<4t$ (aka $h/|t|<4$), 0 otherwise;

If $t<0$, ${\rm Ch}_{-}=-1$ when $h>4t$ (aka $h/|t|>-4$), 0 otherwise.

II) When $h\sim 0$, $\Delta_{1}=\Delta_{2}=-h$,
$\alpha_{1x}=-\alpha_{1y}=-\alpha_{2x}=\alpha_{2y}=t$,
$v_{1x}=-v_{1y}=-v_{2x}=v_{2y}=2t_{s}$, thus ${\rm Ch}_{-}={\rm Ch}_{1-}+{\rm
Ch}_{2-}=-\operatorname{sgn}(-h)=\operatorname{sgn}(h)$;

III) When $h\sim-4t$, $\Delta=-4t-h$, $\alpha_{x}=\alpha_{y}=t$,
$v_{x}=v_{y}=2t_{s}$, thus ${\rm
Ch}_{-}=[\operatorname{sgn}(-4t-h)-\operatorname{sgn}(t)]/2$;

If $t>0$, ${\rm Ch}_{-}=-1$ when $h>-4t$ (aka $h/|t|>-4$), 0 otherwise;

If $t<0$, ${\rm Ch}_{-}=+1$ when $h<-4t$ (aka $h/|t|<4$), 0 otherwise.

These I),II),III) are indeed consistent with Eq.(8) achieved on a lattice.
Notice that ${\rm Ch}_{-}$, if it is not zero, only depends on
$\operatorname{sgn}(h)$, but independent of $\operatorname{sgn}(t)$ and
$\operatorname{sgn}(t_{s})$.

The $\alpha_{x}\alpha_{y}>0$ case corresponding to $|h/t|\sim 4$ transition,
and $\alpha_{x}\alpha_{y}<0$ case corresponding to $|h/t|\sim 0$ transition.
Since the sign of $\alpha_{x}\alpha_{y}$ makes the topological property
dramatically different, in the following, we will discuss
$\alpha_{x}\alpha_{y}>0$ and $\alpha_{x}\alpha_{y}<0$ separately.

###  III.1  The bulk topological transitions near $|h/t|=4$ (the
$\alpha_{x}\alpha_{y}>0$ case)

Without loss of generality, we consider $h/t\sim-4$ and near
$\mathbf{K}_{0}=(0,0)$ case, which belongs to the $\alpha_{x}\alpha_{y}>0$
case,

$\displaystyle
H_{0}(\mathbf{k})=[\Delta+\alpha(k_{x}^{2}+k_{y}^{2})]\sigma_{z}+vk_{x}\sigma_{x}+vk_{y}\sigma_{y}-ck_{y}\sigma_{0}$
(34)

where $\Delta=-(h+4t)$, $\alpha=t$, $v=2t_{s}$, $c=2t_{b}$. Diagonalization of
Eq.(34) leads to two bands

$\displaystyle\epsilon_{\pm}(\mathbf{k})=\pm\sqrt{v^{2}k^{2}+(\Delta+\alpha
k^{2})^{2}}-ck_{y}$ (35)

As shown in Fig.11, when $\alpha\Delta>0$, the critical velocity
$c_{0}=\sqrt{v^{2}+4\alpha\Delta}$. When $\alpha\Delta<0$, the critical
velocity $c_{0}=|v|$. The phase diagram of Eq.(34) is given in Fig.12.

At a fixed $\Delta$, when $c\to c_{0}$, energy bands overlap. There is an
electron pocket near $\mathbf{k}=(0,\sqrt{|\Delta/\alpha|})$ with $z=2$ in the
$k_{y}>0$ regime and the corresponding hole pocket near
$\mathbf{k}=(0,-\sqrt{|\Delta/\alpha|})$ with $z=2$ in the $k_{y}<0$ regime.
When $|c-c_{0}|\ll v$, the gap in Eq.19 vanishes linearly as $c\to c_{0}$ with

$\displaystyle\tilde{\Delta}=\min_{k}\epsilon_{+}(k)=\sqrt{|\Delta/\alpha|}(c-c_{0})+\cdots$
(36)

At a fixed $c<v$, when $\Delta\to 0$, then the upper band and lower band conic
touch at $k=0$ with $z=1$. When $|\Delta|\ll v^{2}/\alpha$, the gap vanishes
also linearly as $\Delta\to 0$ with

$\displaystyle\tilde{\Delta}=\min_{k}\epsilon_{+}(k)=\sqrt{1-c^{2}/v^{2}}|\Delta|+\cdots$
(37)

Both Eq.36 and Eq.37 will be used in evaluating various thermodynamic
quantities in Sec.III-A-3.

Figure 12: The phase diagram of Eq.(34). It is the part near $h/t\sim\pm 4$ in
the global phase diagram in Fig.3 where the Odd Chern metal is the A1 phase.
Thick/thin/dashed line are 2nd/3rd/infinite order Topological phase
transitions (TPTs) respectively. The Chern insulator with
$Ch_{-}=1,\sigma_{H}=1$ to the insulator with $Ch_{-}=0,\sigma_{H}=0$ is 3rd
order TPT due to the linear band touching of the Dirac fermion with $z=1$. The
Dirac fermion at $c/v=0$ has the emergent Lorentz invariance which is broken
by any $0<c/v<1$. The Chern insulator to the Odd Chern metal with
$Ch_{-}=1,\sigma_{H}<1$ and $z=2$ transition can be read from Fig.13. The band
insulator to the band metal transition with $z=2$ can be read from the first 3
boxes in Fig.3. The dashed line from the Odd Chern metal to the band metal
with $Ch_{-}=0,\sigma_{H}=0$ due to the conic band touching between the P- and
the H- FS is infinite order and can be read from the 3rd to the 5th boxes in
Fig.3.  Figure 13:  The dispersion $\epsilon_{\pm}(k)$ as function of $k_{y}$
with fixed $k_{x}=0$ and $v=1$, $\Delta=1/5$, $\alpha=-1/2$,
$c=0,0.5,1.0,1.1$. (a) it has a direct gap at $k_{y}=0$, (b) it has an
indirect gap at $k_{y}\neq 0$, (c) it becomes gapless and shows Fermi points
at $k_{y}=\pm k_{0}$ with the quadratic band touching and $z=2$, (d) it
becomes gapless and show finite Fermi pockets. It is the A1 phase in Fig.3.

####  III.1.1  Hall conductance at zero and finite temperatures

We will also evaluate the Hall conductance at zero and finite T respectively.

(a). Zero temperature Hall conductance

When $c<c_{0}$, the $T=0$ intrinsic Hall conductance in unit $e^{2}/h$ can be
evaluated as
$\sigma_{H}=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\Omega_{-}(\mathbf{k})d^{2}\mathbf{k}$

$\displaystyle\sigma_{H}$ $\displaystyle={\rm
Ch}_{-}=[\operatorname{sgn}(\Delta)-\operatorname{sgn}(\alpha)]/2$
$\displaystyle=\operatorname{sgn}(\Delta)[1-\operatorname{sgn}(\alpha\Delta)]/2$
(38)

which can be obtained by just setting
$\operatorname{sgn}(\alpha_{x})=\operatorname{sgn}(\alpha_{y})$ in Eq.30. It
suggests that $\sigma_{H}=-1$ for $h<-4t$ and $\sigma_{xy}=0$ for $h>-4t$ If
choosing $t>0$, this is consistent with calculation on lattice scale.

When $c>c_{0}$, there is an electron pocket near
$\mathbf{k}=(0,\sqrt{|\Delta/\alpha|})$ in the $k_{y}>0$ regime, and a hole
pocket near $\mathbf{k}=(0,-\sqrt{|\Delta/\alpha|})$ in the $k_{y}<0$ regime (
Fig.13 ). Both $+/-$ band contribute to the Hall conductance as shown in
Eq.32.

$\displaystyle\sigma_{H}=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\Omega_{-}(\mathbf{k})d^{2}\mathbf{k}+\frac{1}{\pi}\int_{\mathbb{R}^{2}}\Omega_{+}(\mathbf{k})\Theta(-\epsilon_{+}(\mathbf{k}))d^{2}\mathbf{k}$
(39)

Because the electron FS is a simple closed loop, Eq.39 can be expressed as the
Berry phase for an adiabatic path around the FS Haldane2004

$\displaystyle\sigma_{H}$ $\displaystyle={\rm Ch}_{-}+\frac{1}{\pi}\oint
d\mathbf{k}_{F}\cdot A_{+}(\mathbf{k}_{F})={\rm
Ch}_{-}+\frac{\phi_{+}}{\pi}\>.$ (40)

Evaluation of the integral Eq.39 or Eq.(40) leads to the same result

$\displaystyle\sigma_{H}$ $\displaystyle={\rm
Ch}_{-}[1+\frac{c-v}{c}\Theta(c-v)\Theta(-\alpha\Delta)]$
$\displaystyle=\operatorname{sgn}(\Delta)\times\min(1,v/c)\quad\text{if
$\alpha\Delta<0$}$ (41)

otherwise $0$. The Hall conductance developed a cusp at the critical velocity
$c=v$ as shown in Fig.14.

Figure 14:  The Hall conductance of Eq.(34) at different $\alpha\Delta/v^{2}$
or $c/v$ values: Varying $\alpha\Delta/v^{2}$ in (a) and (b): (a) fixing
$c/v=0.5$, (b) fixing $c/v=1.2$, There is a jump from Odd Chern metal to the
band metal with the magnitude $v/c=5/6$. However, as indicated in Fig.12, it
is infinite order. Varying $c/v$ in (c) and (d): (c) fixing
$\alpha\Delta/v^{2}=-0.1$, (d) fixing $\alpha\Delta/v^{2}=+0.1$. Again, as
already alerted in Fig.4d, in a generic case, the BM should contributes to a
un-quantized AHE ( See Fig.24 and Table 1 ). But it vanishes in this
particular $n=1$ case due to the fine tuning. They match all the TPTs achieved
on the lattice shown in Fig.4, so the captions of Fig.4 on the universality
classes of the TPTs also apply here.

(b). Finite temperature Hall conductance

When $c<c_{0}$, the intrinsic Hall conductance Qi2006  in unit $e^{2}/h$ can
be evaluated as

$\displaystyle\sigma_{H}(T)=\frac{1}{2\pi}\sum_{s=\pm}\int_{\mathbb{R}^{2}}d^{2}\mathbf{k}\Omega_{s}(\mathbf{k})f(\epsilon_{s}(k))$
(42)

which is just replacing $T^{2}$ in Eq.10 by $R^{2}$.

Due to the C-symmetry, the finite temperature Hall conductance can be
expressed as

$\displaystyle\sigma_{H}(T)$ $\displaystyle={\rm
Ch}_{-}+\frac{1}{\pi}\int_{\mathbb{R}^{2}}d^{2}\mathbf{k}\>\Omega_{+}(\mathbf{k})f(\epsilon_{+}(\mathbf{k}))$
$\displaystyle={\rm Ch}_{-}+\nu(T)$ (43)

which is also just replacing $T^{2}$ in Eq.10 by $R^{2}$.

The $\nu(T=0)$ and ${\rm Ch}_{-}$ has been evaluated in Eq.30 and Eq.33
respectively. At low temperature $T$ where $T$ is the lowest energy scale, one
can get a low temperature expansion of $\nu(T)$. By keeping the leading low
$T$ dependence, assuming $|\alpha\Delta/v^{2}|\ll 1/2$, we have

1) When $\alpha\Delta>0$ in the band insulating phase,
$\nu(T)-\nu(T=0)\propto-\operatorname{sgn}(\Delta)Te^{-|\tilde{\Delta}|/T};~{}~{}~{}-\operatorname{sgn}(\Delta)T^{2};~{}~{}~{}-\operatorname{sgn}(\Delta)T^{2}$
at $c<c_{0},~{}c=c_{0},~{}c>c_{0}$ respectively.

2) When $\alpha\Delta<0$ in the Chern insulating phase,
$\nu(T)-\nu(T=0)\propto-\operatorname{sgn}(\Delta)Te^{-|\tilde{\Delta}|/T};~{}~{}~{}-\operatorname{sgn}(\Delta)T;~{}~{}~{}-\operatorname{sgn}(\Delta)T^{2}$
at $c<c_{0},~{}c=c_{0},~{}c>c_{0}$ respectively.

####  III.1.2  Ground-state energy density and topological phase transitions

When $c<c_{0}$, $\min\epsilon_{+}\geq 0\geq\max\epsilon_{-}$, the lower band
is full occupied and the upper band is complete empty, thus the ground-state
energy density is
$E_{\text{GS}}=(4\pi^{2})^{-1}\int_{\mathbb{R}^{2}}d^{2}\mathbf{k}\>\epsilon_{-}(\mathbf{k})$.
The integral is divergent in $R^{2}$, we need a cut-off $\Lambda$ for $k$. Due
to its odd property, the $ck_{y}$ part in Eq.35 drops off, one obtain

$\displaystyle E_{0}=-\frac{|\Delta|^{3}}{6\pi v^{2}}+E_{\Lambda}$ (44)

where $E_{\Lambda}$ is $\Lambda$, $\alpha\Delta$ and $v$ dependent function
which is differentiable up to 3 degrees of differentiation, but may contain
higher order non-analyticity.

When $c>c_{0}$, $\min\epsilon_{+}<0<\max\epsilon_{-}$, both the lower band and
the upper band are only partially occupied, thus the ground-state energy
density becomes
$E_{\text{GS}}=(4\pi^{2})^{-1}\sum_{s=\pm}\int_{\mathbb{R}^{2}}d^{2}\mathbf{k}\>\epsilon_{s}(\mathbf{k})\Theta(-\epsilon_{s}(\mathbf{k}))$.
Due to the C-symmetry, it can be expressed as
$E_{\text{GS}}=E_{0}+E_{1}\Theta(c-c_{0})$, where the $c$-dependent part is
$E_{1}=(2\pi^{2})^{-1}\int_{\mathbb{R}^{2}}d^{2}\mathbf{k}\>\epsilon_{+}(\mathbf{k})\Theta(-\epsilon_{+}(\mathbf{k}))$.
The integral is convergent, we obtain

$\displaystyle E_{1}=-\frac{1}{16\pi\alpha^{3}}[$
$\displaystyle\frac{1}{4}(c^{2}-c_{0}^{2})(c^{2}+c_{0}^{2}-4v^{2}-8\alpha\Delta)$
$\displaystyle+v^{2}(v^{2}+4\alpha\Delta)\ln(c/c_{0})]$ (45)

In the following, we study the possible TPT encoded in the ground state energy
density.

(a). Band insulator to Chern insulator transition

In the range $c<v$ and tuned by $\Delta$, one only have the $E_{0}$ part. It
is easy to see a third-order non-analytical behaviour at $\Delta=0$,

$\displaystyle E_{\text{GS}}=-\frac{|\Delta|^{3}}{6\pi v^{2}}+\cdots$ (46)

which has the dynamic exponent $z=1$.

(b). Band metal to Odd Chern metal transition:

In the range $c>v$ and tuned by $\Delta$, one need both $E_{0}$ and $E_{1}$
part. It is easy to see the third-order non-analytical behaviour at $\Delta=0$
in the two parts gets canceled:

$\displaystyle E_{\text{GS}}=-\frac{|\Delta|^{3}}{6\pi
v^{2}}+\frac{|\Delta|^{3}}{6\pi v^{2}}+\cdots$ (47)

Carefully treating the exact results
$E_{\text{GS}}=E_{0}+E_{1}\Theta(c-c_{0})$ tells $E_{\text{GS}}$ is smooth at
$\Delta=0$, which is consistent with numerical result’s infinite order
differentiable. This fact explains why the 3rd order TPT across the green line
changes to infinite order across the red dashed line in Fig.12. Even so, as
shown in Fig.13b, there is still a universal jump $\Delta\sigma_{H}=v/c$, so
it is still a TPT.

(c). Band insulator to band metal transition:

In the range $\alpha\Delta>0$ and tuned by either $\Delta$ or $c$. It is easy
to tell a second-order non-analytical behaviour at
$\Delta=(c^{2}-v^{2})/(4\alpha)$ when tuning $\Delta$,

$\displaystyle E_{\text{GS}}\\!=\\!E_{0}-\frac{c^{2}\\!-\\!v^{2}}{4\pi
c^{2}}(\Delta\\!-\\!\frac{c^{2}\\!-\\!v^{2}}{4\alpha})^{2}\Theta(c^{2}\\!-\\!v^{2}\\!-\\!4\alpha\Delta)$
(48)

and also a second-order non-analytical behaviour at
$c=\sqrt{v^{2}+4\alpha\Delta}$ when tuning $c$,

$\displaystyle
E_{\text{GS}}\\!=\\!E_{0}\\!-\\!\frac{\Delta}{4\pi\alpha^{2}}(c\\!-\\!\\!\sqrt{v^{2}\\!+\\!4\alpha\Delta})^{2}\Theta(c^{2}\\!-\\!v^{2}\\!-\\!4\alpha\Delta)$
(49)

which has the dynamic exponent $z=2$.

(d). Chern insulator to Odd Chern metal transition

In the range $\alpha\Delta<0$ and tuned by $c$. It is easy to see a second-
order non-analytical behaviour at $c=v$,

$\displaystyle
E_{\text{GS}}=E_{0}+\frac{\alpha\Delta}{4\pi|\alpha|^{3}}(c-v)^{2}\Theta(c^{2}-v^{2})+\cdots$
(50)

which has the dynamic exponent $z=2$.

In summary, transitions (c) and (d) are second-order QPT with $z=2$; (a) is
third-order QPT with $z=1$. (b) is infinite-order without any non-analyticity.
These are consistent with the results achieved in a lattice in Sec.II-B.

####  III.1.3  Thermodynamic Quantities

(a). The density of states

From the Hamiltonian (25), the Matsubara Green’s function

$\displaystyle
G(\mathbf{k},i\omega_{n})=\frac{P_{+}(\mathbf{k})}{i\omega_{n}-\epsilon_{+}(\mathbf{k})}+\frac{P_{-}(\mathbf{k})}{i\omega_{n}-\epsilon_{-}(\mathbf{k})}$
(51)

where $P_{s}$ are the projection operators onto $s=\pm$ bands and
$P_{+}+P_{-}=1$.

The total DOS is

$\displaystyle
D(\omega)=\int\frac{d^{2}q}{(2\pi)^{2}}[\delta(\omega-\epsilon_{+}(k))+\delta(\omega-\epsilon_{-}(k))]$
(52)

It would be convenient to introduce the DOS for each band
$D(\omega)=D_{+}(\omega)+D_{-}(\omega)$:

$\displaystyle
D_{\pm}(\omega)=\int\frac{d^{2}\mathbf{k}}{4\pi^{2}}\delta(\omega-\epsilon_{\pm}(\mathbf{k})),$
(53)

Due to the C-symmetry, $D_{+}(\omega)=D_{-}(-\omega)$.

$\displaystyle
D_{+}(\omega)=\frac{1}{4\pi|\alpha|}[1+\rho(\omega)]\Theta(\omega-\min\epsilon_{+})$
(54)

For some reason related to the C-symmetry, $\rho(\epsilon)=-\rho(-\epsilon)$
is an odd function. The odd property leads to flat feature of the total DOS
near $\omega=0$ when $\min\epsilon_{+}<0<\max\epsilon_{-}$,

$\displaystyle D(\omega)=\frac{1}{2\pi|\alpha|},\textmd{ if
}\min\epsilon_{+}<\omega<\max\epsilon_{-}$ (55)

The DOS is plotted in Fig.15.

Figure 15: Density of states (DOS) of Eq.(34) at different $c/v$ values: (a)
$c/v=0.5$ in Chern Insulator, (b) $c/v=1.0$ on the QCP between Chern insulator
and Odd Chern metal, (c) $c/v=1.5$ in Odd Chern metal. Here, we fixed $v=1$,
$\alpha=-1$, $\Delta=1$ in Fig.12. Note that $\epsilon_{\pm}(k)$ is quadratic
in $k$ at the band minimum in Fig.13, which leads to a finite DOS at the band
edge and $z=2$. So the DOS at $\omega=0$ in (b) is finite. The flat feature of
total DOS near $\omega=0$ is a result of truncation of $H(k)$ at the $k^{2}$
order. Indeed, the lattice calculations on DOS in Fig.10 shows that it is not
a constant anymore. It matches Fig.10 very well near the low energy
$\omega\sim 0$.

(b). The specific heat and Compressibility

Here, we will make use of the DOS to evaluate the two conserved quantities,
then compute the Wilson ratio. The Helmholtz free energy density (taking
$k_{B}=1$)

$\displaystyle
F(T)=-T\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\ln[2(1+\cosh(\epsilon_{+}(\mathbf{k})/T))]$
(56)

where the zero temperature part of $F(T)$ is nothing but the ground-state
energy density $E_{\text{GS}}$ calculated in Sec.III-A-2. The specific heat
(at constant volume) is given by

$\displaystyle
C_{v}(T)=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\frac{[\epsilon_{+}(\mathbf{k})]^{2}}{T^{2}[1+\cosh(\epsilon_{+}(\mathbf{k})/T)]}$
(57)

The dynamic density-density response function is

$\displaystyle\chi^{00}(q,i\omega_{n})=-T\sum_{iv_{m},k}\operatorname{Tr}[G(i\omega_{n}+iv_{m},k+q)G(iv_{m},k)]$
(58)

Taking the $\omega_{n}\to 0$ , then $q\to 0$ limit in
$\chi^{00}(q,i\omega_{n})$ gives the (isothermal) uniform compressibility

$\displaystyle\kappa_{u}(T)\\!=\\!\\!\sum_{s=\pm}\\!\int\\!\\!\frac{d^{2}k}{(2\pi)^{2}}\frac{\partial
f(\epsilon_{s,k})}{\partial\epsilon_{s,k}}\\!=\\!\\!\int\\!\\!\frac{d\omega
D_{+}(\omega)}{T[1+\cosh(\omega/T)]}$ (59)

and the zero temperature limit $\frac{\partial
f(\epsilon)}{\partial\epsilon}=\delta(\epsilon)$ recovery the well-known
relation between compressibility and DOS $\kappa_{u}(T)=D(0)$.

At low temperature
$T\ll|\Delta|,v^{2}/|\alpha|,c^{2}/|\alpha|,|(c^{2}-v^{2})/\alpha|$, one can
get a low temperature expansion of $C_{v}(T)$ and $\kappa_{u}(T)$ footnote3 .
We do not need to distinguish $\alpha\Delta>0$ or $\alpha\Delta<0$, but need
to care if $\Delta=0$ or not. By keeping the leading low $T$ dependence,
assuming $|\alpha\Delta/v^{2}|<1/2$ to simply the technical calculations
extendtolarge , we have

$\Delta\neq 0$ case in Fig.12

At $c<c_{0}$, we have $\tilde{\Delta}=\min_{k}\epsilon_{+}(k)>0$ and
$D(\omega)=0$ for $|\omega|<\tilde{\Delta}$ and $D(\tilde{\Delta})>0$ at the
gap edge ( Fig.15a ), thus
$C_{v}(T)=2D(\tilde{\Delta})\tilde{\Delta}^{2}T^{-1}e^{-\tilde{\Delta}/T}$.
Similarly, $\kappa_{u}(T)=2D(\tilde{\Delta})Te^{-|\tilde{\Delta}|/T}$.

At $c=c_{0}$, we have $\tilde{\Delta}=\min_{k}\epsilon_{+}(k)=0$ and
$D(\omega)=D(0)+D^{\prime}(0^{+})|\omega|$ for small $|\omega|$ ( Fig.15b ),
thus $C_{v}(T)=\frac{\pi^{2}}{3}D(0)T$. Here $D^{\prime}(0^{+})$ means the
right derivatives of $D(\omega)$ at $0$. Similarly, $\kappa_{u}(T)=D(0)+2\ln
2D^{\prime}(0^{+})T$;

At $c>c_{0}$, we have $\tilde{\Delta}=\min_{k}\epsilon_{+}(k)<0$ and
$D(\omega)=\frac{1}{2\pi|\alpha|}$ for $|\omega|<|\tilde{\Delta}|$ in Eq.55
and $D(\tilde{\Delta})>0$ ( Fig.15c ), thus $C_{v}(T)=\frac{\pi^{2}}{3}D(0)T$.
Similarly,
$\kappa_{u}(T)=D(0)+2[D(\tilde{\Delta})-D(0)]Te^{-|\tilde{\Delta}|/T}$.

$\Delta=0$ case in Fig.12

At $c<c_{0}$, we have $\tilde{\Delta}=\min_{k}\epsilon_{+}(k)=0$ and
$D(\omega)=D^{\prime}(0^{+})|\omega|$ for small $\omega$, thus
$C_{v}(T)=9\zeta(3)D^{\prime}(0^{+})T^{2}$ where $\zeta(3)\approx 1.2021$.
Similarly, $\kappa_{u}(T)=2\ln 2D^{\prime}(0^{+})T$.

At $c=c_{0}$, we have $\tilde{\Delta}=\min_{k}\epsilon_{+}(k)=0$ and
$D(\omega)=D(0)+A|\omega|^{2/3}>0$ for small $\omega$, thus
$C_{v}(T)=\frac{\pi^{2}}{3}D(0)T$. Similarly,
$\kappa_{u}(T)=D(0)+1.1486AT^{5/3}$.

At $c>c_{0}$, we have $\tilde{\Delta}=\min_{k}\epsilon_{+}(k)<0$ and
$D(\omega)=\text{const.}$ for $|\omega|<|\tilde{\Delta}|$, thus
$C_{v}(T)=\frac{\pi^{2}}{3}D(0)T$. Similarly,
$\kappa_{u}(T)=D(0)+2[D(\tilde{\Delta})-D(0)]Te^{-|\tilde{\Delta}|/T}$.

When taking only the leading term in the low-temperature specific heat and
compressibility, we have:

When $\Delta\neq 0$: at $c<c_{0}$,
$C_{v}(T)=2D(\tilde{\Delta})\tilde{\Delta}^{2}T^{-1}e^{-\tilde{\Delta}/T}$ and
$\kappa_{u}(T)=2D(\tilde{\Delta})Te^{-|\tilde{\Delta}|/T}$ ; at $c\geq c_{0}$,
$C_{v}(T)=\frac{\pi^{2}}{3}D(0)T$ and $\kappa_{u}(T)=D(0)$.

When $\Delta=0$: at $c<c_{0}$, $C_{v}(T)=9\zeta(3)D^{\prime}(0^{+})T^{2}$ and
$\kappa_{u}(T)=2\ln 2D^{\prime}(0^{+})T$; at $c\geq c_{0}$,
$C_{v}(T)=\frac{\pi^{2}}{3}D(0)T$ and $\kappa_{u}(T)=D(0)$.

(c). Wilson ratio

The Wilson ratio is defined as $R_{W}=T\kappa_{u}/C_{v}$ which has the
following low temperature behaviours:

$\Delta\neq 0$ case

At $c<c_{0}$, $R_{W}=(T/\tilde{\Delta})^{2}$; At $c\geq c_{0}$,
$R_{W}=3/\pi^{2}$.

$\Delta=0$ case

At $c<c_{0}$, $R_{W}=2\ln 2/(9\zeta(3))\approx 0.1281$; At $c\geq c_{0}$,
$R_{W}=3/\pi^{2}$.

These results are consistent with those achieved directly on the lattice in
Sec.II-C-3 and also listed in the last line of Table-I.

###  III.2  The bulk topological phase transitions near $|h/t|\sim 0$ (the
$\alpha_{x}\alpha_{y}<0$ case).

When $h\sim 0$ and near the two valleys $K_{1}=(\pi,0)$ and $K_{2}=(0,\pi)$,
we have

$\displaystyle H_{1}$
$\displaystyle\\!=\\![\Delta\\!-\\!\alpha(k_{x}^{2}-k_{y}^{2})]\sigma_{z}\\!-\\!vk_{x}\sigma_{x}\\!+\\!vk_{y}\sigma_{y}\\!-\\!ck_{y}\sigma_{0}$
$\displaystyle H_{2}$
$\displaystyle\\!=\\![\Delta\\!+\\!\alpha(k_{x}^{2}-k_{y}^{2})]\sigma_{z}\\!+\\!vk_{x}\sigma_{x}\\!-\\!vk_{y}\sigma_{y}\\!+\\!ck_{y}\sigma_{0}$
(60)

where $\Delta=-h$ and other parameters are the same as the $h\sim 4t$ case
discussed in Sec.III-A. Note the opposite sign of the velocities $v$ between
$k_{x}$ and $k_{y}$ and opposite sign of $\alpha$ between $k_{x}^{2}$ and
$k_{y}^{2}$ indicating $\alpha_{x}\alpha_{y}<0$.

Due to the two valleys, we obtain four bands

$\displaystyle\epsilon_{i,\pm}=\pm\sqrt{v^{2}k^{2}+[\Delta+\eta_{i}\alpha(k_{x}^{2}-k_{y}^{2})]^{2}}+\eta_{i}ck_{y}\>,$
(61)

where $\eta_{i}=(-1)^{i}$ and $i=1,2$. It can be compared to Eq.35 in the
$\alpha_{x}\alpha_{y}>0$ case. So the two cases should have quite different
physical properties.

When $\alpha\Delta<0$, the two critical velocities
$c_{1}=|v|<c_{2}=\sqrt{v^{2}-4\alpha\Delta}$. When
$\sqrt{v^{2}-4\alpha\Delta}>c>|v|$, Thus only $H_{1}$ becomes metal. When
$c>\max(c_{1},c_{2})$, both $H_{1}$ and $H_{2}$ become metal.

When $\alpha\Delta>0$, the two critical velocities
$c_{1}=\sqrt{v^{2}+4\alpha\Delta}>c_{2}=|v|$. When
$\sqrt{v^{2}+4\alpha\Delta}>c>|v|$, Thus only $H_{2}$ becomes metal. When
$c>\max(c_{1},c_{2})$, then both $H_{1}$ and $H_{2}$ become metal.

Another new feature of $\alpha_{x}\alpha_{y}<0$ is that the FS extends to
infinity when $c\geq\sqrt{2}|v|$. The divergent $k_{F}$ suggests a FS
collision between the two valleys, which is consistent with the existence of
phase C1 or C2 in the global lattice phase diagram Fig.3. Below, we will not
discuss the C- phase, so restrict $c<\sqrt{2}|v|$. If $c>\sqrt{2}|v|$, it
leads to the class-3 TPT discussed in weyl  and reviewed in Sec.II.

Figure 16:  The phase diagram of Eq.(60). It is the part near $h/t\sim\pm 0$
in the global phase diagram in Fig.3. Similar to Fig.12, Thick/thin/dashed
line are 2nd/3rd/infinite order Topological phase transitions (TPTs)
respectively. The consecutive Chern insulator to A1 Odd Chern metal, then to
B1 Odd Chern metal transitions can be read from Fig.17. The dashed line from
the B1 Odd Chern metal to the B2 Odd Chern metal can be read from the B1 box
and B1/B2 box on the right in Fig.3. Similar to A1 Odd Chern metal to band
metal transition in Fig.12, it is also induced by the conic touching of the P-
and H- FS and also infinite order.  Figure 17:  The dispersion
$\epsilon_{i,\pm}(k)$ as function of $k_{y}$ with fixed $k_{x}=0$ and $v=1$,
$\Delta=1/5$, $\alpha=-1/2<0$, $c=0,0.5,1.0,\sqrt{1.4},1.3$. (a) both $H_{1}$
and $H_{2}$ has a direct gap at $k_{y}=0$; in Chern insulator phase. (b) Due
to the opposite boost velocity in Eq.61, the dispersion at $K_{1}$ and $K_{2}$
shift to the opposite directions. both $H_{1}$ and $H_{2}$ has an indirect gap
at $k_{y}\neq 0$; (c) $H_{1}$ becomes gapless and shows two Fermi points at
$k_{y}=\pm k_{0}$ with $z=2$, but $H_{2}$ still has an indirect gap. It
corresponds to A1/B1 in Fig.3. (d) $H_{1}$ becomes gapless and show two finite
Fermi pockets, but $H_{2}$ becomes gapless and shows two Fermi points still at
$k_{y}=\pm k_{0}$ also with $z=2$ after subtracting the non-critical $H_{1}$
part; (e) both $H_{1}$ and $H_{2}$ are gapless and show finite Fermi pockets
Nooverlap . It corresponds to B1 in Fig.3.

####  III.2.1  Hall conductance at zero and finite temperatures

(a). Zero temperature

When $c<\min(c_{1},c_{2})$, the Hall conductance $\sigma_{H}$ of each valleys
gives $\sigma_{H}^{(i)}=-\operatorname{sgn}(\Delta)/2$, adding the two
contributions together leads to

$\displaystyle\sigma_{H}=\sigma_{H}^{(1)}+\sigma_{H}^{(2)}=-\operatorname{sgn}(\Delta)$
(62)

which indicates $\sigma_{H}=-1$ for $\Delta>0$ and $\sigma_{H}=+1$ for
$\Delta<0$.

If $c>\min(c_{1},c_{2})$, then we need to discus $\alpha\Delta>0$ or
$\alpha\Delta<0$ separately.

Case $\alpha\Delta<0$ :

If $c_{2}>c>c_{1}=|v|$, $H_{1}$ first develops an instability while $H_{2}$
remains gapped.

$\displaystyle\sigma_{H}^{(1)}$
$\displaystyle=-\operatorname{sgn}(\Delta)/2+\operatorname{sgn}(\Delta)\frac{c-v}{c}\Theta(v-c),\quad$
$\displaystyle\sigma_{H}^{(2)}$ $\displaystyle=-\operatorname{sgn}(\Delta)/2$
(63)

If $c>c_{2}$, both $H_{1}$ and $H_{2}$ are gapless, however,
$\sigma_{xy}^{(2)}$ remains the same as $c=0$ as dictated by Eq.33. Thus, we
arrive at

$\displaystyle\sigma_{H}=\operatorname{sgn}(\Delta)\min(1,v/c)$ (64)

Case $\alpha\Delta>0$ :

If $c_{1}>c>c_{2}=|v|$, $H_{2}$ first develops an instability while $H_{1}$
remains gapped.

$\displaystyle\sigma_{H}^{(1)}$
$\displaystyle=-\operatorname{sgn}(\Delta)/2,\quad$
$\displaystyle\sigma_{H}^{(2)}$
$\displaystyle=-\operatorname{sgn}(\Delta)/2+\operatorname{sgn}(\Delta)\frac{c-v}{c}\Theta(v-c)$
(65)

If $c>c_{1}$, both $H_{1}$ and $H_{2}$ are gapless, however,
$\sigma_{xy}^{(1)}$ remains the same as $c=0$ as also dictated by Eq.33. Thus,
we also arrive at Eq.64.

So we always have the Hall conductance Eq.64, no matter in the insulating
phase or the metallic phase which is independent of many microscopic details.

Figure 18: The Hall conductance of Eq.60 at different $\alpha\Delta/v^{2}$ or
$c/v$ values: varying $\alpha\Delta/v^{2}$ in (a) and (b): (a) fix $c/v=0.5$
(b) fix $c/v=1.2$. There is a jump from B2 to B1 with the magnitude $2\times
v/c=2\times 5/6$, but no change from B1 to A1 Odd Chern metal phase. However,
as indicated in Fig.16, the former is infinite order, the latter is 2nd order
with $z=2$. See also Fig.4b on a lattice. Varying $c/v$ in (c) fix
$\alpha\Delta/v^{2}=-0.1$, $\sigma^{1}_{H}$ stays as a constant. But
$\sigma_{H}=\sigma^{1}_{H}+\sigma^{2}_{H}=v/c$.

(b). Finite temperature Hall conductance

Similar to the $\alpha_{x}\alpha_{y}>0$ case in Eq.43, the finite temperature
Hall conductance at the valley $i$ is

$\displaystyle\sigma_{H}^{(i)}(T)$
$\displaystyle=\sigma_{H}^{(i)}(T=0)+\frac{1}{\pi}\int_{\mathbb{R}^{2}}d^{2}\mathbf{k}\>\Omega_{i,+}(\mathbf{k})$
$\displaystyle\times[f(\epsilon_{i,+}(\mathbf{k}))-\Theta(-\epsilon_{i,+}(\mathbf{k}))]$
$\displaystyle=\mathrm{Ch}_{i,-}+\frac{1}{\pi}\int_{\mathbb{R}^{2}}d^{2}\mathbf{k}\>\Omega_{i,+}(\mathbf{k})f(\epsilon_{i,+}(\mathbf{k}))$
(66)

Thus, the total Hall conductance is:

$\displaystyle\sigma_{H}(T)$
$\displaystyle=\sigma_{H}^{(1)}(T)+\sigma_{H}^{(2)}(T)$
$\displaystyle=\mathrm{Ch}_{1,-}+\mathrm{Ch}_{2,-}+\nu_{1}(T)+\nu_{2}(T)$ (67)

where $\mathrm{Ch}_{i,-}$ and $\nu_{i}(T=0)$ has been studied in Sec.III-A-1.

At a low temperature $T$ which is the lowest energy scale, one can get a low
temperature expansion of $\nu_{i}(T)$. Without loss of generality, we choose
$\alpha\Delta<0$ case, the two critical velocities
$c_{1}=|v|<c_{2}=\sqrt{v^{2}-4\alpha\Delta}=\sqrt{v^{2}+4|\alpha\Delta|}$. By
keeping the leading low $T$ dependence and assuming $|\alpha\Delta/v^{2}|\ll
1/2$, we have

At $c<c_{1}$, $H_{1}$ has a gap $2\tilde{\Delta}_{1}$ and $H_{2}$ has a bigger
gap $2\tilde{\Delta}_{2}>2\tilde{\Delta}_{1}$. Thus,
$\nu_{1}(T)-\nu_{1}(T=0)\propto-\operatorname{sgn}(\Delta)Te^{-|\tilde{\Delta}_{1}|/T}$
and
$\nu_{2}(T)-\nu_{2}(T=0)\propto-\operatorname{sgn}(\Delta)Te^{-|\tilde{\Delta}_{2}|/T}$;

At $c=c_{1}$, $H_{1}$ is critical with $\tilde{\Delta}_{1}=0$ and $H_{2}$
still has a gap $2\tilde{\Delta}_{2}$. Thus,
$\nu_{1}(T)-\nu_{1}(T=0)\propto-\operatorname{sgn}(\Delta)T$ and
$\nu_{2}(T)-\nu_{2}(T=0)\propto-\operatorname{sgn}(\Delta)Te^{-|\tilde{\Delta}_{2}|/T}$;

At $c_{2}>c>c_{1}$, $H_{1}$ is gapless and $H_{2}$ still has a gap
$2\tilde{\Delta}_{2}$. Thus,
$\nu_{1}(T)-\nu_{1}(T=0)\propto-\operatorname{sgn}(\Delta)T^{2}$ and
$\nu_{2}(T)-\nu_{2}(T=0)\propto-\operatorname{sgn}(\Delta)Te^{-|\tilde{\Delta}_{2}|/T}$;

At $c=c_{2}$, $H_{1}$ is gapless and $H_{2}$ is also critical with
$\tilde{\Delta}_{2}=0$. Thus,
$\nu_{1}(T)-\nu_{1}(T=0)\propto-\operatorname{sgn}(\Delta)T^{2}$ and
$\nu_{2}(T)-\nu_{2}(T=0)\propto-\operatorname{sgn}(\Delta)T$;

At $c>c_{2}$, both $H_{1}$ and $H_{2}$ are gapless. Thus,
$\nu_{1}(T)-\nu_{1}(T=0)\propto-\operatorname{sgn}(\Delta)T^{2}$ and
$\nu_{2}(T)-\nu_{2}(T=0)\propto-\operatorname{sgn}(\Delta)T^{2}$.

In summary, if we only need the leading behaviors of
$\nu(T)=\nu_{1}(T)+\nu_{2}(T)$, then we have:

At $c<c_{1}$,
$\nu(T)-\nu(T=0)\propto-\operatorname{sgn}(\Delta)Te^{-|\tilde{\Delta}_{1}|/T}$;
at $c=c_{1}$, $\nu(T)-\nu(T=0)\propto-\operatorname{sgn}(\Delta)T$; at
$c_{2}>c>c_{1}$, $\nu(T)-\nu(T=0)\propto-\operatorname{sgn}(\Delta)T^{2}$; at
$c=c_{2}$, $\nu(T)-\nu(T=0)\propto-\operatorname{sgn}(\Delta)T$; at $c>c_{2}$,
$\nu(T)-\nu(T=0)\propto-\operatorname{sgn}(\Delta)T^{2}$.

####  III.2.2  Ground-state energy density and topological phase transitions

When $c<\min(c_{1},c_{2})$, $\min\epsilon_{i,+}\geq 0\geq\max\epsilon_{i,-}$,
both valley 1&2’s lower band is full occupied and 1&2’s upper band is complete
empty, thus the ground-state energy density is
$E_{\text{GS}}=(4\pi^{2})^{-1}\int_{\mathbb{R}^{2}}d^{2}\mathbf{k}\>[\epsilon_{1-}(\mathbf{k})+\epsilon_{2-}(\mathbf{k})]$.
The integral is divergent, we need a cut-off $\Lambda$ for $k$. Due to its odd
property, the $ck_{y}$ part in Eq.61 drops off, one obtain:

$\displaystyle E_{0}=-\frac{|\Delta|^{3}}{3\pi v^{2}}+E_{\Lambda}$ (68)

where $E_{\Lambda}$ is $\Lambda$, $\alpha\Delta$ and $v$ dependent $C^{3}$
function. Note that $E_{\Lambda}$ in the $\alpha_{x}\alpha_{y}<0$ case is
dramatically different from that in the $\alpha_{x}\alpha_{y}>0$ case listed
in Eq.44.

When $\max(c_{1},c_{2})>c>\min(c_{1},c_{2})$, one of $\epsilon_{1,-}$ and
$\epsilon_{2,-}$ are partially filled, then there is another parts $E_{1}$ and
$E_{\text{GS}}=E_{0}+E_{1}$; When $c>\max(c_{1},c_{2})$, both $\epsilon_{1,-}$
and $\epsilon_{2,-}$ are partially filled, then there is also one more part
$E_{2}$ and $E_{\text{GS}}=E_{0}+E_{1}+E_{2}$, where $E_{1}$ and $E_{2}$ are
slightly more complicated than that in the $\alpha_{x}\alpha_{y}>0$ case
listed in Eq.45:

$\displaystyle E_{1}=\frac{-\pi}{32|\alpha|^{3}}[$ $\displaystyle
8\alpha\Delta v^{2}+2v^{4}(\arcsin\frac{c}{\sqrt{2}v}-\frac{\pi}{4})$
$\displaystyle+c\sqrt{2v^{2}-c^{2}}(v^{2}-8\alpha\Delta-c^{2})]$
$\displaystyle E_{2}=\frac{-\pi}{32|\alpha|^{3}}[$ $\displaystyle
4\alpha\Delta\sqrt{v^{4}-16\alpha^{2}\Delta^{2}}$
$\displaystyle+2v^{4}(\arcsin\frac{c}{\sqrt{2}v}-\frac{1}{2}\arccos\frac{4\alpha\Delta}{v^{2}})$
$\displaystyle+c(v^{2}-8\alpha\Delta-c^{2})\sqrt{2v^{2}-c^{2}}]$ (69)

(a). Chern insulator (+1) to Chern insulator (-1) transition:

In the range $c<v$ and tuned by $\Delta$, one only have the $E_{0}$ part which
has a third-order non-analytical behaviour at $\Delta=0$,

$\displaystyle E_{\text{GS}}=-\frac{|\Delta|^{3}}{3\pi v^{2}}+\cdots$ (70)

which has the dynamic exponent $z=1$.

(b). B1 Odd Chern metal to B2 Odd Chern metal transition:

In the range $c>v$ and the driven parameter is $\Delta$. Now we have both
$E_{0}$ and $E_{1}+E_{2}$ part, the third-order non-analytical term at
$\Delta=0$ from the two parts gets canceled,

$\displaystyle E_{\text{GS}}=-\frac{|\Delta|^{3}}{3\pi
v^{2}}+\frac{|\Delta|^{3}}{3\pi v^{2}}+\cdots$ (71)

which explainers why the 3rd order TPT across the green line changes to
infinite order across the red dashed line in Fig.16. Even so, as shown in
Fig.17b, there is still a universal jump $\Delta\sigma_{H}=2v/c$, so it is
still a TPT.

(c). Odd Chern A metal to Odd Chern B metal transition

In the range $\alpha\Delta>0$ and tuned by either $\Delta$ or $c$, one finds a
second-order non-analytical behaviour at $\Delta=(c^{2}-v^{2})/(4\alpha)$
tuned by $\Delta$,

$\displaystyle E_{\text{GS}}$
$\displaystyle=\frac{v^{2}-c^{2}}{4\pi\alpha\sqrt{c^{2}(2v^{2}-c^{2})}}[\Delta-(c^{2}-v^{2})/(4\alpha)]^{2}$
$\displaystyle\times\Theta(c^{2}-v^{2}-4\alpha\Delta)+\cdots$ (72)

and also a second-order non-analytical behaviour at
$c=\sqrt{v^{2}+4\alpha\Delta}$ tuned by $c$,

$\displaystyle E_{\text{GS}}$
$\displaystyle=-\frac{\Delta\sqrt{v^{2}+4\alpha\Delta}}{4\pi\alpha\sqrt{v^{2}-4\alpha\Delta}}(c-\sqrt{v^{2}+4\alpha\Delta})^{2}$
$\displaystyle\times\Theta(c^{2}-v^{2}-4\alpha\Delta)+\cdots$ (73)

which has the dynamic exponent $z=2$.

(d). Chern insulator to A Odd Chern metal transition:

In the range $\alpha\Delta>0$ and tuned by $c$, one finds a second-order non-
analytical behaviour at $c=v$,

$\displaystyle
E_{\text{GS}}=E_{0}-\frac{(c-v)^{2}\Delta}{4\pi\alpha^{2}}\Theta(c-v)+\cdots$
(74)

which has the dynamic exponent $z=2$.

In summary, (c) and (d) are 2nd-order with $z=2$; (a) is 3rd-order with $z=1$;
(b) is infinite-order.

####  III.2.3  Thermodynamic quantities

(a). The density of states

From the Hamiltonian (25), the Matsubara Green’s function

$\displaystyle
G_{i}(\mathbf{k},i\omega_{n})=\frac{P_{i,+}(\mathbf{k})}{i\omega_{n}-\epsilon_{i,+}(\mathbf{k})}+\frac{P_{i,-}(\mathbf{k})}{i\omega_{n}-\epsilon_{i,-}(\mathbf{k})}$
(75)

where $P_{i,s}$ are the projection operators onto $s=\pm$ bands.

The total DOS for each flavor is

$\displaystyle
D_{i}(\omega)=\int\frac{d^{2}q}{(2\pi)^{2}}[\delta(\omega-\epsilon_{i,+}(k))+\delta(\omega-\epsilon_{i,-}(k))]$
(76)

It would be convenient to introduce the DOS for each band
$D_{i}(\omega)=D_{i,+}(\omega)+D_{i,-}(\omega)$:

$\displaystyle
D_{i,\pm}(\omega)=\int\frac{d^{2}\mathbf{k}}{4\pi^{2}}\delta(\omega-\epsilon_{i,\pm}(\mathbf{k})),$
(77)

Due to the C-symmetry, $D_{i,+}(\omega)=D_{i,-}(-\omega)$. Similar to Eq.54,
we also have

$\displaystyle
D_{i,+}\\!(\omega)\\!=\\!\frac{1}{4\pi|\alpha|}\sqrt{\\!\frac{c^{2}}{2v^{2}\\!-\\!c^{2}}}[1\\!+\\!\rho_{i}(\omega)]\Theta(\omega\\!-\\!\min\epsilon_{i,+}\\!)$
(78)

where $\rho_{i}(\omega)=-\rho_{i}(-\omega)$ is an odd function. When
$\min\epsilon_{i,+}<0<\max\epsilon_{i,-}$, the odd property leads to the flat
feature of total DOS $D(\omega)=D_{1}(\omega)+D_{2}(\omega)$ for $H_{i}$ near
$\omega=0$ ( Fig.19 ):

$\displaystyle
D_{i}(\omega)\\!=\\!\frac{1}{2\pi|\alpha|}\sqrt{\frac{c^{2}}{2v^{2}\\!-c^{2}}},\min\epsilon_{i,+}<\omega<\max\epsilon_{i,-}$
(79)

where as alerted below Eq.61, we restrict $c<\sqrt{2}v$, so that the DOS
remain finite. Otherwise, one must resort to the lattice calculations in
Sec.II.

Figure 19: The Density of states $D_{1}(\omega)$ and $D_{2}(\omega)$ of
Eq.(60) at different $c/v$ values: (a) $c/v=0.5$, (b) $c/v=1.0$, (c)
$c/v=\sqrt{1.4}\approx 1.18$, (d) $c/v=1.4$. Here, we fixed $v=1$,
$\alpha=1/2$, $\Delta=1/5$. In (c) and (d), the central plateau DOS value is
given in Eq.79. Again, similar to Fig.15, the flat feature of total DOS near
$\omega=0$ is a result of truncation of $H(k)$ at $k^{2}$ order, and the
lattice DOS shown in Fig.10 is not a constant anymore. The total DOS
$D(\omega)=D_{1}(\omega)+D_{2}(\omega)$ is just the sum of the two.

(b). The specific heat and compressibility

The Helmholtz free energy density $F(T)=F_{1}(T)+F_{2}(T)$:

$\displaystyle
F_{i}(T)=E_{i,\text{GS}}+\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\Big{(}|\epsilon_{i,+}(\mathbf{k})|$
$\displaystyle-T\ln[2(1+\cosh(\epsilon_{i,+}(\mathbf{k})/T))]\Big{)}$ (80)

where the zero temperature part of $F(T)$ is nothing but the ground-state
energy density $E_{\text{GS}}$ calculated in Sec.III-B-2. The specific heat
(at constant volume) is given by $C_{v}(T)=C_{v,1}(T)+C_{v,2}(T)$:

$\displaystyle
C_{v}(T)=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\sum_{i=1,2}\frac{[\epsilon_{i,+}(\mathbf{k})]^{2}}{T^{2}[1+\cosh(\epsilon_{i,+}(\mathbf{k})/T)]}$
(81)

Similarly, the (isothermal) uniform compressibility
$\kappa_{u}(T)=\kappa_{u,1}(T)+\kappa_{u,2}(T)$ is

$\displaystyle\kappa_{u}(T)$
$\displaystyle=\sum_{i,s}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{\partial
f(\epsilon_{i,s}(k))}{\partial\epsilon_{i,s}(k)}=\int\frac{d\omega
D(\omega)}{2T(1+\cosh(\omega/T))}$ (82)

where $D_{i}(\omega),D(\omega)$ are listed in Eq.79.

Its zero temperature limit $\frac{\partial
f(\epsilon)}{\partial\epsilon}=\delta(\epsilon)$ recovers the well-known
relation between the compressibility and the DOS $\kappa_{u}(T)=D(0)$.

At low temperature $T$ which is the lowest energy scale, one can get a low
temperature expansion of $C_{v}(T)$ and $\kappa_{u}(T)$. Without loss of
generality, we choose $\alpha\Delta<0$ case in Fig.16, the two critical
velocities
$c_{1}=|v|<c_{2}=\sqrt{v^{2}-4\alpha\Delta}=\sqrt{v^{2}+4|\alpha\Delta|}$. By
keeping the leading low $T$ dependence and assuming
$|\alpha\Delta/v^{2}|<1/2$, we have

$\Delta\neq 0$ cases:

At $c<c_{1}$, $H_{1}$ has a gap $2\tilde{\Delta}_{1}>0$ and $H_{2}$ has a
bigger gap $2\tilde{\Delta}_{2}>2\tilde{\Delta}_{1}$, $D_{i}(\omega)=0$ for
$|\omega|<\tilde{\Delta}_{i}$ ( Fig.19a ). Thus,
$C_{v,i}(T)=2D_{i}(\tilde{\Delta}_{i})\tilde{\Delta}_{i}^{2}T^{-1}e^{-|\tilde{\Delta}_{i}|/T}$,
so
$C_{v}(T)=2D_{1}(\tilde{\Delta}_{1})\tilde{\Delta}_{1}^{2}T^{-1}e^{-|\tilde{\Delta}_{1}|/T}$.
Similarly,
$\kappa_{u,i}(T)=2D_{i}(\tilde{\Delta}_{i})Te^{-\tilde{\Delta}_{i}/T}$, so
$\kappa_{u}(T)=2D_{1}(\tilde{\Delta}_{1})Te^{-\tilde{\Delta}_{1}/T}$. Both
$C_{v}(T)$ and $\kappa_{u}(T)$ are dominated by the node 1.

At $c=c_{1}$, $H_{1}$ is critical with $\tilde{\Delta}_{1}=0$ and $H_{2}$
still has a gap $2\tilde{\Delta}_{2}>0$,
$D_{1}(\omega)=D_{1}(0)+D_{1}^{\prime}(0^{+})|\omega|$ for small $|\omega|$,
$D_{2}(\omega)=0$ for $|\omega|<\tilde{\Delta}_{2}$ ( Fig.19b ). Thus,
$C_{v,1}(T)=\frac{\pi^{2}}{3}D_{1}(0)T$ and
$C_{v,2}(T)=2D_{2}(\tilde{\Delta}_{2})\tilde{\Delta}_{2}^{2}T^{-1}e^{-|\tilde{\Delta}_{2}|/T}$,
so $C_{v}(T)=\frac{\pi^{2}}{3}D_{1}(0)T$. Similarly,
$\kappa_{u,1}(T)=D_{1}(0)+2\ln 2D^{\prime}_{1}(0^{+})T$ and
$\kappa_{u,2}(T)=2D_{2}(\tilde{\Delta}_{2})Te^{-\tilde{\Delta}_{2}/T}$, so
$\kappa_{u}(T)=D_{1}(0)+2\ln 2D^{\prime}_{1}(0^{+})T$. Both $C_{v}(T)$ and
$\kappa_{u}(T)$ are still dominated by the node 1.

At $c_{2}>c>c_{1}$, $H_{1}$ is gapless with $\tilde{\Delta}_{1}<0$ and $H_{2}$
still has a gap $2\tilde{\Delta}_{2}>0$,
$D_{1}(\omega)=\frac{1}{2\pi|\alpha|}\sqrt{\frac{c^{2}}{2v^{2}-c^{2}}}$ in
Eq.79 for $|\omega|<|\tilde{\Delta}_{1}|$, $D_{2}(\omega)=0$ for
$|\omega|<\tilde{\Delta}_{2}$. Thus, $C_{v,1}(T)=\frac{\pi^{2}}{3}D_{1}(0)T$
and
$C_{v,2}(T)=2D_{2}(\tilde{\Delta}_{2})\tilde{\Delta}_{2}^{2}T^{-1}e^{-|\tilde{\Delta}_{2}|/T}$,
so $C_{v}(T)=\frac{\pi^{2}}{3}D_{1}(0)T$. Similarly,
$\kappa_{u,1}(T)=D_{1}(0)+2[D_{1}(\tilde{\Delta}_{1})-D_{1}(0)]Te^{-|\tilde{\Delta}_{1}|/T}$
and $\kappa_{u,2}(T)=2D_{2}(\tilde{\Delta}_{2})Te^{-|\tilde{\Delta}_{2}|/T}$,
so
$\kappa_{u}(T)=D_{1}(0)+2[D_{1}(\tilde{\Delta}_{1})-D_{1}(0)]Te^{-|\tilde{\Delta}_{1}|/T}+2D_{2}(\tilde{\Delta}_{2})Te^{-|\tilde{\Delta}_{2}|/T}$.
However, the relative magnitude of $|\tilde{\Delta}_{1}|$ and
$|\tilde{\Delta}_{2}|$ depends on $c$, so one can not tell which is smaller in
general.

At $c=c_{2}$, $H_{1}$ is gapless with $\tilde{\Delta}_{1}<0$ and $H_{2}$
becomes critical with $\tilde{\Delta}_{2}=0$,
$D_{1}(\omega)=\frac{1}{2\pi|\alpha|}\sqrt{\frac{c^{2}}{2v^{2}-c^{2}}}$ in
Eq.79 for $|\omega|<|\tilde{\Delta}_{1}|$,
$D_{2}(\omega)=D_{2}(0)+D_{2}^{\prime}(0^{+})|\omega|$ for small $|\omega|$ (
Fig.19c ). Thus, $C_{v,1}(T)=\frac{\pi^{2}}{3}D_{1}(0)T$ and
$C_{v,2}(T)=\frac{\pi^{2}}{3}D_{2}(0)T$, so $C_{v}(T)=\frac{\pi^{2}}{3}D(0)T$.
Similarly,
$\kappa_{u,1}(T)=D_{1}(0)+2[D_{1}(\tilde{\Delta}_{1})-D_{1}(0)]Te^{-|\tilde{\Delta}_{1}|/T}$
and $\kappa_{u,2}(T)=D_{2}(0)+2\ln 2D_{2}^{\prime}(0)T$, so
$\kappa_{u}(T)=D(0)+2\ln 2D_{2}^{\prime}(0)T$.

At $c>c_{2}$, $H_{1}$ is gapless with $\tilde{\Delta}_{1}<0$ and $H_{2}$ is
also gapless with $\tilde{\Delta}_{2}<0$,
$D_{1}(\omega)=\frac{1}{2\pi|\alpha|}\sqrt{\frac{c^{2}}{2v^{2}-c^{2}}}$ in
Eq.79 for small $|\omega|<|\tilde{\Delta}_{1}|$,
$D_{2}(\omega)=\frac{1}{2\pi|\alpha|}\sqrt{\frac{c^{2}}{2v^{2}-c^{2}}}$ in
Eq.79 for small $|\omega|<|\tilde{\Delta}_{2}|$ ( Fig.19d ). Thus,
$C_{v,1}(T)=\frac{\pi^{2}}{3}D_{1}(0)T$ and
$C_{v,2}(T)=\frac{\pi^{2}}{3}D_{2}(0)T$, so $C_{v}(T)=\frac{\pi^{2}}{3}D(0)T$.
Similarly,
$\kappa_{u,1}(T)=D_{1}(0)+2[D_{1}(\tilde{\Delta}_{1})-D_{1}(0)]Te^{-|\tilde{\Delta}_{1}|/T}$
and
$\kappa_{u,2}(T)=D_{2}(0)+2[D_{2}(\tilde{\Delta}_{1})-D_{2}(0)]Te^{-|\tilde{\Delta}_{2}|/T}$.
so
$\kappa_{u}(T)=D(0)+2[D_{2}(\tilde{\Delta}_{1})-D_{2}(0)]Te^{-|\tilde{\Delta}_{2}|/T}$.

$\Delta=0$ cases:

In this case, $c_{1}=c_{2}=c$ and $H_{1}$ and $H_{2}$ have exactly the same
spectrum.

At $c<v$, $D_{i}(\omega)=D^{\prime}_{i}(0)|\omega|$ for small $|\omega|$, thus
$C_{v,1}(T)=C_{v,2}(T)=9\zeta(3)D^{\prime}_{i}(0^{+})T^{2}$ and
$C_{v}(T)=9\zeta(3)D^{\prime}(0^{+})T^{2}$. Similarly,
$\kappa_{u,1}(T)=\kappa_{u,2}(T)=2\ln 2D_{i}^{\prime}(0)T$ and
$\kappa_{u}(T)=2\ln 2D^{\prime}(0)T$.

At $c=v$ and $\Delta=0$, $D_{i}(\omega)=D_{i}(0)+A_{i}(0)|\omega|^{2/3}$ for
small $|\omega|$, thus $C_{v,1}(T)=C_{v,2}(T)=\frac{\pi^{2}}{3}D_{i}(0)T$ and
$C_{v}(T)=\frac{\pi^{2}}{3}D(0)T$. Similarly,
$\kappa_{u,1}(T)=\kappa_{u,2}(T)=D_{i}(0)+1.1486A_{i}T^{5/3}$ and
$\kappa_{u}(T)=D(0)+1.1486AT^{5/3}$.

At $c>v$ and $\Delta=0$, $D_{i}(\omega)=D_{i}(0)$ for small $|\omega|$,
$C_{v,1}(T)=C_{v,2}(T)=\frac{\pi^{2}}{3}D_{i}(0)T$ and
$C_{v}(T)=\frac{\pi^{2}}{3}D(0)T$. Similarly,
$\kappa_{u,1}(T)=\kappa_{u,2}(T)=D_{i}(0)+2[D_{i}(\tilde{\Delta}_{i})-D_{i}(0)]Te^{-|\tilde{\Delta}_{i}|/T}$
and $\kappa_{u}(T)=D(0)+2[D(\tilde{\Delta})-D(0)]Te^{-|\tilde{\Delta}|/T}$.

If we only consider the leading low temperature behaviors of
$C_{v}(T)=C_{v,1}(T)+C_{v,2}(T)$ and
$\kappa_{u}(T)=\kappa_{u,1}(T)+\kappa_{u,2}(T)$, one can summarize these
results as:

When $\Delta\neq 0$ : at $c<c_{1}$,
$C_{v}(T)=2D_{1}(\tilde{\Delta}_{1})\tilde{\Delta}_{1}^{2}T^{-1}e^{-|\tilde{\Delta}_{1}|/T}$
and $\kappa_{u}(T)=2D_{1}(\tilde{\Delta}_{1})Te^{-\tilde{\Delta}_{1}/T}$; at
$c\geq c_{1}$, $C_{v}(T)=\frac{\pi^{2}}{3}D(0)T$ and $\kappa_{u}(T)=D(0)$.

When $\Delta=0$ : at $c<v$, $C_{v}(T)=9\zeta(3)D^{\prime}(0^{+})T$ and
$\kappa_{u}(T)=2\ln 2D^{\prime}(0)T$; at $c\geq v$,
$C_{v}(T)=\frac{\pi^{2}}{3}D(0)T^{2}$ and $\kappa_{u}(T)=D(0)$.

(c). The Wilson ratio

The Wilson ratio is defined as $R_{W}=T\kappa_{u}/C_{v}$ which has the
following low temperature behaviours.

$\Delta\neq 0$ case:

When $c<c_{0}$, $R_{W}=(T/\tilde{\Delta})^{2}$; when $c\geq c_{0}$,
$R_{W}=3/\pi^{2}$.

$\Delta=0$ case:

When $c<c_{0}$, $R_{W}=2\ln 2/(9\zeta(3))\approx 0.1281$; when $c\geq c_{0}$,
$R_{W}=3/\pi^{2}$.

These results are consistent with those achieved directly on the lattice in
Sec.II-C-3 and also listed in the last line of Table-I.

##  IV  The chiral edge properties in a strip geometry

Following the approach used in the bulk properties, we will first study the
edge properties from the lattice system, then investigate them from the
continuum effective theory, then contrast the two complementary approaches.

###  IV.1  Edge states from the microscopic lattice theory

For the periodic boundary condition in the $y$-direction and open boundary
condition in the $x$-direction, $k_{y}$ is a good quantum number, the
Hamiltonian in the mixed $(i,k_{y})$ representation becomes

$\displaystyle
H\\!=\\!\\!\sum_{k_{y},i,j}\\!\\!c_{i,k_{y}}^{\dagger}\\!\\{[-(h\\!+\\!2t\cos
k_{y})\sigma_{z}\\!\\!+\\!2t_{s}\sin k_{y}\sigma_{y}\\!\\!+\\!2t_{b}\sin
k_{y}\sigma_{0}]\delta_{i,j}$
$\displaystyle+(t\sigma_{z}\\!-\\!it_{s}\sigma_{y})\delta_{i,j+1}+(t\sigma_{z}\\!+\\!it_{s}\sigma_{y})\delta_{i,j-1}\\}c_{j,k_{y}}$
(83)

For the periodic boundary condition in the $x$-direction and open boundary
condition in the $y$-direction, $k_{x}$ is a good quantum number, the
Hamiltonian in the mixed $(k_{x},i)$ representation becomes:

$\displaystyle
H\\!=\\!\\!\sum_{k_{x},i,j}\\!c_{k_{x},i}^{\dagger}\\{[-(h+2t\cos
k_{x})\sigma_{z}+t_{s}\sin k_{x}\sigma_{x}]\delta_{i,j}$
$\displaystyle\\!+\\!(t\sigma_{z}\\!-\\!it_{s}\sigma_{y}\\!-\\!it_{b}\sigma_{0})\delta_{i,j+1}\\!+\\!(t\sigma_{z}\\!+\\!it_{s}\sigma_{y}\\!+\\!it_{b}\sigma_{0})\delta_{i,j-1}\\}c_{k_{x},j}$
(84)

In the Fig.20 and 21, we show the numerical results on the lattice edge
states.

Figure 20: The edge states of the lattice Hamiltonian in a strip geometry. We
fixed $h=-0.5$. From left to right, the parameter $t_{b}/t_{s}$ is
$0,0.5,1.0,1.1$, respectively. (Top) Longitudinal injection: With the periodic
boundary condition in the $y$-direction and the open boundary condition in the
$x$-direction. The edge modes always exist, but undergoes the edge
reconstruction at $t_{s}/t=1$. The two edges move along the opposite
directions when $t_{s}/t<1$ in (a1) and (b1) in the CI, then one edge becomes
flat at $t_{s}/t=1$ in (c1), then two edges move along the same  direction
when $t_{s}/t>1$ in (d1) in the odd CM, (Bottom) Transverse injection:
Exchanging the role of $x$ and $y$ direction. The edge modes exist only when
$t_{s}/t<1$ in (a2) and (b2), but squeezed out at $t_{s}/t=1$ in (c2) where
the direct bulk gap closes, completely disappear when $t_{s}/t>1$ in (d2) in
the odd CM. Although the edge modes show quite different behaviours in the
line 1 and the line 2, there seems a one to one Longitudinal/Transvese edge-
edge correspondence between them. In both figures, one can shift $k\to k+\pi$
to reach $h=+0.5$ results. See also Fig.22 and Fig.23 for the continuum
calculations. See also Fig.S1 for the expanded figure.  Figure 21:  The same
situation as in Fig.20 except at a fixed $h=-3.5$. In both figures, one can
shift $k\to k+\pi$ to reach $h=+3.5$ results. It shows qualitatively the same
edge TPTs as those in Fig.20. See also Fig.S2 for the expanded figure.

####  IV.1.1  Interpretation of the Hall conductance in terms of the edge
states, enriched bulk-edge and new L/edge-T/edge correspondences

The bulk-edge correspondence in a static frame is also enriched under the
injection or in a moving sample: Relative to the injection or boost, there is
a longitudinal or transverse edge, so the original bulk-edge correspondence is
enriched to the bulk to longitudinal/or transverse edge correspondence, then
the longitudinal edge to transverse edge correspondence.

In the longitudinal injecting case, the edge state always exist, so its
contribution to $\sigma_{H}$ remains quantized as $\sigma_{H}=Ch_{-}=1$
throughout the TPT at (c1). However, there is no bulk contributions in the
Chern insulator, but the bulk starts to contribute $\sigma_{H}=\nu_{b}=v/c-1$
in the Odd Chern metal phase. The two parts lead back to Eq.32 evaluated in
the bulk:

$\displaystyle\sigma_{H}$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\Omega_{-}(\mathbf{k})d^{2}\mathbf{k}+\frac{1}{\pi}\int_{\mathbb{R}^{2}}\Omega_{+}(\mathbf{k})\Theta(-\epsilon_{+}(\mathbf{k}))d^{2}\mathbf{k}$
$\displaystyle={\rm Ch}_{-}+\nu_{b}$ (85)

In the transverse injecting case, the edge states exist only before the TPT at
(c2), so its contribution to $\sigma_{H}$ remains quantized as
$\sigma_{H}=Ch_{-}$ only before the TPT at (c2). There is no bulk
contributions in the Chern insulator before (c2). However, after (c2), the
edge states emerge into the bulk and disappear. All the contributions come
from the bulk as $\sigma_{H}=v/c<1$.

###  IV.2  Solving the edge states from the effective theory in the continuum

We will solve the edge states with the periodic boundary condition in the
$y$-direction and the open boundary condition in the $x$-direction, then vice
versa.

####  IV.2.1  Solving the edge states in the longitudinal injection

We will solve the model in a strip geometry topoSF  with the periodic boundary
condition in the $y$-direction and open boundary condition in the
$x$-direction. The continuum Hamiltonian in the mixed $(\partial_{x},k_{y})$
representation:

$\displaystyle H(\partial_{x},k_{y})$
$\displaystyle=(\Delta-\alpha_{x}\partial_{x}^{2})\sigma_{z}-iv_{x}\partial_{x}\sigma_{x}$
$\displaystyle\phantom{=}+\alpha_{y}k_{y}^{2}\sigma_{z}+v_{y}k_{y}\sigma_{y}-ck_{y}\sigma_{0}$
(86)

The problem will first be studied in the $k_{y}=0$ limit where
$H_{1D}=(\Delta-\alpha_{x}\partial_{x}^{2})\sigma_{z}-iv_{x}\partial_{x}\sigma_{x}$,
and then extended to finite $k_{y}\neq 0$. Due to the C-symmetry, the edge
mode $\psi(x,k_{y}=0)$ is expected to exist at zero energy, therefore
$H_{1D}\psi=0$. Multiplying both side by $\sigma_{z}$ gives
$[(\Delta-\alpha_{x}\partial_{x}^{2})+v_{x}\partial_{x}\sigma_{y}]\psi=0$.
Choosing $\sigma_{y}\psi_{\pm}=\pm\psi_{\pm}$ with
$\psi_{\pm}=\tfrac{1}{\sqrt{2}}(\phi,\pm i\phi)^{\intercal}$, or
$\psi_{\pm}=\phi\chi_{\pm}$ with

$\displaystyle\chi_{\pm}=\tfrac{1}{\sqrt{2}}(\begin{smallmatrix}1\\\ \pm
i\end{smallmatrix})$ (87)

then the coupled differential equations can be reduced to a second order
homogeneous ordinary differential equation

$[(\Delta-\alpha_{x}\partial_{x}^{2})\pm v_{x}\partial_{x}]\phi=0\>.$ (88)

Substituting the ansatz $\phi\propto e^{-\lambda x}$ leads to a character
equation $\alpha_{x}\lambda^{2}\pm v_{x}\lambda-\Delta=0$ with the solutions:

$\displaystyle\lambda_{1,2}^{(s)}$
$\displaystyle=\frac{-sv_{x}\pm\sqrt{v_{x}^{2}+4\alpha_{x}\Delta}}{2\alpha_{x}},$
(89)

where $s=\pm$ corresponding to the choice of $\pm$ in Eq.88. Thus, the general
solution can be written as

$\phi=c_{1}e^{-\lambda_{1}x}+c_{2}e^{-\lambda_{2}x}\>.$ (90)

Due to the C-symmetry, one only need to consider one edge. Choosing the left
edge and imposing the wave function to vanish at $x=0$ and $x=+\infty$
requires $c_{1}=-c_{2}$ and $\mathfrak{Re}\lambda_{1,2}>0$. Thus, we have
$\phi_{L}(x)\propto(e^{-\lambda_{1}x}-e^{-\lambda_{2}x})$, where
$\lambda_{1,2}$ denote either $\lambda_{1,2}^{(+)}$ or $\lambda_{1,2}^{(-)}$
in Eq.89 whichever have positive real parts.

The condition for $\mathfrak{Re}\lambda_{1,2}>0$ is analyzed as:

If $v_{x}>0$, $\alpha_{x}>0$, $\Delta<0$, the localized edge state is
$\psi_{L}(x)\propto(e^{-\lambda_{1}^{(-)}x}-e^{-\lambda_{2}^{(-)}x})\chi_{-}$.

If $v_{x}>0$, $\alpha_{x}<0$, $\Delta>0$, the localized edge state is
$\psi_{L}(x)\propto(e^{-\lambda_{1}^{(+)}x}-e^{-\lambda_{2}^{(+)}x})\chi_{+}$.

If $v_{x}<0$, $\alpha_{x}>0$, $\Delta<0$, the localized edge state is
$\psi_{L}(x)\propto(e^{-\lambda_{1}^{(+)}x}-e^{-\lambda_{2}^{(+)}x})\chi_{+}$.

If $v_{x}<0$, $\alpha_{x}<0$, $\Delta>0$, the localized edge state is
$\psi_{L}(x)\propto(e^{-\lambda_{1}^{(-)}x}-e^{-\lambda_{2}^{(-)}x})\chi_{-}$.

Otherwise, there is not any localized edge state.

In short, the left localized edge state exists when $\alpha_{x}\Delta<0$ and
$\psi_{L}=+\operatorname{sgn}(v_{x}\Delta)\psi_{L}$. The C-symmetry indicates
that the right localized edge state also exists when $\alpha_{x}\Delta<0$ and
$\sigma_{y}\psi_{R}=-\operatorname{sgn}(v_{x}\Delta)\psi_{R}$.

When extending the results at $k_{y}=0$ to finite $k_{y}$ case, one can just
replace $\Delta\to\Delta+\alpha k_{y}^{2}$ in Eq.89, and then
$\psi_{L}(x,y)\propto\psi_{L}(x)e^{ik_{y}y}$ is an eigenstate of
$H(\partial_{x},k_{y})$. [It is obvious to see that the exact spectrum is the
same as that achieved by treating $k_{y}$-dependent part perturbatively. Note
that $\chi_{\pm}$ is not eigenstate of $\alpha_{x}k_{x}^{2}\sigma_{z}$.]
Therefore, we get the edge effective Hamiltonian including both left
$\psi_{L}$ and right $\psi_{R}$ edge:

$\displaystyle H_{\text{edge}}(k_{y})$
$\displaystyle\\!=\\!\operatorname{sgn}(v_{x}\Delta)[(v_{y}\\!-\\!c)k_{y}\psi_{L}^{\dagger}\psi_{L}\\!-\\!(v_{y}\\!+\\!c)k_{y}\psi_{R}^{\dagger}\psi_{R}]$
(91)

which shows the dispersion relations for the edge states at the open
$x$-boundary are

$\displaystyle\epsilon_{L}(k_{y})$
$\displaystyle=+\operatorname{sgn}(v_{x}\Delta)(v_{y}-c)k_{y},\quad$
$\displaystyle\epsilon_{R}(k_{y})$
$\displaystyle=-\operatorname{sgn}(v_{x}\Delta)(v_{y}+c)k_{y}$ (92)

Meanwhile, the bulk spectrum is continuous, which is given by

$\displaystyle\epsilon_{\text{bulk}}^{\pm}(k_{y}\\!)\\!=\\!\pm\\!\sqrt{\\!(\\!\Delta\\!+\\!\alpha_{x}k_{x}^{2}\\!\\!+\\!\alpha_{y}k_{y}^{2})^{2}\\!\\!+\\!v_{x}^{2}k_{x}^{2}\\!+\\!v_{y}^{2}k_{y}^{2}}\\!-\\!ck_{y}$
(93)

where $k_{x}$ is a continuous real parameter.

The edge state extends in a finite regime around $k_{y}=0$ which can be
estimated by when the energy of edge state first enter into the bulk spectrum.
Solving $\min_{k_{x}}\epsilon_{\text{bulk}}^{+}(k_{y})=\epsilon_{L}(k_{y})$
gives the max $k_{x}$. Thus when $\alpha\Delta<0$, the edge state survives in
the regime $|k_{y}|<\sqrt{-\Delta/\alpha}$ which is also independent of boost
$c$.

Which shows one edge state becomes zero slope at the QPT, then reverses its
slope after. We plot bulk states and edge states in Fig.22.

Putting $c=0$ in Eq.91 recovers the edge theory without the injection. Then
directly performing a Galileo boost along the edge leads to Eq.91.

Figure 22: Longitudinal (L-) edge: The spectrum of the Hamiltonian
$H_{1D}(\partial_{x},k_{y})$ in Eq.86 with periodic boundary condition in the
$y$-direction and open boundary condition in the $x$-direction. From left to
right, the parameter $c$ is $0,0.5,1.0,1.1$, respectively. Other parameters
are fixed as $v_{x}=v_{y}=1$, $\alpha_{x,y}=1$, $\Delta=-0.1$. The left/right
edge mode is highlighted in the red/green color. The results are consistent
with those on the lattice in Fig.20, 21 with the same boundary conditions.

####  IV.2.2  Solving the edge states in the transverse injection

Similarly, we can also consider the model in a strip geometry topoSF  with the
periodic boundary condition in the $x$-direction and open boundary condition
in the $y$-direction. The continuum Hamiltonian in the mixed
$(k_{x},\partial_{y})$ representation:

$\displaystyle H(k_{x},\partial_{y})$
$\displaystyle=(\Delta-\alpha_{y}\partial_{y}^{2})\sigma_{z}-iv_{y}\partial_{y}\sigma_{y}+ic\partial_{y}\sigma_{0}$
$\displaystyle\phantom{=}+\alpha_{x}k_{x}^{2}\sigma_{z}+v_{x}k_{x}\sigma_{x}$
(94)

The problem will first be studied in the $k_{x}=0$ limit where
$H_{1D}=(\Delta-\alpha_{y}\partial_{y}^{2})\sigma_{z}-iv_{y}\partial_{y}\sigma_{y}+ic\partial_{y}\sigma_{0}$,
and then extended to finite $k_{x}\neq 0$. Due to the C-symmetry, the edge
mode $\psi(k_{x}=0,y)$ is expected to exist at zero energy, thus
$H_{1D}\psi=0$. Multiplying both side by $\sigma_{z}$ gives
$[\Delta+(ic\sigma_{z}-v_{y}\sigma_{x})\partial_{y}-\alpha\partial_{y}^{2}]\psi=0$.
Choosing
$(ic\sigma_{z}-v_{y}\sigma_{x})\psi_{\pm}=\pm\sqrt{\smash[b]{v_{y}^{2}-c^{2}}}\psi_{\pm}$
with $\psi_{\pm}=\phi\chi_{\pm}$ where the two-component spinor is

$\chi_{\pm}=\tfrac{1}{\sqrt{2}}(\begin{smallmatrix}1\\\ \xi_{\pm}\\\
\end{smallmatrix}),~{}~{}~{}\xi_{\pm}=(ic\mp\sqrt{\smash[b]{v_{y}^{2}-c^{2}}})/v_{y}$
(95)

which depends on the transverse boost $c$ explicitly. Then the coupled
differential equation also can be reduced to a second order ordinary
differential equation

$(\Delta\pm\sqrt{\smash[b]{v_{y}^{2}-c^{2}}}\partial_{y}-\alpha_{y}\partial_{y}^{2})\phi=0,$
(96)

where $\pm$ corresponding to the choice of $\psi_{\pm}$. Substituting the
ansatz $\phi=e^{-\lambda y}$ leads to
$\alpha_{y}\lambda^{2}\pm\sqrt{\vphantom{v_{x}}\smash[b]{v_{y}^{2}-c^{2}}}\lambda-\Delta=0$
with the roots:

$\displaystyle\lambda_{1,2}^{(s)}$
$\displaystyle=\frac{-s\sqrt{\vphantom{V_{x}^{o}}\smash[b]{v_{y}^{2}-c^{2}}}\pm\sqrt{\vphantom{v_{x}}\smash[b]{v_{y}^{2}-c^{2}+4\alpha_{y}\Delta}}}{2\alpha_{y}},$
(97)

where $s=\pm$ corresponds to the choice of $\pm$ in Eq.96. Thus, the general
solution can be written as

$\phi=c_{1}e^{-\lambda_{1}y}+c_{2}e^{-\lambda_{2}y}$ (98)

Due to the C-symmetry, one only need consider one edge. Choosing the bottom
edge, imposing the wave function to vanish at $y=0$ and $y=+\infty$ requires
$c_{1}=-c_{2}$ and $\mathfrak{Re}\lambda_{1,2}>0$. Thus, we have
$\phi_{B}(y)\propto(e^{-\lambda_{1}y}-e^{-\lambda_{2}y})$ where
$\lambda_{1,2}$ denote $\lambda_{1,2}^{(+)}$ or $\lambda_{1,2}^{(-)}$ in Eq.97
whichever have positive real parts.

The condition for $\mathfrak{Re}\lambda_{1,2}>0$ is analyzed as:

If $v_{y}^{2}>c^{2}$, $\alpha_{y}>0$, $\Delta<0$, the localized edge state is
$\psi_{B}(y)\propto(e^{-\lambda_{1}^{(-)}y}-e^{-\lambda_{2}^{(-)}y})\chi_{-}$;

If $v_{y}^{2}>c^{2}$, $\alpha_{y}<0$, $\Delta>0$, the localized edge state is
$\psi_{B}(y)\propto(e^{-\lambda_{1}^{(+)}y}-e^{-\lambda_{2}^{(+)}y})\chi_{+}$;

Otherwise, there is not any localized edge state.

In short, the Bottom localized edge state exists when $\alpha_{y}\Delta<0$ and
$c^{2}<v_{y}^{2}$, and
$(ic\sigma_{z}-v_{y}\sigma_{x})\psi_{B}=+\operatorname{sgn}(\Delta)\sqrt{\smash[b]{v_{y}^{2}-c^{2}}}\psi_{B}$.
The C-symmetry indicates that the Top localized edge state exists when
$\alpha_{y}\Delta<0$ and $c^{2}<v_{y}^{2}$, and
$(ic\sigma_{z}-v_{y}\sigma_{x})\psi_{T}=-\operatorname{sgn}(\Delta)\sqrt{\smash[b]{v_{y}^{2}-c^{2}}}\psi_{T}$.

When extending to finite $k_{x}$ case, one need replace
$\lambda_{1,2}\to\lambda_{1,2}(k_{x})$ and $\chi_{\pm}\to\chi_{\pm}(k_{x})$,
then redo the eigenvalue problem. [After tedious algebra, we find it gives the
same spectrum as treated the $k_{x}$-dependent part as perturbation.]
Alternatively, one may just take $\psi_{B}(x,y)=\psi_{B}(y)e^{ik_{x}x}$ and
use the first order perturbation theory. Due to
$\langle\chi_{\pm}|\sigma_{x}|\chi_{\pm}\rangle=\mp\sqrt{\smash[b]{v_{y}^{2}-c^{2}}}/v_{y}$
and $\langle\chi_{\pm}|\sigma_{z}|\chi_{\pm}\rangle=0$, we find the edge
effective Hamiltonian including both the bottom $\psi_{B}$ and the top
$\psi_{T}$ edge state

$\displaystyle
H_{\text{edge}}(k_{x})=-\operatorname{sgn}(v_{y}\Delta)v_{x}\sqrt{1-c^{2}/v_{y}^{2}}k_{x}[\psi_{B}^{\dagger}\psi_{B}-\psi_{T}^{\dagger}\psi_{T}]$
(99)

which indicates the dispersion relations for the edge states at the
$y$-boundary:

$\displaystyle\epsilon_{B}(k_{x})$
$\displaystyle=-\operatorname{sgn}(v_{y}\Delta)v_{x}\sqrt{1-c^{2}/v_{y}^{2}}k_{x},\quad$
$\displaystyle\epsilon_{T}(k_{x})$
$\displaystyle=+\operatorname{sgn}(v_{y}\Delta)v_{x}\sqrt{1-c^{2}/v_{y}^{2}}k_{x}$
(100)

Meanwhile, the bulk spectrum is continuous, which is given by

$\displaystyle\epsilon_{\text{bulk}}^{\pm}(k_{x}\\!)\\!=\\!\pm\\!\sqrt{\\!(\\!\Delta\\!+\\!\alpha_{x}k_{x}^{2}\\!\\!+\\!\alpha_{y}k_{y}^{2})^{2}\\!\\!+\\!v_{x}^{2}k_{x}^{2}\\!+\\!v_{y}^{2}k_{y}^{2}}\\!-\\!ck_{y}$
(101)

where $k_{y}$ is a continuous real parameter. The edge state usual extent in a
finit regime around $k_{x}=0$, which can be estimated by when the energy of
edge state first enter into the bulk spectrum. Solving
$\min_{k_{y}}\epsilon_{\text{bulk}}^{+}(k_{x})=\epsilon_{B}(k_{x})$ gives the
max $k_{x}$. Thus when $\alpha\Delta<0$ and $|v_{y}|>|c|$, the edge state
survives in the regime $|k_{x}|<\sqrt{(c^{2}/v_{y}^{2}-1)\Delta/\alpha}$, and
the regime is decrease as increasing the boost $c$.

Which shows edge states do not exist anymore after the bulk QPT. We plot bulk
states and edge states in Fig.23.

Putting $c=0$ in Eq.99 recovers the edge theory without the injection.
However, in contrast to Eq.91, one may not achieve Eq.99 by directly
performing a Galileo boost. Naively, a direct boost may lead to
$\sqrt{v^{2}+c^{2}}$ instead of $\sqrt{v^{2}-c^{2}}$. To achieve Eq.99, one
may still need to solve the bulk + the transverse boundary condition as done
here and find that one must substitute $c\to ic$ in the naive Galileo boost
results to reach the correct transverse boot result. While the $i$ in the
substitution in $c\to ic$ stands for the decay of edge mode along the boost
direction. Eq.99 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 which breaks the GI explicitly. Indeed, its
2-component spinor $\chi_{\pm}$ does depend on the transverse boost
sensitively.

Figure 23: The transverse (T-) edge: spectrum of the Hamiltonian
$H_{1D}(k_{x},\partial_{y})$ in Eq.94. with periodic boundary condition in the
$x$-direction and open boundary condition in the $y$-direction. From left to
right, the parameter $c$ is $0,0.5,1.0,1.1$, respectively. The T-edge
disappears at the same time as the bulk TPT with its velocity vanishing as
$\sqrt{v^{2}-c^{2}}$. Other parameters are fixed as $v_{x}=v_{y}=1$,
$\alpha_{x,y}=1$, $\Delta=-0.1$. The Top/Bottom edge mode is highlighted in
the red/green color. The results are consistent with those on the lattice in
Fig.20, 21 with the same boundary conditions.

###  IV.3  The bulk-edge correspondence from the continuum edge theory

We will discuss the bulk-edge correspondence near $h\sim-4t$ and $h\sim 0$
respectively.

####  IV.3.1  The bulk-edge correspondence near $h\sim-4t$ case

In the $h\sim-4t$ case, there is only one valley at $K_{0}=(0,0)$, the bulk
effective Hamiltonian is:

$\displaystyle H_{0}$
$\displaystyle\\!=\\![\Delta+\alpha(k_{x}^{2}\\!+\\!k_{y}^{2})]\sigma_{z}\\!+\\!vk_{x}\sigma_{x}\\!+\\!vk_{y}\sigma_{y}\\!-\\!ck_{y}\sigma_{0},$
(102)

where $\Delta=-4t-h$, $\alpha=t$, $v=2t_{s}$, $c=2t_{b}$,

Since we choose $t,t_{s},t_{b}>0$, thus $h<-4t$ leads to $\Delta>0$ and
$\alpha\Delta>0$, no edge states. $h>-4t$ leads to $\Delta<0$ and
$\alpha\Delta<0$. Edge state is possible.

(a) Open boundary condition in the $x$-direction: longitudinal injection

When $h<-4t$, there is no edge state;

When $h>-4t$, there is always one localized edge state which is given by the
continuum theory near $K_{0}=(0,0)$ and exists near $K_{0y}=0$.

(b) Open boundary condition in the $y$-direction: transverse injection

If $\alpha\Delta<0$ and $v^{2}>c^{2}$, then $H_{0}$ has one localized edge
state.

Thus, $h<-4t$, there is no edge state;

When $h>-4t$, if $c^{2}<v^{2}$, there is one localized edge state which is
given by the continuum theory near $K_{0}=(0,0)$ and exists near $K_{0x}=0$.
If $c^{2}>v^{2}$, there is still no edge state.

These results in (a) and (b) are consistent with those on a lattice reached in
Sec.IV-A.

####  IV.3.2 The bulk-edge correspondence near $h\sim 0$ case

In the $h\sim 0$ case, there are two valleys at $K_{1}$, $K_{2}$, the bulk
effective Hamiltonian are:

$\displaystyle H_{1}$
$\displaystyle\\!=\\![\Delta-\alpha(k_{x}^{2}-k_{y}^{2})]\sigma_{z}\\!-\\!vk_{x}\sigma_{x}\\!+\\!vk_{y}\sigma_{y}\\!-\\!ck_{y}\sigma_{0}$
$\displaystyle H_{2}$
$\displaystyle\\!=\\![\Delta+\alpha(k_{x}^{2}-k_{y}^{2})]\sigma_{z}\\!+\\!vk_{x}\sigma_{x}\\!-\\!vk_{y}\sigma_{y}\\!+\\!ck_{y}\sigma_{0}$
(103)

where $\Delta=-h$, $\alpha=t$, $v=2t_{s}$, $c=2t_{b}$, written in the generic
form of Eq.25:

$\displaystyle H_{1}$
$\displaystyle\\!=\\!(\Delta\\!+\\!\alpha_{1x}k_{x}^{2}+\alpha_{1y}k_{y}^{2})\sigma_{z}\\!+\\!v_{1x}k_{x}\sigma_{x}\\!+\\!v_{1y}k_{y}\sigma_{y}\\!+\\!c_{1}k_{y}\sigma_{0}$
$\displaystyle H_{2}$
$\displaystyle\\!=\\!(\Delta+\alpha_{2x}k_{x}^{2}\\!+\\!\alpha_{2y}k_{y}^{2})\sigma_{z}\\!+\\!v_{2x}k_{x}\sigma_{x}\\!+\\!v_{2y}k_{y}\sigma_{y}\\!+\\!c_{2}k_{y}\sigma_{0}$
(104)

which leads to the relations:

$\displaystyle\alpha_{1x}\\!=\\!-\alpha,\>\alpha_{1y}\\!=\\!+\alpha,\>v_{1x}\\!=\\!-v,\>v_{1y}\\!=\\!+v,\>c_{1}\\!=\\!-c$
$\displaystyle\alpha_{2x}\\!=\\!+\alpha,\>\alpha_{2y}\\!=\\!-\alpha,\>v_{2x}\\!=\\!+v,\>v_{2y}\\!=\\!-v,\>c_{2}\\!=\\!+c$
(105)

If $\alpha_{1x}\Delta>0$, then $\alpha_{2x}\Delta<0$,
$\alpha_{1y}\Delta<0,\alpha_{2y}\Delta>0$, which means only one valley may
have one edge state for a given type of boundary. Since we choose
$t,t_{s},t_{b}>0$, thus $h>0$ leads to $\Delta<0$ and $\alpha\Delta<0$; $h<0$
leads to $\Delta>0$ and $\alpha\Delta>0$.

(a) Open boundary condition in the $x$-direction: longitudinal injection

If $\alpha\Delta>0$, then $\alpha_{1x}\Delta<0$ which means $H_{1}$ has one
localized edge state;

If $\alpha\Delta<0$, then $\alpha_{2x}\Delta<0$ which means $H_{2}$ has one
localized edge state.

Thus, there is always one edge state. When $h>0$, there is one edge state
given by the continuum theory near $K_{2}=(0,\pi)$ and exists near
$K_{2y}=\pi$. This is expected, because it is smoothly connected to the edge
state near $h<4t$ where there is only one valley at $K_{3}=(\pi,\pi)$ and one
edge states exists near $K_{3y}=\pi$.

When $h<0$, there is one edge state given by the continuum theory near
$K_{1}=(\pi,0)$ and exists near $K_{1y}=0$. This is expected, because it is
smoothly connected to the edge state near $h>-4t$ where there is only one
valley at $K_{0}=(0,0)$ and one edge states exists near $K_{0y}=0$.

(b) Open boundary condition in the $y$-direction: transverse injection

If $c^{2}>v^{2}$, then there is not any localized edge state. If $c^{2}<v^{2}$
and $\alpha\Delta>0$, then $\alpha_{2y}\Delta<0$ which means $H_{2}$ has one
localized edge state; if $c^{2}<v^{2}$ and $\alpha\Delta<0$, then
$\alpha_{1y}\Delta<0$ which means $H_{1}$ has one localized edge state.

Thus, there is one edge state only if $t_{b}<t_{s}$.

When $h>0$, there is one edge state given by the continuum theory near
$K_{1}=(\pi,0)$ and exists near $K_{1x}=\pi$. This is expected, because it is
smoothly connected to the edge state near $h<4t$ where there is only one
valley at $K_{3}=(\pi,\pi)$ and one edge states exists near $K_{3x}=\pi$.

When $h<0$, there is one edge state given by the continuum theory near
$K_{2}=(0,\pi)$ and exists near $K_{2x}=0$. This is expected, because it is
smoothly connected to the edge state near $h>-4t$ where there is only one
valley at $K_{0}=(0,0)$ and one edge states exists near $K_{0x}=0$.

These results in (a) and (b) are consistent with those on a lattice reached in
Sec.IV-A.

##  V  Gauge invariant current: bulk properties

Injecting currents into the system could result in the following form:

$H_{inj,x}=-[\kappa_{b1}\sum_{i}J_{ix}+i\sum^{\infty}_{n=2}(t_{bn}/n)c^{\dagger}_{i}c_{i+nx}]+h.c.$
(106)

where $J_{ix}$ is the NN gauge-invariant current, the other terms
$n=2,3,\cdots$ are NNN, NNNN,…. $\kappa_{b1}$ is dimensionaless and
$t_{bn},n\geq 2$ carry the same dimension as the hopping.

To capture the physics, one only need to include the NN and the $n=2$ NNN term
( which can also be called higher order current ). We still take the “divide
and conquer” strategy to treat the two terms separately and differently. As
shown in Fig.19-22, for the $n=1$ case, the bulk TPT and the edge
reconstruction happens at the same time. But this coincidence could be due to
the $n=1$ case which maybe a fine tuning phase. It is not protected by any
symmetry. So they can split in a more general case. If so, the edge
reconstruction must happen earlier than the bulk TPT, not the other way
around. Then, there must be an intermediate phase between the bulk TPT and the
edge reconstruction, we call such a phase an odd Chern insulator phase which
has the same bulk properties as the ordinary Chern insulator, but different
edge properties: Its longitudinal edge modes satisfy the exotic relation
$v_{L}v_{R}>0$, its transverse edge modes satisfy the conventional relation
$v_{L}v_{R}<0$. In the longitudinal edge, the edge mode undergoes its own edge
TPT with an longitudinal edge dynamic exponent $z_{L}=3$ instead of being
exactly flat for the $n=1$ even inside the Chern insulator before the bulk
TPT. So there is an even enriched surface TPT before the bulk TPT. As to be
shown in Sec.VII and Sec.VIII, this case also provide another example of odd
Chern metal. The BM also leads to a non-vanishing AHE. Remarkably, the
$\sigma_{H}$ jump from the odd Chern metal to the BM remains the same
universal non-integer number as $n=1$.

###  V.1  The NN gauge-invariant current and the NNN current term

Due to the commutation relation

$\displaystyle[c_{i,\sigma}^{\dagger}c_{i,\sigma},\\!\sum_{j}\\!c_{j,\alpha}^{\dagger}T_{\alpha\beta}c_{j+\delta,\beta}]\\!=\\!c_{i,\alpha}^{\dagger}T_{\alpha\beta}c_{i+\delta,\beta}\\!-\\!c_{i-\delta,\alpha}^{\dagger}T_{\alpha\beta}c_{i,\beta}$
(107)

where the summation over spin indices $\sigma,\alpha,\beta$ is implied.

Given a general Hubbard Hamiltonian with SOC on a square lattice

$\displaystyle H_{0}$
$\displaystyle=\sum_{j\delta}[c_{j,\alpha}^{\dagger}T_{\alpha\beta}^{\delta}c_{j+\delta,\beta}+h.c.]+\sum_{j}Un_{j}^{2}-\mu\sum_{j}n_{j}$
$\displaystyle-h\sum_{j}(n_{j\uparrow}-n_{j\downarrow})$ (108)

where $\delta=\hat{x},\hat{y}$ and $U$ is Hubbard interaction. The quantum
anomalous Hall model Eq.1 is just a special case of the general Hamiltonian in
Eq.(108), with $T^{x}=-t_{0}\sigma_{z}+it_{s0}\sigma^{x}$ and
$T^{y}=-t_{0}\sigma_{z}+it_{s0}\sigma^{y}$.

Combining the Heisenberg equation of motion and the continuity equation

$\displaystyle\frac{d}{dt}n_{i}=i[H_{0},n_{i}]=-\nabla\cdot\mathbf{J}_{i}$
$\displaystyle
i[H_{0},n_{i}]=\sum_{\alpha\beta\delta}[i(c_{i,\alpha}^{\dagger}T_{\alpha\beta}^{\delta}c_{i+\delta,\beta}-c_{i-\delta,\alpha}^{\dagger}T_{\alpha\beta}^{\delta}c_{i,\beta})+h.c.]$
$\displaystyle\nabla\cdot\mathbf{J}_{i}=J_{i}^{x}-J_{i-x}^{x}+J_{i}^{y}-J_{i-y}^{y}$
(109)

Thus one can identify the gauge-invariant current as

$\displaystyle J_{i}^{x}$
$\displaystyle=-ic_{i,\alpha}^{\dagger}T_{\alpha\beta}^{x}c_{i+x,\beta}+h.c.$
$\displaystyle J_{i}^{y}$
$\displaystyle=-ic_{i,\alpha}^{\dagger}T_{\alpha\beta}^{y}c_{i+y,\beta}+h.c.$
(110)

which is different from the injecting current Eq.2.

The total number conservation follows:

$\displaystyle\frac{d}{dt}\sum_{i}n_{i}=\sum_{i}(J_{i}^{x}-J_{i-x}^{x}+J_{i}^{y}-J_{i-y}^{y})=0$
(111)

However, $\sum_{i}J_{i}^{x}\neq 0$ and $\sum_{i}J_{i}^{y}\neq 0$.

Consider a new Hamiltonian
$H_{inj}=H_{0}-\sum_{i}(\kappa_{b1,x}J_{i}^{x}+\kappa_{b1,y}J_{i}^{y})$,

$\displaystyle H_{inj}$
$\displaystyle=\sum_{j\delta}[\kappa_{\delta}e^{i\phi_{\delta}}c_{j,\alpha}^{\dagger}T_{\alpha\beta}^{\delta}c_{j+\delta,\beta}+h.c.]+\sum_{j}Un_{j}^{2}$
$\displaystyle-\mu\sum_{j}n_{j}-h\sum_{j}(n_{j\uparrow}-n_{j\downarrow})$
(112)

where $\kappa_{\delta}=\sqrt{1+\kappa_{b1,\delta}^{2}}$ and
$\phi_{\delta}=\arg(1+i\kappa_{b1,\delta}),tan\phi_{\delta}=\kappa_{b1,\delta}$.
The phase factor $e^{i\phi_{\delta}}$ can be transformed away via:

$\tilde{c}_{j}=e^{-i(j_{x}\phi_{x}+j_{y}\phi_{y})}c_{j}$ (113)

This is expected based on the fact that the gauge-invariant current has the
same structure as the hopping and the SOC term, so can be absorbed by a
transformation like Eq.113, but leave the interaction $U$ and the chemical
potential $\mu$ un-touched. This absorbtion does not happen for the injecting
current Eq.2.

Now we study the Hamiltonian in the $\tilde{c}_{j}$ basis and incorporate the
NNN $n=2$ boost term

$\displaystyle
H_{\text{inj}}=\tilde{H}_{0}+\sum_{i}[i(t_{b}/n)\tilde{c}_{i}^{\dagger}\tilde{c}_{i+ny}+h.c.]$
(114)

where $t=t_{0}\sqrt{1+\kappa^{2}_{b1}},t_{s}=t_{s0}\sqrt{1+\kappa^{2}_{b1}}$.
For simplicity, we assume $t_{0}=t_{s0}$, so

$t=t_{s}=t_{0}\sqrt{1+\kappa^{2}_{b1}}$ (115)

Then in the $\tilde{c}_{j}$ basis, the $n=2$ term $t_{b}$ becomes effectively
as:

$t_{b}=\frac{1-\kappa^{2}_{b1}}{1+\kappa^{2}_{b1}}t_{b2}$ (116)

where $t_{b}$ has the same sign as $t_{b2}$ only when $0\leq\kappa_{b1}<1$,