††thanks: The two authors contributed equally to this work

# Generalized Kramers-Wanier Duality from Bilinear Phase Map

Han Yan (闫寒)  hanyan<EMAIL_ADDRESS>Institute for Solid State
Physics, The University of Tokyo. Kashiwa, Chiba 277-8581, Japan  Linhao Li
<EMAIL_ADDRESS>Institute for Solid State Physics, The University of
Tokyo. Kashiwa, Chiba 277-8581, Japan  Department of Physics and Astronomy,
Ghent University, Krijgslaan 281, S9, B-9000 Ghent, Belgium

###### Abstract

We present the Bilinear Phase Map (BPM), a concept that extends the Kramers-
Wannier (KW) transformation to investigate unconventional gapped phases, their
dualities, and phase transitions. Defined by a matrix of $\mathbb{Z}_{2}$
elements, the BPM not only encapsulates the essence of KW duality but also
enables exploration of a broader spectrum of generalized quantum phases and
dualities. By analyzing the BPM’s linear algebraic properties, we elucidate
the loss of unitarity in duality transformations and derive general non-
invertible fusion rules. Applying this framework to (1+1)D systems yields the
discovery of new dualities, shedding light on the interplay between various
Symmetry Protected Topological (SPT) and Spontaneous Symmetry Breaking (SSB)
phases. Additionally, we construct a duality web that interconnects these
phases and their transitions, offering valuable insights into relations
between different quantum phases.

Introduction —  Identifying distinct quantum phases of matter and
understanding the dualities and phase transitions between them stand as a
central challenge in quantum many-body physics. Recent decades have witnessed
the discovery of a multitude of exotic gapped phases, e.g., symmetry protected
topological (SPT) phase [1, 2, 3, 4], topological orders [5, 6, 7, 8], fracton
orders [9, 10, 11] and spontaneous symmetry breaking (SSB) phases.
Intriguingly, some gapped phases are interconnected to each other through
duality transformations, even though they exhibit vastly different physical
properties. The most well-known example is Kramers-Wannier (KW)
transformation, which relates the paramagnetic phase and ferromagnetic phases
of the transverse field Ising chain [12, 13, 12]. As these two phases have
different ground state degeneracies, the KW transformation is realized by a
nonunitary operator, which satisfies the noninvertible “Ising-category” fusion
rule [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. The loss of
unitarity can be recovered by introducing symmetry twisted boundary conditions
(TBC) [23, 26, 27, 28, 29]. Moreover, when the system is self-dual, the KW
duality becomes a anomalous noninvertible symmetry [30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 62, 63], which forbids gapped phases with a unique
ground state (that we call uniquely gapped phases for short) [64, 65, 66, 67,
68, 69, 70, 71, 72]. Thus the self-dual point must be a first-order or
continuous phase transition between the duality-related phases.

In this work, we propose a generalization of KW transformation, which is
denoted as Bilinear Phase Map (BPM). The BPM is characterized by a matrix of
$\mathbb{Z}_{2}$ numbers. Notably, the matrix not only captures all the
essential information of original KW duality in its plain linear algebra
properties, but also engenders a wider array of exotic phases and phase
transitions. For general BPMs, we present a systematic approach to address
ground state degeneracy and the loss of unitarity by considering more twisted
boundary conditions, and also derive general non-invertible fusion rules, all
by simply examining the linear algebra features of its matrix.

As an application, we construct two new BPMs in (1+1)D, denoted as
$\mathcal{N}_{\text{3-KW}}$ and $\mathcal{N}_{\text{4-KW}}$, which are
associated with three-site-interacting and four-site-interacting Ising spin
chain respectively [73, 74, 75, 76, 77, 78, 79, 80, 81]. Notably, for self-
dual systems, we prove $\mathcal{N}_{\text{4-KW}}$ is anomalous while
$\mathcal{N}_{\text{3-KW}}$ is anomaly-free, allowing an SPT phase to exist.
We also find a generalized Kennedy-Tasaki (KT) duality [82, 83, 84, 85, 86,
87, 88, 89, 90, 23, 91, 92, 93] between this SPT phase and an SSB phase. Based
on these results, we propose one web of duality in Fig. 1 connecting gapped
phases and another web in Fig. 2 between related phase transitions.

The Kramers-Wanier Duality —  We first briefly review the KW duality of the
spin-1/2 chains, as a preparation for the generalized KW duality to be
discussed in the next section.

Let us consider a closed spin-1/2 chain with $L$ sites. On each site $i$ sits
a spin-1/2 variable $s_{i}\in\\{0,1\\}$. We also consider the $\mathbb{Z}_{2}$
symmetry generated by $U=\prod_{j}X_{j}$, which flips all spins, namely
$s_{j}\to s_{j}+1$. The KW transformation is realized by gauging the
$\mathbb{Z}_{2}$ symmetry for the entire Hilbert space. On the (1+1)D lattice,
the $\mathbb{Z}_{2}$ gauge field is defined as dual spins
$\\{\hat{s}_{i-\frac{1}{2}}\\}$ on the link. Therefore, the spins
$\\{s_{i}\\}$ on the original lattice are mapped to dual spins
$\\{\hat{s}_{i-\frac{1}{2}}\\}$ under KW transformation. In addition, we use
$(-1)^{\hat{u}}$ to denote the eigenvalue of the dual symmetry
$\hat{U}:=\prod_{i=1}^{L}\hat{X}_{i-\frac{1}{2}}$, and $\hat{t}$ to denote the
boundary condition
${\hat{s}_{i-\frac{1}{2}+L}}={\hat{s}_{i-\frac{1}{2}}+\hat{t}}$.

The KW transformation is realized by an operator $\mathcal{N}$ that maps the
basis state to a state of the gauge field spins,

$\displaystyle\begin{split}&\mathcal{N}\ket{\\{s_{i}\\}}\\\
&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i+\frac{1}{2}}\\}}(-1)^{\sum_{j=1}^{L}(s_{j-1}+s_{j})\hat{s}_{j-\frac{1}{2}}+\hat{t}s_{L}}\ket{\\{\hat{s}_{i+\frac{1}{2}}\\}}.\end{split}$
(1)

The exponents in (1) are reminiscent of the minimal coupling of the gauge
fields. The boundary terms in the exponents are chosen to give the correct
mapping of symmetry-twist sectors.

The KW duality is particularly useful in understanding the physics of 1D spin
chains with the same global $\mathbb{Z}_{2}$ symmetry, such as the Hamiltonian

$\mathcal{H}=-\sum_{i,i+1}Z_{i}Z_{i+1}-h\sum_{i}X_{i}.$ (2)

KW Duality from the Bilinear Phase Map —  We now introduce the concept of
Bilinear Phase Map (BPM), which will be the core of the generalized duality.

Note that if we consider periodic boundary condition (PBC), namely
$\hat{t}=0$, Eq. (1) can be written in a more compact form after shifting
$\hat{s}_{j-\frac{1}{2}}$ to $\hat{s}_{j}$ as

$\begin{split}\mathcal{N}\ket{\\{s_{i}\\}}&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j=1}^{L}(s_{j-1}+s_{j})\hat{s}_{j-1}}\ket{\\{\hat{s}_{i}\\}}\,\\\
&\equiv\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j,k=1}^{L}s_{j}A_{jk}\hat{s}_{k}}\ket{\\{\hat{s}_{i}\\}}\end{split}$
(3)

where ${\bf\it A}$ is a $\mathbb{Z}_{2}$ valued $L\times L$ matrix,

${\bf\it A}=\left(\begin{array}[]{ccccc}1&1&0&\cdots&0\\\ 0&1&1&0&\cdots\\\
0&0&1&1&\cdots\\\ \cdots&\cdots&\cdots&\cdots&\cdots\\\
1&0&\cdots&0&1\end{array}\right).$ (4)

A crucial feature is that the rank of ${\bf\it A}^{\text{T}}$ is $L-1$, and it
has a non-trivial kernel

${\bf\it a}\equiv\ker{\bf\it A}=\ker{\bf\it
A}^{\text{T}}=\left(1,1,\cdots,1\right).$ (5)

The kernel is the root of several key properties of the KW duality. First, we
have

$\begin{split}&\mathcal{N}\ket{\\{s_{i}\\}}=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j,k=1}^{L}s_{j}A_{jk}\hat{s}_{k}}\ket{\\{\hat{s}_{i}\\}}\\\
&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j,k=1}^{L}(s_{j}+a_{j})A_{jk}\hat{s}_{k}}\ket{\\{\hat{s}_{i}\\}}\\\
&=\mathcal{N}\ket{\\{s_{i}+1\\}}.\end{split}$ (6)

That is, the duality mapping does not distinguish $\ket{\\{s_{i}\\}}$ and
$\ket{\\{s_{i}+1\\}}$. Both states are mapped to the same state of the dual
spins $\hat{s}_{i}$. Therefore, if we start from a system with an SSB phase
with two ground states $\ket{\\{s_{i}\\}}$ and $\ket{\\{s_{i}+1\\}}$, the KW
duality will map them to the same state. For the same reason, the KW duality
maps the states in the odd sector of the $\mathbb{Z}_{2}$ symmetry (i.e.,
$\ket{\\{s_{i}\\}}-\ket{\\{s_{i}+1\\}}$) to zero. Hence, the kernel leads to a
loss of unitarity of the KW duality.

Similarly, since we also have

$\begin{split}&\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j,k=1}^{L}s_{j}A_{jk}\hat{s}_{k}}\ket{\\{\hat{s}_{i}\\}}\\\
&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j,k=1}^{L}s_{j}A_{jk}\hat{s}_{k}}\ket{\\{\hat{s}_{i}+1\\}}.\end{split}$
(7)

That is, the state mapped to by the KW duality is always in the even sector of
dual $\mathbb{Z}_{2}$ operation defined by the kernel
$\hat{s}_{i}\to\hat{s}_{i}+a_{i}$.

Another feature associated with the kernel is the collection of states that
are mapped into the paramagnetic state
$\ket{\\{\hat{s}_{i}\\}}=\ket{\rightarrow\rightarrow\dots}$. To start, by
definition of the mapping, it is always true that $\ket{{\bf\it
0}}\equiv\ket{\\{s_{i}=0\\}}$ is mapped into
$\ket{\rightarrow\rightarrow\dots}$ because $\sum_{j,k=1}^{L}s_{j}A_{jk}=0$.
It then follows that $\ket{\\{0+a_{i}\\}}=\ket{\\{1,1,\dots\\}}$ satisfies the
same condition, and is mapped to the paramagnetic state.

Finally, the boundary condition terms $s_{L}\hat{t}$ can be understood in the
context of the phase map too. It is simply making the replacement

$\sum_{j,k=1}^{L}s_{j}A_{jk}\hat{s}_{k}\longrightarrow\sum_{j,k=1}^{L}s_{j}A_{jk}\hat{s}_{k}+{\bf\it
s}\cdot\hat{{\bf\it t}},$ (8)

where $\hat{{\bf\it t}}=(0,\dots,0,1)$ for the twisted boundary condition. The
reason why this term works is that it distinguishes the $\mathbb{Z}_{2}$ dual
two states $\ket{\\{s_{i}\\}}$ and $\ket{\\{s_{i}+a_{i}\\}}$, i.e.,
$\hat{{\bf\it t}}\cdot{{\bf\it s}}\neq\hat{{\bf\it t}}\cdot({{\bf\it
s}}+{{\bf\it a}})$, so the problem of ${\bf\it A}$ being rank $L-1$ and hence
the map being non-unitary is resolved. Based on this, we can actually
introduce other general $\hat{{\bf\it t}}$ with odd number of element $1$ that
achieves the same purpose. Physically, such $\hat{{\bf\it t}}$’s correspond to
twisting the spins odd times on the chain. That is, the kernel defines the
twisted boundary condition that recovers the unitarity of KW duality.

Generalized Bilinear Phase Map —  We now turn to 1D systems with other types
of global symmetries — here “global” is defined as the symmetry operation
grows linearly with the system size. One such example is the symmetry of
flipping the even or odd spins only on the spin chain. While it is possible to
construct generalized KW duality for such models, it is not straightforward to
see its properties such as the loss of unitarity and sectors of different
boundary conditions.

This is exactly the problem solved by the Bilinear Phase Map construction: for
each generalized KW duality, one simply needs to analyze its corresponding
matrix ${\bf\it A}$ to straightforwardly derive these properties of the
generalized duality.

Table 1:  Property of generalized KW duality from Bilinear phase map Generalized KW duality | Bilinear Phase Map ${\bf\it A}$
---|---
non-unitary | rank-deficient
underlying global symmetry | kernel of ${\bf\it A}^{T}$
two states map to the same state | two states’ difference is the kernel of ${\bf\it A}^{T}$
states map to paramagnet state | zero state adding kernel of ${\bf\it A}^{T}$
boundary terms recovering unitarity | linear terms differentiating kernels of ${\bf\it A}^{T}$

We consider a generalized KW duality map under PBC described by BPM as

$\begin{split}\mathcal{N}_{\text{BPM}}\ket{\\{s_{i}\\}}&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j,k=1}^{L}s_{j}A_{jk}\hat{s}_{k}}\ket{\\{\hat{s}_{i}\\}},\\\
\mathcal{N}^{\dagger}_{\text{BPM}}\ket{\\{\hat{s}_{i}\\}}&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{s_{i}\\}}(-1)^{\sum_{j,k=1}^{L}s_{j}A_{jk}\hat{s}_{k}}\ket{\\{s_{i}\\}},\end{split}$
(9)

where ${\bf\it A}$ can be any $L\times L$ matrix defined on a ring with $L$
sites and each element is $\mathbb{Z}_{2}$ value, namely $A_{jk}=0,1$. In
practice, we are more interested in ${\bf\it A}$ with reasonable properties
such as translational invariance and locality. The properties of the
corresponding BPM can be then easily read out from linear algebra of ${\bf\it
A}$, as summarized in the Table 1.

Suppose the matrix ${\bf\it A}^{\text{T}}$ have $N$ linearly independent
kernel vectors:

${\bf\it b}^{m}\in\ker A^{\text{T}}~{},m=1,\cdots,N.$ (10)

Then the BPM duality mapping does not distinguish state $\ket{\\{s_{i}\\}}$
and $\ket{\\{s_{i}+b^{m}_{i}\\}}$ due to

$\begin{split}&\mathcal{N}_{\text{BPM}}\ket{\\{s_{i}\\}}=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j,k=1}^{L}s_{j}A_{jk}\hat{s}_{k}}\ket{\\{\hat{s}_{i}\\}}\\\
&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j,k=1}^{L}(s_{j}+b^{m}_{j})A_{jk}\hat{s}_{k}}\ket{\\{\hat{s}_{i}\\}}\\\
&=\mathcal{N}_{\text{BPM}}\ket{\\{s_{i}+b^{m}_{i}\\}},\forall
m=1,\cdots,N,\end{split}$ (11)

which shows explicitly that the mapping is not unitary. Moreover, the group
$G$ generated by $U_{m}:\\{s_{i}\\}\to\\{s_{i}+b^{m}_{i}\\}$ will be mapped to
the identity group acting on the dual spins under BPM.

This motivates us to consider the Hamiltonian invariant under $G$ since the
Hamiltonian for $\hat{s}_{i}$ after BPM naturally commutes with identity. In
particular, if the $\\{s_{i}\\}$-system is in $G$-SSB phase with $2^{N}$
degenerate ground state (these states are $\ket{{\bf\it 0}}$, $\ket{{\bf\it
0}+{\bf\it b}^{1}}$, $\ket{{\bf\it 0}+{\bf\it b}^{2}}$, $\ket{{\bf\it
0}+{\bf\it b}^{1}+{\bf\it b}^{2}}$ , …), the dual system is in the trivial or
SPT phase with a unique ground state.

Recovering the unitarity can be achieved by additional terms
${\bf\it\hat{t}}\cdot{\bf\it s}$ to the BPM to distinguish the kernel states
$\ket{\\{s_{i}={b}^{m}_{i}\\}}$ and the state $\ket{\\{s_{i}=0\\}}$ (several
terms with different ${\bf\it\hat{t}}$’s may be needed if there are several
kernels).

Finally, the fusion rule of BPM and its conjugation can be directly computed
by

$\begin{split}&\mathcal{N}^{\dagger}_{\text{BPM}}\mathcal{N}_{\text{BPM}}\ket{\\{s_{i}\\}}\\\
&=\prod^{N}_{m=1}(1+U_{m})\ket{s_{i}}=\sum_{g\in G}g\ket{s_{i}}.\end{split}$
(12)

In particular, when the matrices ${\bf\it A}$ and ${\bf\it A}^{T}$ are related
by translation over $n$ sites

$\begin{split}A_{j,k}=A_{k,j+n},\end{split}$ (13)

we can further calculate the fusion of two BPMs:

$\begin{split}&\mathcal{N}_{\text{BPM}}\mathcal{N}_{\text{BPM}}\ket{\\{s_{i}\\}}=T^{n}(\sum_{g\in
G}g)\ket{s_{i}}.\end{split}$ (14)

where $T$ is the translation operator:
$T\ket{\\{s_{i}\\}}=\ket{\\{s^{\prime}_{i}=s_{i-1}\\}}$ [94, 26].

Examples of Generalized KW duality —  As the first example of generalized BPM
and the KW duality, we consider the following three-site interacting Ising
chain of chain length $L\in 3\mathbb{Z}$,

$\begin{split}H_{\text{3-Ising}}=-\sum^{L}_{i=1}(hX_{i}+Z_{i}Z_{i+1}Z_{i+2}).\end{split}$
(15)

When $h=\infty$, this system is in a trivially gapped phase with a
paramagnetic ground state $\ket{\rightarrow\rightarrow\cdots}$. When $h=0$,
this system has SSB ground states that we need to understand.

Similar to the usual two-site Ising model with the transverse field, these two
phases can be related by a generalized KW duality [73], under PBC given by

$\begin{split}\mathcal{N}_{\text{3-KW}}\ket{\\{s_{i}\\}}&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j=1}^{L}s_{j}(\hat{s}_{j}+\hat{s}_{j+1}+\hat{s}_{j+2})}\ket{\\{\hat{s}_{i}\\}}\\\
&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j=1}^{L}\hat{s}_{j}(s_{j-2}+s_{j-1}+s_{j})}\ket{\\{\hat{s}_{i}\\}}.\end{split}$
(16)

Its BPM matrix is

${\bf\it A}_{\text{3-KW}}=\left(\begin{array}[]{cccccc}1&1&1&0&\cdots&0\\\
0&1&1&1&\cdots&0\\\ 0&0&1&1&\cdots&0\\\ \cdots&\cdots&\cdots&\cdots&\cdots\\\
1&0&\cdots&0&1&1\\\ 1&1&0&\cdots&0&1\\\ \end{array}\right).$ (17)

Then the properties of symmetry and SSB ground state degeneracy can be
directly derived from the expression of ${\bf\it A}_{\text{3-KW}}$. The
associated ${\bf\it A}_{\text{3-KW}}^{T}$ matrix has a two dimension kernel
space generated by

$\begin{split}&{\bf\it b}^{1}=(1,1,0,1,1,0,\cdots),\\\ &{\bf\it
b}^{2}=(0,1,1,0,1,1,\cdots),\end{split}$ (18)

which shows that the system has a $(\mathbb{Z}_{2})^{2}$ symmetry:

$U_{A}\ket{\\{s_{i}\\}}=\ket{\\{s_{i}+b^{1}_{i}\\}},\;U_{G}\ket{\\{s_{i}\\}}=\ket{\\{s_{i}+b^{2}_{i}\\}}.$
(19)

In operator form, they are written as

$U_{A}=\prod^{L/3}_{i=1}X_{3i+1}X_{3i+2},\quad
U_{G}=\prod^{L/3}_{i=1}X_{3i+2}X_{3i+3},$ (20)

which satisfies the algebra with translation
$TU_{A}T^{-1}=U_{G},T^{2}U_{G}T^{-2}=U_{A}$. The states can be organized into
eigenstates of $U_{A/G}$ with eigenvalue $(-1)^{u_{A/G}}=\pm 1$, i.e.
$u_{A/G}=0,1$.

From the all spin-up state, this $(\mathbb{Z}_{2})^{2}$ symmetry can generate
all ground states of SSB phases as $\ket{{\bf\it 0}}$, $\ket{{\bf\it
0}+{\bf\it b}^{1}}$, $\ket{{\bf\it 0}+{\bf\it b}^{2}}$, $\ket{{\bf\it
0}+{\bf\it b}^{1}+{\bf\it b}^{2}}$:

$\begin{split}&\ket{\text{GS}}_{1}=\ket{\uparrow\uparrow\uparrow\cdots},\quad\ket{\text{GS}}_{2}=\ket{\downarrow\downarrow\uparrow\cdots},\\\
&\ket{\text{GS}}_{3}=\ket{\uparrow\downarrow\downarrow\cdots},\quad\ket{\text{GS}}_{4}=\ket{\downarrow\uparrow\downarrow\cdots},\end{split}$
(21)

and all these ground states are mapped to
$|\rightarrow\rightarrow\cdots\rangle$ by $N_{\text{3-KW}}$ under PBC.

Moreover, one can directly check this BPM induces the following transformation
of Pauli operators

$\begin{split}&\mathcal{N}_{\text{3-KW}}X_{i}=\hat{Z}_{i}\hat{Z}_{i+1}\hat{Z}_{i+2}\mathcal{N}_{\text{3-KW}},\\\
&\mathcal{N}_{\text{3-KW}}Z_{i-2}Z_{i-1}Z_{i}=\hat{X}_{i}\mathcal{N}_{\text{3-KW}}.\end{split}$
(22)

and map the transverse field $h$ in eq. (15) to $1/h$. Thus the dual model
also has a $(\mathbb{Z}_{2})^{2}$ symmetry generated by

$\hat{U}_{A}=\prod^{L/3}_{i=1}\hat{X}_{3i+1}\hat{X}_{3i+2},\quad\hat{U}_{G}=\prod^{L/3}_{i=1}\hat{X}_{3i+2}\hat{X}_{3i+3}.$
(23)

Likewise, the dual Hilbert space can also be organized into four symmetry
sectors labeled by $(\hat{u}_{A},\hat{u}_{G})\in\\{0,1\\}^{2}$. By acting the
products of operators on a general state, we further find the fusion rules
under PBC 111Unlike the conventional KW duality, the translation and its
square, in this case, are not reduced to identity in the low energy limit, as
they satisfy the nontrivial algebra with $(\mathbb{Z}_{2})^{2}$ symmetry
operators.:

$\begin{split}&\mathcal{N}_{\text{3-KW}}\times
U_{A/G}=\mathcal{N}_{\text{3-KW}},\quad\hat{U}_{A/G}\times\mathcal{N}_{\text{3-KW}}=\mathcal{N}_{\text{3-KW}},\\\
&\mathcal{N}^{\dagger}_{\text{3-KW}}\times\mathcal{N}_{\text{3-KW}}=(1+U_{A})(1+U_{G}),\\\
&\mathcal{N}_{\text{3-KW}}\times\mathcal{N}_{\text{3-KW}}=(1+U_{A})(1+U_{G})T^{2}.\end{split}$
(24)

Now, let us discuss the unitarity problem of BPM by introducing boundary spins
$(\hat{t}_{A},\hat{t}_{G})\in\\{0,1\\}^{2}$ in $\\{\hat{s}_{i}\\}$-system,
which corresponds to the untwisted/twisted boundary conditions of
$(\mathbb{Z}_{2})^{2}$ symmetry [96, 97, 97, 98, 99]:

$\begin{split}&\hat{s}_{L+3k+1}=\hat{s}_{3k+1}+\hat{t}_{A},\hat{s}_{L+3k+2}=\hat{s}_{3k+2}+(\hat{t}_{A}+\hat{t}_{G}),\\\
&\hat{s}_{L+3k}=\hat{s}_{3k}+\hat{t}_{G}.\end{split}$ (25)

Then we can modify the BPM as follows:

$\begin{split}&\mathcal{N}_{\text{3-KW}}\ket{\\{s_{i}\\}}\\\
&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j=1}^{L}\hat{s}_{j}(s_{j-2}+s_{j-1}+s_{j})+\hat{t}_{G}s_{L}+\hat{t}_{A}s_{L-1}}\ket{\\{\hat{s}_{i}\\}}.\end{split}$
(26)

This modified BPM can distinguish four SSB ground states, satisfying that
$s_{j-2}+s_{j-1}+s_{j}=0$. The BPM maps them to the same state

$\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{s_{L-1}\hat{t}_{A}+s_{L}\hat{t}_{G}}\ket{\\{\hat{s}_{i}\\}},$
(27)

that is the paramagnetic state $|\rightarrow\rightarrow\cdots\rangle$ with a
phase $(-1)^{\hat{t}_{A}s_{L-1}+\hat{t}_{G}s_{L}}$. When
$\hat{t}_{A}=\hat{t}_{G}=0$, this phase is trivial and only linear combination
$\sum^{4}_{i=1}\ket{\text{GS}}_{i}$ with $u_{1}=u_{2}=0$ survives. But when
$\hat{t}_{A}=1$ and $\hat{t}_{G}=0$, two ground states with $s_{1}=1$ will
have additional $-1$ sign after mapping. Then only linear combination
$(\ket{\text{GS}}_{1}+\ket{\text{GS}}_{4}-\ket{\text{GS}}_{2}-\ket{\text{GS}}_{3})$
survives under BPM duality. This combination has symmetry charge
$u_{A}=u_{G}=1$. On the other hand, when $\hat{t}_{A}=0$ and $\hat{t}_{G}=1$,
two ground states with $s_{L}=1$ will have additional $-1$ sign after mapping.
Only linear combination
$(\ket{\text{GS}}_{1}-\ket{\text{GS}}_{4}+\ket{\text{GS}}_{2}-\ket{\text{GS}}_{3})$
survives under BPM, which has symmetry charge $u_{A}=0,u_{G}=1$. This
statement above is also consistent with symmetry-twisted sector mapping in the
appendix.

Generalized duality triangle —  The self-dual point at $h=1$ is a continuous
phase transition belonging to four-state Potts university class with center
charge $c=1$ [81], where the duality transformation becomes an emergent
symmetry. However, unlike the usual KW duality (1) symmetry which is
anomalous, the BPM (16) is anomaly free, namely it allows the self-dual unique
gapped phases, e.g., the $\mathbb{Z}^{A}_{2}\times\mathbb{Z}^{G}_{2}$ SPT
phase. A solvable Hamiltonian is given by

$H_{\text{SPT}}=-\sum^{L}_{i=1}a_{i},\quad a_{i}=(-1)^{i}Z_{i-1}Y_{i}Z_{i+1}.$
(28)

Such SPT Hamiltonian can be constructed by decorated domain wall (DW) method
[100, 101, 102]. As one can check, the product of two nearest neighbored terms
is $Z_{i-1}X_{i}X_{i+1}Z_{i+2}$, which comes from decorating the domain wall
term $Z_{i-1}Z_{i+2}$ with charge operator $X_{i}X_{i+1}$. For example, if we
assume $i=1\text{(mod 3)}$, the $Z_{i-1}Z_{i+2}$ is a domain wall term of
$U_{G}$ and the charge operator $X_{i}X_{i+1}$ is associated with $U_{A}$
222More precisely, the ground state is an eigenstate of
$Z_{i-1}X_{i}X_{i+1}Z_{i+2}$ with eigenvalue 1 and thus has the SPT feature.
The reason why not choosing $H=-\sum^{L}_{i=1}Z_{i-1}X_{i}X_{i+1}Z_{i+2}$ is
that this Hamiltonian has an emergent symmetry $\prod_{i}X_{i}$ and is in the
corresponding SSB phase.. Such construction can be implemented by a unitary
transformation $U_{\text{3-DW}}$, which can map the SPT Hamiltonian to
trivially gapped Hamiltonian:

$U^{\dagger}_{\text{3-DW}}H_{\text{SPT}}U_{\text{3-DW}}=-\sum^{L}_{i=1}X_{i}\equiv
H_{\text{triv}}.$ (29)

The detail of the SPT phase is shown in the appendix.

Now, since there are SPT, trivially gapped and SSB phases, we can construct a
web of duality connecting them, which is summarized in Fig. 1.

Figure 1:  Three gapped phases with $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
symmetry and the dualities between them.

Here $U_{\text{3-KT}}$ is a generalized Kennedy-Tasaki transformation [23,
91]:

$U_{\text{3-KT}}=U_{\text{3-DW}}U_{\text{3-KW}}U^{\dagger}_{\text{3-DW}}.$
(30)

Moreover, such a web of duality can also connect phase transitions between two
different gapped phases, as shown in Fig. 2. These duality-related three phase
transitions have the same center charge $c=1$.

Figure 2: Three phase transitions between two different gapped phases with
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry and $c=1$ and the dualities
between them.

Lastly, we have constructed another BPM duality example that connects $H=-\sum
X_{i}$ and $H=-\sum Z_{i}Z_{i+1}Z_{i+2}Z_{i+3}$ in the appendix.
Interestingly, on self-dual points, this duality becomes an anomalous
symmetry, which guarantees the self-dual theory must be either a first-order
or continuous phase transition.

Summary and Discussion —  In this paper, we present the Bilinear Phase Map
(BPM), a handy tool in understanding quantum phase transitions and exotic
gapped phases. Our approach, expands the Kramers-Wannier (KW) transformation,
and explores a broader spectrum of quantum phases, addressing the challenge of
unitarity loss in duality transformations. Our analysis leads to the
derivation of general non-invertible fusion rules and the discovery of new
BPMs in (1+1)D systems, which uncover intricate relationships between SPT and
SSB phases. Looking forward, this work opens up several intriguing questions
and potential research directions. For example, we plan to extend this
analysis to from $\mathbb{Z}_{2}$ to $\mathbb{Z}_{N}$ BPMs in a future work.
It also paves the way for exploring the applicability of BPM in higher-
dimensional systems and its implications in symmetries of quantum systems.
Additionally, the anomaly characteristics of BPMs present a fertile ground for
further theoretical exploration, potentially leading to the discovery of new
quantum phases and transitions.

###### Acknowledgements.

We thank Yunqin Zheng, Yuan Miao, Xiao Wang, Masaki Oshikawa, and Weiguang Cao
for the helpful discussions.

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##  Appendix A Symmetry-twisting mapping of three-site BPM duality

In this appendix, we will derive the symmetry-twist sectors of BPM
$\mathcal{N}_{\text{3-KW}}$. Similar to the boundary spins in
$\\{\hat{s}_{i}\\}$-system, we also introduce boundary spins
$(t_{1},t_{2})\in\\{0,1\\}^{2}$ in $\\{s_{i}\\}$-system:

$\begin{split}&s_{L+3k+1}=s_{3k+1}+t_{A},\quad
s_{L+3k+2}=s_{3k+2}+(t_{A}+t_{G}),\quad s_{L+3k}=s_{3k}+t_{G}.\end{split}$
(31)

Then we find a consistent modified expression of $\mathcal{N}_{\text{3-KW}}$
in $\\{s_{i}\\}$ and $\\{\hat{s}_{i}\\}$ systems:

$\begin{split}\mathcal{N}_{\text{3-KW}}\ket{\\{s_{i}\\}}&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j=1}^{L}s_{j}(\hat{s}_{j}+\hat{s}_{j+1}+\hat{s}_{j+2})+t_{A}\hat{s}_{1}+t_{G}\hat{s}_{2}}\ket{\\{\hat{s}_{i}\\}}\\\
&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j=1}^{L}\hat{s}_{j}(s_{j-2}+s_{j-1}+s_{j})+\hat{t}_{G}s_{L}+\hat{t}_{A}s_{L-1}}\ket{\\{\hat{s}_{i}\\}}.\end{split}$
(32)

From this formula, it is straightforward to check the symmetry-twist mapping:

$\begin{split}[(\hat{u}_{A},\hat{t}_{A}),(\hat{u}_{G},\hat{t}_{G})]=[(t_{A}+t_{G},u_{A}),(t_{G},u_{A}+u_{G})].\end{split}$
(33)

Let us first acts $\hat{U}_{A}\times\mathcal{N}_{\text{3-KW}}$ and
$\hat{U}_{G}\times\mathcal{N}_{\text{3-KW}}$ on the state $\ket{\\{s_{i}\\}}$:

$\begin{split}&\hat{U}_{A}\mathcal{N}_{\text{3-KW}}\ket{\\{s_{i}\\}}=(-1)^{t_{A}+t_{G}}\mathcal{N}_{\text{3-KW}}\ket{\\{s_{i}\\}},\\\
&\hat{U}_{G}\mathcal{N}_{\text{3-KW}}\ket{\\{s_{i}\\}}=(-1)^{t_{G}}\mathcal{N}_{\text{3-KW}}\ket{\\{s_{i}\\}}.\end{split}$
(34)

This holds for any state $N_{\text{3-KW}}|\psi\rangle$, where $|\psi\rangle$
is a general state in $\\{s_{i}\\}$ system
$\ket{\psi}=\sum_{\\{s_{i}\\}}\psi_{\\{s_{i}\\}}\ket{\\{s_{i}\\}}$. Thus any
state obtained by acting $N_{\text{3-KW}}$ must be eigenstate of $\hat{U}_{A}$
and $\hat{U}_{G}$ with eigenvalue
$(\hat{u}_{A},\hat{u}_{G})=(t_{A}+t_{G},t_{G})$.

Next, we continue to consider $\mathcal{N}_{\text{3-KW}}\times U_{1}$ and
$\mathcal{N}_{\text{3-KW}}\times U_{G}$:

$\begin{split}&\mathcal{N}_{\text{3-KW}}U_{A}\ket{\\{s_{i}\\}}=(-1)^{\hat{t}_{A}}\mathcal{N}_{\text{3-KW}}\ket{\\{s_{i}\\}},\\\
&\mathcal{N}_{\text{3-KW}}U_{G}\ket{\\{s_{i}\\}}=(-1)^{\hat{t}_{A}+\hat{t}_{G}}\mathcal{N}_{\text{3-KW}}\ket{\\{s_{i}\\}}.\end{split}$
(35)

Similarly, this is valid for general state $|\psi\rangle$. In particular, we
can consider an eigenstate $\ket{\Psi}$ of $(U_{A},U_{G})$ with eigenvalue
$(u_{A},u_{G})$. Thus we have

$\begin{split}&\mathcal{N}_{\text{3-KW}}U_{A}\ket{\Psi}=(-1)^{\hat{t}_{A}}N_{\text{3-KW}}\ket{\Psi}=N_{\text{3-KW}}(-1)^{u_{A}}\ket{\Psi},\\\
&\mathcal{N}_{\text{3-KW}}U_{G}\ket{\Psi}=(-1)^{\hat{t}_{A}+\hat{t}_{G}}N_{\text{3-KW}}\ket{\Psi}=N_{\text{3-KW}}(-1)^{u_{G}}\ket{\Psi},\end{split}$
(36)

namely,

$(u_{A},u_{G})=(\hat{t}_{A},\hat{t}_{A}+\hat{t}_{G}).$ (37)

Then it follows that $(\hat{t}_{A},\hat{t}_{G})=(u_{A},u_{A}+u_{G})$.

##  Appendix B SPT phase invariant under three-site BPM duality and Kennedy-
Tasaki duality

In this appendix, we will discuss the SPT phase invariant under
$\mathcal{N}_{\text{3-KW}}$ and the Kennedy-Tasaki duality between SPT phase
and SSB phase.

###  B.1 The Hamiltonian of SPT phase

Let us begin our discussion with an exactly solvable Hamiltonian with $L\in
6\mathbb{Z}$:

$H_{\text{SPT}}=-\sum^{L}_{i=1}a_{i},\quad a_{i}=(-1)^{i}Z_{i-1}Y_{i}Z_{i+1}.$
(38)

It is straightforward to show this Hamiltonian has the $(\mathbb{Z}_{2})^{2}$
symmetry (20) and is invariant under three-site BPM. By a $U_{\text{3-DW}}$
transformation, this SPT Hamiltonian is mapped to a Hamiltonian belonging to
the trivially gapped phase:

$U^{\dagger}_{\text{3-DW}}H_{\text{SPT}}U_{\text{3-DW}}=-\sum^{L}_{i=1}X_{i}=H_{\text{triv}},$
(39)

where

$U_{\text{3-DW}}=\prod^{L}_{i=1}\exp(-\frac{\pi
i}{4}(-1)^{i}Z_{i})\prod^{L}_{i=1}\exp[\frac{\pi i}{4}(1-Z_{i})(1-Z_{i+1})]T,$
(40)

and $T$ is one-site lattice translation. The dual Hamiltonian has a unique
ground state, thus the Hamiltonian (38) also has a unique gapped ground state.

###  B.2 String order parameters, ground state charge under twisted boundary
condition and edge modes

In this section, we will detect the SPT order by different methods. The first
method is the string order parameter:

$\begin{split}\langle S_{U_{A}}\rangle&=(-1)^{m-n+1}\langle
Z_{3n}\prod^{m}_{k=n}(X_{3k+1}X_{3k+2})Z_{3m+3}\rangle=\langle\prod^{m}_{k=n}a_{3k+1}a_{3k+2}\rangle=1,\\\
\langle S_{U_{G}}\rangle&=(-1)^{m-n+1}\langle
Z_{3n-2}\prod^{m}_{k=n}(X_{3k-1}X_{3k})Z_{3m+1}\rangle=\langle\prod^{m}_{k=n}a_{3k-1}a_{3k}\rangle=1.\end{split}$
(41)

The string order parameter $S_{U_{A}}$ ($S_{U_{G}}$) is obtained by dressing
the string operator of $U_{A}$ ($U_{G}$) symmetry with charged operator of
$U_{G}$ ($U_{A}$) symmetry at endpoints, which is consistent with decorated
domain wall construction.

The second way to probe the SPT order is ground state charge under twisted
boundary conditions on the closed chains. For simplicity, we assume $L\in
6\mathbb{Z}$. Let us first twist the boundary condition using the
$\mathbb{Z}_{2}^{A}$ symmetry (labeled by $\mathbb{Z}_{2}^{A}$-TBC), and
measure the $\mathbb{Z}_{2}^{G}$ charge of the ground state. Twisting the
boundary condition by $\mathbb{Z}_{2}^{A}$ means imposing a domain wall
between sites $L-1$ and $1$ by changing the sign of the term
$Z_{L-1}Y_{L}Z_{1}$. The SPT Hamiltonian (38) becomes

$H^{\mathbb{Z}_{2}^{A}}_{\text{SPT}}=\sum^{L-1}_{i=1}a_{i}-a_{L}.$ (42)

We note that the twisted and untwisted SPT Hamiltonian are related by a
unitary transformation
$H_{\text{SPT}}^{\mathbb{Z}_{2}^{A}}=Z_{L}H_{\text{SPT}}Z_{L}$. Denote the
ground state under PBC as $\ket{\text{GS}}$, and that under
$\mathbb{Z}_{2}^{A}$-TBC as
$\ket{\text{GS}}_{\text{tw}}^{\mathbb{Z}_{2}^{A}}$. We have

$\displaystyle\ket{\text{GS}}_{\text{tw}}^{\mathbb{Z}_{2}^{A}}=Z_{L}\ket{\text{GS}}.$
(43)

It follows that

$\displaystyle
U_{G}\ket{\text{GS}}_{\text{tw}}^{\mathbb{Z}_{2}^{A}}=U_{G}Z_{L}\ket{\text{GS}}=-Z_{L}\ket{\text{GS}}=-\ket{\text{GS}}_{\text{tw}}^{\mathbb{Z}_{2}^{A}}$
(44)

which shows that $\ket{\text{GS}}_{\text{tw}}^{\mathbb{Z}_{2}^{A}}$ has
$\mathbb{Z}_{2}^{G}$ charge 1. Here we used the fact that the ground state
under PBC is neutral under $\mathbb{Z}_{2}^{G}$. We can alternatively twist
the boundary condition using $\mathbb{Z}_{2}^{G}$ symmetry (labeled by
$\mathbb{Z}_{2}^{G}$-TBC), and measure the $\mathbb{Z}_{2}^{A}$ charge of the
ground state. By the same method, one can show that the ground state
$\ket{\text{GS}}_{\text{tw}}^{\mathbb{Z}_{2}^{G}}$ has odd
$\mathbb{Z}_{2}^{A}$ charge.

At last, we will derive how symmetry fractionalizes on edge modes. Let us
place the spin system on an open chain with $1\leq i\leq L$ and choose the OBC
such that only the interactions completely supported on the chain are kept.
The Hamiltonian is

$\displaystyle H_{\text{SPT}}^{\text{OBC}}=-\sum^{L-1}_{j=2}a_{j}.$ (45)

There are two boundary terms on each edge: $Z_{1}$, $X_{1}Z_{2}$, $Z_{L}$,
$Z_{L-1}X_{L}$. All the operators commute with bulk Hamiltonian and the
anticommutation relation of two terms on each edge gives rise to 2-fold
degenerate subspace.

To see the symmetry fractionalization, we note that for ground states

$\begin{split}&\prod^{L/3-1}_{k=1}a_{3k+1}a_{3k+2}=-Z_{3}\prod^{L/3-1}_{k=1}(X_{3k+1}X_{3k+2})Z_{L}=1,\\\
&\prod^{L/3-1}_{k=1}a_{3k-1}a_{3k}=-Z_{1}\prod^{L/3-1}_{k=1}(X_{3k-1}X_{3k})Z_{L-2}=1.\end{split}$
(46)

This implies the symmetry operator fractionalizes as
$U_{A/G}=-\mathcal{L}_{A/G}\mathcal{R}_{A/G}$ where

$\begin{split}&\mathcal{L}_{A}=X_{1}X_{2}Z_{3},\mathcal{R}_{A}=Z_{L},\\\
&\mathcal{L}_{A}=Z_{1},\mathcal{R}_{G}=Z_{L-2}X_{L-1}X_{L}.\end{split}$ (47)

On each edge, the projective representation of $\mathcal{L}$ and $\mathcal{R}$
gives rise to the edge modes. Such symmetry fractionalizes and the resulting
edge modes are robust as long as the bulk gap is not closed.

At last, we remark that when $L\notin 3$, the Hamiltonian (38) does not
respect $(\mathbb{Z}_{2})^{2}$ symmetry. This is because BPM under PBC only
has the trivial kernel $(0,0,0,\cdots)$ and is not associated with $U_{A}$ and
$U_{G}$ in this case. Thus the $\mathcal{N}_{3-\text{KW}}$ is a unitary
transformation and the Hamiltonian (15) with $h=0$ has a unique ground state.

###  B.3 The generalized Kennedy-Tasaki transformation

Similar to the reference [23], we can also construct a generalized Kennedy-
Tasaki transformation, which can relate the SPT phase (38) and the SSB model
$H_{\text{SSB}}=-\sum^{L}_{i=1}Z_{i}Z_{i+1}Z_{i+2}$:

$U_{\text{3-KT}}=U_{\text{3-DW}}U_{\text{3-KW}}U^{\dagger}_{\text{3-DW}}.$
(48)

It is straightforward to derive the fusion rule of this KT transformation:

$\begin{split}U_{\text{3-KT}}U_{\text{3-KT}}&=U_{\text{3-DW}}U_{\text{3-KW}}U_{\text{3-KW}}U^{\dagger}_{\text{3-DW}}\\\
&=U_{\text{3-DW}}(1+U_{A})(1+U_{G})T^{2}U^{\dagger}_{\text{3-DW}}\\\
&=(1+U_{A})(1+U_{G})T^{2}\end{split}$ (49)

where we use the fact that $U_{\text{3-DW}}$ commutes with two-site
translation and $U_{A}$ and $U_{G}$ operators.

##  Appendix C Four-site BPM duality transformation

###  C.1 BPM duality under PBC and fusion rules

In this appendix, we discuss the BPM duality which is related to the following
four-site Ising chain with $L\in 4\mathbb{Z}$:

$\begin{split}H_{\text{4-Ising}}=-\sum^{L}_{i=1}(hX_{i}+Z_{i}Z_{i+1}Z_{i+2}Z_{i+3}).\end{split}$
(50)

This system has a unique ground state with all $X_{i}=1$ when $h=\infty$,
while it is in the SSB phase when $h=0$. To understand the duality between
these two phases, we can construct a generalized KW duality under PBC:

$\begin{split}\mathcal{N}_{\text{4-KW}}\ket{\\{s_{i}\\}}&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j=1}^{L}s_{j}(\hat{s}_{j}+\hat{s}_{j+1}+\hat{s}_{j+2}+\hat{s}_{j+3})}\ket{\\{\hat{s}_{i}\\}}\\\
&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j=1}^{L}\hat{s}_{j}(s_{j-3}+s_{j-2}+s_{j-1}+s_{j})}\ket{\\{\hat{s}_{i}\\}},\end{split}$
(51)

whose BPM matrix is :

${\bf\it A}_{\text{4-KW}}=\left(\begin{array}[]{ccccccc}1&1&1&1&0&\cdots&0\\\
0&1&1&1&1&\cdots&0\\\ 0&0&1&1&1&\cdots&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots\\\ 1&1&0&\cdots&0&1&1\\\
1&1&1&0&\cdots&0&1\\\ \end{array}\right).$ (52)

Similarly, the properties of symmetry and SSB ground state degeneracy can be
derived from the kernels of ${\bf\it A}^{T}_{\text{4-KW}}$:

$\displaystyle{\bf\it b}^{1}=(1,0,1,0,1,0,1,0,\cdots),\quad{\bf\it
b}^{2}=(0,1,0,1,0,1,0,1,\cdots),\quad{\bf\it b}^{3}=(1,1,0,0,1,1,0,0,\cdots).$
(53)

This shows the system has a $(\mathbb{Z}_{2})^{3}$ symmetry:

$\displaystyle U_{o}\ket{\\{s_{i}\\}}=\ket{\\{s_{i}+b^{1}_{i}\\}},\quad
U_{e}\ket{\\{s_{i}\\}}=\ket{\\{s_{i}+b^{2}_{i}\\}},\quad
U_{1}\ket{\\{s_{i}\\}}=\ket{\\{s_{i}+b^{3}_{i}\\}}.$ (54)

In operator form, they are given by:

$\begin{split}&U_{e}=\prod^{L/2}_{i=1}X_{2i},\quad
U_{o}=\prod^{L/2}_{i=1}X_{2i+1},\quad
U_{1}=\prod^{L/4}_{i=1}X_{4i+1}X_{4i+2}.\end{split}$ (55)

Thus all states can be organized into eigenstates of $U_{e/o/1}$ with
eigenvalue $(-1)^{u_{e/o/1}}=\pm 1$, i.e. $u_{e/o/1}=0,1$.

The eight ground states of the SSB phase can be generated by these symmetry
operators from the all spin-up state:

$\begin{split}&\ket{\text{GS}}_{1}=\ket{\uparrow\uparrow\uparrow\uparrow\cdots},\quad\ket{\text{GS}}_{2}=\ket{\uparrow\downarrow\uparrow\downarrow\cdots},\\\
&\ket{\text{GS}}_{3}=\ket{\downarrow\uparrow\downarrow\uparrow\cdots},\quad\ket{\text{GS}}_{4}=\ket{\downarrow\downarrow\downarrow\downarrow\cdots},\\\
&\ket{\text{GS}}_{5}=\ket{\downarrow\downarrow\uparrow\uparrow\cdots},\quad\ket{\text{GS}}_{6}=\ket{\downarrow\uparrow\uparrow\downarrow\cdots},\\\
&\ket{\text{GS}}_{7}=\ket{\uparrow\downarrow\downarrow\uparrow\cdots},\quad\ket{\text{GS}}_{8}=\ket{\uparrow\uparrow\downarrow\downarrow\cdots}.\end{split}$
(56)

This BPM duality induces the transformation of Pauli operators

$\begin{split}&\mathcal{N}_{\text{4-KW}}X_{i}=\hat{Z}_{i}\hat{Z}_{i+1}\hat{Z}_{i+2}\hat{Z}_{i+3}\mathcal{N}_{\text{4-KW}},\\\
&\mathcal{N}_{\text{4-KW}}Z_{i-3}Z_{i-2}Z_{i-1}Z_{i}=\hat{X}_{i}\mathcal{N}_{\text{4-KW}},\end{split}$
(57)

and thus exchanges transverse field term and four-site lsing term. Likewise,
the dual Hilbert space can also be organized into four symmetry sectors
labeled by $(\hat{u}_{o},\hat{u}_{e},\hat{u}_{1})\in\\{0,1\\}^{3}$.

We can also determine fusion rules by acting the product of
$\hat{U}_{e/o/1}\times\mathcal{N}_{\text{4-KW}}$,
$\mathcal{N}_{\text{4-KW}}\times U_{e/o/1}$ and
$\mathcal{N}_{\text{4-KW}}\times\mathcal{N}_{\text{4-KW}}$ on a general state:

$\begin{split}&\mathcal{N}_{\text{4-KW}}\times
U_{e/o/1}=\mathcal{N}_{\text{4-KW}},\quad\hat{U}_{e/o/1}\times\mathcal{N}_{\text{4-KW}}=\mathcal{N}_{\text{4-KW}},\\\
&\mathcal{N}_{\text{4-KW}}\times\mathcal{N}_{\text{4-KW}}=(1+U_{e})(1+U_{o})(1+U_{1})T^{3}\,.\end{split}$
(58)

###  C.2 Unitarity problem and symmetry-twist transformation

To solve this unitarity problem in this case, we need to add three additional
boundary spins $t_{e}$, $t_{o}$ and $t_{1}$ in $\\{s_{i}\\}$-systems and
another three spins $\hat{t}_{e}$, $\hat{t}_{o}$ and $\hat{t}_{1}$ in
$\\{\hat{s}_{i}\\}$-systems, i.e. the untwisted/twisted boundary condition of
$(\mathbb{Z}_{2})^{3}$ symmetry:

$\begin{split}&s_{L+4k+1}=s_{4k+1}+(t_{o}+t_{1}),s_{L+4k+2}=s_{4k+2}+(t_{e}+t_{1}),\\\
&s_{L+4k+3}=s_{4k+3}+t_{o},\quad s_{L+4k}=s_{4k}+t_{e}.\\\
&\hat{s}_{L+4k+1}=\hat{s}_{4k+1}+(\hat{t}_{o}+\hat{t}_{1}),\hat{s}_{L+4k+2}=\hat{s}_{4k+2}+(t_{e}+t_{1}),\\\
&\hat{s}_{L+4k+3}=\hat{s}_{4k+3}+\hat{t}_{o},\quad\hat{s}_{L+4k}=\hat{s}_{4k}+\hat{t}_{e}.\end{split}$
(59)

We also find a consistent modification of BPM:

$\begin{split}&\mathcal{N}_{\text{4-KW}}\ket{\\{s_{i}\\}}=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j=1}^{L}s_{j}(\hat{s}_{j}+\hat{s}_{j+1}+\hat{s}_{j+2}+\hat{s}_{j+3})+t_{e}(\hat{s}_{2}+\hat{s}_{3})+t_{o}(\hat{s}_{1}+\hat{s}_{2})+t_{1}\hat{s}_{1}}\ket{\\{\hat{s}_{i}\\}}\\\
&=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j=1}^{L}\hat{s}_{j}(s_{j-3}+s_{j-2}+s_{j-1}+s_{j})+\hat{t}_{o}(s_{L-1}+s_{L-2})+\hat{t}_{e}(s_{L}+s_{L-1})+\hat{t}_{1}s_{L-2}}\ket{\\{\hat{s}_{i}\\}}.\end{split}$
(60)

By a similar method, one can find the symmetry-twist mapping from this
formula:

$\begin{split}&[(\hat{u}_{o},\hat{t}_{o}),(\hat{u}_{e},\hat{t}_{e}),(\hat{u}_{1},\hat{t}_{1})]\\\
=&[(t_{o}+t_{e}+t_{1},u_{e}+u_{o}+u_{1}),(t_{o}+t_{e},u_{e}+u_{1}),(t_{1}+t_{e},u_{o}+u_{e})].\end{split}$
(61)

We can apply this modified BPM to fix the unitarity problem for SSB ground
states, which satisfies that $s_{j-3}+s_{j-2}+s_{j-1}+s_{j}=0$. The BPM maps
them to the paramagnetic state with $\hat{X}_{i}=1$:

$\frac{1}{2^{\frac{L}{2}}}(-1)^{\hat{t}_{o}(s_{L-1}+s_{L-2})+\hat{t}_{e}(s_{L}+s_{L-1})+\hat{t}_{1}s_{L-2}}\sum_{\\{\hat{s}_{i}\\}}\ket{\\{\hat{s}_{i}\\}}.$
(62)

When $\hat{t}_{o}=\hat{t}_{e}=\hat{t}_{1}=0$, the phase is trivial and only
linear combination $\sum^{8}_{i=1}\ket{\text{GS}}_{i}$ with all $u=0$
survives. But when $\hat{t}_{o}=1$ and $\hat{t}_{e}=\hat{t}_{1}=0$, four
ground states with $s_{L-1}+s_{L-2}=1$ will be mapped with additional $-1$
sign. Then only linear combination
$(\ket{\text{GS}}_{1}+\ket{\text{GS}}_{4}+\ket{\text{GS}}_{6}+\ket{\text{GS}}_{7}-\ket{\text{GS}}_{2}-\ket{\text{GS}}_{3}-\ket{\text{GS}}_{5}-\ket{\text{GS}}_{8})$
survives. This combination has symmetry charge $u_{o}=u_{e}=u_{1}=1$, which is
the solution of Eq. (LABEL:eq:sym-twist_map). It is straightforward to check
other cases and linear combinations of SSB ground states with different
symmetry eigenvalues will be mapped to the paramagnetic state under different
boundary conditions, which satisfies the rule of symmetry-twist mapping
(LABEL:eq:sym-twist_map).

###  C.3 Anomaly of four-site BPM duality symmetry

On the self-dual point $h=1$, the model $H_{\text{4-Ising}}$ is at a first-
order phase transition between SSB phase and the trivial phase [74, 80]. The
BPM duality $\mathcal{N}_{\text{4-KW}}$ also becomes an emergent non-
invertible symmetry for the self-dual theory. Such an emergent symmetry is
anomalous in the sense that it cannot allow a symmetric uniquely gapped phase
under any symmetric perturbations and hence self-dual theories must be always
at first-order or second-order first phase transitions.

To prove the anomaly of $\mathcal{N}_{\text{4-KW}}$, let us first show this
duality operator can be decomposed as the product under PBC:
$\mathcal{N}_{\text{4-KW}}=\frac{1}{2}\mathcal{N}_{\text{KW'}}\times(\mathcal{N}_{\text{KW}})^{\dagger}\times\mathcal{N}_{\text{KW'}}$.
Here the $\mathcal{N}_{\text{KW}}$ is the usual KW transformation (1) and
$\mathcal{N}_{\text{KW'}}$ is the combination of two KW transformations acting
on even and odd sites:

$\begin{split}&\mathcal{N}_{\text{KW'}}\ket{\\{s_{i}\\}}=\frac{1}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum_{j,k=1}^{L}(s_{j-2}+s_{j})\hat{s}_{k}}\ket{\\{\hat{s}_{i}\\}}.\end{split}$
(63)

We can directly check this result

$\begin{split}&\mathcal{N}_{\text{KW'}}\times(\mathcal{N}_{\text{KW}})^{\dagger}\times\mathcal{N}_{\text{KW'}}\ket{\\{s_{i}\\}}\\\
=&\frac{1}{2^{\frac{3L}{2}}}\sum_{\\{s^{\prime}_{i},s^{\prime\prime}_{i},\hat{s}_{i}\\}}(-1)^{\sum^{L}_{j=1}s^{\prime}_{j}(s_{j-2}+s_{j})+s^{\prime}_{j}(s^{\prime\prime}_{j-1}+s^{\prime\prime}_{j})+\hat{s}_{j}(s^{\prime\prime}_{j-2}+s^{\prime\prime}_{j})}\ket{\\{\hat{s}_{i}\\}}\\\
=&\frac{1}{2^{\frac{L}{2}}}\sum_{\\{s^{\prime\prime}_{i},\hat{s}_{i}\\}}\delta(s_{j-2}+s_{j}+s^{\prime\prime}_{j-1}+s^{\prime\prime}_{j})(-1)^{\sum^{L}_{j=1}\hat{s}_{j}(s^{\prime\prime}_{j-2}+s^{\prime\prime}_{j})}\ket{\\{\hat{s}_{i}\\}}\\\
=&\frac{1}{2^{\frac{L}{2}}}\sum_{\\{s^{\prime\prime}_{i}=s_{i}+s_{i-1}+0/1,\hat{s}_{i}\\}}(-1)^{\sum^{L}_{j=1}\hat{s}_{j}(s^{\prime\prime}_{j-2}+s^{\prime\prime}_{j})}\ket{\\{\hat{s}_{i}\\}}\\\
=&\frac{2}{2^{\frac{L}{2}}}\sum_{\\{\hat{s}_{i}\\}}(-1)^{\sum^{L}_{j=1}\hat{s}_{j}(s_{j-3}+s_{j-2}+s_{j-1}+s_{j})}\ket{\\{\hat{s}_{i}\\}}\\\
=&2\mathcal{N}_{\text{4-KW}}\ket{\\{\hat{s}_{i}\\}}.\end{split}$ (64)

Now, let us prove the anomaly by the contraction method. We first assume a
uniquely gapped system is self-dual under PBC and its ground state
$|\psi\rangle$ should be short-range entangled (SRE). Due to symmetry-twist
mapping, $|\psi\rangle$ should be even under each $\mathbb{Z}_{2}$ symmetry.
If we focus on the $\mathbb{Z}^{e}_{2}\times\mathbb{Z}^{o}_{2}$ symmetry, the
possible uniquely gapped phase can only be the
$\mathbb{Z}^{e}_{2}\times\mathbb{Z}^{o}_{2}$ SPT phase, since trivially gapped
phase is mapped to an SSB phase under this four-site BPM. Then we can perform
$(\mathcal{N}_{\text{KW'}})^{\dagger}$ or $\mathcal{N}_{\text{KW'}}$ which
both keep the SPT phase invariant [23]. Thus
$\mathcal{N}_{\text{KW'}}^{\dagger}|\psi\rangle$ and
$\mathcal{N}_{\text{KW'}}|\psi\rangle$ are still ground states of the
$\mathbb{Z}^{e}_{2}\times\mathbb{Z}^{o}_{2}$ SPT systems and thus SRE. On the
other hand, due to (64), we have

$\begin{split}\langle\psi|\frac{1}{2}\mathcal{N}_{\text{KW'}}\times\frac{1}{2\sqrt{2}}\mathcal{N}_{\text{KW}}^{\dagger}\mathcal{N}_{\text{KW'}}|\psi\rangle=\langle\psi|\frac{1}{2\sqrt{2}}\mathcal{N}_{\text{4-KW}}|\psi\rangle=e^{i\theta}\end{split}$
(65)

where we multiply the normalized coefficient $\frac{1}{2\sqrt{2}}$. That is

$\frac{1}{2\sqrt{2}}\mathcal{N}_{\text{KW}}^{\dagger}\mathcal{N}_{\text{KW'}}|\psi\rangle=e^{i\theta}\frac{1}{2}\mathcal{N}_{\text{KW'}}^{\dagger}|\psi\rangle.$
(66)

However, the $\mathcal{N}^{\dagger}_{\text{KW}}$ maps the
$\mathbb{Z}^{e}_{2}\times\mathbb{Z}^{o}_{2}$ SPT phase to an SSB phase of
global spin flip $U_{e}U_{o}$. Thus
$\frac{1}{2\sqrt{2}}\mathcal{N}^{\dagger}_{\text{KW}}\mathcal{N}_{\text{KW'}}|\psi\rangle$
is a cat state of SSB phase with even charge of $U_{e}U_{o}$ which is not SRE
and that finishes our proof.