#  Topological pump of ${SU(Q)}$ quantum chain and Diophantine equation

Yasuhiro Hatsugai https://patricia.ph.tsukuba.ac.jp/~hatsugai  Department of
Physics, University of Tsukuba, Tsukuba 305-8571  Yoshihito Kuno  Graduate
School of Engineering Science, Akita University, Akita 010-8502

###### Abstract

A topological pump of the ${SU(Q)}$ quantum chain is proposed associated with
a current due to a local $[U(1)]^{\otimes{Q}}$ gauge invariance of colored
fermions. The ${SU(Q)}$ invariant dimer phases are characterized by the
$Z_{Q}$ Berry phases as a topological order parameter with a $d$-dimensional
twist space ($d={Q}-1$) as a synthetic Brillouin zone. By inclusion of the
symmetry breaking perturbation specified by a rational parameter $\Phi=P/Q$,
the pump, that encloses around the phase boundary, is characterized by the
${Q}$ Chern numbers associated with the currents due to uniform infinitesimal
twists. The analysis of the systems under the open/periodic/twisted boundary
conditions clarifies the bulk-edge correspondence of the pump where the large
gauge transformation generated by the center of mass (CoM) plays a central
role. An explicit formula for the Chern number is given by using the
Diophanine equation.

Numerical demonstration by the exact diagonalization and the DMRG for finite
systems (${Q}=3,4$ and $5$) have been presented to confirm the general
discussions for low energy spectra, edge states, CoM’s, Chern numbers and the
bulk-edge correspondence. A modified Lieb-Schultz-Mattis type argument for the
general ${SU(Q)}$ quantum chain is also mentioned.

##  I Introduction

$U(1)$ gauge invariance is a key ingredient for the quantum Hall effects as
pointed out by Laughlin [1] and is true for the Chern insulators where the
conserved current associated with the gauge field is closely related with the
topological numbers: the Chern number of the gapped many-body state and the
TKNN integers of the one-particle bands [2, 3, 4, 5, 6]. It guarantees
topological stability of the phase without any further symmetry protection.
Existence of the non-trivial phases with non-zero Chern number is only allowed
with time-reversal symmetry breaking that suggests chiral nature of the phase.

As for most of the topological phases, the bulk is hidden, in a sense that the
topological number of the bulk without boundaries is not a physical
observable. What have been observed experimentally are low energy localized
modes as the chiral edge states localized near the boundaries [7, 8, 9]. The
edge states reflect the topological number of the bulk as the bulk-edge
correspondence [10, 11, 12, 13, 14, 15, 16, 17] where $U(1)$ local gauge field
associated with the Aharonov-Bohm flux is crucial due to the Laughlin
argument. The effective theory also justifies the bulk-edge correspondence for
topologically non-trivial systems [18].

Focused studies in the decades reveal various chiral modes in quantum and non-
quantum phenomena in quite different phenomena have a topological origin and
under the control of the bulk-edge correspondence where the Chern number of
the bulk predicts the direction and the number of the chiral modes. The first
non-quantum example can be one-way propagating modes of a gyromagnetic
photonic crystal [19, 20, 21], mechanical chiral modes in microtubes [22] and
coupled optical resonators [23] are also governed by the bulk-edge
correspondence. The concept is applied for a wide variety of phenomena in
photonics [24, 25, 26] and topological circuits [27, 28]. It also includes
mechanical systems [29, 30] and cold atoms [31]. The bulk-edge correspondence
is also a key concept in the focused studies of topological insulators in the
decades [32, 33, 34, 35, 36]. One of the recent surprises is that equatorial
waves near the equator of the earth, that is well known in geophysics, are the
chiral edge modes associated with the non zero-Chern number [37]. There exist
chiral edge modes in evolutionary game theory [38] and biological flows in
neural progenitor cells [39]. The bulk-edge correspondence is universal. In
these classical phenomena, chiral edge states are topologically stable and
protected by the bulk gap, although the Chern numbers of the bulk are never
observed but guarantee the existence of the edge states. Even in these
phenomena, one can introduce $U(1)$ gauge fields to the governing equation by
the minimal coupling to the spatial derivative. The $U(1)$ gauge field is
fictitious and never observed but it predicts the chiral edge states
associated with the Laughlin argument, which implies the bulk-edge
correspondence in classical systems as well.

The local $U(1)$ gauge invariance in one-dimension implies an adiabatic charge
transport of a gapped quantum chain associated with time as an additional
synthetic dimension. This is a topological pump originally proposed by
Thouless [40, 41, 42, 43, 44] in (1+1) dimensions where the transported charge
is quantized as is written by the Chern number. Similar pumping for other
degrees of freedom have been also proposed [45, 46, 47]. Here we do not need
time-reversal symmetry breaking in the 1D system for the non-trivial
topological pump. This idea of the topological pump is old but only after more
than three decades’ experimental trials, the topological pump has been finally
realized in cold atom experiments [48, 49]. The pump is real and topological.
These discoveries motivate to clarify the effects of edge states in the
topological pump [50]. The topological pump is an adiabatic transfer of the
charge. As for an open system with boundaries, the adiabatic cycle of the pump
implies everything is going back to the original state after the period. It
implies nothing is transported in total. What occurs is that the contributions
due to the bulk and edges are cancelled. That is, the pumped charge due to
bulk is given by a back action of the edge states. In a suitable
normalization, a localization length of the edge states scales to zero in a
large system size limit. It implies a quantization of contribution due to edge
states in a large system. It guarantees quantization of the pumped charge due
to bulk. This is the key idea of a topological nature of the topological pump
and the bulk-edge correspondence of the pump [50]. The bulk-edge
correspondence is special in the topological pump, that is, the edge is hidden
and the bulk gives a physical observable, a center of mass that is a time
integral of the current [51, 52, 50, 53]. The pumped $U(1)$ charge due to the
edge states is never experimentally observed due to the gapless nature of the
edge states. It implies the breakdown of the adiabaticity. In other words,
contribution of the edge states can not be measured experimentally in a finite
speed pump since the adiabatic condition can not be satisfied. What is
measured is that of the bulk. In the experiment, one measures a motion of the
center of mass of the system. Its derivative is the current.

Recently this bulk-edge correspondence of the topological pump is also
investigated for interacting fermions [52, 54], quantum spins [55, 56] and
bosons [57, 58, 59, 60]. This cancellation mechanism is applied for fractional
quantum Hall states [61]. A phase transition point between gapped symmetry
protected topological phases (SPT) [62, 63] is the source of non-trivial
topology. This topological transition is characterized by the quantized Berry
phases [64, 65, 66, 67, 68, 69]. The role of the edge states of spin pumping
is also discussed in a mathematically rigorous way for the AKLT Hamiltonian
[70] and its modifications [71]. Note that experimental studies for the
topological pump using cold atoms are rapidly developing [72, 73, 74].

We here investigate topological pump of the ${SU(Q)}$ quantum chain. It is
surprising rich and the bulk-edge correspondence clarifies all details of the
rich structures. One of the surprises is the Diophantine equation, that has
been successfully explains the TKNN integer of the quantum Hall effect (Harper
equation) on a lattice, also used to explain the Chern numbers of the
${SU(Q)}$ pump analytically.

The paper is organized as follows. After the introduction, in Sec.II, the
${SU(Q)}$ quantum chain due to Affleck is described as a generalization of the
$S=1$ bilinear-biquadratic quantum chain. By using a colored fermion
representation, symmetries of the system are described. Especially $Z_{Q}$ and
large gauge transformation due to ${Q}$ gauge symmetries are introduced. A
gapped ${SU(Q)}$ symmetric dimer phase is discussed for a periodic system that
is a source of the non-trivial topological pump. By using a time as a
synthetic dimension and introducing the ${SU(Q)}$ symmetry breaking term, an
${SU(Q)}$ topological pump is proposed, that goes around the gap closing dimer
transition of the ${SU(Q)}$ quantum chain. In Sec.III, ${Q}$ currents
associated with the gauge symmetries are introduced and the center of mass
(CoM), that generates the large gauge transformation is defined where
open/twisted/periodic boundary conditions are carefully discussed. The
$d={Q}+1$ dimensional synthetic Brillouin zone as a parameter space to define
the current and $Q$-closed paths passing through the $Z_{Q}$ symmetric point
are introduced where the averaged currents along the paths are used to define
the topological pump in the adiabatic approximation. The averaged CoM along
the path for the open boundary condition is not continuous and a topological
number of the edge states is defined by using the discontinuities. The bulk-
edge correspondence is proposed for the ${Q}$ different topological numbers of
bulk and edges. In Sec.IV, $Z_{Q}$ quantization of the Berry phases is defined
by the loop in the synthetic Brillouin zone and discussed in details. A
modified Lieb-Schultz-Mattis type argument is also given associated with the
(anti)-translational symmetry of the uniform problem. In Sec.V, $Z_{Q}$
symmetry of systems with odd number of sites are discussed in relation to the
edge states with numerical justifications. Also $Z_{Q}\times Z_{Q}$ emergent
symmetry for systems in the open systems after taking an infinite size limit
is discussed based on the numerical calculations of low energy spectra. The
topological numbers of the system with edges are given due to this emergent
symmetry. Based on the bulk-edge correspondence, an explicit analytical
formula of the ${Q}$ Chern numbers of bulk is given by using the Diophantine
equation of the TKNN for the quantum Hall effect on the lattice. In Sec.VI,
numerical evaluation of the topological numbers of edges and bulk are
explicitly given by using the exact diagonalization and DMRG calculation for
${Q}=3,4$ and $5$ systems, that justify the consistency of the discussion.

##  II ${SU(Q)}$ quantum chain

###  II.1 Fermion representation

Let us start considering an $S=1$ quantum spin chain with nearest neighbor
bilinear-biquadratic interaction

$\displaystyle H_{S}(\omega_{S})$
$\displaystyle=\sum_{j}\big{[}\cos\omega_{S}(\bm{S}_{j}\cdot\bm{S}_{j+1})+\sin\omega_{S}(\bm{S}_{j}\cdot\bm{S}_{j+1})^{2}\big{]},$

where
${[}S_{j,\alpha},S_{j,\beta}{]}=i\sum_{\gamma}\epsilon_{\alpha\beta\gamma}S_{j,\gamma}$,
($\alpha,\beta=1,2,3$) and $\bm{S}_{j}^{2}=S(S+1)$, ($S=1$). It has a long
history of study [75, 76, 77, 78, 79]. We discuss its non-uniform ${SU(Q)}$
extension by a fermion representation due to Affleck [80, 81, 82, 83] (See
also appendix A)

$\displaystyle\bm{S}_{j}\cdot\bm{S}_{j+1}$
$\displaystyle=H^{(1)}(\\{1\\})\big{|}_{{Q}=3}+(const),$ (1)
$\displaystyle(\bm{S}_{j}\cdot\bm{S}_{j+1})^{2}$
$\displaystyle=H^{(2)}(\\{1\\})\big{|}_{{Q}=3}+(const),$ (2)

where $H^{(1,2)}$ are defined for general ${Q}$ as

$\displaystyle H^{(1,2)}(\\{J^{(1,2)}_{j,\alpha\beta}\\})$
$\displaystyle=\sum_{j}h^{(1,2)}_{j,j+1},$ $\displaystyle h_{j,j+1}^{(1)}$
$\displaystyle=\sum_{\alpha\beta}^{Q}J^{(1)}_{j,\alpha\beta}c_{j,\alpha}^{\dagger}c_{j+1,\beta}^{\dagger}c_{j+1,\alpha}c_{j,\beta}$
$\displaystyle=\sum_{\alpha\beta}^{Q}J^{(1)}_{j,\alpha\beta}h^{ex}_{j,\alpha;j+1,\beta},$
(3) $\displaystyle h_{j,j+1}^{(2)}$
$\displaystyle=\sum_{\alpha\beta}^{Q}J^{(2)}_{j,\alpha\beta}c_{j,\alpha}^{\dagger}c_{j+1,\alpha}^{\dagger}c_{j+1,\beta}c_{j,\beta}$
$\displaystyle=\sum_{\alpha\beta}^{Q}J^{(2)}_{j,\alpha\beta}h^{ph}_{j,\alpha;j\beta},$
(4) $\displaystyle h^{ph}_{j,\alpha;j\beta}$
$\displaystyle=\psi_{j,\alpha;j+1,\alpha}^{\dagger}\psi_{j,\beta;j+1,\beta},$
(5)

where $({\bm{J}^{(1,2)}_{j}})^{\dagger}=\bm{J}^{(1,2)}_{j}$,
$\big{(}\bm{J}^{(1,2)}_{j}\big{)}_{\alpha,\beta}=J^{(1,2)}_{j,\alpha\beta}$
and $c_{j,\alpha}$, ($\alpha=1,\cdots,{Q}$) is a canonical fermion
annihilation operator for a color $\alpha=1,\cdots,{Q}$ at the site $j$,
${\\{}c_{j,\alpha},c_{j^{\prime},\beta}^{\dagger}{\\}}=\delta_{jj^{\prime}}\delta_{\alpha\beta}$
with a constraint $\sum_{\alpha}\hat{n}_{j,\alpha}=1$,
$\hat{n}_{j,\alpha}=c_{j,\alpha}^{\dagger}c_{j,\alpha}$ at each site
${}^{\forall}j$. The exchange of colors, $h^{ex}_{i,\alpha;j,\beta}$ and the
pair hopping, $h^{ph}_{i,\alpha;j,\beta}$ at the link $i,j$ are defined as

$\displaystyle h^{ex}_{i,\alpha;j,\beta}$
$\displaystyle=c_{i,\alpha}^{\dagger}c_{j,\beta}^{\dagger}c_{j,\alpha}c_{i,\beta},$
(6) $\displaystyle h^{ph}_{i,\alpha;j,\beta}$
$\displaystyle=\psi_{i,\alpha;j,\alpha}^{\dagger}\psi_{i,\beta;j,\beta},$ (7)

where $\psi_{i,\alpha;j,\beta}=c_{i,\alpha}c_{j,\beta}$ is a pairing
amplitude. They operate as

$\displaystyle h^{ex}_{i,\alpha;j,\beta}|\beta_{i}\alpha_{j}\rangle$
$\displaystyle=|\alpha_{i}\beta_{j}\rangle,$ $\displaystyle
h^{ph}_{i,\alpha;j,\beta}|\beta_{i}\beta_{j}\rangle$
$\displaystyle=|\alpha_{i}\alpha_{j}\rangle,$

where
$|\alpha_{i}\beta_{j}\rangle=c_{i,\alpha}^{\dagger}c_{j,\beta}^{\dagger}|0\rangle$.

Up to constant, $H_{S}$ reduces to the sum of $H^{(1)}$ and $H^{(2)}$ when
$Q=3$. The spin-1 operators at the site $j$ is written by a generator of the
$SO(3)$ spatial rotation as
$S_{j,\alpha}=\sum_{\beta\gamma}c_{j,\beta}^{\dagger}S^{\alpha}_{\beta\gamma}c_{j,\gamma}$,
where $S^{\alpha}_{\beta\gamma}=-i\epsilon_{\alpha\beta\gamma}$,
($\alpha=1,2,3)$. See also appendix A.

As for the boundary condition, we discuss both of the open boundary condition
and the periodic boundary condition ($c_{L+1,\alpha}\equiv c_{1,\alpha}$)
assuming the lattice sites are labeled as $j=1,\cdots,L$ unless otherwise
specified. We discuss each of the $H^{(1,2)}$ separately or $H$ at
$\omega_{S}=0,\frac{\pi}{2}$ since the transformation properties are
different.

###  II.2 ${SU(Q)}$ and $Z_{Q}$ Symmetries

When the coupling is color independent,
$J^{(1,2)}_{j,\alpha\beta}=J_{j}^{(1,2)}$, the Hamiltonian $H^{(1,2)}$ is
invariant for the global $SU(Q)$ transformation respectively,

$\displaystyle{\cal U}^{(1)}H^{(1)}({\cal U}^{(1)})^{\dagger}$
$\displaystyle=H^{(1)},$ $\displaystyle{\cal U}^{(2)}H^{(2)}({\cal
U}^{(2)})^{\dagger}$ $\displaystyle=H^{(2)},$ $\displaystyle{\cal U}^{(1)}$
$\displaystyle=e^{-i\sum_{j,\alpha\beta}c_{j,\alpha}^{\dagger}u_{\alpha\beta}c_{j,\beta}},$
(8) $\displaystyle{\cal U}^{(2)}$
$\displaystyle=e^{-i\sum_{j,\alpha\beta}(-1)^{j-1}c_{j,\alpha}^{\dagger}u_{\alpha\beta}c_{j,\beta}},$
(9)

where $u$ is a generator of $Q\times Q$ traceless hermitian matrix (${\rm
Tr}\,u=0$ and $u^{\dagger}=u$) 111 When $Q=3$, it is spanned by the Gell-Mann
matrices  $\displaystyle\lambda_{1}$
$\displaystyle=\left[\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&0\end{array}\right],\lambda_{2}=\left[\begin{array}[]{ccc}0&-i&0\\\
i&0&0\\\
0&0&0\end{array}\right]=S^{3},\lambda_{3}=\left[\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\ 0&0&0\end{array}\right],$ $\displaystyle\lambda_{4}$
$\displaystyle=\left[\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\
1&0&0\end{array}\right],\lambda_{5}=\left[\begin{array}[]{ccc}0&0&-i\\\
0&0&0\\\
i&0&0\end{array}\right]=-S^{2},\lambda_{6}=\left[\begin{array}[]{ccc}0&0&0\\\
0&0&1\\\ 0&1&0\end{array}\right],$ $\displaystyle\lambda_{7}$
$\displaystyle=\left[\begin{array}[]{ccc}0&0&0\\\ 0&0&-i\\\
0&i&0\end{array}\right]=S^{1},\lambda_{8}=\left[\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&-2\end{array}\right]/\sqrt{3}.$ . Note that the fermions
transform as ($g=(g^{*})^{-1}=e^{iu}\in{SU(Q)}$ 222 For an Hermite matrix
$\bm{G}$, let us define ${\cal G}=\bm{c}^{\dagger}\bm{G}\bm{c}$ where
$\bm{c}^{\dagger}=(c_{1}^{\dagger},c_{2}^{\dagger},\cdots)$. Assuming that
$\bm{G}$ is diagonalized by a unitary matrix $\bm{U}$ as
$\bm{G}=\bm{U}\bm{g}\bm{U}^{\dagger}$, $\bm{g}={\rm
diag}\,(g_{1},g_{2},\cdots)$, $g_{i}\in\mathbb{R}$, we have ${\cal
G}=\sum_{i}g_{i}d_{i}^{\dagger}d_{i}$ where
$\bm{d}\equiv\bm{U}^{\dagger}\bm{c},\bm{c}=\bm{U}\bm{d}$. Then for ${\cal
U}=e^{-i\theta{\cal G}},\ {\cal U}d_{i}{\cal U}^{\dagger}=e^{-i\theta
g_{i}d_{i}^{\dagger}d_{i}}d_{i}e^{i\theta
g_{i}d_{i}^{\dagger}d_{i}}=e^{i\theta g_{i}}d_{i}$ ($\theta\in\mathbb{R}$), it
reads ${\cal U}\bm{c}\,{\cal U}^{\dagger}=\bm{U}{\cal U}\bm{d}\,{\cal
U}^{\dagger}=\bm{U}e^{i\theta\bm{g}}\bm{d}=\bm{U}e^{i\theta\bm{g}}\bm{U}^{\dagger}\bm{U}\bm{d}=e^{i\theta\bm{G}}\bm{c}$.
Also it gives ${\cal U}\bm{c}^{\dagger}{\cal
U}^{\dagger}=\bm{c}^{\dagger}e^{-i\theta\bm{G}}$ )

$\displaystyle{\cal U}^{(1)}c_{j,\alpha}({\cal U}^{(1)})^{\dagger}$
$\displaystyle=\sum_{\beta}g_{\alpha\beta}c_{j,\beta},$ (10)
$\displaystyle{\cal U}^{(2)}c_{j,\alpha}({\cal U}^{(2)})^{\dagger}$
$\displaystyle=\left\\{\begin{array}[]{cc}\sum_{\beta}g_{\alpha\beta}c_{j,\beta}&j:\text{{odd}}\\\
\sum_{\beta}g^{*}_{\alpha\beta}c_{j,\beta}&j:\text{{even}}\end{array}\right..$
(13)

Especially $Z_{Q}\subset{SU(Q)}$ symmetry is important for the following
discussion from topological view points. Although the twists introduced later
break $SU({Q})$ symmetry in general, this $Z_{Q}$ still remains as a symmetry
at the high symmetric twists (denoted by $G$, see below) which is a
generalized anti-periodic boundary condition for ${Q}=2$ [86].

$Z_{Q}$ is given by the global cyclic shift of the fermion colors as

$\displaystyle c_{j,\alpha}$ $\displaystyle\to c_{j,\alpha-1,\text{ mod
}{Q}}={\cal U}_{Z_{Q}}c_{j,\alpha}{\cal U}_{Z_{Q}}^{\dagger},$ (14)
$\displaystyle\left(\begin{array}[]{c}c_{j,1}\\\ c_{j,2}\\\ \vdots\\\
c_{j,{Q}}\end{array}\right)$
$\displaystyle\to\left(\begin{array}[]{c}c_{j,{Q}}\\\ c_{j,1}\\\ \vdots\\\
c_{j,{Q}-1}\end{array}\right),$ (23)
$\displaystyle=Z_{Q}\left(\begin{array}[]{c}c_{j,1}\\\ c_{j,2}\\\ \vdots\\\
c_{j,{Q}}\end{array}\right)={\cal U}_{Z_{Q}}\left(\begin{array}[]{c}c_{j,1}\\\
c_{j,2}\\\ \vdots\\\ c_{j,{Q}}\end{array}\right){\cal U}_{Z_{Q}}^{\dagger},$
(32)

where

$\displaystyle Z_{Q}$
$\displaystyle={\left(\begin{array}[]{cccc}0&0&\cdots&1\\\ 1&0&0&\vdots\\\
\vdots&\ddots&\ddots&0\\\ 0&\cdots&1&0\end{array}\right)}=e^{iz_{Q}}\in
SU(Q),$ (37) $\displaystyle{\cal U}_{Z_{Q}}$
$\displaystyle=\prod_{j}e^{-ic_{j,\alpha}^{\dagger}(z_{Q})_{\alpha\beta}c_{j,\beta}}.$
(38)

See Sec.III.1.

It implies that all of the eigen states are labeled by the eigen values of the
unitary transformation ${\cal U}_{Z_{Q}}$ as

$\displaystyle{\cal U}_{Z_{Q}}|\omega^{n}\rangle$
$\displaystyle=|\omega^{n}\rangle\omega^{n}.$

###  II.3 Gauge symmetry

Further the Hamiltonians have a (local) $[U(1)]^{Q}$ gauge invariance for
$\varphi_{j,\alpha}\in\mathbb{R}$, ($\forall j$ and ${}^{\forall}\alpha$), as

$\displaystyle{\cal U}_{G}^{(1)}H^{(1)}(\\{J_{j,\alpha\beta}^{(1)}\\})({\cal
U}_{G}^{(1)})^{\dagger}$
$\displaystyle=H^{(1)}(\\{\bar{J}_{j,\alpha\beta}^{(1)}\\}),$
$\displaystyle{\cal U}_{G}^{(2)}H^{(2)}(\\{J_{j,\alpha\beta}^{(2)}\\})({\cal
U}_{G}^{(2)})^{\dagger}$
$\displaystyle=H^{(2)}(\\{\bar{J}_{j,\alpha\beta}^{(2)}\\}),$
$\displaystyle{\cal U}_{G}^{(1)}c_{j,\alpha}({\cal U}_{G}^{(1)})^{\dagger}$
$\displaystyle=e^{i\varphi_{j,\alpha}}c_{j,\alpha},$ (39) $\displaystyle{\cal
U}_{G}^{(2)}c_{j,\alpha}({\cal U}_{G}^{(2)})^{\dagger}$
$\displaystyle=e^{i(-1)^{{j-1}}\varphi_{j,\alpha}}c_{j,\alpha},$ (40)

where

$\displaystyle{\cal U}_{G}^{(1)}$
$\displaystyle=e^{-i\sum_{j,\alpha}\varphi_{j,\alpha}\hat{n}_{j,\alpha}},$
(41) $\displaystyle{\cal U}_{G}^{(2)}$
$\displaystyle=e^{-i\sum_{j,\alpha}(-1)^{{j-1}}\varphi_{j,\alpha}\hat{n}_{j,\alpha}}.$
(42)

Then, the couplings in the Hamiltonian of Eq.(2) are transformed as

$\displaystyle\bar{J}_{j,\alpha\beta}^{(1)}$
$\displaystyle=e^{i\Omega_{j,\alpha;j+1,\beta}}J_{j,\alpha\beta}^{(1)},$ (43)
$\displaystyle\bar{J}_{j,\alpha\beta}^{(2)}$
$\displaystyle=e^{i(-1)^{{j-1}}\Omega_{j,\alpha;j+1,\beta}}J_{j,\alpha\beta}^{(2)},$
(44) $\displaystyle\Omega_{j,\alpha;j+1,\beta}$
$\displaystyle=-(\varphi_{j,\alpha}-\varphi_{j,\beta})+(\varphi_{j+1,\alpha}-\varphi_{j+1,\beta}),$
(45)

where $j=1,\cdots,L-1$ for the open boundary condition and
$\varphi_{L+1,\alpha}\equiv\varphi_{1,\alpha}$ for the periodic boundary
condition. We always assume the system size $L$ is even for the discussion of
the periodic boundary condition.

Taking all local gauge parameters constant,
$\varphi_{j,\alpha}=\phi_{\alpha}$, one has

$\displaystyle
e^{-i\phi_{\alpha}N_{\alpha}}H^{(1)}e^{i\phi_{\alpha}N_{\alpha}}$
$\displaystyle=H^{(1)},$ (46) $\displaystyle
e^{-i\phi_{\alpha}\bar{N}_{\alpha}}H^{(2)}e^{i\phi_{\alpha}\bar{N}_{\alpha}}$
$\displaystyle=H^{(2)},$ (47)

where $N_{\alpha}=\sum_{j}\hat{n}_{j,\alpha}$ and
$\bar{N}_{\alpha}=\sum_{j}(-1)^{j{-1}}\hat{n}_{j,\alpha}$. Differentiation by
$\phi_{\alpha}$ implies $Q$ conservation laws ($\alpha=1,\cdots,{Q}$)

$\displaystyle{[}N_{\alpha},H^{(1)}{]}$
$\displaystyle=0,\quad{[}\bar{N}_{\alpha},H^{(2)}{]}=0,$ (48)

where ${[}N_{\alpha},N_{\beta}{]}=0$ and
${[}\bar{N}_{\alpha},\bar{N}_{\beta}{]}=0$.

Since the $Z_{Q}$ operation shifts these quantum numbers as
$\bar{N}_{\alpha}\to\bar{N}_{\alpha-1}$, it results in degeneracy if

$\displaystyle(\bar{N}_{1},\cdots,\bar{N}_{Q})$
$\displaystyle\neq(\bar{N}_{Q},\bar{N}_{1},\cdots,\bar{N}_{{Q}-1}).$

###  II.4 Large gauge transformation

Taking the gauge parameters as

$\displaystyle\varphi_{j,\alpha}$ $\displaystyle=x_{j}\varphi_{\alpha},$ (49)
$\displaystyle x_{j}$
$\displaystyle=\frac{j-j_{0}}{L}\in[-\frac{1}{2},\frac{1}{2}],\quad
j_{0}=\frac{L+1}{2},$ (50) $\displaystyle\Omega_{j,\alpha;j+1\beta}$
$\displaystyle=\left\\{\begin{array}[]{cc}\frac{1}{L}({\varphi_{\alpha}-\varphi_{\beta}})&j=1,\cdots,L-1\\\
(\frac{1}{L}-1)({\varphi_{\alpha}-\varphi_{\beta}})&j=L\end{array}\right.,$
(53)

the $[U(1)]^{Q}$ large gauge transformation ${\cal U}_{LG}^{(1,2)}$ is defined
by

$\displaystyle{\cal U}_{LG}^{(1)}$
$\displaystyle=e^{-i\sum_{\alpha}\varphi_{\alpha}{\cal P}_{\alpha}^{(1)}},$
(54) $\displaystyle{\cal U}_{LG}^{(2)}$
$\displaystyle=e^{-i\sum_{\alpha}\varphi_{\alpha}{\cal P}_{\alpha}^{(2)}},$
(55)

where

$\displaystyle{\cal P}_{\alpha}^{(1)}$
$\displaystyle=\sum_{j}x_{j}\hat{n}_{j,\alpha},\ \ {\cal
P}_{\alpha}^{(2)}=\sum_{j}(-1)^{j{-1}}x_{j}\hat{n}_{j,\alpha},$ (56)

are the center of mass (CoM) [87, 88, 89]. They are generators of the large
gauge transformations [50, 53]. They induce changes in the couplings as

$\displaystyle{\cal U}_{LG}^{(1)}H^{(1)}(\\{J_{j,\alpha\beta}^{(1)}\\})({\cal
U}_{LG}^{(1)})^{\dagger}$
$\displaystyle=H^{(1)}(\\{\bar{J}_{j,\alpha\beta}^{(1),G}\\}),$
$\displaystyle{\cal U}_{LG}^{(2)}H^{(2)}(\\{J_{j,\alpha\beta}^{(2)}\\})({\cal
U}_{LG}^{(2)})^{\dagger}$
$\displaystyle=H^{(2)}(\\{\bar{J}_{j,\alpha\beta}^{(2),G}\\}),$
$\displaystyle\bar{J}_{j,\alpha\beta}^{(1),G}$
$\displaystyle=e^{i\frac{\varphi_{\alpha}-\varphi_{\beta}}{L}}J_{j,\alpha\beta}^{(1)}\left\\{\begin{array}[]{cc}1&j=1,\cdots,L-1\\\
e^{-i(\varphi_{\alpha}-\varphi_{\beta})}&j=L\end{array}\right.,$ (59)
$\displaystyle\bar{J}_{j,\alpha\beta}^{(2),G}$
$\displaystyle=e^{i(-1)^{{j-1}}\frac{\varphi_{\alpha}-\varphi_{\beta}}{L}}J_{j,\alpha\beta}^{(2)}\left\\{\begin{array}[]{cc}1&j=1,\cdots,L-1\\\
e^{+i(\varphi_{\alpha}-\varphi_{\beta})}&j=L:\text{even}\end{array}\right.,$
(62)

where

$\displaystyle{\cal U}_{LG}^{(1)}c_{j,\alpha}({\cal U}_{LG}^{(1)})^{\dagger}$
$\displaystyle=e^{i\varphi_{\alpha}x_{j}}c_{j,\alpha},$ (63)
$\displaystyle{\cal U}_{LG}^{(2)}c_{j,\alpha}({\cal U}_{LG}^{(2)})^{\dagger}$
$\displaystyle=e^{i\varphi_{\alpha}(-1)^{{j-1}}x_{j}}c_{j,\alpha}.$ (64)

Note that the constraint $\sum_{\alpha}\hat{n}_{j,\alpha}=1$ implies 333 When
$L$ is even,
$1-2+-\cdots+(L-1)-L=(1-2)+(3-4)+\cdots+((L-1)-L)=-1\frac{L}{2}=-L/2$ and
$\sum_{j}(-1)^{j-1}x_{j}=L^{-1}\sum_{j}(-1)^{j-1}(j-j_{0})=-\frac{1}{2}$. When
$L$ is odd,
$1-2+-\cdots-(L-1)+L=(1-2)+(3-4)+\cdots+(L-2)-(L-1))+L=-\frac{L-1}{2}+L=\frac{L+1}{2}$
and
$\sum_{j}(-1)^{j-1}x_{j}=L^{-1}(\sum_{j}(-1)^{j-1}j-j_{0})=L^{-1}(\frac{L+1}{2}-\frac{L+1}{2})=0$

$\displaystyle\sum_{\alpha=1}^{Q}{\cal P}_{\alpha}^{(1)}$
$\displaystyle=\sum_{j}x_{j}=0,$ (65) $\displaystyle\sum_{\alpha=1}^{Q}{\cal
P}_{\alpha}^{(2)}$
$\displaystyle=\left\\{\begin{array}[]{cc}-\frac{1}{2}&L:\text{even}\\\
0&L:\text{odd}\end{array}\right..$ (68)

###  II.5 Periodic system: gapped ground state of dimers

To realize a topological pump, we require a gapped unique ground state for a
periodic boundary condition and also with non-trivial edge states for a system
with edges. See examples [50, 54, 56, 91, 92, 93]

As for the ${Q}=3$ case, the spectra and eigen states of the two site systems
$H^{(1,2)}_{a,b}$ are listed in Tables 1 and 2 ($J_{\alpha\beta,1}=J$)
respectively ($|0\rangle$ is a fermion vacuum). Generic ${Q}$ case for
$H^{(2)}_{a,b}$ is summarized in Table 3. The sites are labeled by $a$ (:odd)
and $b$ (:even). They are consistent with the decomposition of the
representations, $3\otimes 3=\bar{3}\oplus 6$ and $3\otimes\bar{3}=1\oplus 8$.

Table 1: Energies and eigen states of $H^{(1)}$ (${Q}=3$). where $|\alpha_{a}\beta_{b}\rangle=c_{i\alpha}^{\dagger}c_{j\beta}^{\dagger}|0\rangle$. The sites are labeled by $a$ (:odd) and $b$ (:even).  state | $E$ | $(N_{1},N_{2},N_{3})$
---|---|---
$|1_{a}1_{b}\rangle$ | $J$ | $(2,0,0)$
$|2_{a}2_{b}\rangle$ | $J$ | $(0,2,0)$
$|3_{a}3_{b}\rangle$ | $J$ | $(0,0,2)$
$(|1_{a}2_{b}\rangle+|2_{a}1_{b}\rangle)/\sqrt{2}$ | $J$ | $(1,1,0)$
$(|1_{a}2_{b}\rangle-|2_{a}1_{b}\rangle)/\sqrt{2}$ | $-J$ | $(1,1,0)$
$(|2_{a}3_{b}\rangle+|3_{a}2_{b}\rangle)/\sqrt{2}$ | $J$ | $(0,1,1)$
$(|3_{a}2_{b}\rangle-|2_{a}3_{b}\rangle)/\sqrt{2}$ | $-J$ | $(0,1,1)$
$(|3_{a}1_{b}\rangle+|1_{a}3_{b}\rangle)/\sqrt{2}$ | $J$ | $(1,0,1)$
$(|3_{a}1_{b}\rangle-|1_{a}3_{b}\rangle)/\sqrt{2}$ | $-J$ | $(1,0,1)$
Table 2: Energy and eigen states of $H^{(2)}$ (${Q}=3$). The sites are labeled by $a$ (:odd) and $b$ (:even) ($\omega=\frac{-1+i\sqrt{3}}{2}$).  state | $E$ | $(\bar{N}_{1},\bar{N}_{2},\bar{N}_{3})$
---|---|---
$|S_{ab}\rangle=(|1_{a}1_{b}\rangle+|2_{a}2_{b}\rangle+|3_{a}3_{b}\rangle)/\sqrt{3}$ | $3J$ | $(0,0,0)$
$|\omega_{ab}\rangle=(|1_{a}1_{b}\rangle+\omega|2_{a}2_{b}\rangle+\omega^{2}|3_{a}3_{b}\rangle)/\sqrt{3}$ | $0$ | $(0,0,0)$
$|\omega_{ab}^{2}\rangle=(|1_{a}1_{b}\rangle+\omega^{2}|2_{a}2_{b}\rangle+\omega^{4}|3_{a}3_{b}\rangle)/\sqrt{3}$ | $0$ | $(0,0,0)$
$|1_{a}2_{b}\rangle$ | $0$ | $(1,-1,0)$
$|2_{a}1_{b}\rangle$ | $0$ | $(-1,1,0)$
$|2_{a}3_{b}\rangle$ | $0$ | $(0,1,-1)$
$|3_{a}2_{b}\rangle$ | $0$ | $(0,-1,1)$
$|3_{a}1_{b}\rangle$ | $0$ | $(-1,0,1)$
$|1_{a}3_{b}\rangle$ | $0$ | $(1,0,-1)$
Table 3:  Energies and eigen states of $H^{(2)}$ where $|\omega_{ij}^{n}\rangle$, $n=1,\cdots,Q-1$, $|\alpha_{a}\beta_{b}\rangle$, $\alpha\neq\beta=1,\cdots,Q$, ($\omega=e^{i\frac{2\pi}{Q}})$. The sites are labeled by $a$ (:odd) and $b$ (:even). Generic case.  state | $E$ | $\bar{N}_{\alpha}$
---|---|---
$|S_{a,b}\rangle=\psi_{a,b}^{\dagger}|0\rangle$ | $QJ$ | ${}^{\forall}\bar{N}_{\alpha}=0$
$|\omega_{ab}^{n}\rangle=Q^{-1/2}\sum_{\alpha=1}^{Q}\omega^{\alpha n}|\alpha_{a}\alpha_{b}\rangle$ | $0$ | ${}^{\forall}\bar{N}_{\alpha}=0$
$|\alpha_{a}\beta_{b}\rangle,\ \alpha\neq\beta$ | $0$ | $\bar{N}_{\alpha}=-\bar{N}_{\beta}=1,\bar{N}_{\gamma\neq\alpha,\beta}=0$

Since we need a unique (singlet) ground state for the two site problem as a
dimer in the following, we discuss $H^{(2)}$ or $H$ of
$\omega_{S}=\frac{\pi}{4}$

$\displaystyle J_{j}$
$\displaystyle=\left\\{\begin{array}[]{cc}J_{o}&j:\text{odd}\\\
J_{e}&j:\text{even}\end{array}\right.,\quad J_{o},J_{e}\leq 0.$

The extension to the ${SU(Q)}$ case (Table 3) is straightforward and due to
the decomposition ${Q}\otimes\bar{Q}=1\oplus({Q}^{2}-1)$ [94]. It is a SPT
protected by $Z_{Q}$ symmetry [64, 95]. The two site Hamiltonian for
$J_{j}=J<0$ is

$\displaystyle H_{ab}$ $\displaystyle=J\psi_{a,b}^{\dagger}\psi_{a,b}$

where $\psi_{a,b}={Q}^{-1/2}\sum_{\alpha}\psi_{a,\alpha;b,\alpha}$, ($a\neq
b$).

The singlet is given by $|S_{ab}\rangle=\psi_{a,b}^{\dagger}|0\rangle$ with
its energy ${Q}J$ and $\bar{N}_{\alpha}=0,^{\forall}\alpha$, since
${[}\psi_{a,b},\psi_{a,b}^{\dagger}{]}|0\rangle=1$ 444
${[}\psi_{a,b},\psi_{a,b}^{\dagger}{]}=1-{Q}^{-1}\sum_{\alpha}(n_{a,\alpha}+n_{b,\alpha})$
. The rest of zero energy ${Q}^{2}-1$ states are given by the ${Q}-1$ states,
$|\omega_{ij}^{n}\rangle={Q}^{-1/2}\sum_{\alpha}\omega^{n\alpha}|\alpha_{i}\alpha_{j}\rangle$,
$n=1,\cdots,{Q}-1$ with $\bar{N}_{\alpha}=0$ and ${Q}({Q}-1)$ states,
$|\alpha_{i}\beta_{j}\rangle$, ($\alpha\neq\beta$) with
$\bar{N}_{\alpha}=1,\bar{N}_{\beta}=-1,\bar{N}_{\gamma}=0,(\gamma\neq\alpha,\beta)$
where we assume $a$ is odd and $b$ is even.

Noting this two-site problem, we have two different unique gapped ground
states for the periodic system ($c_{L+1,\alpha}\equiv c_{1,\alpha}$) with
different dimer limits $J_{o}=0$ and $J_{e}=0$ as

$\displaystyle|g_{{\rm pe},eo}\rangle$
$\displaystyle=\prod_{j=1}^{L/2}\psi_{2j+1,2j}^{\dagger}|0\rangle,\
(J_{o}=0),$ (69) $\displaystyle|g_{{\rm pe},oe}\rangle$
$\displaystyle=\prod_{j=1}^{L/2}\psi_{2j-1,2j}^{\dagger}|0\rangle,\
(J_{e}=0).$ (70)

Note that both states are labeled by the occupations, $\bar{N}_{\alpha}=0$,
$\alpha=1,\cdots,{Q}$.

Due to the adiabatic continuity, the ground state is gapped and unique if the
interaction between the dimers ($J_{j,\alpha\beta}$, $j$:even) is finite but
weak enough ($J_{2}\neq 0,|J_{2}|\ll|J_{1}|$). It is a SPT phase protected by
$Z_{Q}$ symmetry associated with the quantized Berry phase [95, 67]. See
Sec.III.1. As is clear in the Table, we do not require ${SU(Q)}$ symmetry. We
may allow $Z_{Q}$ invariant twists at any links as is introduced later. At
$J_{o}=J_{e}$, the energy gap (of a finite system) closes as is clear from the
discontinuous change of the quantized Berry phase (discussed later). This gap
closing point is a source of the non-trivial topology which we discuss in this
paper.

##  III Currents, center of mass and bulk-edge correspondence

###  III.1 Current and Synthetic Brillouin zone (twist space)

Noting that the large gauge transformation, let us start considering a
dimerized Hamiltonian $H^{(2)}(\\{J^{(2)}_{j,\alpha\beta}\\})$ by

$\displaystyle J_{j,\alpha\beta}$
$\displaystyle=\left\\{\begin{array}[]{cl}J_{j}&j=1,\cdots,L-1\\\
J_{j}e^{-i(\varphi_{\alpha}-\varphi_{\beta})}&\text{PBC},\ j=L:\text{even}\\\
0&\text{OBC},\ j=L\end{array}\right.,$ (74) $\displaystyle J_{i}$
$\displaystyle=J_{0}+\delta J(-1)^{j}\cos\frac{2\pi t}{T}\in\mathbb{R},$ (75)

where $\ J_{0},\delta J\in\mathbb{R}$ and $t$ is a time with a period $T$. To
be explicit, we take $J_{0}<0$ and $\delta J>0$ in the following numerical
demonstration. We also include a symmetry breaking term to realize the
topological pump. To be concrete, let us consider a following term for the
generic ${SU(Q)}$ case (it reduces to the staggered potential for $SU(2)$
case)

$\displaystyle H_{B}(t)$
$\displaystyle=\sum_{j,\alpha}\hat{n}_{j,\alpha}\Delta_{\alpha}(t),$ (76)
$\displaystyle\Delta_{\alpha}(t)$ $\displaystyle=\Delta\sin
2\pi(\frac{t}{T}+\Phi{\alpha}),$ (77)

where $\Phi=\frac{P}{Q}$ and $\Delta$ is a strength of a symmetry breaking.
The integers $P$ and $Q$ are mutually co-prime. We omit the superscript “(2)”
unless explicitly specified and both of the periodic/open systems are
discussed. Noting that Eq.(74), the Hamiltonian $H_{{\rm
op}}(\\{J_{j,\alpha\beta}\\})$ for the open system is
$\varphi_{\alpha}$-independent and $H_{{\rm
pe}}(\\{J_{j,\alpha\beta}\\})\equiv H_{\rm tw}$, for the periodic system, is
with a twisted boundary condition. The difference between $H_{\rm op}$ and
$H_{\rm tw}$ is only at the boundary link, $j=L$. Then the large gauge
transformation by ${\cal U}_{LG}$ induces ${\cal O}(L^{-1})$ twists for each
link as

$\displaystyle\bar{H}_{{\rm op},{\rm pe}}$ $\displaystyle=H_{{\rm op},{\rm
pe}}(\\{\bar{J}_{j,\alpha\beta}\\}),$ (78)
$\displaystyle\bar{J}_{j,\alpha\beta}$
$\displaystyle=e^{i{(-1)^{j-1}}\frac{\varphi_{\alpha}-\varphi_{\beta}}{L}}J_{j}\
\left\\{\begin{array}[]{cc}(j=1,\cdots,L-1)&:\text{open}\\\
(j=1,\cdots,L)&:\text{periodic}\end{array}\right..$ (81)

Explicitly for the open/periodic cases, they are written as

$\displaystyle\bar{H}_{\rm op}$ $\displaystyle={\cal U}_{LG}H_{\rm op}{\cal
U}_{LG}^{\dagger},$ (82) $\displaystyle\bar{H}_{\rm pe}$ $\displaystyle={\cal
U}_{LG}H_{\rm tw}{\cal U}_{LG}^{\dagger},$ (83) $\displaystyle{\cal U}_{LG}$
$\displaystyle=e^{-i\sum_{\alpha}\varphi_{\alpha}{\cal P}_{\alpha}}.$ (84)

The twists are uniform both for the periodic $\bar{H}_{\rm pe}$ and the open
$\bar{H}_{\rm op}$ Hamiltonians. The periodic system is translational
invariant by the period $2$ with dimerization.

Without dimerization, uniformity of the $H_{\rm pe}$ is written as $J_{j}=J$
($j$-independent). This is inherited as anti-translation invariance of
$\bar{H}_{\rm pe}$ as

$\displaystyle{\cal A}_{\cal T}\bar{H}_{\rm pe}{\cal A}_{\cal T}^{-1}$
$\displaystyle=\bar{H}_{\rm pe},$ (85) $\displaystyle{\cal A}_{\cal T}$
$\displaystyle={\cal U}_{\cal T}{\cal K},$ (86) $\displaystyle{\cal U}_{\cal
T}c_{j,\alpha}{\cal U}_{\cal T}^{\dagger}$ $\displaystyle=c_{j+1,\alpha},$
(87)

where ${\cal U}_{\cal T}$ is unitary and ${\cal A}_{\cal T}$ is anti-unitary
(${\cal K}$ is a complex conjugate). It implies
$\bar{J}_{j,\alpha\beta}=(\bar{J}_{j\pm 1,\alpha\beta})^{*}$.

In the following, the Hamiltonian is extended by adding a symmetry breaking
term $H_{B}$, that is gauge invariant as ${\cal U}_{LG}H_{B}{\cal
U}_{LG}^{\dagger}=H_{B}$. See Table 4.

Table 4: Hamiltonians, energies and states with different boundary conditions.

|  $\theta$ dependence | on $T^{d}(\Theta)$ | Energy on $T^{d}(\Theta)$ | ground state | current
---|---|---|---|---|---
$H_{\rm op}$ | independent | independent | independent | $|g_{0,{\rm op}}\rangle$ | -
$\bar{H}_{\rm op}$ | uniform | ill-defined∗1 | independent | $|g_{\rm op}\rangle={\cal U}_{LG}|g_{0,{\rm op}}\rangle$ | $j_{\rm op}=\hbar^{-1}\langle g_{\rm op}|\partial_{\theta}\bar{H}_{\rm op}|g_{\rm op}\rangle$
$H_{\rm tw}$ | at the boundaries | well-defined | well-defined∗2 | $|g_{0,{\rm tw}}\rangle$ | -
$\bar{H}_{\rm pe}$ | uniform | ill-defined∗1 | well-defined∗2 | $|g_{\rm pe}\rangle={\cal U}_{LG}|g_{0,{\rm tw}}\rangle$ | $j_{\rm pe}=\hbar^{-1}\langle g_{\rm pe}|\partial_{\theta}\bar{H}_{\rm pe}|g_{\rm pe}\rangle$

∗1: Does not satisfy the periodicity in $\Theta$. ∗2: Assuming the unique
ground state on $\Theta$, $E_{\rm tw}=E_{\rm pe}$.

To define a current, let us introduce a $d$-dimensional twist space
($d$-dimensional torus),
$T^{d}=\\{(\theta_{1},\theta_{2},\cdots,\theta_{d})|\,\theta_{\alpha}\in[0,2\pi],\alpha=1,\cdots,d\\}$
( $d=Q-1$). This is a synthetic Brillouin zone [95], which introduces twist
for the Hamiltonian $H_{\rm tw}$ by

$\displaystyle\varphi_{1}$ $\displaystyle=\theta_{1},$
$\displaystyle\varphi_{2}$ $\displaystyle=\theta_{1}+\theta_{2},$
$\displaystyle\vdots$ $\displaystyle\varphi_{\alpha}$
$\displaystyle=\theta_{1}+\cdots+\theta_{\alpha},$ $\displaystyle\vdots$
$\displaystyle\varphi_{d}$
$\displaystyle=\theta_{1}+\theta_{2}+\cdots+\theta_{d},$
$\displaystyle\varphi_{Q}$ $\displaystyle\equiv\varphi_{0}=0$

that is,

$\displaystyle\varphi_{1}-\varphi_{0}$ $\displaystyle=\theta_{1},$
$\displaystyle\varphi_{2}-\varphi_{1}$ $\displaystyle=\theta_{2},$
$\displaystyle\vdots$ $\displaystyle\varphi_{\alpha}-\varphi_{\alpha-1}$
$\displaystyle=\theta_{\alpha},$ $\displaystyle\vdots$
$\displaystyle\varphi_{d}-\varphi_{d-1}$ $\displaystyle=\theta_{d},$
$\displaystyle\varphi_{Q}-\varphi_{Q-1}$
$\displaystyle=-\theta_{1}-\theta_{2}\cdots-\theta_{d}\equiv\theta_{Q},$

where $\theta_{\alpha}$ is defined in modulo $2\pi$, that is,
$\theta_{\alpha}=0$ and $2\pi$ are identified. It implies a formal relation
$\theta_{1}+\cdots+\theta_{d}+\theta_{Q}=0$. Note that the Hamiltonian depends
on

$\displaystyle\varphi_{\alpha}-\varphi_{\beta}$
$\displaystyle=\theta_{\alpha}+\cdots+\theta_{\beta+1},\
^{\forall}\alpha,\beta,\ \alpha\geq\beta+1.$

Since the Hamiltonian is invariant for the $Z_{Q}$ shift of the fermions,
$c_{j,\alpha}\to c_{j,\alpha-1}$, which induces a shift
$\varphi_{\alpha}\to\varphi_{\alpha+1}$, and also for the constant shift of
${}^{\forall}\varphi_{\alpha}$ by subtracting $\theta_{1}$ denoted by $\sim$
as

$\displaystyle\varphi_{1}$
$\displaystyle\to\varphi_{2}=\theta_{1}+\theta_{2}\sim\theta_{2},$
$\displaystyle\varphi_{2}$
$\displaystyle\to\varphi_{3}=\theta_{1}+\theta_{2}+\theta_{3}\sim\theta_{2}+\theta_{3},$
$\displaystyle\vdots$ $\displaystyle\varphi_{\alpha}$
$\displaystyle\to\varphi_{\alpha+1}=\theta_{1}+\cdots+\theta_{\alpha+1}\sim\theta_{2}+\cdots+\theta_{\alpha+1},$
$\displaystyle\vdots$ $\displaystyle\varphi_{d-1}$
$\displaystyle\to\varphi_{d}=\theta_{1}+\theta_{2}+\cdots+\theta_{d}\sim\theta_{2}+\cdots+\theta_{d},$
$\displaystyle\varphi_{d}$
$\displaystyle\to\varphi_{d+1}=\varphi_{Q}=\theta_{1}+\theta_{2}+\cdots+\theta_{d}+\theta_{Q},$
$\displaystyle\qquad\sim\theta_{2}+\cdots+\theta_{d}+\theta_{Q},$
$\displaystyle\varphi_{Q}$
$\displaystyle=\varphi_{0}\to\varphi_{1}=\theta_{1}\sim 0.$

It is given by the cyclic shift of the parameter space supplemented by
$\theta_{Q}$

$\displaystyle(\theta_{1},\cdots,\theta_{d},\theta_{Q})$
$\displaystyle\to(\theta_{2},\cdots,\theta_{Q},\theta_{1}).$ (88)

It implies $Z_{Q}$ equivalence of loops $\ell_{V_{\alpha}GV_{\alpha+1}}$,
$\alpha=1,\cdots,Q$ as shown later.

It is useful to express this parameter space as shown in Fig.1 and Fig.2. Let
us start $Q=d+1$ equivalent points $V_{\alpha}$, ($\alpha=1,\cdots,Q\equiv 0$)
on a $(d-1)$-dimensional sphere $S^{d-1}$, which is constructed recursively
from the $0$-dimensional sphere (2 points).
$G=(\frac{2\pi}{Q},\cdots,\frac{2\pi}{Q})$ is a center of mass of all vertices
$V_{0},\cdots,V_{d}$, which is a center of the sphere on which all vertices
lie. See ref. [95] for the details. Its low dimensional examples are 2
vertices of a line ($S^{0}$, $d=1$, Fig. 2(a)), 3 vertices of a triangle on a
circle ($S^{1}$, $d=2$, Fig. 2(b)) and 4 vertices of a tetrahedron on a sphere
in 3 dimension ($S^{2}$, $d=3$). $T^{d}$ is spanned by the $d$ vectors,
$\vec{e}_{j}=\overrightarrow{V_{0}V_{\alpha}}/2\pi$, $\alpha=1,\cdots,d$ as
$\Theta=\theta_{1}\vec{e}_{1}+\cdots+\theta_{d}\vec{e}_{d}$, which is
abbreviated as $\Theta=(\theta_{1},\cdots,\theta_{d})$.

Figure 1:  $d$-dimensional parameter space of the twist and $Q$ loops
$\ell_{V_{0}GV_{1}},\ell_{V_{1}GV_{2}},\cdots,\ell_{V_{d}GV_{0}}$ on torus
$T^{d}$. Figure 2:  Examples for the paths in the parameter spaces for (a)
$d=1,{Q}=2$ and (b) $d=2,{Q}=3$ where $G$ is a $Z_{Q}$ symmetric point. The
twist at $G$, ${Q}=2$ corresponds to the anti-periodic boundary condition.

It defines a synthetic Brillouin zone for the twisted Hamiltonian

$\displaystyle H_{\rm tw}(\Theta)=H_{\rm
tw}(\Theta+2\pi\vec{e}_{\alpha}),\alpha=1,\cdots,d.$ (89)

that is, all vertices ${}^{\forall}V_{\alpha}$ are identified.

Note that the $Q$ paths

$\displaystyle\ell_{V_{\alpha}GV_{\alpha+1}}$
$\displaystyle=-\overrightarrow{GV_{\alpha}}+\overrightarrow{GV_{\alpha+1}},\
\alpha=1,\cdots,Q,$ (90)

in Fig.1, forms loops for $H_{\rm tw}(\Theta)$. Note that $\bar{H}_{\rm
pe}(\Theta)$ does not satisfies this periodicity.

By taking any path
$\ell=\\{\Theta|(\Theta=\Theta(\theta),\theta\in[0,2\pi]\\}$ parameterized by
$\theta$ in the parameter space, let us define a current

$\displaystyle{\cal J}^{\ell}$
$\displaystyle\equiv\hbar^{-1}\partial_{\theta}\bar{H}=\hbar^{-1}\partial_{\theta}H(\\{\bar{J}_{j,\alpha\beta}\\}).$
(91)

Eqs.(82)and (83) imply respectively

$\displaystyle{{\cal J}}_{{\rm op}}$
$\displaystyle=\hbar^{-1}\partial_{\theta}\bar{H}_{\rm
op}=-i\hbar^{-1}\sum_{\alpha}\partial_{\theta}\varphi_{\alpha}{[}{\cal
P}_{\alpha},\bar{H}_{\rm op}{]},$ (92) $\displaystyle{{\cal J}}_{{\rm pe}}$
$\displaystyle=\hbar^{-1}\partial_{\theta}\bar{H}_{\rm
pe}=-i\hbar^{-1}\sum_{\alpha}\partial_{\theta}\varphi_{\alpha}{[}{\cal
P}_{\alpha},\bar{H}_{\rm pe}{]}$
$\displaystyle\qquad\qquad\qquad\qquad+\hbar^{-1}{\cal
U}_{LG}\partial_{\theta}H_{\rm tw}{\cal U}_{LG}^{\dagger},$ (93)

Then the evaluation by the time-dependent state $|G_{{\rm op}}(t)\rangle$ for
the open boundary condition that obeys the Schrödinger equation,
$i\hbar\partial_{t}|G_{{\rm op}}(t)\rangle=\bar{H}_{{\rm op}}|G_{{\rm
op}}(t)\rangle$, ($\langle G_{{\rm op}}|G_{{\rm op}}\rangle=1$) gives

$\displaystyle{j}_{\rm op}^{G}$ $\displaystyle\equiv\langle G_{{\rm
op}}(t)|{{\cal J}}_{\rm op}|G_{{\rm
op}}(t)\rangle=\sum_{\alpha}\partial_{\theta}\varphi_{\alpha}\partial_{t}{P}^{G}_{{\rm
op},\alpha},$ (94)

where

$\displaystyle P^{G}_{{\rm op},\alpha}(t)$ $\displaystyle=\langle G_{{\rm
op}}(t)|{\cal P}_{\alpha}|G_{{\rm op}}(t)\rangle,$ (95)

is a center of mass of $|G_{\rm op}(t)\rangle$. Note that the similar
discussion for the periodic boundary condition is not simple. Eq.(93) has
extra term ($H_{\rm tw}$ depends on $\varphi_{\alpha}$) and also the operator
${\cal P}_{\alpha}$ itself does not have a physical meaning since the origin
of the reduced coordinate $x_{j}$ is arbitrary for the periodic boundary
condition.

###  III.2 Adiabatic current

Let us start by a general discussion of the time-dependent evolution of a
state by the adiabatic approximation assuming the initial state $|G(0)\rangle$
is a gapped unique ground state $|g(0)\rangle$ of the snapshot Hamiltonian
$\bar{H}(t)$,

$\displaystyle|G(0)\rangle$ $\displaystyle=|g(0)\rangle,$
$\displaystyle\bar{H}(t)|n(t)\rangle$ $\displaystyle=|n(t)\rangle E_{n}(t),$

where $n=0,1,2,\cdots,$ and $|g\rangle=|n\rangle,n=0$, $\langle
n|n^{\prime}\rangle=\delta_{nn^{\prime}}$. We further assume the snapshot
Hamiltonian is always gapped $E_{n}(t)>E_{g}(t)$, ($n\neq g$). When the time
modulation of the Hamiltonian is slow enough, the adiabatic approximation is
justified (See appendix C) as

$\displaystyle|G\rangle$ $\displaystyle\approx
C\big{[}|g\rangle+i\hbar\sum_{n\neq g}\frac{|n\rangle\langle
n|\partial_{t}g\rangle}{E_{n}-E_{g}}\big{]},$ (96)

where $C$ is a time dependent phase factor ($|C|=1$). In the present
discussion, $\bar{H}$ is $\bar{H}_{\rm op}$ or $\bar{H}_{\rm pe}$.

Under this adiabatic approximation, the observed current is given by [40] (See
also appendix C)

$\displaystyle j^{G}$ $\displaystyle=\langle G|{\cal J}|G\rangle\approx j,$
(97) $\displaystyle j$ $\displaystyle=\hbar^{-1}\langle
g|\partial_{\theta}\bar{H}|g\rangle-iB,$ (98) $\displaystyle B$
$\displaystyle=\partial_{\theta}A_{t}-\partial_{t}A_{\theta},$ (99)

where $A_{\mu}$ ($\mu=\theta,t$) is the Berry connection

$\displaystyle A_{\mu}$ $\displaystyle=\langle g|\partial_{\mu}g\rangle.$
(100)

Here we assume the ground state $|g\rangle$ as a function of $\theta$ and $t$
by $\bar{H}(\theta,t)|g(\theta,t)\rangle=|g(\theta,t)\rangle E_{g}(\theta,t)$.
It is covariant for the phase transformation, $|g^{\prime}\rangle=|g\rangle
e^{i\chi}$, ($\chi=\chi(\theta,t)$) that induces the gauge transformation for
the Berry connection

$\displaystyle A^{\prime}_{\mu}$ $\displaystyle=A_{\mu}+\partial_{\mu}\chi,$
(101)

although the field strength $B$ and the current $j$ are gauge invariant.

Let us here define the average current along the path $\ell$ such as
$\ell_{V_{\alpha}GV_{\alpha+1}}$ connecting two equivalent points in the
parameter space (parameterized by $\theta\in[0,2\pi]$)

$\displaystyle\bar{j}^{\ell}(t)$
$\displaystyle=\int_{0}^{2\pi}\frac{d\theta}{2\pi}\,j(\theta,t).$ (102)

Due to the Feynmann’s theorem, the first term Eq.(98) is written as $\langle
g|\partial_{\theta}\bar{H}|g\rangle=\partial_{\theta}E(\theta)$,
$E_{\theta}=\langle g|\bar{H}|g\rangle$. It vanishes for the open boundary
condition, since $E_{\rm op}$ is $\theta$ independent. As for the periodic
boundary condition, again $\int_{0}^{2\pi}d\theta\partial_{\theta}E_{\rm
pe}(\theta)=0$ since $E_{\rm pe}(\theta)=E_{\rm tw}(\theta)$ is periodic for
any closed path connecting the equivalent points in the period $T$. See Table
4. It results in

$\displaystyle\bar{j}^{\ell}(t)$
$\displaystyle=-i\int_{\ell}\frac{d\theta}{2\pi}\,B.$ (103)

Note that this is valid both for open and periodic boundary conditions. Since
the current is carried by bulk, the effect of the boundaries is ${\cal
O}(L^{-1})$. One can expect, in the infinite size limit,

$\displaystyle\bar{j}^{\ell}_{\rm pe}$ $\displaystyle=\bar{j}^{\ell}_{\rm
op},\ (L\to\infty).$ (104)

This is the bulk-edge correspondence for the adiabatic current.

###  III.3 Temporal gauge and discontinuity of CoM

Adiabatic pump is a periodic transfer of charge between the time period
$[0,T]$ assuming that the periodicity of the Hamiltonian as
$\bar{H}(t+T)=\bar{H}(t)$ under the adiabatic condition. Here $\bar{H}$
denotes $\bar{H}_{\rm pe}$ or $\bar{H}_{\rm op}$ respectively. As for the
gauge fixing of the Berry connection, let us take a temporal gauge
$A_{t}^{(t)}(\theta,t)=0$ by

$\displaystyle A_{\theta}^{(t)}(\theta,t)$
$\displaystyle=-\int_{0}^{t}d\tau\,B(\theta,\tau).$ (105)

This is apparently gauge invariant and one may check
$B=\partial_{\theta}A_{t}^{(t)}-\partial_{t}A_{\theta}^{(t)}=-\partial_{t}A_{\theta}^{(t)}$.
Note that $A_{\theta}^{(t)}$ is not periodic in time, $A^{(t)}(\theta,t+T)\neq
A^{(t)}(\theta,t)$, although $\bar{H}$ is periodic in time (period $T$). It is
further written by the Berry connection in a generic gauge as [50]

$\displaystyle A_{\theta}^{(t)}(\theta,t)$
$\displaystyle=-\int_{0}^{t}d\tau\,\big{[}\partial_{\theta}A_{t}(\theta,\tau)-\partial_{t}A_{\theta}(\theta,\tau)\big{]}$
$\displaystyle=A_{\theta}(\theta,t)-\partial_{\theta}\int_{0}^{t}d\tau\,A_{t}(\theta,\tau)-A_{\theta}(\theta,0).$
(106)

The average current in the temporal gauge is written as

$\displaystyle\bar{j}^{\ell}(t)$
$\displaystyle=-i\int_{\ell}\frac{d\theta}{2\pi}B=i\partial_{t}\bar{A}_{\theta}^{(t)}(t),$
(107) $\displaystyle\bar{A}_{\theta}^{(t),\ell}(t)$
$\displaystyle=\int_{\ell}\frac{d\theta}{2\pi}\,A_{\theta}^{(t)}(\theta,t)$
$\displaystyle=\frac{-1}{2\pi}\int_{\ell}d\theta\,\int_{0}^{t}d\tau
B(\theta,\tau).$ (108)

This is valid both for open and periodic boundary condition.

As for the open boundary condition, the parameter dependence is only due to
the large gauge transformation ${\cal U}_{LG}$ as $|g_{\rm op}\rangle={\cal
U}_{LG}|g_{0,{\rm op}}\rangle$ where $|g_{0,{\rm op}}\rangle$, is a ground
state of $H_{\rm op}$, ($H_{\rm op}|g_{0,{\rm op}}\rangle=|g_{0,{\rm
op}}\rangle E_{g}$). We safely assume $|g_{0,{\rm op}}\rangle$ is $\theta$
independent, $\partial_{\theta}|g_{0,{\rm op}}\rangle=0$. It implies

$\displaystyle A_{t,{\rm op}}$ $\displaystyle=\langle g_{\rm
op}|\partial_{t}g_{\rm op}\rangle=\langle g_{0,{\rm op}}|\partial_{t}g_{0,{\rm
op}}\rangle,\ \partial_{\theta}A_{t,{\rm op}}=0,$ (109) $\displaystyle
A_{\theta,{\rm op}}$ $\displaystyle=\langle g_{\rm op}|\partial_{\theta}g_{\rm
op}\rangle=\langle g_{0,{\rm op}}|{\cal
U}_{LG}^{\dagger}\partial_{\theta}{\cal U}_{LG}|g_{0,{\rm op}}\rangle$
$\displaystyle\qquad\qquad\qquad=-i\sum_{\alpha}\partial_{\theta}\varphi_{\alpha}P_{\alpha}(t),$
(110)

where

$\displaystyle P_{\alpha}(t)$ $\displaystyle=\langle g_{0,{\rm op}}|{\cal
P}_{\alpha}|g_{0,{\rm op}}\rangle={\sum_{j}(-1)^{j{-1}}x_{j}n_{j,\alpha}(t)},$
(111)

is a center of mass of the $\alpha$-particle without twists
($n_{j,\alpha}(t)=\langle g_{0,{\rm op}}(t)|\hat{n}_{j,\alpha}|g_{0,{\rm
op}}(t)\rangle$). The Berry connection in the temporal gauge is written as

$\displaystyle A_{\theta,{\rm op}}^{(t)}(\theta,t)$
$\displaystyle=-i\sum_{\alpha}\partial_{\theta}\varphi_{\alpha}\big{[}P_{\alpha}(t)-P_{\alpha}(0)\big{]}.$
(112)

The current averaged over the path $\ell$ is written as

$\displaystyle\bar{j}^{\ell}$
$\displaystyle=\bigg{[}\int_{\ell}\frac{d\theta}{2\pi}\sum_{\alpha}\partial_{\alpha}\varphi_{\alpha}\bigg{]}\partial_{t}P_{\alpha}(t)$
$\displaystyle=\partial_{t}\bar{P}^{\ell}(t),$ (113)

where

$\displaystyle\bar{P}^{\ell}(t)$
$\displaystyle=\sum_{\alpha}\frac{\Delta\varphi_{\alpha}}{2\pi}P_{\alpha}(t),$
(114) $\displaystyle\Delta\varphi_{\alpha}$
$\displaystyle=\varphi_{\alpha}(2\pi)-\varphi_{\alpha}(0).$ (115)

Along the path $\ell_{V_{0}GV_{1}}$, $\theta_{1}:0\to 2\pi$ and
$\theta_{\alpha}:0\to 0$, $\alpha=2,\cdots,d$. It implies
$\varphi_{\alpha}:0\to 2\pi$, $\Delta\varphi_{\alpha}=2\pi$ for all
$\alpha=1,\cdots,d$. Also $\varphi_{0}=\varphi_{Q}=0$. Therefore the averaged
current $\bar{j}_{\rm op}^{\ell_{V_{0}GV_{1}}}$ is written as

$\displaystyle\bar{j}_{\rm op}^{\ell_{V_{0}GV_{1}}}$
$\displaystyle=\partial_{t}\bar{P}^{\ell_{V_{0}GV_{1}}},$ (116)
$\displaystyle\bar{P}^{\ell_{V_{0}GV_{1}}}$ $\displaystyle=-P_{0},$ (117)

where $P_{0}\equiv-\sum_{\alpha=1}^{d}P_{\alpha}$. (Note that $P_{0}=P_{Q}$,
($L$: odd) and $P_{0}=P_{Q}+1/2$, ($L$: even) due to the constraint Eq.(68).
As for ${Q}=3$, $P_{0}=-P_{1}-P_{2}$. (See Fig.2). Along the path
$\ell_{V_{1}GV_{2}}$, $\theta_{1}:2\pi\to 0$, $\theta_{2}:0\to 2\pi$,
$\theta_{\alpha}:0\to 0$, ($\alpha=3,\cdots,d$). It implies
$\Delta\varphi_{1}=-2\pi$, $\Delta\varphi_{\alpha}=0$, ($\alpha=2,\cdots,d$)
and

$\displaystyle\bar{j}_{\rm op}^{\ell_{V_{1}GV_{2}}}$
$\displaystyle=\partial_{t}\bar{P}^{\ell_{V_{1}GV_{2}}},$ (118)
$\displaystyle\bar{P}^{\ell_{V_{1}GV_{2}}}$ $\displaystyle=-P_{1}.$ (119)

Similarly along the path $\ell_{V_{\alpha}GV_{\alpha+1}}$,
($\alpha=1,\cdots,d-1$), $\theta_{\alpha}:2\pi\to 0$ and
$\theta_{\alpha+1}:0\to 2\pi$. It implies $\Delta\varphi_{\alpha}=-2\pi$,
$\Delta\varphi_{\beta}=0$, ($\beta\neq\alpha$). Then, in general,

$\displaystyle\bar{j}_{\rm op}^{\ell_{V_{\alpha}GV_{\alpha+1}}}$
$\displaystyle=\partial_{t}\bar{P}^{\ell_{V_{\alpha}GV_{\alpha+1}}},$ (120)
$\displaystyle\bar{P}^{\ell_{V_{\alpha}GV_{\alpha+1}}}$
$\displaystyle=-P_{\alpha},\ \alpha=1,\cdots,d-1,d.$ (121)

Note that this is justified also for $\alpha=d$, since along the last path
$\ell_{V_{d}GV_{Q}}=\ell_{V_{d}GV_{0}}$, $\theta_{d}:2\pi\to 0$ and
$\Delta\varphi_{d}=-2\pi$. Since the center of mass $P_{\alpha}$ is a physical
observable of the snapshot ground state for the open boundary condition, it is
periodic in time, $P_{\alpha}(t+T)=P_{\alpha}(t)$, as the Hamiltonian is
periodic. An important observation is that $\bar{P}^{\ell}(t)$ is not
continuous ($L\to\infty$) and has discontinuities at $t=t_{i}$, $i=1,2,\cdots$
of the jump $\pm\frac{1}{2}$ due to edge states (as shown later). Then the
pumped charge $Q^{\ell}_{\rm op}$ in the cycle $T$ due to the current
$\bar{j}^{\ell}_{\rm op}$ is written as

$\displaystyle Q^{\ell}_{\rm op}$
$\displaystyle=\int_{0}^{T}dt\,\bar{j}^{\ell}_{\rm
op}=\sum_{i}\int_{t_{i}}^{t_{i+1}}dt\,\bar{j}_{\rm op}^{\ell}$
$\displaystyle=\sum_{i}\int_{t_{i}+0}^{t_{i+1}-0}dt\,\partial_{t}\bar{P}^{\ell}(t)=\sum_{i}\bar{P}^{\ell}(t)\big{|}_{t_{i}+0}^{t_{i+1}-0}$
$\displaystyle=-\sum_{i}\bar{P}^{\ell}(t)\big{|}_{t_{i}-0}^{t_{i}+0}=-\sum_{i}\Delta\bar{P}^{\ell}(t_{i})=I^{\ell}.$
(122)

The discontinuities $I^{\ell}$ is defined by

$\displaystyle-I^{\ell}$ $\displaystyle=\sum_{i}\Delta\bar{P}^{\ell}(t_{i}),$
(123) $\displaystyle\Delta\bar{P}^{\ell}(t_{i})$
$\displaystyle=\bar{P}^{\ell}(t)\big{|}_{t_{i}-0}^{t_{i}+0}=\pm\frac{1}{2},$
(124)

where $\bar{P}^{\ell}(t)$ is not continuous at $t=t_{i}$ ($i=1,\cdots$). The
sign is determined by the behavior of the edge state that causes the jump.
Since the localization length (typical length scale) of the edge states is
finite, it scales to zero in the rescaled coordinate $x_{j}$. It implies the
contribution of the edge states localized near one of the boundaries is
$\pm\frac{1}{2}$ (See appendix G). Due to the conservation of the charge, the
number of the discontinuities is even. It implies the sum of the
discontinuities, $I$, is an integer. This is the quantization of the pumped
charge.

The physical current $\bar{j}_{\rm op}^{\ell}$ is carried by the bulk even
with the open boundary condition and is determined by the discontinuities due
to the edge states by the back action based on the periodicity of
$\bar{P}^{\ell}$ in time.

###  III.4 Bulk-edge correspondence

As for the periodic boundary condition, $\bar{A}_{\theta}^{(t),\ell}$ is
smooth and the pumped charge averaged along the path $\ell$ is given by

$\displaystyle Q_{\rm pe}^{\ell}$
$\displaystyle\equiv\int_{0}^{T}dt\,\bar{j}^{\ell}_{\rm pe}$
$\displaystyle=\frac{1}{2\pi i}\int_{0}^{T}dt\,\int_{\ell}d\theta\,B_{\rm
pe}(\theta,t),$ (125) $\displaystyle B_{\rm pe}(\theta,t)$
$\displaystyle=\partial_{\theta}A_{t,{\rm pe}}-\partial_{t}A_{\theta,{\rm
pe}},$ (126) $\displaystyle A_{\mu,{\rm pe}}$ $\displaystyle=\langle g_{\rm
pe}|\partial_{\mu}g_{\rm pe}\rangle,$ (127)

where $\bar{H}_{\rm pe}|g_{\rm pe}\rangle=|g_{\rm pe}\rangle E_{g}$. As for
the periodic boundary condition, the Berry connection $A_{\mu,{\rm pe}}$ and
thus $B_{\rm pe}$ is also defined by the Hamiltonian $\bar{H}_{\rm pe}$,
Eq.(83), that is not periodic/invariant by the shift
$\Theta\to\Theta+2\pi\vec{e}_{\alpha}$. See Table 4. It implies that the
periodic Hamiltonian $\bar{H}_{\rm pe}$, Eq.(83), is not defined on the torus
$T^{d}$. On the other hand, the ground state, $|g_{0}\rangle$, of the twisted
Hamiltonian, $H_{\rm tw}$, Eq.(82), is periodic by the shift and well defined
on the torus $T^{d}$. Noting that $|g_{\rm pe}\rangle={\cal U}_{LG}|g_{0,{\rm
tw}}\rangle$ and $H_{\rm tw}|g_{0,{\rm tw}}\rangle=|g_{0,{\rm tw}}\rangle
E_{g}^{\rm tw}$, one has

$\displaystyle A_{\theta,{\rm pe}}$
$\displaystyle=-i\sum_{\alpha}\partial_{\theta}\varphi_{\alpha}P_{\alpha,{\rm
tw}}+A_{\theta,{\rm tw}},$ (128) $\displaystyle A_{t,{\rm pe}}$
$\displaystyle=A_{t,{\rm tw}},$ (129)

where $P_{\alpha,{\rm tw}}=\langle g_{0,{\rm tw}}|{\cal P}_{\alpha}|g_{0,{\rm
tw}}\rangle$.

It results in

$\displaystyle B_{\rm pe}$ $\displaystyle=B_{\rm
tw}+i\sum_{\alpha}\partial_{\theta}\varphi_{\alpha}\partial_{t}P_{\alpha,{\rm
tw}}.$ (130)

Since $P_{\alpha,{\rm tw}}$ is smooth and periodic in time, the last term does
not contribute to the total pumped charge. Then

$\displaystyle Q^{\ell}_{\rm pe}$ $\displaystyle=\frac{1}{2\pi
i}\int_{\ell}d\theta\,\int_{0}^{T}dt\,B_{\rm tw}(\theta,t)\equiv C^{\ell}.$
(131)

This integral is over a torus $T^{2}=\\{(\theta,t)|\theta\in\ell,t\in[0,T]\\}$
without boundaries. It gives the Chern number $C$ that is integer. Now due to
Eq.(122), we have

$\displaystyle Q^{\ell}_{\rm pe}$ $\displaystyle=Q^{\ell}_{\rm
tw}=Q^{\ell}_{\rm op},$ (132) $\displaystyle I^{\ell}$
$\displaystyle=C^{\ell}.$ (133)

This is the bulk-edge correspondence of the topological pump.

As for the canonical path $\ell_{V_{\alpha}GV_{\alpha+1}}$, $\alpha=0,\cdots
d={Q}-1$, it is given by

$\displaystyle I^{\alpha}$ $\displaystyle=C^{\alpha},\ \alpha=0,\cdots,{Q}-1,$
(134)

where

$\displaystyle I^{\alpha}$ $\displaystyle\equiv
I^{\ell_{V_{\alpha}GV_{\alpha+1}}}=\sum_{i}\Delta P_{\alpha}(t_{i}),$ (135)

is a sum of the discontinuities of the $\alpha$-particle in the cycle and
$C^{\alpha}\equiv C^{\ell_{V_{\alpha}GV_{\alpha+1}}}$ is the Chern number
defined on a torus $T^{2}=[0,T]\times V_{\alpha}GV_{\alpha+1}$.

##  IV $Z_{Q}$ Berry phase, symmetry and gap closing

###  IV.1 $Z_{Q}$ quantization

In this section, let us discuss the Hamiltonian without symmetry breaking term
($\Delta=0$). Using the Hamiltonian $H_{\rm tw}$, the Berry phase
$\gamma_{\ell}$ is defined since the path $\ell$ forms a loop for $H_{\rm tw}$
as

$\displaystyle i\gamma_{\ell}$ $\displaystyle=\int_{\ell}d\theta\,A_{\theta}.$
(136)

The $Z_{Q}$ shift ${\cal U}_{Z_{Q}}$ as a shift in the parameter space as
shown in Sec.II.2 induces a map of the Hamiltonians in the parameter space
$\varphi_{\alpha}\to\varphi_{\alpha+1}$ (See Eq.(88)) and the canonical loop
$\ell_{V_{\alpha}GV_{\alpha+1}}$

$\displaystyle{\cal U}_{Z_{Q}}\bar{H}_{\rm tw}(\bm{\theta}){\cal
U}_{Z_{Q}}^{\dagger}$ $\displaystyle=\bar{H}_{\rm tw}(\bm{\theta}^{\prime}),$
(137)

where 2 points $\bm{\theta}$ and $\bm{\theta}^{\prime}$ in $\Theta$ are
parameterized by the same $\theta$ as

$\displaystyle\bm{\theta}$
$\displaystyle\in\ell_{V_{\alpha-1}GV_{\alpha}}(\theta),$
$\displaystyle\bm{\theta}^{\prime}$
$\displaystyle\in\ell_{V_{\alpha}GV_{\alpha+1}}(\theta).$

It implies that we may take

$\displaystyle|g_{0,{\rm tw}}(\bm{\theta}^{\prime})\rangle$
$\displaystyle={\cal U}_{Z_{Q}}|g_{0,{\rm tw}}(\bm{\theta})\rangle,$ (138)

and

$\displaystyle\gamma_{\ell_{V_{\alpha-1}GV_{\alpha}}}$
$\displaystyle=\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}\equiv\gamma_{Q},\
\alpha=1,\cdots{Q}$ (139)

since ${\cal U}_{Z_{Q}}$ does not depend on the parameter and thus $\langle
g_{0,{\rm tw}}(\bm{\theta}^{\prime})|\partial_{\theta}g_{0,{\rm
tw}}(\bm{\theta}^{\prime})\rangle=\langle g_{0,{\rm
tw}}(\bm{\theta})|\partial_{\theta}g_{0,{\rm tw}}(\bm{\theta})\rangle$. Then
using the fact,

$\displaystyle\sum_{\alpha=1}^{Q}\ell_{V_{\alpha}GV_{\alpha+1}}$
$\displaystyle=0,$ (140)

it results $Q\gamma_{Q}=0$, (mod $2\pi$). It implies $Z_{Q}$ quantization

$\displaystyle\gamma_{Q}$ $\displaystyle=\frac{2\pi n}{Q},\ \ n\in\mathbb{Z}.$
(141)

This $Z_{Q}$ Berry phase characterizes a symmetry protected topological phase
[64, 97, 95, 67]. It is a generalized the $Z_{2}$ Berry phase characterizes a
singlet pair or a covalent bond [64] 555 This $Z_{2}$ quantization for the
Berry phase for the non-interacting systems was also discussed before [130,
131].. Similarly $\gamma_{Q}$ characterizes the ${SU(Q)}$ dimer phase. The
dimer limit is characterized by this $Z_{Q}$ Berry phase and the symmetry
protection and adiabatic continuity guarantee the quantization. Unless the gap
closes by the deformation to the dimer limit $J_{i}=0$ ($i$:odd), the Berry
phase is given by (See Fig.3 and appendix E)

$\displaystyle\gamma_{Q}$ $\displaystyle=+\frac{2\pi}{{Q}},\ \text{mod }2\pi.$
(142)

It is topologically stable unless the gap closes against finite coupling
$J_{p}<J_{e}<0$ and $J_{e}<J_{o}<0$. It also gives

$\displaystyle\gamma_{V_{\alpha}GV_{\beta^{\prime}}}$
$\displaystyle=+\frac{2\pi}{{Q}}(\alpha-\beta),\ \text{mod }2\pi.$ (143)

Figure 3:  $Z_{Q}$ Berry phase $\gamma_{Q}$ for ${Q}=3,4$ and $5$ as a
function of $J_{e}/J_{o}$ ($L=8$). We have confirmed that the results are
reproduced by the symmetry indicators Eq.(153).

###  IV.2 Symmetry indicators

Let us first note that the Berry phase needs to be evaluated by a single gauge
fixing [99, 100, 64, 97] (see Appendix D). Here we assume that
$|g_{\phi}\rangle$ is gauge fixed over the loop
$\ell_{V_{\alpha-1}GV_{\alpha}}=\ell_{V_{\alpha-1}G}-\ell_{V_{\alpha}G}$ by a
single gauge fixing by $|\phi\rangle$ as

$\displaystyle|g_{\phi}\rangle$ $\displaystyle=P|\phi\rangle/\sqrt{N_{\phi}},$
(144)

where $P=|g_{0,{\rm tw}}\rangle\langle g_{0,{\rm tw}}|$ and
$N_{\phi}=|\langle\phi|g_{0,{\rm tw}}\rangle|^{2}\neq 0$ 666If
$N_{\phi}=|\langle\phi|g_{0,{\rm tw}}\rangle|=0$, slightly modify
$|\phi\rangle$, then $N_{\phi}\neq 0$. It is always possible if the dimension
of the Hilbert space is larger than $1$..

Noting that

$\displaystyle H_{\rm tw}(\bm{\theta}^{\prime})$ $\displaystyle={\cal
U}_{Z_{Q}}H_{\rm tw}(\bm{\theta}){\cal U}_{Z_{Q}}^{\dagger},$ (145)

where $\bm{\theta}\in\ell_{V_{\alpha-1}G}(\theta)$, and
$\bm{\theta}^{\prime}\in\ell_{V_{\alpha}G}(\theta)$, one has

$\displaystyle|g_{\phi}(\bm{\theta}^{\prime})\rangle$ $\displaystyle={\cal
U}_{Z_{Q}}|g_{\phi}(\bm{\theta})\rangle e^{-i^{\exists}\Omega(\bm{\theta})}.$
(146)

This extra phase factor is due to the fact that the phase convention of the
state $|g_{\phi}(\bm{\theta}^{\prime})\rangle$ by $|\phi\rangle$ at
$\bm{\theta}^{\prime}$ is, in general, different from that of ${\cal
U}_{Z_{Q}}|g_{\phi}(\bm{\theta})\rangle$.

Since ${\cal U}_{Z_{Q}}$ is independent of the parameter, the Berry phase is
written as

$\displaystyle\gamma_{Q}$
$\displaystyle=\gamma_{\ell_{V_{\alpha-1}GV_{\alpha}}}=-i\int_{\ell_{V_{\alpha-1}GV_{\alpha}}}d\theta\,\langle
g_{\phi}|\partial_{\theta}g_{\phi}\rangle$
$\displaystyle=-i\int_{\ell_{V_{\alpha-1}G}}d\theta\,\langle
g_{\phi}|\partial_{\theta}g_{\phi}\rangle+i\int_{\ell_{V_{\alpha}G}}d\theta\,\langle
g_{\phi}|\partial_{\theta}g_{\phi}\rangle$
$\displaystyle={+}\int_{\ell_{V_{\alpha-1}G}}d\theta\,\partial_{\theta}\Omega=\Omega(G)-\Omega(O),$
(147)

where all vertices $V_{\alpha}$ are identified to the origin $O$ in modulo
$2\pi$. Supplementing $\theta_{Q}=-\sum_{\alpha=1}^{d}\theta_{\alpha}$, at the
vertices $O$ and $G$, the parameters are

$\displaystyle\bm{\theta}={O}$
$\displaystyle:(\theta_{1},\cdots,\theta_{Q})=(0,\cdots,0),\qquad\text{mod}\,2\pi$
(148) $\displaystyle\bm{\theta}=G$
$\displaystyle:(\theta_{1},\cdots,\theta_{Q})=(\frac{2\pi}{Q},\cdots,\frac{2\pi}{Q}),\
\text{mod}\,2\pi.$ (149)

It implies the Hamiltonian is invariant by the shift of the fermions by ${\cal
U}_{Z_{Q}}$ at $O$ and $G$ as

$\displaystyle{[}H_{\rm tw}(\bm{\theta}),{\cal U}_{Z_{Q}}{]}$
$\displaystyle=0,\ \ \bm{\theta}=O,G.$ (150)

Therefore ${\cal U}_{Z_{Q}}$ is a symmetry of the Hamiltonian and
$e^{i\Omega}$ is an eigen value of the symmetry operation (symmetry indicator)
as

$\displaystyle{\cal U}_{Z_{Q}}|g_{\phi}(O)\rangle$
$\displaystyle=|g_{\phi}(O)\rangle e^{{+}i\Omega(O)},$ (151)
$\displaystyle{\cal U}_{Z_{Q}}|g_{\phi}(G)\rangle$
$\displaystyle=|g_{\phi}(G)\rangle e^{+{}i\Omega(G)}.$ (152)

Since ${\cal U}_{Z_{Q}}$ is trivial at $O$ and $\Omega(O)=0$, it results in

$\displaystyle\gamma_{Q}$ $\displaystyle=\Omega(G)=\text{Arg}\,\langle
g_{\phi}(G)|{\cal U}_{Z_{Q}}|g_{\phi}(G)\rangle.$ (153)

Physical meaning of these quantities is clear by the adiabatic deformation to
the dimer limit by the $Z_{Q}$ Berry phase. It also implies the gap closing at
$G$ associated with a topological transition due to the discrete change of the
$Z_{Q}$ Berry phase.

It is also directly observed by the discretized formula of the Berry phase
(discretizing the path $V_{0}G$ into $M$ segments
$\theta_{m}={\frac{2\pi}{{Q}}}\frac{m}{M}$, ($m=0,\cdots,M$) as

$\displaystyle\gamma_{Q}$
$\displaystyle=\lim_{M\to\infty}\text{Arg}\,\big{(}\langle\theta_{0}|\theta_{1}\rangle\langle\theta_{1}|\theta_{2}\rangle\cdots\langle\theta_{M-1}|\theta_{M}\rangle$
$\displaystyle\qquad\qquad\times\langle\theta_{M}|\theta_{M-1}^{\prime}\rangle\cdots\langle\theta_{2}^{\prime}|\theta_{1}^{\prime}\rangle\langle\theta_{1}^{\prime}|\theta_{0}\rangle\big{)}.$

where $|\theta_{m}^{\prime}\rangle={\cal U}_{Z_{Q}}|\theta_{m}\rangle$,
$|\theta_{m}\rangle=|g_{{\phi}}(\theta_{m})\rangle$. Since ${\cal
U}_{Z_{Q}}|\theta_{M}\rangle=|\theta_{M}\rangle e^{i\Omega(G)}$ and ${\cal
U}_{Z_{Q}}|\theta_{0}\rangle=|\theta_{0}\rangle e^{i\Omega(O)}$, it is written
as

$\displaystyle\gamma_{Q}$
$\displaystyle=\lim_{M\to\infty}\text{Arg}\,\big{(}\langle\theta_{0}|\theta_{1}\rangle\langle\theta_{1}|\theta_{2}\rangle\cdots\langle\theta_{M-1}|\theta_{M}\rangle$
$\displaystyle\qquad\qquad\times\langle\theta_{M}|{\cal
U}_{Z_{Q}}|\theta_{M-1}\rangle\cdots\langle\theta_{2}|\theta_{1}\rangle\langle\theta_{1}|{\cal
U}^{\dagger}_{Z_{Q}}|\theta_{0}\rangle\big{)}$
$\displaystyle=\Omega(G)-\Omega(O),$

where ${\cal U}_{Z_{Q}}^{\dagger}|\theta_{0}\rangle=|\theta_{0}\rangle
e^{-i\Omega(O)}$ and $\langle\theta_{M}|{\cal
U}_{Z_{Q}}=e^{i\Omega(G)}\langle\theta_{M}|$ due to Eqs.(151) and (152).

###  IV.3 Modified Lieb-Schultz-Mattis (LSM) argument

As is clear, the system of the dimer limit is gapped. This gap is stable for
inclusion of finite coupling between the dimers, at least, for a finite size
system. One may naturally expect this gap converges to some finite values by
taking an infinite size limit $L\to\infty$ assuming the ground state is
adiabatically connected to a set of disconnected dimers. As for a uniform
system, existence of the gap is unclear and the problem has a long history of
studies. Some of the recent studies are topological. Especially in relation
with the Haldane conjecture for the $S=1$ Heisenberg model [102]. Since
$H^{(2)}$ for ${Q}=2$, is equivalent to the standard $S=1/2$ Heisenberg model,
$H^{(1)}$ (appendix B), well-known Lieb-Schultz-Mattis (LSM) theorem [103,
104] is applied and the energy gap of the finite system with the periodic
boundary condition vanishes when $L\to\infty$. This is consistent with the
existence of the gapless excitation as the des Cloizeaux and Pearson mode of
the ${Q}=2$ case [105]. Note that the LSM theorem also allows existence of the
finite size gap between the states which become degenerate in the
thermodynamic limit associated with the symmetry breaking. On the other hand,
for the ${Q}=3$ case, a series of studies [106, 107, 108] has clarified that
the uniform system has doubly degenerate dimerized ground states in the
$L\to\infty$ limit. The case, ${Q}>3$, is also discussed by Affleck suggesting
a similar conclusion (double degeneracy due to dimerization) [109]. In this
subsection, we give a topological argument for the gap closing for even
${Q}\geq 2$ of the finite size system.

The gap of the finite system under the twist is strongly constrained by
considering the Berry phase, which works as a topological order parameter
responding to the local twist as an external perturbation [64, 95, 110, 111,
112, 113]. In Ref. [114], the standard $S=1/2$ Heisenberg model, $H^{(1)}$
(${Q}=2$) was considered. If the gap remains open for all values of the twist,
one can prove that the Berry phase pattern, associated with the local twist at
the link, needs to be alternating in this $S=1/2$ case. This clearly
contradicts with the uniformity of the system. It results in that the Berry
phase can not be defined, that is, the gap closing of the system at some twist
[114]. The argument can be extended to the present system ${Q}>2$ as shown
here. The claim is that, as for a finite system of even ${Q}\geq 2$, the
energy gap between the ground state and the next one under the twisted
boundary condition vanishes at some twist.

Up to this point, we have discussed Berry phases associated with the twist at
the boundary link $L$ and $1$. Let us write it as $\gamma^{L}_{\ell}$. In a
similar way, one may also define the Berry phase $\gamma^{L-1}$ associated
with the twist at $L-1$ and $L$. Let us write the Hamiltonians with the twists
as

$\displaystyle H_{\rm tw}^{L}$
$\displaystyle=J_{e}\sum_{\alpha,\beta}e^{-i(\varphi_{\alpha}-\varphi_{\beta})}c_{L,\alpha}^{\dagger}c_{1,\alpha}^{\dagger}c_{1,\beta}c_{L,\beta}$
$\displaystyle+J_{o}\sum_{\alpha,\beta}c_{L-1,\alpha}^{\dagger}c_{L,\alpha}^{\dagger}c_{L,\beta}c_{L-1,\beta}+\cdots,$
(154) $\displaystyle H_{\rm tw}^{L-1}$
$\displaystyle=J_{e}\sum_{\alpha,\beta}c_{L,\alpha}^{\dagger}c_{1,\alpha}^{\dagger}c_{1,\beta}c_{L,\beta}$
$\displaystyle+J_{o}\sum_{\alpha,\beta}e^{{+}i(\varphi_{\alpha}-\varphi_{\beta})}c_{L-1,\alpha}^{\dagger}c_{L,\alpha}^{\dagger}c_{L,\beta}c_{L-1,\beta}+\cdots,$
(155)

where $\cdots$ does not include $c_{L,\alpha}$. Note that the sign of the
twist is reversed. They are related with each other by the the gauge
transformation ${\cal
U}_{L}=e^{{-}i\sum_{\alpha}\varphi_{\alpha}\hat{n}_{L,\alpha}}$, ${\cal
U}_{L}^{\dagger}c_{L,\alpha}{\cal
U}_{L}=e^{{+}i\varphi_{\alpha}\hat{n}_{L,\alpha}}c_{L,\alpha}$ (See also
appendix E) as

$\displaystyle H_{\rm tw}^{L}$ $\displaystyle={\cal U}_{L}H_{\rm
tw}^{L-1}{\cal U}_{L}^{\dagger}.$ (156)

The Berry phases are defined by $|g^{L}\rangle$ and $|g^{L-1}\rangle$, which
are the ground states of $H^{L}_{\rm tw}$ and $H^{L-1}_{\rm tw}$ respectively
as

$\displaystyle i\gamma^{L}$ $\displaystyle=\int_{\ell}d\theta A^{L},\
A^{L}=\langle g^{L}|\partial_{\theta}g^{L}\rangle,$ (157) $\displaystyle
i\gamma^{L-1}$ $\displaystyle=\int_{\ell}d\theta A^{L-1},\ A^{L-1}=\langle
g^{L-1}|\partial_{\theta}g^{L-1}\rangle,$ (158)

where $H^{L}|g^{L}\rangle=|g^{L}\rangle E$ and
$H^{L-1}|g^{L-1}\rangle=|g^{L-1}\rangle E$. Noting that $|g^{L}\rangle={\cal
U}_{L}|g^{L-1}\rangle$ 777 We take a global single gauge for $|g^{L-1}\rangle$
(appendix D). Then the phase of $|g^{L}\rangle\equiv{\cal
U}_{L}|g^{L-1}\rangle$ is also globally unique. It implies Eq.(160) is without
$2\pi$ ambiguity. , it induces

$\displaystyle A^{L}$
$\displaystyle=A^{L-1}{-}i\sum_{\alpha}(\partial_{\theta}\varphi_{\alpha})\langle
g^{L-1}|\hat{n}_{L,\alpha}|g^{L-1}\rangle.$ (159)

Generically, with the twist, ${SU(Q)}$ symmetry is (slightly) broken even
without explicit symmetry breaking term $H_{B}$, that is, the fermions with
different colors are not equivalent and $\langle
g^{L-1}|\hat{n}_{L,\alpha}|g^{L-1}\rangle\neq\frac{1}{{Q}}$. However, this
symmetry breaking effect due to the twist is not localized at the twisted
link. The large gauge transformation Eqs.(55) and (84), ${\cal U}_{LG}$, maps
the periodic system with ${\cal O}(1/L)$ twist to the system with twisted
boundary condition preserving the local charge density remains unchanged
because (${[}{\cal U}_{{LG}},\hat{n}_{j,\alpha}{]}=0)$. Then $\langle
g^{L-1}|\hat{n}_{j,\alpha}|g^{L-1}\rangle$ is $j$ independent both for the
periodic/twisted system. It implies the effects are of the order of $L^{-1}$
as $\langle g^{L-1}|\hat{n}_{L,\alpha}|g^{L-1}\rangle=\frac{1}{{Q}}+{\cal
O}(L^{-1})$. Then integrating Eq.(159) over the loop
$\ell_{V_{\alpha}GV_{\alpha+1}}$, we have for a sufficiently large system

$\displaystyle\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L}$
$\displaystyle=\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L-1}{+}\frac{2\pi}{{Q}},$
(160)

since $\Delta\varphi_{\beta}=-2\pi(\beta=\alpha),0(\beta\neq\alpha)$. Note
that the possible ${\cal O}(L^{-1})$ extra term vanishes after the integration
to be consistent with the $Z_{Q}$ quantization of the Berry phases. This
constraint needs to be satisfied for any systems even with site dependent
$J_{i}$’s. The two dimer limits,
$\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L}=0$,
$\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L-1}={-}\frac{2\pi}{Q}$ and
$\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L}={+}\frac{2\pi}{Q}$,
$\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L-1}=0$, are consistent with
Eq.(160).

If the system is uniform, $J_{o}=J_{e}$, the anti-translation invariance of
the system with the twist, $H^{L}_{\rm tw}={\cal A}_{{\cal T}}H_{\rm
tw}^{L-1}{\cal A}_{{\cal T}}^{-1}$, implies

$\displaystyle\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L}$
$\displaystyle=-\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L-1},\ \text{mod
}2\pi.$ (161)

By Eqs.(160) and (161), we have constraints for the Berry phase for a uniform
system as

$\displaystyle\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L}$
$\displaystyle=-\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L-1}={+}\frac{\pi}{Q},\
\text{mod }2\pi$ (162)

or

$\displaystyle\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L}$
$\displaystyle=\pi\frac{Q{+}1}{Q},\
\gamma_{\ell_{V_{\alpha}GV_{\alpha+1}}}^{L-1}=\pi\frac{Q{-}1}{Q},\ \text{mod
}2\pi.$ (163)

As for the even ${Q}$, these constraints, Eq.(162) and Eq.(163), contradict
the $Z_{Q}$ quantization of the Berry phases $\gamma=2\pi\frac{n}{Q},\
n\in\mathbb{Z}$. This contradiction implies that the Berry phase can not be
well-defined. It is only possible when the gap of the (finite size) system
closes. A level crossing between the ground state and the next one occurs at
some twist parameter. Assuming the degenerate dimer states for the infinite
size system, the gap between the linear combinations of the dimer states of
the finite size system closes at the twisted parameters. As for the odd ${Q}$,
the second case, Eq.(163), is compatible with the $Z_{Q}$ quantization ($Q\pm
1$ is even), although these quantized values are different from that of the
dimer limit 888 It implies the gap closing at some twist by inclusion of the
coupling between the dimers.. In principle, it allows a unique gapped ground
state of the uniform system for any value of the twist. Although it does not
occur in the present numerical calculations shown in Fig.3, inclusion of long
range couplings and additional terms which respect $Z_{Q}$ symmetry may
realize such a ground state.

##  V Emergent $Z_{Q}\times Z_{Q}$ symmetry and explicit Chern numbers

###  V.1 Open system: edge states and low energy spectrum

Figure 4:  Dimer configurations of open systems for even/odd sites.

Although most of the discussion in the paper is for even $L$ systems, let us
consider, in this section, even/odd $L$ systems separately, especially near
the dimer limits $|J_{o}|\ll|J_{e}|$ and $|J_{e}|\ll|J_{o}|$.

$L$:odd.  When the system size $L$ is odd, the ground states are given for
each dimer limits $J_{o}=0$ and $J_{e}=0$ as (See Fig.4)

$\displaystyle|g_{1,eo}^{L:\text{odd}},\alpha\rangle$
$\displaystyle=c_{1,\alpha}^{\dagger}\prod_{j=1}^{(L-1)/2}\psi_{2j+1,2j}^{\dagger}|0\rangle,\
(J_{o}=0),$ (164) $\displaystyle|g_{L,eo}^{L:\text{odd}},\alpha\rangle$
$\displaystyle=c_{L,\alpha}^{\dagger}\prod_{j=1}^{(L-1)/2}\psi_{2j-1,2j}^{\dagger}|0\rangle,\
(J_{e}=0),$ (165)

where $\bar{N}_{\beta}={1}$ for $\beta=\alpha$ and ${0}$ otherwise. It implies
${Q}$-fold degeneracy of the ground states. Their charge distributions are

$\displaystyle\langle
g_{1,eo}^{L:\text{odd}},\alpha|n_{j,\beta}|g_{1,eo}^{L:\text{odd}},\alpha\rangle$
$\displaystyle=\left\\{\begin{array}[]{cc}\frac{1}{Q}&j\neq 1\\\
0&j=1,\beta\neq\alpha\\\ 1&j=1,\beta=\alpha\end{array}\right.,$ (169)
$\displaystyle\langle
g_{L,eo}^{L:\text{odd}},\alpha|n_{j,\beta}|g_{L,eo}^{L:\text{odd}},\alpha\rangle$
$\displaystyle=\left\\{\begin{array}[]{cc}\frac{1}{Q}&j\neq L\\\
0&j=1,\beta\neq\alpha\\\ 1&j=1,\beta=\alpha\end{array}\right.,$ (173)

where $|g_{1,eo}^{L:\text{odd}},\alpha\rangle$ is a product of the bulk and
completely localized state at $j=1$ with the color $\alpha$. Similarly
$|g_{L,eo}^{L:\text{odd}},\alpha\rangle$ is a product of the bulk and
completely localized state at $j=L$ with the color $\alpha$. This degeneracy
is stable for inclusion of a finite coupling $J_{o}$ and $J_{e}$ since finite
matrix elements with different quantum numbers $\bar{N}_{\alpha}$’s are
prohibited due to the symmetry. It implies the charge distributions are
modified continuously for a finite coupling. The numerical results for $L=9$
systems with dimerization $\bar{N}_{\alpha}:(1,0,0,0,0)$ are obtained by the
exact diagonalization and shown in Fig.5. They are consistent with the present
picture. The ground state is given by the gapped bulk and boundary states
(edge states at both ends) localized near the boundaries.

Figure 5:  Charge distribution $\langle n_{j,\alpha}\rangle$ of the unique
ground state of the odd $L$ system ($L=9$) with ${Q}=5$,
$\bar{N}_{\alpha}:(1,0,0,0,0)$ (a): $J_{o}=-0.8$, $J_{e}=-1.2$ and (b):
$J_{o}=-1.2$, $J_{e}=-0.8$.

$L$:even. As for the system with $L$: even, the ground states of the dimer
limit are again given by

$\displaystyle|g_{{\rm op},eo}^{L:\text{even}}(\alpha,\beta)\rangle$
$\displaystyle=c_{1,\alpha}^{\dagger}c_{L,\beta}^{\dagger}\prod_{j=1}^{L/2-1}\psi_{2j+1,2j}^{\dagger}|0\rangle,(J_{o}=0)$
(174) $\displaystyle|g_{{\rm op},oe}^{L:\text{even}}\rangle$
$\displaystyle=\prod_{j=1}^{L/2}\psi_{2j-1,2j}^{\dagger}|0\rangle,\ (J_{e}=0)$
(175)

where $\bar{N}_{\gamma}=0,^{\forall}\\!\gamma$ for both cases. It implies
${Q}^{2}$-fold degeneracy for $J_{o}=0$ and gapped unique ground state for
$J_{e}=0$. If $|J_{o}|>|J_{e}|$, the unique gapped ground state,
$|g_{eo}^{L:\text{even}}\rangle$ is stable for inclusion of the finite
coupling $J_{e}$. However, as for $|J_{o}|<|J_{e}|$, in contrast to the
$L$:odd case, the ${Q}^{2}$-fold degeneracy of the ground states is unstable
for the finite size systems. The degeneracy is lifted for the finite coupling
due to the residual interaction between the edge states at both ends. As for a
chain of the finite length, we expect an effective coupling between the two
boundary states at both ends. It is a generalization of the Kennedy’s
discussion [117, 118, 119, 120]. This effective coupling, $J_{\rm eff}$, is
expected to behave as $e^{-L/\xi}$ for $L$ where $\xi$ is a correlation length
between the edge states, which can be proportional to the inverse of the bulk
energy gap. This is confirmed numerically for ${Q}=3$ and $4$ in Fig.6 and
Fig.7.

Figure 6:  Energy gap of the $SU(3)$ symmetric Hamiltonian $H^{(2)}$, (a):
periodic boundary condition and (b) open boundary condition. The system sizes
are $L=6,8,10,12,14,16$ and $J_{j}=-0.8(j:\text{odd})$ and
$J_{j}=-1.2(j:\text{even})$. The data of the open boundary condition for
$L\geq 10$ are fitted by the localization length $\xi=1.52$.

Figure 7:  Energy gap of the $SU(5)$ symmetric Hamiltonian $H^{(2)}$, (a):
periodic boundary condition and (b) open boundary condition. The system sizes
are $L=4,6,8,10$ and $J_{j}=-0.8(j:\text{odd})$ and
$J_{j}=-1.2(j:\text{even})$. The data of the open boundary condition for
$L\geq 6$ are fitted by the localization length $\xi=0.91$.  Table 5: Lowest
15 energies of $H^{(2)}$ with ${Q}=3$ ($L=6$ and $8$).

| Periodic | Open | Open
---|---|---|---
$(J_{1},J_{2})$ | $(-0.8,-1.2),(-1.2,-0.8)$ | $(-1.2,-0.8)$ | $(-0.8,-1.2)$
| $L=6$ | $L=8$ | $L=6$ | $L=8$ | $L=6$ | $L=8$
1 | -12.00000000 | -15.91335471 | -11.54504463 | -15.51898367 | -9.19845500 | -12.87602132
2 | -9.32998146 | -13.40936964 | -8.73554793 | -12.84032668 | -8.65117764 | -12.67776804
3 | -9.32998146 | -13.40936964 | -8.73554793 | -12.84032668 | -8.65117764 | -12.67776804
4 | -9.32998146 | -13.40936964 | -8.73554793 | -12.84032668 | -8.65117764 | -12.67776804
5 | -9.32998146 | -13.40936964 | -8.73554793 | -12.84032668 | -8.65117764 | -12.67776804
6 | -9.32998146 | -13.40936964 | -8.73554793 | -12.84032668 | -8.65117764 | -12.67776804
7 | -9.32998146 | -13.40936964 | -8.73554793 | -12.84032668 | -8.65117764 | -12.67776804
8 | -9.32998146 | -13.40936964 | -8.73554793 | -12.84032668 | -8.65117764 | -12.67776804
9 | -9.32998146 | -13.40936964 | -8.73554793 | -12.84032668 | -8.65117764 | -12.67776804
10 | -8.40000000 | -12.45492049 | -8.01759677 | -12.29865018 | -7.32249080 | -11.45644523
11 | -8.04273842 | -12.45492049 | -8.01759677 | -12.29865018 | -6.72949546 | -10.84747170
12 | -8.04273842 | -12.45492049 | -8.01759677 | -12.29865018 | -6.72949546 | -10.84747170
13 | -8.04273842 | -12.45492049 | -8.01759677 | -12.29865018 | -6.72949546 | -10.84747170
14 | -8.04273842 | -12.45492049 | -8.01759677 | -12.29865018 | -6.72949546 | -10.84747170
15 | -8.04273842 | -12.45492049 | -8.01759677 | -12.29865018 | -6.72949546 | -10.84747170

That is, we expect an exponentially small coupling between the both ends which
operates for the ${Q}^{2}$-fold low energy multiplet of the edge states. It is
described by the effective Hamiltonian of the effective boundary fermions
(assuming that they live at $j=1$ and $L$) $c_{1,\alpha}$ and $c_{L,\alpha}$
($\alpha=1,\cdots,N$ ) as

$\displaystyle h_{\text{eff}}$
$\displaystyle=J_{\text{eff}}c_{1,\alpha}^{\dagger}c_{1,\beta}c_{L,\alpha}^{\dagger}c_{L,\beta}={Q}J_{\text{eff}}\psi_{1,L}^{\dagger}\psi_{1,L}.$
(176)

The energy spectrum of the low energy multiplet is given by the decomposition
of the tensor product of ${SU(Q)}$ representation as
${Q}\otimes\bar{Q}=1\oplus({Q}^{2}-1)$ as a generalization of singlet-triplet
decomposition for the Kennedy’s case. The unique ground state among the
multiple is a singlet approximately given by

$\displaystyle|S_{1,L}\rangle$
$\displaystyle=\psi_{1,L}^{\dagger}\otimes\prod_{j}^{L/2-1}\psi_{2j+1,2j}^{\dagger}|0\rangle,$
(177)

with its energy ${Q}J_{\text{eff}}$. This is a generalization of the Kenndey’s
singlet and triplet for the $S=1$ Haldane chain [117, 118, 119, 120]. Note
that the state $|S_{1,L}\rangle$ is interpreted as a tensor product of a
gapped bulk and edge states. Due to the uniqueness and the ${SU(Q)}$
invariance, it implies that the one point function is constant as $\langle
S_{1,L}|n_{j\alpha}|S_{1,L}\rangle=1/Q$ for all $\alpha$ when the average is
defined by the trace over the degenerate states. Anything localized is not
observed in the charge distribution. This is to be compared with the results
shown in Fig.5 for the odd $L$ case. The other ${Q}^{2}-1(=8,N=3)$ are at the
zero energy. The lowest 15 energies of ${Q}=3$ are listed in Table 5. System
size dependencies of the gap for the ${Q}=3$ and ${Q}=5$ are shown in the
insets of Fig.6 and Fig.7. They show $J_{eff}\propto e^{-L/\xi}$, which imply
that the low energy multiplets are described by the edge states. Assuming this
behavior, we may assume exact ${Q}^{2}$ degeneracy for the infinite system
with boundaries, that is, taking an infinite size limit for the open system.
This ${Q}^{2}$-fold degeneracy is exact only in the $L\to\infty$ limit. In
this sense, the ${Q}^{2}$-fold degeneracy implies that emergence of
$Z_{Q}^{\text{left}}\times Z_{Q}^{\text{right}}$ symmetry in the infinite
chain, which was originally mentioned in a chiral symmetric fermion system
[121]. This corresponds to the $2^{2}$-fold degeneracy and $Z_{2}\times Z_{2}$
symmetry of the Haldane chain [122] and dimer phases of $S=1/2$ quantum spin
chain [123]. See Table 5 as well.

###  V.2 Low energy multiplet of edge states with symmetry breaking

Let us consider a dimerized system $|J_{e}|\neq|J_{o}|$, ($J_{i}<0$) with open
boundary condition. We assume the system size $L$ is even assuming it is
sufficiently large (compared with the gap). When $|J_{o}|<|J_{e}|$, the low
energy sector of the system is composed of a ${Q}$-fold degenerate multiplet
with edge states localized near $j=1$ and $j=L$ which is spanned by degenerate
${Q}^{2}$ low energy states Eq.(174),
$\\{|g_{{eo};\alpha,\beta}^{L:\text{even}}\rangle|\alpha,\beta=1,\cdots,{Q}\\}$.
This multiplet is separated from the other states by the bulk gap. The
interaction between the both ends are negligibly small since we assume the
system sized is large. Then the low energy multiplet is ${Q}^{2}$-fold
degenerate and the $Z_{Q}$ symmetry breaking term, $H_{B}$, Eq.(77), operates
within this multiplet perturbatively assuming that the symmetry breaking is
small compared with the bulk gap. This perturbative discussion is exact as for
the level crossing (selection rule) within the multiplet assuming the gap
between the multiplet and else (global spectral structure) is finite where the
energy scale of the splitting is governed by the gap of the bulk. When the
dimerization pattern is reversed, $|J_{o}|>|J_{e}|$, the system is gapped even
for the open boundary condition and the ground state is adiabatically
connected to the unique gapped one, Eq.(175).

Let us extend the time dependence of the pump by shifting the timing of the
dimerization as follows

$\displaystyle J_{i}$ $\displaystyle=J_{0}+(-1)^{j}\delta
J\cos\big{(}2\pi(\frac{t-t_{0}}{T})\big{)},\ J_{0}<0,\delta J>0,$ (178)
$\displaystyle\Delta_{\alpha}$
$\displaystyle=\Delta\sin\big{(}2\pi(\Phi\alpha+\frac{t}{T})\big{)}.$ (179)

As for the open boundary condition, the edge states only appear when the
coupling is weak at the both boundaries ($L$: even), that is,
$|J_{o}|<|J_{e}|$ ($J_{0}<0,\delta J>0$). This period is specified by

$\displaystyle\frac{t_{0}}{T}+\frac{1}{4}+n<\frac{t}{T}<\frac{t_{0}}{T}+\frac{3}{4}+n,\
^{\exists}n\in\mathbb{Z}.$ (180)

Let us first discuss energies of the symmetry breaking Hamiltonian $H_{B}$,
Eq.(77), within the ${Q}$-fold degenerate multiplet
$|\alpha,\beta\rangle=|g_{{\rm op},eo}^{L:\text{even}}(\alpha,\beta)\rangle$,
$\alpha,\beta=1,\cdots,{Q}$

$\displaystyle H_{B}(t)|g_{{\rm op},eo}^{L:\text{even}}(\alpha,\beta)\rangle$
$\displaystyle=|g_{{\rm op},eo}^{L:\text{even}}(\alpha,\beta)\rangle
E_{\alpha\beta}(t),$ $\displaystyle E_{\alpha\beta}(t)$
$\displaystyle=\Delta_{\alpha}(t)+\Delta_{\beta}(t).$

See Table 6 and Fig.8. Assuming the system size is sufficiently large
$L\to\infty$, the center of masses (CoM) $P_{\alpha}$ and the quantum numbers
$\bar{N}_{\alpha}$ are also shown. For example, as for the state,
$|1_{1},1_{L}\rangle=|g_{{\rm op},eo}^{L:\text{even}}(1,1)\rangle$, the CoM’s
are $P_{1}=x_{1}\cdot(+1)+x_{L}\cdot(-1)\to-1$, $P_{2}=0$ and $P_{3}=0$. As
for the state, $|1_{1},2_{L}\rangle=|g_{{\rm
op},eo}^{L:\text{even}}(1,2)\rangle$, $P_{1}=x_{1}\cdot(+1)\to-\frac{1}{2}$,
$P_{2}=x_{L}\cdot(-1)\to-\frac{1}{2}$ and $P_{3}=0$. Generically, as for the
state, $|\alpha_{1},\alpha_{L}\rangle$, $\bar{N}_{\alpha}=(0,\cdots,0)$. Its
CoM’s are $P_{\alpha}=-1$ and $P_{\gamma}=0$ for any $\gamma\neq\alpha$. As
for the state, $|\alpha,\beta\rangle$, ($\alpha\neq\beta$),
$\bar{N}_{\alpha}=+1$, $\bar{N}_{\beta}=-1$, $\bar{N}_{\gamma}=0$ for any
$\gamma\neq\alpha,\beta$. Its CoM’s are $P_{\alpha}=P_{\beta}=-\frac{1}{2}$,
$P_{\gamma}=0$ $(\gamma\neq\alpha,\beta$). Explicit examples for ${Q}=3$ are
shown in Table 6.

Table 6: Energies, $E_{\alpha\beta}(t)=\Delta_{\alpha}(t)+\Delta_{\beta}(t)$, quantum numbers, $\bar{N}_{\alpha}$, and CoM’s, $P_{\alpha}$, of the symmetry breaking term $H_{B}$ for the multiplet of the boundary spins, $\Delta_{\alpha}=\Delta\sin 2\pi(\Phi\alpha+\frac{t}{T}),\ \alpha=1,2,3$ where $\Phi=P/Q,P=1,Q=3$. The generic ${Q}$ case is discussed in a straightforward way.  Multiplet | $E_{\alpha}+E_{\beta}$ | $\bar{N}_{\alpha}$ |  $P_{\alpha}$, ($L\to\infty$)
---|---|---|---
$|1_{1},1_{L}\rangle$ | $E_{11}=2\Delta_{1}$ | $(0,0,0)$ | $(-1,0,0)$
$|2_{1},2_{L}\rangle$ | $E_{22}=2\Delta_{2}$ | $(0,0,0)$ | $(0,-1,0)$
$|3_{1},3_{L}\rangle$ | $E_{33}=2\Delta_{3}$ | $(0,0,0)$ | $(0,0,-1))$
$|1_{1},2_{L}\rangle$ | $E_{12}=\Delta_{1}+\Delta_{2}$ | $(+1,-1,0)$ | $(-\frac{1}{2},-\frac{1}{2},0)$
$|2_{1},1_{L}\rangle$ | $E_{21}=\Delta_{2}+\Delta_{1}$ | $(-1,+1,0)$ | $(-\frac{1}{2},-\frac{1}{2},0)$
$|1_{1},3_{L}\rangle$ | $E_{13}=\Delta_{1}+\Delta_{2}$ | $(+1,0,-1)$ | $(-\frac{1}{2},-\frac{1}{2},0)$
$|3_{1},1_{L}\rangle$ | $E_{31}=\Delta_{3}+\Delta_{1}$ | $(-1,0,+1)$ | $(-\frac{1}{2},0,-\frac{1}{2})$
$|2_{1},3_{L}\rangle$ | $E_{23}=\Delta_{2}+\Delta_{3}$ | $(0,+1,-1)$ | $(0,-\frac{1}{2},-\frac{1}{2})$
$|3_{1},2_{L}\rangle$ | $E_{32}=\Delta_{3}+\Delta_{2}$ | $(0,-1,+1)$ | $(0,-\frac{1}{2},-\frac{1}{2})$

Figure 8:  Normalized low energy spectra of the symmetry breaking term
$H_{B}(t)$ for the multiplet $|\alpha_{L}\beta_{R}\rangle$ where $L$ and $R$
are effective spins at both ends,
$E_{\alpha\beta}(t)=\Delta_{\alpha}(t)+\Delta_{\beta}(t)$ as a function of
$t$. The gray rectangles show the region $|J_{o}|<|J_{e}|$, that is specified
by Eq.(180) for $t_{0}/T=0.01$. Colored lines are $E_{\alpha\alpha}$ and the
numbers shown in the bottom denote $\alpha$. They are for $P=1$, $\Phi=1/Q$
and the ones in the parenthesis is for $P=2$, $\Phi=2/Q$. (a) ${Q}=3$ (b)
${Q}=4$ and (c) ${Q}=5$.  Figure 9:  Lowest 15 energy levels of the $SU(3)$
symmetric Hamiltonian of the 12 site system with open boundary condition
within $\bar{N}_{\alpha}$ sectors, where $\delta=0.5,\Delta=0.00001$. To avoid
numerical instability of the diagonalization, $\Delta=0.0$ is not used. The
black lines are for $\bar{N}_{\alpha}:(0,0,0)$, the red ones are for
$(1,-1,0)$ and $(-1,1,0)$, the orange ones are for $(0,1,-1)$ and $(0,-1,1)$,
and the blue ones are for $(-1,0,1)$ and $(1,0,-1)$. The colored lines are
degenerate with the black ones and hidden within the width of the lines. The
gray region is for $|J_{0}(t)|<|J_{e}(t)|$ where the low energy spectrum is
composed of approximately degenerate $3^{2}$ edge states of the multiplet.
This justifies that the calculation for the sector, $\bar{N}_{\alpha}=0$
(${}^{\forall}\alpha$) can be enough as for the low energy spectra.

Examples of the time dependence of $E_{\alpha\beta}(t)$ for ${Q}=3$, ${Q}=4$
and ${Q}=5$ are shown in Fig.8. Generically the lowest energies are always
given by $E_{\alpha\alpha}(t)$ if $\Delta>0$. Then the $j$-th level crossing
between $E_{\alpha\alpha}(t)$ and $E_{\beta\beta}(t)$, ($\alpha\neq\beta$), at
$t=t_{j}$, ($j=1,\cdots,Q$) occurs when

$\displaystyle\sin 2\pi(\frac{P}{Q}\alpha+\frac{t_{j}}{T})$
$\displaystyle=\sin 2\pi(\frac{P}{Q}\beta+\frac{t_{j}}{T}),$ (181)
$\displaystyle{P}\alpha$ $\displaystyle\equiv j\qquad(\text{mod}\,Q),$ (182)
$\displaystyle{P}\beta$ $\displaystyle\equiv j+1\ (\text{mod}\,Q).$ (183)

Let us write as $\alpha=\tau_{j}$ and $\beta=\tau_{j+1}$ where the Diophantine
equation due to TKNN [2, 124] is

$\displaystyle j$ $\displaystyle=\tau_{j}P+s_{j}Q,$ (184)

where $\tau_{j},s_{j}\in\mathbb{Z}$ is used in modulo $Q$.

It implies that the level crossing from the energy $E_{\tau_{j},\tau_{j}}$ to
$E_{\tau_{j+1},\tau_{j+1}}$ occurs at $t_{j}$ (see Fig.8) 999
$\displaystyle\sin 2\pi(\frac{j}{Q}+\frac{t_{j}}{T})-\sin
2\pi(\frac{j+1}{Q}+\frac{t_{j}}{T})$ $\displaystyle=2\sin 2\pi\frac{1}{2Q}\cos
2\pi(\frac{j+\frac{1}{2}}{Q}+\frac{t_{j}}{T})=0.$ It implies
$\frac{1}{4}\text{ or }\frac{3}{4}=\frac{j+\frac{1}{2}}{Q}+\frac{t_{j}}{T}$.
Here we choose $\frac{3}{4}$ since $\frac{1}{4}$ gives the level crossing at
the positive energies.

$\displaystyle\frac{t_{j}}{T}$
$\displaystyle=\frac{3}{4}-\frac{j+\frac{1}{2}}{Q},\ \text{mod}\,1.$ (185)

Let us discuss the shift, $t_{0}$, dependence of the low energy spectrum of
the Hamiltonians with open boundary condition. The low energy spectrum without
symmetry breaking perturbation for ${Q}=3$ is shown in Fig.9. It shows
(approximate) ${Q}^{2}=9$ fold degeneracy due to edge states when the coupling
at the boundary is weak as specified in the period by Eq.(180). Generically
the degeneracy is ${Q}^{2}$. Assuming the emergent ${Q}\times{Q}$ symmetry of
the infinite chain with boundaries, this degeneracy is lifted by the symmetry
breaking perturbation $H_{B}$. The low energy spectra for ${Q}=3$ and ${Q}=5$
are shown in Fig.10 and Fig.11.

As the results indicate, the hybridization of the edge states at both
boundaries is negligibly small. Then approximate level crossings in the
figures are identified only by the spectrum of the symmetry breaking
Hamiltonian $H_{B}$ between the edge states, which are explicitly shown in
Fig.10 and Fig.11. For example, in Fig.10(a), there is a three fold
(approximate) level crossing at $t=t_{3}$. This should be compared with the
level crossing in Fig.8(a) at $t=t_{3}$. As for the spectrum of $H_{B}$, the
three fold degeneracy is given by the change of the ground state from the
state with the energy $E_{11}$ (red) to that with $E_{33}$ (blue). At the
level crossing, some other states with energies $E_{\alpha\beta}$
($\alpha\neq\beta$) also pass through the level crossing. Correspondingly, in
Fig.10(a), the ground state is given by the state of the sector
$N_{\alpha}:(0,0,0)$ and the else $(-1,1,0)$ is passing through. The behavior
of the ground state energy is cusp like. However, it should not be a rigorous
level crossing due to the the mixture of the edge states at both ends. It
induces tiny (exponentially small as a function of the system size, $L$) level
repulsion. This level repulsion vanishes by taking $L\to\infty$. This is
negligibly small for the present parameter in Fig.10. The emergent symmetry
$Z^{\rm left}_{Q}\times Z^{\rm right}_{Q}$ ($L\to\infty$) protects this
(asymptotic) level crossing. Just before the level crossing ($t=t_{3}-)$, the
edge state due to $\alpha=1$ near $j=1$ and $\alpha=1$ near $j=L$ are the
ground state. The edge states at both ends contribute to $P_{1}$ by $-1$ in
pair (See Table 6). Note that there is another contribution due to bulk as
well. After the crossing ($t=t_{3}+)$, the edge state changes to the one due
to $\alpha=3$ near $j=1$ and $\alpha=3$ near $j=L$. As for the CoM, $P_{1}$,
contribution from the edge state with energy $E_{11}$ vanishes at $t=t_{3}$.
The bulk contribution remains the same (since it is continuous in time $t$) at
$t=t_{3}$. Then it implies $\Delta P_{1}=+1$. The similar consideration
implies $\Delta P_{3}=-1$. In the present case, there is no further level
crossing in Fig.10 (a). Then according to Eq.(135), $I^{1}=+1$, $I^{2}=0,$ and
$I^{3}=-1$. Assuming this emergent symmetry and the level crossings, each jump
of the CoM $\Delta P_{\alpha}$ is identified for each level crossing as shown
in the caption of Fig.10 and Fig.11 supplemented with the sum of the
discontinuities $I^{\alpha}$. In the next section, CoM’s for ${Q}=3$ case is
directly calculated by using the DMRG calculation. Also direct calculation of
the Chern number $C^{\alpha}$ are compared in the following section. It
enables us to confirm the bulk-edge correspondence Eq.(122) and Eq.(133).

###  V.3 Explicit Chern numbers and Diophantine equation

To be simple let us first consider a system at $t_{0}=+0$ (see Fig.8). In the
pump cycle, one may see a series of the jumps in $P_{\alpha}$. As for the sum
of the discontinuities $I^{\alpha}$, a pair of the jumps except the first and
the last ones is cancelled (See Fig.10 and Fig.11). The last jump is due to
the level crossing from some state with energy $E_{\alpha\alpha}$ to the state
with $E_{{Q}{Q}}$. It results in $I^{Q}=-1$. Similarly the first jump is a
level crossing from the state with energy $E_{\alpha\alpha}$
($\alpha=\tau_{\frac{{Q}-1}{2}}$) to the state with energy $E_{\beta\beta}$
($\beta=\tau_{\frac{{Q}-1}{2}-1}$) for ${Q}$:odd and from $E_{\alpha\alpha}$
($\alpha=\tau_{\frac{{Q}}{2}}$) to $E_{\beta\beta}$
($\beta=\tau_{\frac{{Q}}{2}-1}$) for ${Q}$:even. It results in
$I^{\tau_{\frac{{Q}-1}{2}}}=+1$ for ${Q}$ (odd) and
$I^{\tau_{\frac{{Q}}{2}}}=+1$ for ${Q}$ (even).

In a similar way as for the time dependence Eq.(179), the edge states appear
at $t/T=t_{\text{ini}}/T\equiv t_{0}/T+1/4$ and vanishes
$t/T=t_{\text{fin}}/T\equiv t_{0}/T+{3}/{4}$ for each $t_{0}$. Within the
period, $[t_{\text{ini}},t_{\text{fin}}]$, the level crossings due to the edge
state $\beta=\tau_{j+1}$ to $\alpha=\tau_{j}$ occur at $t=t_{j}$, Eq.(185),
that cause the jumps in $\Delta\bar{P}^{\tau_{j}}=-1$ and
$\Delta\bar{P}^{\tau_{j+1}}=+1$ for all $j$’s that satisfy
$t_{\text{ini}}<t_{j}<t_{\text{fin}}$ (See Fig.8). Since the paired jumps
inside the period cancel with each other, the first one $t_{j_{\text{ini}}}$,
$t_{\text{ini}}<t_{j_{\text{ini}}}<t_{\text{ini}}+1/Q$ gives the sum of the
discontinuity, $I^{\tau_{j_{\text{ini}+1}}}=+1$. Similarly the last one
$t_{j_{\text{fin}}}$, $t_{\text{fin}}-1/Q<t_{j_{\text{fin}}}<t_{\text{fin}}$
gives to the sum of the discontinuity $I^{\tau_{j_{\text{fin}}}}=-1$.
Otherwise $I^{\alpha}=0$
($\alpha\notin\\{\tau_{j_{\text{ini}}+1},\tau_{j_{\text{fin}}}\\}$). The
conditions are written as 101010 The first condition is written as
$\displaystyle\frac{t_{0}}{T}+\frac{1}{4}<\frac{3}{4}-\frac{j_{\text{ini}}+\frac{1}{2}}{Q}<\frac{t_{0}}{T}+\frac{1}{4}+\frac{1}{Q}$
$\displaystyle\frac{t_{0}}{T}<\frac{1}{2}-\frac{j_{\text{ini}}+\frac{1}{2}}{Q}<\frac{t_{0}}{T}+\frac{1}{Q}$
$\displaystyle
Q\frac{t_{0}}{T}<\frac{Q}{2}-({j_{\text{ini}}+\frac{1}{2}})<Q\frac{t_{0}}{T}+1$
$\displaystyle
Q(\frac{1}{2}-\frac{t_{0}}{T})-\frac{3}{2}<j_{\text{ini}}<Q(\frac{1}{2}-\frac{t_{0}}{T})-\frac{1}{2}$
Similarly the last one is written as follows.
$\displaystyle\frac{t_{0}}{T}+\frac{3}{4}-\frac{1}{Q}<\frac{3}{4}-\frac{j_{\text{fin}}+\frac{1}{2}}{Q}<\frac{t_{0}}{T}+\frac{3}{4}$
$\displaystyle\frac{t_{0}}{T}-\frac{1}{Q}<-\frac{j_{\text{fin}}+\frac{1}{2}}{Q}<\frac{t_{0}}{T}$
$\displaystyle
Q\frac{t_{0}}{T}-1<-({j_{\text{fin}}+\frac{1}{2}})<Q\frac{t_{0}}{T}$
$\displaystyle-Q\frac{t_{0}}{T}-\frac{1}{2}<j_{\text{fin}}<-Q\frac{t_{0}}{T}+\frac{1}{2}$
Then it implies  $\displaystyle j_{\text{ini}}+1$ $\displaystyle=\lfloor
Q(\frac{1}{2}-\frac{t_{0}}{T})+\frac{1}{2}\rfloor$ $\displaystyle
j_{\text{fin}}$ $\displaystyle=\lfloor-Q\frac{t_{0}}{T}+\frac{1}{2}\rfloor.$

$\displaystyle\frac{t_{0}}{T}+\frac{1}{4}<\frac{3}{4}-\frac{j_{\text{ini}}+\frac{1}{2}}{Q}<\frac{t_{0}}{T}+\frac{1}{4}+\frac{1}{Q},$
(186)
$\displaystyle\frac{t_{0}}{T}+\frac{3}{4}-\frac{1}{Q}<\frac{3}{4}-\frac{j_{\text{fin}}+\frac{1}{2}}{Q}<\frac{t_{0}}{T}+\frac{3}{4}.$
(187)

It implies

$\displaystyle j_{\text{ini}}+1$ $\displaystyle=\lfloor
Q(\frac{1}{2}-\frac{t_{0}}{T})+\frac{1}{2}\rfloor,$ (188) $\displaystyle
j_{\text{fin}}$ $\displaystyle=\lfloor-Q\frac{t_{0}}{T}+\frac{1}{2}\rfloor,$
(189)

where $\lfloor x\rfloor$ is the largest integer less than $x$. Finally we have
with using the bulk-edge correspondence

$\displaystyle I^{\alpha}(t_{0})$
$\displaystyle=C^{\alpha}(t_{0})=\left\\{\begin{array}[]{cl}{-1}&\alpha\equiv\tau_{\lfloor
Q(1-\frac{t_{0}}{T})+\frac{1}{2}\rfloor}\ \text{mod }Q\\\
{+1}&\alpha\equiv\tau_{\lfloor
Q(\frac{1}{2}-\frac{t_{0}}{T})+\frac{1}{2}\rfloor}\ \text{mod }Q\\\
0&{\text{otherwise}}\end{array}\right.,$ (193)

It implies a series of topological transitions associated with the shift of
the dimerization $t_{0}$.

The Chern numbers of the generic path $\ell_{V_{\alpha}GV_{\beta}}$,
($1\leq\alpha<\beta\leq Q$), $C^{\ell_{V_{\alpha}GV_{\beta}}}$ is simply given
by

$\displaystyle C^{\ell_{V_{\alpha}GV_{\beta}}}$
$\displaystyle=\sum_{\gamma=\alpha}^{\beta-1}C^{\gamma}.$ (194)

##  VI Numerical evaluation of Topological numbers

In this section, extensive data for the numerical evaluation of the low energy
spectra, the CoMs, and the topological numbers (the sum of the jumps and the
Chern numbers) are shown. They are given by the DMRG and the exact
diagonalization.

###  VI.1 Low energy spectra of the finite size systems

Low energy spectra of the ${Q}=3$, $L=12$ system and the ${Q}=5$, $L=10$
system are shown in Fig.10 and Fig.11. The results for the ${Q}=3$ case are
consistently compared with the DMRG calculation shown in Sec.VI.2.

(a) $\Delta P_{1}(t_{3})=+1$ and $\Delta P_{3}(t_{3})=-1$ implies
$I^{\alpha}:(+1,0,-1)$.

(b) $\Delta P_{1}(t_{3})=+1$, $\Delta P_{3}(t_{3})=-\Delta P_{3}(t_{2})=-1$
and $\Delta P_{2}(t_{2})=-1$ implies $I^{\alpha}:(+1,-1,0)$.

(c) $\Delta P_{3}(t_{2})=+1$, and $\Delta P_{2}(t_{2})=-1$, implies
$I^{\alpha}:(0,-1,+1)$.

(d) $\Delta P_{3}(t_{2})=+1$, $\Delta P_{2}(t_{2})=-\Delta P_{2}(t_{1})=-1$
and $\Delta P_{1}(t_{1})=-1$ implies $I^{\alpha}:(-1,0,+1)$.

(e) $\Delta P_{2}(t_{1})=+1$, $\Delta P_{1}(t_{1})=-1$ implies
$I^{\alpha}:(-1,+1,0)$.

(f) $\Delta P_{2}(t_{1})=+1$, $\Delta P_{1}(t_{1})=-\Delta P_{1}(t_{3})=+1$
and $\Delta P_{3}(t_{3})=-1$ implies $I^{\alpha}:(0,+1,-1)$.

Figure 10:  $t_{0}$ dependence of the lowest 15 energy levels of the $SU(3)$
symmetric Hamiltonian of the 12 site system with open boundary condition
within $\bar{N}_{\alpha}$ sectors, where $\delta=0.5,\Delta=0.5$ and
$\Phi=1/3$ (${Q}=3$). The black lines are for $\bar{N}_{\alpha}:(0,0,0)$, the
red ones are for $(1,-1,0)$ and $(-1,1,0)$, the orange ones are for $(0,1,-1)$
and $(0,-1,1)$, and the blue ones are for $(-1,0,1)$ and $(1,0,-1)$. The gray
region is for $|J_{0}(t)|<|J_{e}(t)|$ where the low energy spectrum is
composed of the multiplet of the edge states of the dimension ${Q}^{2}$. One
expects ${Q}\times{Q}$ emergent symmetry, in the $L\to\infty$ limit, that is
responsible for the level crossings within the multiplet. The lowest eigen
state is identified by $E_{\alpha\alpha}$ assuming the emergent ${Q}\times{Q}$
symmetry for the infinite system. The dimensions of the Hilbert spaces are
$35169$ for $(0,0,0)$ and $27888$ for the else (Compare with the Fig.8). Due
to Eq.(135), $I^{\alpha}=\sum_{i}\Delta P^{\alpha}(t_{i})$.

(a) ($t_{0}=0.0+0$) $\Delta P_{2}(t_{1})=+1$, $\Delta P_{1}(t_{1})=-\Delta
P_{1}(t_{5})=-1$ and $\Delta P_{5}(t_{5})=-1$, implies
$I^{\alpha}:(0,+1,0,0,-1)$.

(b) ($t_{0}=0.2+0$) $\Delta P_{1}(t_{5})=+1$, $\Delta P_{5}(t_{5})=-\Delta
P_{5}(t_{4})=-1$ and $\Delta P_{4}(t_{4})=-1$, implies
$I^{\alpha}:(+1,0,0,-1,0)$.

(c) ($t_{0}=0.4+0$) $\Delta P_{5}(t_{4})=+1$, $\Delta P_{4}(t_{4})=-\Delta
P_{4}(t_{3})=-1$ and $\Delta P_{3}(t_{3})=-1$, implies
$I^{\alpha}:(0,0,-1,0,+1)$.

(d) ($t_{0}=0.6+0$) $\Delta P_{4}(t_{3})=+1$, $\Delta P_{3}(t_{3})=-\Delta
P_{3}(t_{2})=-1$ and $\Delta P_{2}(t_{2})=-1$, implies
$I^{\alpha}:(0,-1,0,+1,0)$.

(e) ($t_{0}=0.8+0$) $\Delta P_{3}(t_{2})=+1$, $\Delta P_{2}(t_{2})=-\Delta
P_{2}(t_{1})=-1$ and $\Delta P_{(}t_{1})=-1$, implies
$I^{\alpha}:(-1,0,+1,0,0)$.

Figure 11:  $t_{0}$ dependence of the lowest 15 energy levels of the $SU(5)$
symmetric Hamiltonian of the 10 site system with open boundary condition
within $\bar{N}_{\alpha}$ sectors, where $\delta=0.5,\Delta=0.5$ and
$\Phi=1/5$ (${Q}=5$). Only the date for $\bar{N}_{\alpha}:(0,0,0,0,0)$ are
shown, which give ground state energies. The gray region is for
$|J_{0}(t)|<|J_{e}(t)|$ where the low energy spectrum is composed of the
multiplet of the edge states of the dimension ${Q}^{2}$. One expects
${Q}\times{Q}$ emergent symmetry, in the $L\to\infty$ limit, that is
responsible for the level crossings within the multiplet. The lowest eigen
state is identified by $E_{\alpha\alpha}$ assuming the emergent ${Q}\times{Q}$
symmetry for the infinite system. The dimensions of the Hilbert spaces are
$127905$ (Compare with the Fig.8). Due to Eq.(135), $I^{\alpha}=\sum_{i}\Delta
P^{\alpha}(t_{i})$.

###  VI.2 Discontinuities of CoM by DMRG

In this section, the CoM’s as discussed are directly calculated by using the
DMRG method. The $Q=3$ fermion system can be simulated by mapping to the
following biquadratic spin model with open boundary condition (See appendix A,
Eq.(198)),

$\displaystyle H_{S}=H^{(2)}_{S}+H^{S}_{B},$ (195)

where

$\displaystyle
H^{(2)}_{S}=\sum^{L/2}_{\ell=1}\biggl{[}J_{o}(t)(\vec{S}_{2\ell-1}\cdot\vec{S}_{2\ell})^{2}+J_{e}(t)(\vec{S}_{2\ell}\cdot\vec{S}_{2\ell+1})^{2}\biggr{]},$
$\displaystyle
H^{S}_{B}(t)=-\sum_{j,\alpha}\Delta_{\alpha}(S^{\alpha}_{j})^{2}.$ (196)

Here $J_{o}=J_{i\in\text{odd}}$, $J_{e}=J_{i\in\text{even}}$ and
$\Delta_{\alpha}$ is given by Eq.(179).

We calculate the staggered quadratic spin center of mass (CoM), given by

$\displaystyle P^{s}_{1}(t)$ $\displaystyle=$
$\displaystyle\sum^{L}_{j=1}(-1)^{j}x_{j}\langle
S^{x2}_{j}\rangle{={\delta_{L}}+P_{1}},$ $\displaystyle P^{s}_{2}(t)$
$\displaystyle=$ $\displaystyle\sum^{L}_{j=1}(-1)^{j}x_{j}\langle
S^{y2}_{j}\rangle{={\delta_{L}}+P_{2}},$ $\displaystyle P^{s}_{3}(t)$
$\displaystyle=$ $\displaystyle\sum^{L}_{j=1}(-1)^{j}x_{j}\langle
S^{z2}_{j}\rangle{={\delta_{L}}+P_{3}},$ (197)

where $\delta_{L}=0\,(L:\text{odd})$, $1/2\,(L:\text{even})$, $j_{0}=(L+1)/2$
and $\langle\cdot\rangle$ means taking the expectation value for the
groundstate at the time $t$. The CoM of the spins $P^{s}_{\alpha}$,
$(\alpha=1,2,3)$ is directly related to that of fermions, $P_{\alpha}$. Note
that the factor $(-1)^{j-1}$ in Eq.(111) and the summation over $j$. In the
following, we fix the parameters as $J_{0}=-1$, $\delta J=0.5$ and
$\Delta=0.5$, same to the parameter set of the results in Fig.8-10.

To calculate the behavior of the CoM, we employ DMRG algorithm in TeNPy
package[127]. The numerical results for various $t_{0}$ are shown in Fig. 12.
We observe the behavior of the CoM in all data divides into two parts,
continuous part and jump part. The time evolution of the CoM in the continuous
part indicates the presence of the bulk current. We verify that each jump
point are identical to the level crossing points ($t_{1}$, $t_{2}$ and
$t_{3}$), expected from Fig.8 and Fig.10. The jumps of the CoM, $P_{\alpha}$,
and the sum them, $I^{\alpha}$, are summarized in Table 7. This is consistent
to the low energy spectra shown in Fig.10 (see its caption).

As for a finite size effect mentioned in the appendix G, we have shown a
concrete example, in Fig. 13. The jump of $\Delta P^{s}_{1}$ and $\Delta
P^{s}_{3}$ approach to 1 and -1 for $L\to\infty$. This agrees to the
exponential localization of the edge states as discussed.

Figure 12: The behavior of the sq-CoM’s $P^{s}_{\alpha}$ for various $t_{0}$ and components. The system size is $L=64$ with open boundary condition. The data shows $t_{0}/T=1/12$[(a)], $3/12$[(b)], $5/12$[(c)], $7/12$[(d)], $9/12$[(e)] and $11/12$[(f)]. Table 7:  Jumps of the CoM, the sum of them in Fig. 12 and values of the numerically obtained Chern numbers picked up from Fig.14(a),(b) and (c). $t_{0}$ | $(\Delta P_{1},\Delta P_{2},\Delta P_{3})|_{t=t_{1}}$ | $(\Delta P_{1},\Delta P_{2},\Delta P_{3})|_{t=t_{2}}$ | $(\Delta P_{1},\Delta P_{2},\Delta P_{3})|_{t=t_{3}}$ | $(I^{1},I^{2},I^{3})$ | $(C^{1},C^{2},C^{3}=C^{0})$
---|---|---|---|---|---
$\frac{1}{12}T$ | $(0,0,0)$ | $(0,0,0)$ | $(+1,0,-1)$ | $(+1,0,-1)$ | $(+1,0,-1)$
$\frac{3}{12}T$ | $(0,0,0)$ | $(0,-1,+1)$ | $(+1,0,-1)$ | $(+1,-1,0)$ | $(+1,-1,0)$
$\frac{5}{12}T$ | $(0,0,0)$ | $(0,-1,+1)$ | $(0,0,0)$ | $(0,-1,+1)$ | $(0,-1,+1)$
$\frac{7}{12}T$ | $(-1,+1,0)$ | $(0,-1,+1)$ | $(0,0,0)$ | $(-1,0,+1)$ | $(-1,0,+1)$
$\frac{9}{12}T$ | $(-1,+1,0)$ | $(0,0,0)$ | $(0,0,0)$ | $(-1,+1,0)$ | $(-1,+1,0)$
$\frac{11}{12}T$ | $(-1,+1,0)$ | $(0,0,0)$ | $(+1,0,-1)$ | $(0,+1,-1)$ | $(0,+1,-1)$
Figure 13: System size dependence of the jumps of CoM’s at $t=t_{3}$ and
$t_{0}/T=1/12$. Results for (a) $\Delta P^{s}_{1}$ and (b) $\Delta P^{s}_{3}$.

###  VI.3 Direct evaluation of Chern numbers

Using the bulk-edge correspondence of the topological pump, the Chern number,
that is a total pumped charge of the bulk should be the same to the
discontinuity of the CoMs, Eq.(193). We have confirmed it by a direct
evaluation of the Chern number Eq.(131) obtained by the integral of the field
strength $B_{\rm tw}$. This is done by using the Fukui-Hatsugai-Suzuki formula
[128] for ${Q}=3$, $\Phi=1/3,2/3$, ${Q}=4$, $\Phi=1/4,3/4$ and ${Q}=5$,
$\Phi=1/5,2/5,3/5,4/5$. We have plotted the Chern numbers as a function of
$t_{0}$. The results are shown in Figs.14, 15 and 16. The analytic formula
Eq.(193) is plotted by the solid lines and the numerical values obtained by
discretized integration formula by the Fukui-Hatsugai-Suzuki formula are shown
in the red circles. They agree with each other almost completely except a few
points for ${Q}=5$, $\Phi=2/5$ and $\Phi=3/5$. They are near the topological
phase transitions where any numerical calculation can be unstable.

Figure 14:  $t_{0}$ dependence of the Chern numbers. ${Q}=3,\Phi=\frac{1}{3}$
and $\frac{2}{3}$. $J_{0}=-1.0,\delta=0.5$ and $\Delta=0.5$. The dimension of
the Hilbert space is 4653 ($L=10$). The data points are for
$t_{0}=(i-0.5)/51,i=1,\cdots,51$. Red points are numerical evaluation of
Eq.(131) by Fukui-Hatsugai-Suzuki formula [128] and the lines are analytical
results, Eq.(193).

Figure 15:  $t_{0}$ dependence of the Chern numbers. ${Q}=4,\Phi=\frac{1}{4}$
and $\frac{3}{4}$. $J_{0}=-1.0,\delta=0.5$ and $\Delta=0.5$. The dimension of
the Hilbert space is 2716 ($L=8$). The data points are for
$t_{0}=(i-0.5)/51,i=1,\cdots,51$. Red points are numerical evaluation of
Eq.(131) by Fukui-Hatsugai-Suzuki formula [128] and the lines are analytical
results, Eq.(193).

Figure 16:  $t_{0}$ dependence of the Chern numbers. ${Q}=5,\Phi=\frac{1}{5}$
and $\frac{2}{5}$. $J_{0}=-1.0,\delta=0.5$ and $\Delta=0.5$. The dimension of
the Hilbert space is 545 ($L=6$). The data points are for
$t_{0}=(i-0.5)/51,i=1,\cdots,51$. Red points are numerical evaluation of
Eq.(131) by Fukui-Hatsugai-Suzuki formula [128] and the lines are analytical
results, Eq.(193).

Figure 17:  $t_{0}$ dependence of the Chern numbers. ${Q}=5,\Phi=\frac{3}{5}$
and $\frac{4}{5}$ $J_{0}=-1.0,\delta=0.5$ and $\Delta=0.5$. The dimension of
the Hilbert space is 545 ($L=6$). The data points are for
$t_{0}=(i-0.5)/51,i=1,\cdots,51$. Red points are numerical evaluation of
Eq.(131) by Fukui-Hatsugai-Suzuki formula [128] and the lines are analytical
results, Eq.(193).

##  VII Summary

A topological pump of the ${SU(Q)}$ invariant quantum chain is proposed where
the gauge invariance of the colored fermions plays a central role. Introducing
a symmetry breaking perturbation, ${Q}$ Chern numbers, which are given by the
integral over the torus defined by the symmetric path in the synthetic
Brillouin zone and the time cycle, characterize the bulk pump topologically.
As for an open boundary condition, the sums of the discontinuities in ${Q}$
different center of masses, which generate the large gauge transformation,
give topological numbers of the pump as well. These discontinuities are
topological numbers due to the edge states. Relations among the
open/periodic/twisted boundary conditions are discussed in details, that
justifies the bulk-edge correspondence. Using this bulk-edge correspondence,
an explicit analytic formula for the ${Q}$ Chern numbers is given associated
with the Diophantine equation due to TKNN. The low energy spectra and the
topological quantities are numerically evaluated, that justify the consistency
of the whole discussion.

###### Acknowledgements.

The work is supported in part by KAKENHI from JSPS Grants No.17H06138 (Y.H.),
Grants No.21K13849 (Y.K.) and CREST JPMJCR19T1 (Y.H.).

##  Appendix A Color description of $S=1$ spins [80, 81]

Let us first summarize a vector representation of $S=1$ angular momentum in
relation to the rotations in 3D as

$\displaystyle R^{x}(\alpha)$ $\displaystyle=\left[\begin{array}[]{ccc}1&&\\\
&\cos\alpha&-\sin\alpha\\\
&\sin\alpha&\cos\alpha\end{array}\right]=1\oplus(\cos\alpha-i\sigma_{y}\sin\alpha)$
$\displaystyle=1\oplus e^{-i\alpha\sigma_{y}}\equiv e^{-i\alpha T^{1}},$
$\displaystyle T^{1}$ $\displaystyle=-i\left[\begin{array}[]{ccc}0&&\\\
&0&1\\\ &-1&0\end{array}\right],\ T^{1}_{ij}=-i\epsilon_{1ij}.$

Similarly

$\displaystyle R_{y}(\beta)$ $\displaystyle=e^{-i\beta T^{2}},$ $\displaystyle
R_{z}(\gamma)$ $\displaystyle=e^{-i\gamma T^{3}},$

where

$\displaystyle T^{a}_{ij}$ $\displaystyle=-i\epsilon_{aij}.$

To summarize, they are explicitly defined by

$\displaystyle T^{1}$ $\displaystyle=\left[\begin{array}[]{ccc}0&0&0\\\
0&0&-i\\\ 0&i&0\end{array}\right],T^{2}=\left[\begin{array}[]{ccc}0&0&i\\\
0&0&0\\\ -i&0&0\end{array}\right],T^{3}=\left[\begin{array}[]{ccc}0&-i&0\\\
i&0&0\\\ 0&0&0\end{array}\right].$

They are spins as

$\displaystyle({[}T^{a},T^{b}{]})_{ij}$
$\displaystyle=-\epsilon_{aik}\epsilon_{bkj}+\epsilon_{bik}\epsilon_{akj}=\epsilon_{aik}\epsilon_{bjk}-\epsilon_{bik}\epsilon_{ajk}$
$\displaystyle=\delta_{ab}\delta_{ij}-\delta_{aj}\delta_{ib}-\delta_{ba}\delta_{ij}+\delta_{bj}\delta_{ai}$
$\displaystyle=\delta_{ai}\delta_{bj}-\delta_{aj}\delta_{ib},$ $\displaystyle
i\epsilon_{abc}T^{c}_{ij}$
$\displaystyle=\epsilon_{abc}\epsilon_{cij}=\epsilon_{abc}\epsilon_{ijc}$
$\displaystyle=\delta_{ai}\delta_{bj}-\delta_{aj}\delta_{bi}.$

Thus

$\displaystyle{[}T^{a},T^{b}{]}$ $\displaystyle=i\epsilon_{abc}T^{c}.$

Further

$\displaystyle(T^{a}T^{a})_{ij}$
$\displaystyle=T^{a}_{ik}T^{a}_{kj}=-\epsilon_{aik}\epsilon_{akj}=\epsilon_{aki}\epsilon_{akj}=2\delta_{ij},$
$\displaystyle T^{a}T^{a}$ $\displaystyle=2E_{3}.$

This implies $S=1$.

Let us consider a bilinear-biquadratic Hamiltonian of $S=1$ quantum spin chain
as

$\displaystyle H$
$\displaystyle=\sum_{n}\big{[}\cos\omega_{S}(\bm{S}_{n}\cdot\bm{S}_{n+1})+\sin\omega(\bm{S}_{n}\cdot\bm{S}_{n+1})^{2}\big{]}.$

The spin 1 operators are written by color fermions,
$c^{\dagger}=(c_{+1}^{\dagger},c_{0}^{\dagger},c_{-1}^{\dagger})$, due to
Affleck as

$\displaystyle S^{a}$
$\displaystyle=c^{\dagger}T^{a}c=c_{\alpha}^{\dagger}T^{a}_{\alpha\beta}c_{\beta},\
\alpha,\beta=1,2,3$

with a constraint

$\displaystyle\sum_{\alpha}n_{\alpha}$ $\displaystyle=1,\
n_{\alpha}=c_{\alpha}^{\dagger}c_{\alpha}.$

Let us check here.

Using a useful relations 111111 $\displaystyle
T^{a}_{\alpha\beta}T^{a}_{\gamma\delta}$
$\displaystyle=-\epsilon_{a\alpha\beta}\epsilon_{a\gamma\delta}=-\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma}$
$\displaystyle(T^{a}T^{b})_{\alpha\beta}(T^{a}T^{b})_{\gamma\delta}$
$\displaystyle=\epsilon_{a\alpha\kappa}\epsilon_{b\kappa\beta}\epsilon_{a\gamma\lambda}\epsilon_{\lambda\delta}=\epsilon_{a\alpha\kappa}\epsilon_{a\gamma\lambda}\epsilon_{b\kappa\beta}\epsilon_{b\lambda\delta}$
$\displaystyle=(\delta_{\alpha\gamma}\delta_{\kappa\lambda}-\delta_{\alpha\lambda}\delta_{\gamma\kappa})(\delta_{\kappa\lambda}\delta_{\beta\delta}-\delta_{\kappa\delta}\delta_{\lambda\beta})$
$\displaystyle=(\delta_{\alpha\gamma}\delta_{\kappa\lambda}-\delta_{\alpha\lambda}\delta_{\gamma\kappa})\delta_{\kappa\lambda}\delta_{\beta\delta}$
$\displaystyle-(\delta_{\alpha\gamma}\delta_{\kappa\lambda}-\delta_{\alpha\lambda}\delta_{\gamma\kappa})\delta_{\kappa\delta}\delta_{\lambda\beta}$
$\displaystyle=(\delta_{\alpha\gamma}\delta_{\lambda\lambda}-\delta_{\alpha\lambda}\delta_{\gamma\lambda})\delta_{\beta\delta}$
$\displaystyle-(\delta_{\alpha\gamma}\delta_{\delta\lambda}-\delta_{\alpha\lambda}\delta_{\gamma\delta})\delta_{\lambda\beta}$
$\displaystyle=(3\delta_{\alpha\gamma}-\delta_{\alpha\gamma})\delta_{\beta\delta}$
$\displaystyle-\delta_{\alpha\gamma}\delta_{\delta\beta}+\delta_{\alpha\beta}\delta_{\gamma\delta}$
$\displaystyle=\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\beta}\delta_{\gamma\delta}$

$\displaystyle T^{a}_{\alpha\beta}T^{a}_{\gamma\delta}$
$\displaystyle=\delta_{\alpha\delta}\delta_{\beta\gamma}-\delta_{\alpha\gamma}\delta_{\beta\delta}$
$\displaystyle(T^{a}T^{b})_{\alpha\beta}(T^{a}T^{b})_{\gamma\delta}$
$\displaystyle=\delta_{\alpha\beta}\delta_{\gamma\delta}+\delta_{\alpha\gamma}\delta_{\beta\delta},$

and noting the constraint, we have

$\displaystyle{[}S^{a},S^{b}{]}$
$\displaystyle=c_{\alpha}^{\dagger}c_{\beta}c_{\gamma}^{\dagger}c_{\delta}({T^{a}}_{\alpha\beta}{T^{b}}_{\gamma\delta}-{T^{b}}_{\alpha\beta}{T^{a}}_{\gamma\delta})$
$\displaystyle=c_{\alpha}^{\dagger}(\delta_{\beta\gamma}-c_{\gamma}^{\dagger}c_{\beta})c_{\delta}({T^{a}}_{\alpha\beta}{T^{b}}_{\gamma\delta}-{T^{b}}_{\alpha\beta}{T^{a}}_{\gamma\delta})$
$\displaystyle=c_{\alpha}^{\dagger}c_{\delta}({T^{a}}_{\alpha\beta}{T^{b}}_{\beta\delta}-{T^{b}}_{\alpha\beta}{T^{a}}_{\beta\delta})$
$\displaystyle=c_{\alpha}^{\dagger}c_{\delta}({[}T^{a},T^{b}{]})_{\alpha\delta}=i\epsilon_{abc}S^{c}$
$\displaystyle\bm{S}^{2}$
$\displaystyle=S^{a}S^{a}=c_{\alpha}^{\dagger}c_{\beta}c_{\gamma}^{\dagger}c_{\delta}{T^{a}}_{\alpha\beta}{T^{a}}_{\gamma\delta}$
$\displaystyle=c_{\alpha}^{\dagger}(\delta_{\beta\gamma}-c_{\gamma}^{\dagger}c_{\beta})c_{\delta}{T^{a}}_{\alpha\beta}{T^{a}}_{\gamma\delta}$
$\displaystyle=c_{\alpha}^{\dagger}c_{\delta}{T^{a}}_{\alpha\beta}{T^{a}}_{\beta\delta}=c_{\alpha}^{\dagger}c_{\beta}({T^{a}}^{2})_{\alpha\beta}=2\sum_{\alpha}n_{\alpha}=2$

As for the interaction between the sites $i,j$, it is written as

$\displaystyle\bm{S}_{i}\cdot\bm{S}_{j}$
$\displaystyle=c_{i,\alpha}^{\dagger}T^{a}_{\alpha\beta}c_{i,\beta}c_{j,\gamma}^{\dagger}T^{a}_{\gamma\delta}c_{j,\delta}$
$\displaystyle=c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\gamma}^{\dagger}c_{j,\delta}T^{a}_{\alpha\beta}T^{a}_{\gamma\delta}$
$\displaystyle=c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\gamma}^{\dagger}c_{j,\delta}(\delta_{\alpha\delta}\delta_{\beta\gamma}-\delta_{\alpha\gamma}\delta_{\beta\delta})$
$\displaystyle=c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\beta}^{\dagger}c_{j,\alpha}-c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\alpha}^{\dagger}c_{j,\beta}$
$\displaystyle(\bm{S}_{i}\cdot\bm{S}_{j})^{2}$
$\displaystyle=(c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\gamma}^{\dagger}c_{j,\delta}T^{a}_{\alpha\beta}T^{a}_{\gamma\delta})(c_{i,\alpha^{\prime}}^{\dagger}c_{i,\beta^{\prime}}c_{j,\gamma^{\prime}}^{\dagger}c_{j,\delta^{\prime}}T^{b}_{\alpha^{\prime}\beta^{\prime}}T^{b}_{\gamma^{\prime}\delta^{\prime}})$
$\displaystyle=(c_{i,\alpha}^{\dagger}c_{i,\beta}c_{i,\alpha^{\prime}}^{\dagger}c_{i,\beta^{\prime}})(c_{j,\gamma}^{\dagger}c_{j,\delta}c_{j,\gamma^{\prime}}^{\dagger}c_{j,\delta^{\prime}})T^{a}_{\alpha\beta}T^{a}_{\gamma\delta}T^{b}_{\alpha^{\prime}\beta^{\prime}}T^{b}_{\gamma^{\prime}\delta^{\prime}}$
$\displaystyle=c_{i,\alpha}^{\dagger}(\delta_{\beta\alpha^{\prime}}-c_{i,\alpha^{\prime}}^{\dagger}c_{i,\beta})c_{i,\beta^{\prime}}\cdot
c_{j,\gamma}^{\dagger}(\delta_{\gamma^{\prime}\delta}-c_{j\gamma^{\prime}}^{\dagger}c_{j,\delta})c_{j,\delta^{\prime}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad
T^{a}_{\alpha\beta}T^{a}_{\gamma\delta}T^{b}_{\alpha^{\prime}\beta^{\prime}}T^{b}_{\gamma^{\prime}\delta^{\prime}}$
$\displaystyle=c_{i,\alpha}^{\dagger}c_{i,\beta^{\prime}}\cdot
c_{j,\gamma}^{\dagger}c_{j,\delta^{\prime}}\delta_{\beta\alpha^{\prime}}\delta_{\gamma^{\prime}\delta}T^{a}_{\alpha\beta}T^{a}_{\gamma\delta}T^{b}_{\alpha^{\prime}\beta^{\prime}}T^{b}_{\gamma^{\prime}\delta^{\prime}}$
$\displaystyle=c_{i,\alpha}^{\dagger}c_{i,\beta^{\prime}}c_{j,\gamma}^{\dagger}c_{j,\delta^{\prime}}T^{a}_{\alpha\beta}T^{a}_{\gamma\delta}T^{b}_{\beta\beta^{\prime}}T^{b}_{\delta\delta^{\prime}}$
$\displaystyle=c_{i,\alpha}^{\dagger}c_{i,\beta^{\prime}}c_{j,\gamma}^{\dagger}c_{j,\delta^{\prime}}(T^{a}T^{b})_{\alpha\beta^{\prime}}(T^{a}T^{b})_{\gamma\delta^{\prime}}$
$\displaystyle=c_{i,\alpha}^{\dagger}c_{i,\beta^{\prime}}c_{j,\gamma}^{\dagger}c_{j,\delta^{\prime}}(\delta_{\alpha\beta^{\prime}}\delta_{\gamma\delta^{\prime}}+\delta_{\alpha\gamma}\delta_{\beta^{\prime}\delta^{\prime}})$
$\displaystyle=c_{i,\alpha}^{\dagger}c_{i,\alpha}c_{j,\gamma}^{\dagger}c_{j,\gamma}+c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\alpha}^{\dagger}c_{j,\beta}$
$\displaystyle=c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\alpha}^{\dagger}c_{j,\beta}+1$

Note that the fermion number operators are written by the spin operators as

$\displaystyle n_{j,\alpha}$ $\displaystyle=1-(S_{j}^{\alpha})^{2}.$ (198)

Then omitting the constant, the Hamiltonian is given as

$\displaystyle H$
$\displaystyle=\sum_{i<j}\cos\omega\,c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\beta}^{\dagger}c_{j,\alpha}+(\sin\omega-\cos\omega)c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\alpha}^{\dagger}c_{j,\beta}$
$\displaystyle=\cos\omega_{S}H^{(1)}(\\{1\\})+(\sin\omega_{S}-\cos\omega_{S})H^{(2)}(\\{1\\})$

where (slightly extending the parameter space) with $J^{1,2}_{\alpha\beta}=1$
as

$\displaystyle H^{(1)}(\\{J^{(1)}_{i,\alpha;j,\beta}\\})$
$\displaystyle=\sum_{i<j}J^{(1)}_{i,\alpha
j;j,\beta}c_{i\alpha}^{\dagger}c_{i,\beta}c_{j,\beta}^{\dagger}c_{j,\alpha}$
$\displaystyle H^{(2)}(\\{J^{(2)}_{i,\alpha j;j,\beta}\\})$
$\displaystyle=\sum_{i<j}J^{(2)}_{i,\alpha
j;j,\beta}c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\alpha}^{\dagger}c_{j,\beta}$

Hermiticity implies

$\displaystyle[H^{(1)}(J^{(1)}_{i,\alpha;j,\beta})]^{\dagger}$
$\displaystyle=\sum_{i<j}(J^{(1)}_{i,\alpha;j\beta})^{*}c_{j,\alpha}^{\dagger}c_{j,\beta}c_{i,\beta}^{\dagger}c_{i,\alpha}$
$\displaystyle=\sum_{i<j}(J^{(1)}_{i,\beta;j,\alpha})^{*}c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\beta}^{\dagger}c_{j,\alpha}$
$\displaystyle J^{(1)}_{i,\alpha;j,\beta}$ $\displaystyle=J^{(1)}_{i,\beta
j,\alpha}$
$\displaystyle[H^{(2)}(\\{J^{(2)}_{i,\alpha;j,\beta}\\})]^{\dagger}$
$\displaystyle=\sum_{i<j}(J^{(2)}_{i,\alpha;j\beta})^{*}c_{j,\beta}^{\dagger}c_{j,\alpha}c_{i,\beta}^{\dagger}c_{i,\alpha}$
$\displaystyle=\sum_{i<j}(J^{(2)}_{i,\beta;j,\alpha})^{*}c_{i,\alpha}^{\dagger}c_{i,\beta}c_{j,\alpha}^{\dagger}c_{j,\beta}$
$\displaystyle J^{(2)}_{i,\alpha;j,\beta}$ $\displaystyle=J^{(2)}_{i,\beta
j,\alpha}$

It is written as a matrix form
$({\bm{J}^{(1,2)}_{ij}})^{\dagger}=\bm{J}^{(1,2)}_{ij}$ where
$\big{(}\bm{J}^{(1,2)}_{ij}\big{)}_{\alpha,\beta}=J^{(1,2)}_{i,\alpha;j,\beta}$.

##  Appendix B $SU(2)$ case

For the $N=2$ case, let us perform a particle-hole transformation for the odd
site $j$ as

$\displaystyle{\cal U}_{\Theta}$
$\displaystyle=\prod^{\leftarrow}_{j:\text{odd}}\xi_{j}=\cdots\xi_{5}\xi_{3}\xi_{1},\
\ {\cal
U}_{\Theta}^{-1}=\prod^{\rightarrow}_{j:\text{odd}}\xi_{j}=\xi_{1}\xi_{3}\xi_{5}\cdots,$
$\displaystyle{\cal U}_{\Theta}c_{j,\alpha}{\cal U}_{\Theta}^{-1}$
$\displaystyle=\left\\{\begin{array}[]{cc}c_{j,\alpha}&j:\text{even}\\\
c_{j,\alpha}^{\dagger}&j:\text{odd}\end{array}\right..$

where $\xi_{j}=c_{j}+c_{j}^{\dagger}$ and $\xi_{j}^{2}=1$. The Hamiltonian
$H^{(2)}$ ($J^{2}_{j,\alpha\beta}=J_{j}^{(2)}$) is transformed as

$\displaystyle{\cal U}_{\Theta}H^{(2)}(\\{J_{j}^{(2)}\\}){\cal U}_{\Theta}$
$\displaystyle=\sum_{j:\text{odd},\alpha\beta}J_{j}^{(2)}c_{j,\alpha}c_{j+1,\alpha}^{\dagger}c_{j+1,\beta}c_{j,\beta}^{\dagger}$
$\displaystyle+\sum_{j:\text{even},\alpha\beta}J_{j}^{(2)}c_{j,\alpha}^{\dagger}c_{j+1,\alpha}c_{j+1,\beta}^{\dagger}c_{j,\beta}$
$\displaystyle=\sum_{j:\text{odd},\alpha\beta}J_{j}^{(2)}(\delta_{\alpha\beta}c_{j+1,\alpha}^{\dagger}c_{j+1,\beta}$
$\displaystyle\ \
-c_{j,\beta}^{\dagger}c_{j+1,\alpha}^{\dagger}c_{j+1,\beta}c_{j,\alpha})$
$\displaystyle-\sum_{j:\text{even},\alpha\beta}J_{j}^{(2)}c_{j,\alpha}^{\dagger}c_{j+1,\beta}^{\dagger}c_{j+1,\alpha}c_{j,\beta}$
$\displaystyle=-H^{(1)}(\\{J_{j}^{(2)}\\})+const.$

Note that $H^{(1)}$ is the $SU(2)$ Heisenberg model as confirmed by writing
$c_{j,\alpha}=a_{\alpha}$ and $c_{j+1,\alpha}=b_{\alpha}$

$\displaystyle\sum_{\alpha\beta}$ $\displaystyle
a_{\alpha}^{\dagger}a_{\beta}b_{\beta}^{\dagger}b_{\alpha}=a_{\uparrow}^{\dagger}a_{\uparrow}b_{\uparrow}^{\dagger}b_{\uparrow}+a_{\uparrow}^{\dagger}a_{\downarrow}b_{\downarrow}^{\dagger}b_{\uparrow}+a_{\downarrow}^{\dagger}a_{\uparrow}b_{\uparrow}^{\dagger}b_{\downarrow}+a_{\downarrow}^{\dagger}a_{\downarrow}b_{\downarrow}^{\dagger}b_{\downarrow}$
$\displaystyle=n^{a}_{\uparrow}n^{b}_{\uparrow}+n^{a}_{\downarrow}n^{b}_{\downarrow}+S^{a}_{+}S^{b}_{-}+S^{a}_{-}S^{b}_{+}=2\bm{S}^{a}\cdot\bm{S}^{b}+\frac{1}{2}$

where
$4S^{a}_{z}S^{b}_{z}=(n^{a}_{\uparrow}-n^{a}_{\downarrow})(n^{b}_{\uparrow}-n^{b}_{\downarrow})=n^{a}_{\uparrow}n^{b}_{\uparrow}+n^{a}_{\downarrow}n^{b}_{\downarrow}-n^{a}_{\uparrow}n^{b}_{\downarrow}-n^{a}_{\downarrow}n^{b}_{\uparrow}=2(n^{a}_{\uparrow}n^{b}_{\uparrow}+n^{a}_{\downarrow}n^{b}_{\downarrow})-(n^{a}_{\uparrow}+n^{a}_{\downarrow})(n^{a}_{\uparrow}+n^{b}_{\downarrow})=2(n^{a}_{\uparrow}n^{b}_{\uparrow}+n^{a}_{\downarrow}n^{b}_{\downarrow})-1$
due to the constraint.

##  Appendix C Adiabatic approximation and the current

Now let us here summarize a derivation of the current $j=\langle
G|\hat{J}|G\rangle$ in the adiabatic approximation [40].

$\displaystyle\hat{J}$ $\displaystyle=$
$\displaystyle\hbar^{-1}\partial_{\theta}H(\theta).$

The many body state $|G\rangle$ is adiabatically evolved from the snap shot
ground state of a time dependent $H(t)$ as

$\displaystyle{\mathrm{i}}\hbar|\dot{G}(t)\rangle$ $\displaystyle=$
$\displaystyle H(t)|G(t)\rangle,\quad|G(0)\rangle=|g\rangle,$

where $|\alpha\rangle=|\alpha(t)\rangle$ is a orthonormalized eigen states of
the snap shot Hamiltonian and $|g\rangle$ is its ground state as

$\displaystyle H(t)|\alpha(t)\rangle$ $\displaystyle=$ $\displaystyle
E_{\alpha}(t)|\alpha(t)\rangle,\quad\langle\alpha|\beta\rangle=\delta_{\alpha\beta}.$

Writing as
$|G\rangle=e^{-({\mathrm{i}}/\hbar)\int_{0}^{t}dt^{\prime}\,E_{g}(t^{\prime})}\sum_{\alpha}|\alpha\rangle
c_{\alpha}$, $c_{g}(0)=1$, $c_{\alpha}(0)=0$, ($\alpha\neq g$), the
Schrodinger equation is written as

$\displaystyle E_{g}\sum_{\alpha}|\alpha\rangle
c_{\alpha}+{\mathrm{i}}\hbar\sum_{\alpha}(|\alpha\rangle\dot{c}_{\alpha}+|\partial_{t}\alpha\rangle
c_{\alpha})$ $\displaystyle=$
$\displaystyle\sum_{\alpha}E_{\alpha}|\alpha\rangle c_{\alpha}.$

Then multiplying $\langle g|$, and noting that $|c_{\alpha}|\ll|c_{g}|$,
$\alpha\neq g$, it reduces to $\dot{c}_{g}+c_{g}\langle
g|\partial_{t}g\rangle\approx 0$, that implies

$\displaystyle c_{g}$ $\displaystyle=$ $\displaystyle
e^{{\mathrm{i}}\gamma(t)},\quad\gamma(t)={\mathrm{i}}\int_{0}^{t}dt^{\prime}\,\langle
g|\partial_{t}g\rangle.$

Also multiplying $\langle\alpha|$, ($\alpha\neq g$), one obtains
$E_{g}c_{\alpha}+{\mathrm{i}}\hbar\langle\alpha|\partial_{t}g\rangle
c_{g}\approx E_{\alpha}c_{\alpha}$, that implies
$c_{\alpha}={\mathrm{i}}\hbar\frac{\langle\alpha|\partial_{t}g\rangle
c_{g}}{E_{\alpha}-E_{g}}$. Now the time dependent ground state is given as

$\displaystyle|G\rangle$ $\displaystyle=$ $\displaystyle
e^{-({\mathrm{i}}/\hbar)\int_{0}^{t}dt^{\prime}E_{g}(t^{\prime})}e^{{\mathrm{i}}\gamma(t)}\bigg{[}|g\rangle+{\mathrm{i}}\hbar\sum_{\alpha\neq
g}\frac{|\alpha\rangle\langle\alpha|\partial_{t}g\rangle}{E_{\alpha}-E_{g}}\bigg{]}.$

Then the expectation value of $\partial_{\theta}H$ is written as

$\displaystyle\langle G|\partial_{\theta}H|G\rangle=\langle
g|\partial_{\theta}H|g\rangle$
$\displaystyle+{\mathrm{i}}\hbar\sum_{\alpha\neq g}\frac{\langle
g|\partial_{\theta}H|\alpha\rangle\langle\alpha|\partial_{t}g\rangle-\langle\partial_{t}g|\alpha\rangle\langle\alpha|\partial_{\theta}H|g\rangle}{E_{\alpha}-E_{g}}.$

Here let us remind a general relation $\langle
g|\partial_{\theta}H|\alpha\rangle=(E_{\alpha}-E_{g})\langle
g|\partial_{\theta}\alpha\rangle$, that obeys from taking a derivative of the
eigen equation $H|\alpha\rangle=E_{\alpha}|\alpha\rangle$ as
$\langle\beta|\partial
H|\alpha\rangle+E_{\beta}\langle\beta|\partial\alpha\rangle=\partial
E_{\alpha}\langle\beta|\alpha\rangle+E_{\alpha}\langle\beta|\partial\alpha\rangle$.

$\displaystyle\delta j_{x}$ $\displaystyle=$ $\displaystyle\langle
G|J|G\rangle-\langle g|J|g\rangle,$ $\displaystyle=$
$\displaystyle{\mathrm{i}}\sum_{\alpha\neq g}\big{(}\langle
g|\partial_{\theta}\alpha\rangle\langle\alpha|\partial_{t}g\rangle+\langle\partial_{t}g|\alpha\rangle\langle\alpha|\partial_{\theta}g\rangle\big{)}$
$\displaystyle=$ $\displaystyle-{\mathrm{i}}\sum_{\alpha\neq
g}\big{(}\langle\partial_{\theta}g|\alpha\rangle\langle\alpha|\partial_{t}g\rangle-\langle\partial_{t}g|\alpha\rangle\langle\alpha|\partial_{\theta}g\rangle\big{)}$
$\displaystyle=$
$\displaystyle-{\mathrm{i}}\sum_{\alpha}\big{(}\langle\partial_{\theta}g|\alpha\rangle\langle\alpha|\partial_{t}g\rangle-\langle\partial_{t}g|\alpha\rangle\langle\alpha|\partial_{\theta}g\rangle\big{)}$
$\displaystyle=$
$\displaystyle-{\mathrm{i}}\big{(}\langle\partial_{\theta}g|\partial_{t}g\rangle-\langle\partial_{t}g|\partial_{\theta}g\rangle\big{)}=-{\mathrm{i}}B,$

where the field strength $B$ and the Berry connection $A_{\mu}$ is defined as

$\displaystyle B$ $\displaystyle=$
$\displaystyle\partial_{\theta}A_{t}-\partial_{t}A_{\theta},\ A_{\mu}=\langle
g|\partial_{\mu}g\rangle,\quad\mu=\theta,t.$

To summarize, in the adiabatic approximation, we have

$\displaystyle|G\rangle$
$\displaystyle=C\bigg{[}|g\rangle+{\mathrm{i}}\hbar\sum_{\alpha\neq
g}\frac{|\alpha\rangle\langle\alpha|\partial_{t}g\rangle}{E_{\alpha}-E_{g}}\bigg{]},$
$\displaystyle\langle G|\hat{J}|G\rangle$ $\displaystyle=\langle
g|\hat{J}|g\rangle-iB,$

where
$C=e^{-({\mathrm{i}}/\hbar)\int_{0}^{t}dt^{\prime}E_{g}(t^{\prime})}e^{{\mathrm{i}}\gamma(t)}$.

##  Appendix D Berry phase and gauge fixing[99, 100, 64, 97]

Let us start a $D$-dimensional Euclidean space
$x=(x_{1},x_{2},\cdots,x_{D})\in\mathbb{R}^{D}$ as a parameter space of the
Hamiltonian $H(x)$. As for the Berry phase associated with a loop $\ell$, we
further assume that its ground state $|g(x)\rangle$,

$\displaystyle H|g\rangle$ $\displaystyle=|g\rangle E_{g},\ \langle
g|g\rangle=1,$

is gapped along the path $\ell$

$\displaystyle H(x)|n(x)\rangle$ $\displaystyle=|n(x)\rangle E_{n}(x),\
E_{n}(x)>E_{g}(x),\ n\neq g,\ ^{\forall}\\!x\in\ell.$

Note that the phase of the snapshot eigenstate is arbitrary

$\displaystyle|g\rangle$ $\displaystyle=|g^{\prime}\rangle e^{i\Theta},\
\Theta\in\mathbb{R}$

where $H|g^{\prime}\rangle=|g^{\prime}\rangle E_{g}$ and $\langle
g^{\prime}|g^{\prime}\rangle=1$.

As is well known the Berry connection $A_{\mu}=\langle
g|\partial_{\mu}g\rangle$, $\mu=1,\cdots,D$ depends of the phase of the ground
state as

$\displaystyle A_{\mu}$ $\displaystyle=\langle
g|\partial_{\mu}g\rangle=A^{\prime}_{\mu}+i\partial_{\mu}\Theta,$

where $A^{\prime}_{\mu}=\langle g^{\prime}|\partial_{\mu}g^{\prime}\rangle$
and we assume that the parameter dependence of $|g\rangle$ and
$|g^{\prime}\rangle$ is smooth and differentiable. The Berry phase is define
as