# Unconventional quantum phase transitions
in a one-dimensional Lieb-Schultz-Mattis system

Wayne Zheng1,3, D. N. Sheng2, and Yuan-Ming Lu1 1Department of Physics, The
Ohio State University, Columbus, Ohio 43210, USA
2Department of Physics and Astronomy, California State University Northridge,
Northridge, California 91330, USA
3Department of Physics, The Chinese University of Hong Kong, Shatin, New
Territories, Hong Kong, China

###### Abstract

We study quantum phases and phase transitions in a one-dimensional interacting
fermion system with a Lieb-Schultz-Mattis (LSM) type anomaly. Specifically,
the inversion symmetry enforces any symmetry-preserving gapped ground state of
the system to be a Kitaev chain, following a Lieb-Schultz-Mattis type theorem
that we prove. Alternatively, via the Jordan-Wigner transformation, this
system describes a spin system whose gapped ground states must break either
the inversion or the Ising symmetry associated with fermion parity. We obtain
a phase diagram using analytical methods and variational matrix product state
simulations, and study the critical behaviors of the quantum phase transitions
therein using entanglement entropy, energy variance and finite size scaling of
order parameters. In particular, we observe continuous phase transitions
between different ordered phases that are beyond the Ginzburg-Landau-Wilson
paradigm, in analogy to the deconfined quantum critical points in two spatial
dimensions. We show this type of 1D deconfined quantum critical point is
described by the Tomonaga-Luttinger liquid theory, and extract the Luttinger
parameter and critical exponents. We also identify a gapless phase between two
ordered phases, which cannot be described by a U(1) Luttinger liquid.

###### Contents

  1. I Introduction
  2. II The model
    1. II.1 Lattice model and its symmetries
    2. II.2 A Lieb-Schultz-Mattis theorem for the Kitaev chain
  3. III Numerical methods
    1. III.1 Spin chain representation and order parameters
    2. III.2 Matrix product state, variance and entanglement
  4. IV The phase diagram
    1. IV.1 The non-interacting limit
    2. IV.2 A representative study for the interacting cases: $\Delta=1.0$
    3. IV.3 Overview of the whole phase diagram
  5. V Characterizing the critical behaviors
    1. V.1 Finite-size analysis of the critical points
    2. V.2 Realizing DQCPs by fusing two dualed order parameters and the possible breakdown of the Tomonaga-Luttinger liquid
    3. V.3 Finite-size analysis of the central charge
  6. VI Concluding remarks
  7. A Symmetry implementations on the fermion chain vs. the spin chain
    1. A.1 Non-local inversion symmetry in the spin representation
    2. A.2 Locality-preserving inversion symmetry in the spin representation
  8. B Zero-mode, entanglement entropy and finite-size analysis in the XY model on even and odd open chains
  9. C Larger bond-dimension test
  10. D A string order parameter that preserves non-local inversion symmetry (77)
  11. E Numerical results on on even lattices as a perturbative analysis
    1. E.1 Critical exponents at the critical point(s)
    2. E.2 Central charge in the gapless phase

##  I Introduction

A paradigm beyond the Landau theory of spontaneous symmetry breaking is the
_deconfined quantum critical point_ (DQCP). It was firstly proposed for the
Neel order to valence bond solid (VBS) transition on a two-dimensional square
lattice Senthil  _et al._ (2004a, b), as a type of continuous quantum phase
transition between two ordered phases that cannot be related by symmetry
breakings. Compared to the Ginzburg-Landau-Wilson paradigm, a DQCP features
many novel aspects such as emergent symmetries and self-duality Wang  _et al._
(2017).

Recently a lot of interests arise for revisiting one spatial dimension (1d) to
realize the deconfined quantum criticality. In particular, the 1d spin-$1/2$
chain with both nearest- and second-neighbor anisotropic exchange interactions
have been extensively studiedMudry  _et al._ (2019); Jiang and Motrunich
(2019); Roberts  _et al._ (2019); Huang  _et al._ (2019), which exhibits a
DQCP between a (anti-)ferromagnetic order and a VBS phase. The critical
behaviors of this DQCP has also been carefully examined and compared to field-
theory predictions. The phase transition between these two gapped orders is a
direct second-order quantum phase transition, whose long-wavelength low-energy
theory is expected to exhibit an emergent $U(1)$ symmetry. On the other hand,
this 1d DQCP is closely related to the Lieb-Schultz-Mattis (LSM) theorem Lieb
_et al._ (1961); Oshikawa (2000); Cheng  _et al._ (2016); Po  _et al._ (2017);
Lu  _et al._ (2020); Hastings (2004); Yang  _et al._ (2018); Lu (2017);
Parameswaran  _et al._ (2013), which forbids a gapped symmetric ground state
that preserves both translation and the discrete $Z_{2}\times Z_{2}$ spin
rotational symmetries Jiang and Motrunich (2019).

In this paper, we study a 1D lattice model of interacting fermions, with a
different LSM-type anomaly. In particular, any gapped ground state that
preserves a site-centered inversion symmetry must be a Kitaev chain, with an
odd number of Majorana bound states on each boundary. Through a Jordan-Wigner
transformation, it becomes a spin-$1/2$ chain, whose gapped ground states must
break either the inversion symmetry or the Ising symmetry associated with the
fermion parity. We prove such a LSM-type theorem, and study a generic 1d
fermion model with nearest-neighbor couplings that preserves this inversion
symmetry. The phase diagram of our model has a rich structure: there are DQCPs
between different ordered phases beyond the Landau theory, as well as stable
gapless phases separating the ordered phases. The rest part of this paper is
organized as follows: In Sec. II, we show our model and discuss its
symmetries. In Sec. III, the numerical methods used to study the model are
discussed. In Sec. IV, the phase diagram of the model is obtained, using
analytical solutions in the non-interacting limit and numerical results for
the interacting model. In Sec. V, the critical behaviors at the phase
boundaries are carefully analyzed, focusing on the DQCP described by the
Luttinger liquid theory, and a stable gapless phase. Finally, the concluding
remarks are given in Sec. VI.

##  II The model

###  II.1 Lattice model and its symmetries

We consider the following 1D model of interacting fermions

$\displaystyle H$
$\displaystyle=\sum_{j}(-)^{j}\left[t\left(c_{j}^{\dagger}c_{j+1}+h.c.\right)+\left(\Delta
c_{j}^{\dagger}c_{j+1}^{\dagger}+h.c.\right)\right]$ (1)
$\displaystyle+\left(\text{i}\Delta^{\prime}c_{j}^{\dagger}c_{j+1}^{\dagger}+h.c.\right)+V\left(n_{j}-\frac{1}{2}\right)\left(n_{j+1}-\frac{1}{2}\right)$

It breaks all the global (onsite) symmetries except for the fermion parity
conservation $\mathcal{P}_{f}=(-1)^{\hat{F}}$. In fact, assuming $t>0$, this
Hamiltonian includes all possible nearest-neighbor (NN) couplings that
preserve a site-centered unitary inversion symmetry defined as follows:

$c_{j}\overset{\mathcal{I}}{\longrightarrow}\hskip 1.0pt\mathrm{i}\hskip
1.0ptc_{-j}^{\dagger}.$ (2)

If $c_{j}\equiv\gamma_{j}+\hskip 1.0pt\mathrm{i}\hskip 1.0pt\eta_{j}$ is
written in terms of Majorana fermions $\gamma_{j}$ and $\eta_{j}$, this very
inversion symmetry actually realize

$\begin{pmatrix}\gamma_{j}\\\
\eta_{j}\end{pmatrix}\overset{\mathcal{I}}{\longrightarrow}\begin{pmatrix}\eta_{-j}\\\
\gamma_{-j}\end{pmatrix}$ (3)

saying permuting the two corresponding Majorana fermions. In a periodic chain
with even sites, the above NN-only model also preserves a magnetic translation
symmetry $\tilde{T}_{x}$ defined as

$\displaystyle\tilde{T}_{x}\equiv(-1)^{\sum_{j}j\hat{n}_{j}}T_{x}\cdot\mathcal{K},~{}~{}~{}n_{j}\equiv
c_{j}^{\dagger}c_{j};$ (4) $\displaystyle
c_{j}\overset{\tilde{T}_{x}}{\longrightarrow}(-1)^{j+1}c_{j+1}.$

where $\mathcal{K}$ represents the complex conjugation. This gives rise to a
link-centered anti-unitary inversion symmetry
$\tilde{\mathcal{I}}=\mathcal{I}\cdot\tilde{T}_{x}$:

$\displaystyle
c_{j}\overset{\tilde{\mathcal{I}}}{\longrightarrow}(-1)^{j+1}\hskip
1.0pt\mathrm{i}\hskip 1.0ptc^{\dagger}_{-j-1}$ (5)

Besides, the model Eq. (1) also exhibits an anti-unitary particle-hole
symmetry

$\tilde{\mathcal{C}}=\mathcal{C}\cdot\mathcal{K}:~{}~{}~{}c_{j}\rightarrow(-1)^{j}c_{j}^{\dagger}$
(6)

where $\mathcal{C}$ represents the unitary particle-hole transformation.

The generators $\\{\mathcal{I},\tilde{\mathcal{I}}\\}$ of the symmetry group
satisfy the following algebra:

$\displaystyle\tilde{\mathcal{I}}^{2}=1,$ (7)
$\displaystyle\mathcal{I}(-1)^{\hat{F}}\mathcal{I}^{-1}=(-1)^{L}(-1)^{\hat{F}}.$
(8)

where $L\in{\mathbb{Z}}$ is the system size of the 1d chain. In a periodic
chain of length $L=0\mod 2$, the magnetic translation satisfies:

$\displaystyle(\tilde{T}_{x})^{L}=(\mathcal{P}_{f})^{L(L-1)/2}$ (9)

Using the following Jordan-Wigner transformation

$\displaystyle
c_{j}=(-)^{\frac{j(j-1)}{2}}\left[\prod_{k<j}\left(-\sigma_{k}^{z}\right)\right]\sigma_{j}^{-},$
$\displaystyle
S_{j}^{z}\equiv\frac{\sigma^{z}_{j}}{2}=c_{j}^{\dagger}c_{j}-\frac{1}{2}$ (10)

the fermion model Eq. (1) can also be rewritten as a spin-$\frac{1}{2}$ chain:

$\displaystyle
H_{\text{spin}}=\sum_{j}\left(t\sigma_{j}^{+}\sigma_{j+1}^{-}+\Delta\sigma_{j}^{+}\sigma_{j+1}^{+}+h.c.\right)$
(11)
$\displaystyle+(-)^{j}\left(\text{i}\Delta^{\prime}\sigma_{j}^{+}\sigma_{j+1}^{+}+h.c.\right)+VS_{j}^{z}S^{z}_{j+1}$
$\displaystyle=\sum_{j}\sum_{\alpha=x,y,z}J_{\alpha}S_{j}^{\alpha}S_{j+1}^{\alpha}+(-1)^{j}\Gamma(S_{j}^{x}S_{j+1}^{y}+S_{j}^{y}S^{x}_{j+1})$

where the exchange couplings are given by

$\displaystyle J_{x}=2(t+\Delta),~{}~{}J_{y}=2(t-\Delta),~{}~{}J_{z}=V,$
$\displaystyle\Gamma=-2\Delta^{\prime}.$ (12)

This is a familiar XYZ modelMudry  _et al._ (2019), supplemented by an extra
staggered anisotropic exchange coupling of strength $\Gamma$. The introduction
of $\Gamma$ terms has important consequences: it leads to new gapless phases
unseen in the XYZ modelMudry  _et al._ (2019); Jiang and Motrunich (2019).

The symmetry group of the spin model111Strictly speaking, the symmetry
operators (16) in the spin model differs from the inversion symmetry (2) in
the fermion model, since the latter is not a locality-preserving unitary. This
subtlety is discussed in detail in Appendix A.  are generated by

$\displaystyle\mathcal{P}_{f}=(-1)^{\hat{F}}=\prod_{r}(-\sigma^{z}_{r}),$ (13)
$\displaystyle\tilde{T_{x}}=T_{x}\cdot\mathcal{K},$ (14)
$\displaystyle\tilde{\mathcal{C}}=(\prod_{r}\sigma^{x}_{r})\cdot\mathcal{K},$
(15) $\displaystyle\mathcal{I}=(\prod_{r}\sigma^{x}_{r})\cdot\mathcal{O}_{I}.$
(16)

where $\mathcal{O}_{I}$ is the spatial inversion operator, and the Pauli
matrix $\sigma^{x}_{r}=2S_{r}^{x}$. In particular, the inversion symmetry
$\mathcal{I}$ anticommutes with the Ising symmetry Eq. (13), on a spin chain
of an odd length.

###  II.2 A Lieb-Schultz-Mattis theorem for the Kitaev chain

One significant consequence of the inversion symmetry Eq. (2) is the following
theorem of Lieb-Schultz-Mattis Lieb  _et al._ (1961); Hastings (2005);
Oshikawa (2000); Parameswaran  _et al._ (2013); Zaletel and Vishwanath (2015);
Else and Thorngren (2020) (LSM) type:

Theorem: _In an one-dimensional spinless fermion system preserving the
inversion symmetry ( 2), any gapped symmetric ground state must be a Kitaev
chain._

This theorem is closely related to the family of LSM theorems for symmetry
protected topological (SPT) phases Chen  _et al._ (2013); Senthil (2015)
discussed recently Yang  _et al._ (2018); Lu  _et al._ (2020); Lu (2017);
Jiang  _et al._ (2019), but differs in the sense that here any gapped
symmetric ground state is enforced to be a nontrivial invertible phase Freed
and Hopkins (2016), i.e. the Kitaev chain Kitaev (2001), rather than SPT
phases.

Below we prove the theorem in two aspects. First we show that when restricted
to a non-interacting fermion system with translation symmetry, we can use the
polarization formula of BdG bands to show a gapped ground state must have a
nontrivial $\mathbb{Z}_{2}$ topological invariant, hence belonging to a Kitaev
chain. Next, we will show that in a generic interacting open chain with
inversion symmetry, a gapped ground state must be a Kitaev chain with Majorana
zero modes at the boundary.

Firstly, we consider a periodic fermion chain with an even number of sites.
Since each unit cell includes two sites, they transform as

$\displaystyle\begin{pmatrix}c_{2r}\\\
c_{2r+1}\end{pmatrix}\overset{\mathcal{I}}{\longrightarrow}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\begin{pmatrix}c^{\dagger}_{-2r}\\\
c^{\dagger}_{-2(r+1)+1}\end{pmatrix}$ (17)

In the basis of Majorana fermions

$\displaystyle c_{2r}=\frac{\chi_{r}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\eta_{r}}{2},~{}~{}c_{2r+1}=\frac{\chi^{\prime}_{r}+\hskip
1.0pt\mathrm{i}\hskip 1.0pt\eta^{\prime}_{r}}{2}.$ (18)

they transform as follows under inversion

$\displaystyle\Phi_{k}\equiv\begin{pmatrix}\chi(k)\\\ \eta(k)\\\
\chi^{\prime}(k)\\\
\eta^{\prime}(k)\end{pmatrix}\overset{\mathcal{I}}{\longrightarrow}\begin{pmatrix}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&e^{\hskip 1.0pt\mathrm{i}\hskip 1.0ptk}\\\ 0&0&e^{\hskip
1.0pt\mathrm{i}\hskip 1.0ptk}&0\end{pmatrix}\Phi_{-k}$ (19)

in the momentum space. In other words, the inversion symmetry is implemented
by unitary rotation

$\displaystyle R_{\mathcal{I}}(k)=\begin{pmatrix}1&0\\\ 0&e^{\hskip
1.0pt\mathrm{i}\hskip 1.0ptk}\end{pmatrix}_{\vec{\tau}}\otimes\mu_{x}$ (20)

where $\vec{\tau}$ and $\vec{\mu}$ are Pauli matrices for the sublattice and
Majorana indices respectively. A generic quadratic Bogoliubov-de Gennes (BdG)
Hamiltonian has the following form

$\displaystyle\hat{H}_{\text{free}}=\sum_{0\leq
k\leq\pi}\psi_{-k}^{T}h(k)\psi_{k},$ (21)

where Hermitian matrix $h(k)$ satisfies particle-hole and inversion
symmetries:

$\displaystyle h^{T}(k)=h^{\ast}(k)=-h(-k),$ (22) $\displaystyle
R_{\mathcal{I}}(k)h(k)R_{\mathcal{I}}^{-1}(k)=h(-k).$ (23)

The $\mathbb{Z}_{2}$-valued topological invariant Schnyder  _et al._ (2008);
Budich and Ardonne (2013) for such a BdG Hamiltonian in symmetry class D is
given by the quantized polarization of the filled BdG bands:

$\displaystyle\nu=e^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\int\text{d}kA(k)}=\pm
1,$ (24) $\displaystyle A(k)=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{\epsilon_{n}(k)<0}\langle\psi_{n}(k)|\partial_{k}\psi_{n}(k)\rangle.$

where $A(k)$ is the Berry connection of the filled bands. The quantized
$\nu=\pm 1$ is a consequence of the particle-hole symmetry (22), and inversion
symmetry (23) constrains the Berry curvature as follows:

$\displaystyle A(-k)=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{\epsilon<0}\langle\psi_{n}(k)|R_{\mathcal{I}}^{\dagger}(k)\partial_{k}R_{\mathcal{I}}(k)|\psi_{n}(k)\rangle-A(k)$
$\displaystyle=\sum_{\epsilon<0}\langle\psi_{n}(k)|\frac{1+\tau_{z}}{2}|\psi_{n}(k)\rangle-A(k)$
(25)

Particle-hole symmetry (22) further indicates that

$\displaystyle\sum_{\epsilon<0}\langle\psi_{n}(k)|\frac{1+\tau_{z}}{2}|\psi_{n}(k)\rangle=\sum_{\epsilon>0}\langle\psi_{n}(-k)|\frac{1+\tau_{z}}{2}|\psi_{n}(-k)\rangle$

and hence

$\displaystyle\int\text{d}kA(k)=\pi\text{Tr}(1+\tau_{z})-\int\text{d}kA(k)\Longrightarrow$
$\displaystyle\int\text{d}kA(k)=\pi\mod 2\pi$ (26)
$\displaystyle\Longrightarrow\nu=e^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int\text{d}kA(k)}=-1.$ (27)

Therefore, for a gapped BdG Hamiltonian with a well-defined polarization, the
$\mathbb{Z}_{2}$ invariant must be nontrivial, and hence it must be a Kitaev
chain with Majorana edge modes.

Secondly, we consider a generic interacting Hamiltonian preserving inversion
symmetry Eq. (2), on an open chain with an odd number of sites $L=1\mod 2$. In
this case, there is one single inversion center on the middle site, and the
inversion symmetry acts as a supersymmetry which changes the fermion parity:

$\displaystyle\mathcal{I}(-1)^{\hat{F}}\mathcal{I}^{-1}=-(-1)^{\hat{F}}$ (28)

This implies at least 2-fold degeneracy for all energy levels, hence two
degenerate ground states on an open chain with opposite fermion parities. If
the bulk is gapped, this necessarily leads to zero modes on the edge. Since
there is no extra global symmetry in the system to protect the edge modes,
they can only be an odd number of Majorana zero modes (MZMs) on each edge.
This indicates the ground state is an open Kitaev chain. And its total fermion
parity is flipped by the inversion symmetry, which exchanges the MZMs on the
two edges.

Therefore, we have shown that a gapped ground state preserving inversion
symmetry Eq. (2) must be a Kitaev chain with an odd number of MZMs on each
open boundary. This LSM theorem for Kitaev chain in the fermion context can
also be translated into the spin chain language, via Jordan-Wigner
transformation Eq. (10). In the spin language, it manifests as the more
familiar LSM theorem:

Theorem: _In an one-dimensional spin- $1/2$ chain with both Ising symmetry Eq.
(13) and inversion symmetry Eq. (16), its ground state is either gapless or
spontaneously breaks symmetries._

In other words, the spin-$1/2$ chain with both Ising and inversion symmetries
do not admit any short-range entangled ground state.

##  III Numerical methods

###  III.1 Spin chain representation and order parameters

For the convenience of numerical simulation, we use the traditional Jordan-
Wigner transformation:

$\displaystyle c_{j}$
$\displaystyle=\left[\prod_{k=0}^{j-1}\left(-\sigma_{k}^{z}\right)\right]\sigma_{j}^{-},$
(29) $\displaystyle\sigma_{j}^{z}$ $\displaystyle=2c_{j}^{\dagger}c_{j}-1,$

to rewrite the fermionic model Eq. (1) as a spin chain

$\displaystyle H$
$\displaystyle=\sum_{j}(-)^{j}\left(t\sigma_{j}^{+}\sigma_{j+1}^{-}+h.c.\right)$
(30)
$\displaystyle+\left(\left[(-)^{j}\Delta+\text{i}\Delta^{\prime}\right]\sigma_{j}^{+}\sigma_{j+1}^{+}+h.c.\right)$
$\displaystyle+\frac{1}{4}V\sigma_{j}^{z}\sigma_{j+1}^{z},$

where $\sigma^{\pm}=(\sigma^{x}\pm\text{i}\sigma^{y})/2$. $\sigma^{x,y,z}$ are
Pauli matrices. Note that this Jordan-Wigner transformation differs from Eq.
(10) by a $(-1)^{j(j-1)}/2$ sign, hence a different form of the Hamiltonian
(30) compared to (11). In this representation, the symmetry generators of
Hamiltonian (30) writes:

$\displaystyle\mathcal{P}_{f}=(-1)^{\hat{F}}=\prod_{r}(-\sigma^{z}_{r}),$
$\displaystyle\tilde{T_{x}}=(\prod_{r=\text{odd}}\sigma^{z}_{r})T_{x}\cdot\mathcal{K},$
$\displaystyle\tilde{\mathcal{C}}=(\prod_{r}\sigma^{x}_{r})\cdot\mathcal{K},$
$\displaystyle\mathcal{I}=(\prod_{r}\sigma^{x}_{r})\cdot(\prod_{r=\text{odd}}\sigma^{z}_{r})\cdot\mathcal{O}_{I}.$

The on-site matrix product operator (MPO) for the Hamiltonian Eq. (11) can be
written as a $5\times 5$ matrix $V^{[j]}=$

$\begin{pmatrix}\mathbbm{1}&0&0&0&0\\\ \sigma^{+}&0&0&0&0\\\
\sigma^{-}&0&0&0&0\\\ \sigma^{z}&0&0&0&0\\\
0&t_{j}\sigma^{-}+\tilde{\Delta}_{j}\sigma^{+}&t_{j}\sigma^{+}+\tilde{\Delta}_{j}^{*}\sigma^{-}&\frac{1}{4}V\sigma^{z}&\mathbbm{1}\end{pmatrix},$
(31)

where we define
$t_{j}\equiv(-)^{j}t,\tilde{\Delta}_{j}\equiv(-)^{j}\Delta+\text{i}\Delta^{\prime}$.
The boundary vectors for the open boundary condition (OBC) are
$v_{L}=(0,0,0,0,\mathbbm{1}),v_{R}=(\mathbbm{1},0,0,0,0)^{T}$.

Recall that $\mathcal{P}_{f}\rightarrow-\mathcal{P}_{f}$ under the inversion
operation $\mathcal{I}$ in an open chain of an odd length, according to (28).
As indicated by the LSM theorem earlier, a gapped ground state either breaks
inversion $\mathcal{I}$ or the parity symmetry $\mathcal{P}_{f}$. If
$\mathcal{P}_{f}$ is preserved while $\mathcal{I}$ is broken, it can be
characterized by the non-vanishing order parameters such as

$\displaystyle M_{\text{FM-}z}$
$\displaystyle\equiv\frac{1}{L}\sum_{j=-N}^{N}\sigma_{j}^{z},$ (32)
$\displaystyle M_{\text{AFM-}z}$
$\displaystyle\equiv\frac{1}{L}\sum_{j=-N}^{N}(-)^{j}\sigma_{j}^{z},$

which are invariant under the symmetry $\mathcal{P}_{f}$ whereas
$\mathcal{I}^{-1}M_{\text{(A)FM-}z}\mathcal{I}=-M_{\text{(A)FM-}z}$. Their
MPOs can be written in the form as

$V_{\text{FM-}z}^{[j]}=\begin{pmatrix}\mathbbm{1}&0\\\
\sigma^{z}&\mathbbm{1}\end{pmatrix},\quad
V_{\text{AFM-}z}^{[j]}=\begin{pmatrix}\mathbbm{1}&0\\\
e^{\text{i}\pi{j}}\sigma^{z}&\mathbbm{1}\end{pmatrix}.$ (33)

To detect possible symmetry breaking of $\mathcal{P}_{f}$, we use the
following inversion-symmetric order parameter

$M_{x}=\frac{1}{L}\sum_{j=-N}^{N}(-)^{\frac{j(j+1)}{2}}\sigma_{j}^{x}.$ (34)

Since $\mathcal{P}_{f}^{-1}M_{x}\mathcal{P}_{f}=-M_{x}$, non-vanishing
$\langle{M}_{x}\rangle$ implies the spontaneous symmetry breaking of
$\mathcal{P}_{f}$. Although spontaneous symmetry breaking (SSB) cannot really
occur on a finite chain, DMRG tends to select a minimally entangled ground
state, which means $|\langle{M}\rangle|$ could be a good estimation for the
SSB. However, it can be very unstable near a quantum critical point, where a
macroscopic superposed cat state becomes possible. To overcome this
difficulty, we can add an explicit symmetry-breaking term such as
$H_{h}=h\left(\sigma_{-N}^{z}+\sigma_{N}^{z}\right)$ on the boundary to split
the degenerate ground states. Alternatively, we can use root mean square order
parameter $\sqrt{\langle{M}^{2}\rangle}$ instead of the order parameter
$|\langle{M}\rangle|$ itself to detect SSB Binder and Heermann (2010); Pang
_et al._ (2019). In addition to these order parameters, another alternative
way to detect spontaneous symmetry breaking is to look at the two point
correlation function associated with the corresponding order parameter Beekman
_et al._ (2019). The advantage of the correlation function is that it can be
used in finite systems where the expectation value of the order parameter is
zero.

###  III.2 Matrix product state, variance and entanglement

For Eq. (11) on a finite chain consisting the number of sites $L$ under OBC,
its many-body wavefunction can be represented by a matrix product state (MPS)
as

$|\psi\rangle=\sum_{\left\\{s\right\\}}\left(A^{s_{0}}A^{s_{1}}\dots
A^{s_{L-1}}\right)|s_{0},s_{1},\dots,s_{L-1}\rangle.$ (35)

$A^{s_{0},s_{L-1}}$ are two boundary vectors with dimensions $1\times{d}$ and
$d\times{1}$, respectively. Here $d=2$ denotes the dimension of the local
Hilbert space. The _exact_ MPS requires the largest bond dimension
$\chi_{\text{max}}=\sqrt{d}^{L}$ at the center of the chain. However, it is
not practically achievable therefore we need to put a fixed cut-off $\chi$ to
compress the wavefunction. Written in the Schmidt basis Li and Haldane (2008)
in terms of two spatial parts $A$ and $B$, the wavefunction looks like

$|\psi\rangle=\sum_{\alpha=0}^{\chi-1}e^{-\frac{\omega_{\alpha}}{2}}|\alpha\rangle_{A}\otimes|\alpha\rangle_{B},$
(36)

where $\\{\omega\\}$ is the _entanglement spectrum_ , which is the eigenvalue
spectrum of the local entanglement Hamiltonian $\mathcal{H}_{A}$ defined by
the reduced density matrix $\rho_{A}={e}^{-\mathcal{H}_{A}}$.
$|\alpha\rangle_{A,B}$ are the orthogonal Schmidt basis. By using the
variational method Schollwöck (2005, 2011), we can obtain a ground state of
the corresponding Hamiltonian iteratively. We have tested that both random
initial MPS and the “infinite” method to initialize the system provide
accurate ground states. An generic and effective numerical criterion to
estimate how accurately the wavefunction is approximated by an MPS with a
fixed bond dimension is the so-called _variance_ McCulloch (2007); Hubig  _et
al._ (2018)

$v^{2}=\langle\psi|H^{2}-E_{0}^{2}|\psi\rangle,$ (37)

which is easy to compute in the MPS-MPO framework Schollwöck (2011). It is
known that $v$ is proportional to the truncation error in the density matrix
renormalization group (DMRG) method McCulloch (2007). Moreover $v$ is
essentially determined by the distribution of the entanglement spectrum
Calabrese and Lefevre (2008). For a gapped state, it tends to be exponentially
small given a sufficient but not too large $\chi$. When the system is
approaching criticality and becoming long-range entangled, $v$ would increase
rapidly since the correlation length $\xi$ becomes as large as the system size
and the entanglement spectrum is much more densely distributed, resulting in a
much larger entanglement entropy (EE)
$S_{A}=-\text{tr}\left(\rho_{A}\ln\rho_{A}\right)\propto\ln\xi$ in comparison
to gapped ones Calabrese and Cardy (2004); Pollmann  _et al._ (2009). Variance
can be used to distinguish different phases and identify the critical points
between them.

Because of the translational symmetry breaking by the OBC on a finite lattice,
the open ends can induce dimerization and hence oscillations in bond energy
and EE in 1d quantum many-body systems Laflorencie  _et al._ (2006); Affleck
_et al._ (2009). Therefore, the bipartite EE for subsystem $A$ has the
following form Mudry  _et al._ (2019)

$S_{A}(l)=S_{A}^{u}(l)+(-)^{l}S_{A}^{o}(l)+S_{0},$ (38)

where $l=0,\dots,L-2$ denotes the links. $S_{A}^{u}(l)$ is the uniform part
and $S_{A}^{o}(l)$ is the oscillation part of the EE. $S_{A}^{o}(l)$ is
phenomenologically proportional to the oscillatory part of the bond energy as
$S_{A}^{o}(l)=\alpha E^{o}(l)$ Laflorencie  _et al._ (2006). The bond energy
behaves as $E_{b}(l)\equiv\langle{h_{l}}\rangle=E_{b}^{u}+(-)^{l}E_{b}^{o}(l)$
where $E_{b}^{u}$ is a constant, which can be extracted as
$E_{b}^{u}=\frac{1}{2}\left[E_{b}(L/2)+E_{b}(L/2+1)\right]$. Once
$S_{A}^{u}(l)$ is extracted by fitting and finding the optimal $\alpha$, on a
finite lattice with the number of bonds $\mathcal{L}=L-1$, Cardy’s formula
under OBC reads Calabrese and Cardy (2004)

$S_{A}^{u}(l)=\frac{c}{6}\ln\left[\frac{2\mathcal{L}}{\pi}\sin\frac{\pi(l+1/2)}{\mathcal{L}}\right]+S_{0},$
(39)

where $c$ is the central charge characterizing the corresponding conformal
field theory (CFT), and $l=0,\dots,\mathcal{L}-1$. However, in a 1d chain
consisting of an odd number of sites, EE will develop plateaus stemming from
the incommensurate oscillations Deschner and Sørensen (2013) and exact-zero
modes in our LSM system, which may cause an underestimated central charge. We
demonstrate this issue with a simpler example, the 1d XY model, in the
Appendix B.

##  IV The phase diagram

Before discussing the phase diagram of the 1d system, we first restrict the
phase space to be studied by symmetry analysis. In the fermion model Eq. (1)
parametrized by $(t,\Delta,\Delta^{\prime},V)$, it is straightforward to
verify the following symmetries:

$\displaystyle
H^{\ast}(t,\Delta,\Delta^{\prime},V)=H(t,\Delta,-\Delta^{\prime},V),$
$\displaystyle
T_{x}H(t,\Delta,\Delta^{\prime},V)T_{x}^{-1}=H(-t,-\Delta,\Delta^{\prime},V),$
$\displaystyle e^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{\pi}{2}\sum_{r}\hat{n}_{r}}H(t,\Delta,\Delta^{\prime},V)e^{-\hskip
1.0pt\mathrm{i}\hskip
1.0pt\frac{\pi}{2}\sum_{r}\hat{n}_{r}}=H(t,-\Delta,-\Delta^{\prime},V).$

In other words, changing the sign of hopping $t$, or real pairing $\Delta$, or
imaginary pairing $\Delta^{\prime}$ does not affect the spectrum of the 1d
chain, in the thermodynamic limit. Therefore, we set $t=1$ to be a positive
constant, and restrict our numerical studies to the parameter regime
$\Delta>0,~{}\Delta^{\prime}>0$. Below we present our results on the phase
diagram of 1d model as Eq. (1) or (11).

###  IV.1 The non-interacting limit

If $V=0$, Eq. (1) is free-fermion model that can be solved exactly, and it
provides a good starting point to understand the full phase diagram of the
interacting model. We consider a closed 1d chain of $\mathcal{N}=2N$ sites (or
$N$ unit cells) Under periodic boundary condition (PBC), where we label the
sites by $j\in\\{0,\dots,\mathcal{N}-1\\}$. Note that there are two sites
$j=0,\mathcal{N}/2$ as inversion centers in this 1d chain. There are
$N=\mathcal{N}/2$ unit cells, labeled as $l=0,\cdots,N-1$, with the site index
$j=2l+\alpha$, $l=0,\dots,N-1$, $\alpha=0,1$. Therefore the free fermion
Hamiltonian can be rewritten as

$\displaystyle H_{0}$
$\displaystyle=\sum_{\alpha,l}\left[(-)^{\alpha}tc_{2l+\alpha}^{\dagger}c_{2l+\alpha+1}+h.c.\right]$
(40)
$\displaystyle+\sum_{\alpha,l}\left(\left[(-)^{\alpha}\Delta+\text{i}\Delta^{\prime}\right]c_{2l+\alpha}^{\dagger}c_{2l+\alpha+1}^{\dagger}+h.c.\right).$

Fourier transformations are defined as

$d_{k,\alpha}=\frac{1}{\sqrt{N}}\sum_{l=0}^{N-1}e^{\text{i}kl}c_{2l+\alpha}$
(41)

In the spinor basis of
$\eta_{k}=(d_{k,0},d_{-k,0}^{\dagger},d_{k,1},d_{-k,1}^{\dagger})^{T}$,
$H_{0}$ has the following form in momentum space

$H_{0}=\sum_{k\geqslant 0}\eta_{k}^{\dagger}\Gamma(k)\eta_{k},$ (42)

with

$\Gamma(k)=\begin{pmatrix}0&0&t(k)&\Delta(k)\\\ 0&0&-\Delta^{*}(-k)&-t(k)\\\
t^{*}(k)&-\Delta(-k)&0&0\\\ \Delta^{*}(k)&-t^{*}(k)&0&0\end{pmatrix},$ (43)

where we define $t(k)\equiv
t(1-e^{\text{i}k})=-2\text{i}t\sin\left(\frac{k}{2}\right)e^{\text{i}k/2}$ and
$\Delta(k)\equiv\left[\Delta\left(1+e^{\text{i}k}\right)+\text{i}\Delta^{\prime}\left(1-e^{\text{i}k}\right)\right]=2\left[\Delta\cos\left(\frac{k}{2}\right)+\Delta^{\prime}\sin\left(\frac{k}{2}\right)\right]e^{\text{i}k/2}$.
Note that $t(-k)=t^{*}(k)$. We can obtain the four-band dispersion relations
as $\pm\epsilon(\pm k)$, where

$\epsilon(k)=2\left[\sqrt{t^{2}\sin^{2}\left(\frac{k}{2}\right)+\Delta^{2}\cos^{2}\left(\frac{k}{2}\right)}+\Delta^{\prime}\sin\left(\frac{k}{2}\right)\right].$
(44)

We denote the two positive eigenvalues as
$\epsilon_{0}(k)\geqslant\epsilon_{1}(k)\geqslant 0,~{}\forall{k}\geqslant 0$.
Therefore the Hamiltonian will be diagonalized to the form
$H_{0}=\sum_{k\geqslant 0}\gamma_{k}^{\dagger}\Lambda(k)\gamma_{k}$, in which
$\Lambda(k)=\text{diag}\left\\{\epsilon_{0}(k),-\epsilon_{0}(k),\epsilon_{1}(k),-\epsilon_{1}(k)\right\\},\gamma_{k}=(f_{k,0},f_{-k,0}^{\dagger},f_{k,1},f_{-k,1}^{\dagger})^{T}$.
If we set $t=1.0$ as the energy unit, the non-interacting phase diagram
$(\Delta,\Delta^{\prime})$ for $V=0$ is illustrated in FIG. 1. Some
representative cases are discussed as follows:

  * •

$\Delta=0,\Delta^{\prime}=1.0$ is a special case where the lower band is flat
lying exactly at zero energy. It means that gapless excitations appear for all
$k$. If $\Delta^{\prime}\neq{1.0}$, the system features two linearly
dispersing Majorana modes at $k_{0}=0$: they have different velocities and are
hence not conformally invariant.

  * •

$\Delta\neq 0,\Delta^{\prime}<1.0$ always give us a gapped superconductor,
i.e. a Majorana chain. Particularly, $\Delta=1.0,\Delta^{\prime}=0.0$ features
a flat band spectrum of Bogoliubov quasiparticles.

  * •

$\Delta\neq 0,\Delta^{\prime}=1.0$ is gapless at $k_{0}=\pi$. The dispersion
is expanded as $\pm\epsilon_{1}(k_{0}+\delta k)\approx\pm\frac{1}{4}\delta
k^{2}+\mathcal{O}(\delta k^{2})$, which is quadratic near $k_{0}=\pi$.

  * •

$\Delta\neq 0,\Delta^{\prime}>1.0$ gives rise to a gapless phase with a pair
of linearly dispersing Majorana modes. The gapless point is located at
$k_{0}=2\arctan\frac{\Delta}{\sqrt{\Delta^{\prime 2}-t^{2}}}$. Around this
point, the dispersion relation reads $\pm\epsilon_{1}(k_{0}+\delta
k)\approx\pm{v_{s}}(k_{0})\delta{k}+\mathcal{O}(\delta k^{2})$, which is
linear. The speed of the Majorana mode is

$v_{s}(k_{0})=\frac{t^{2}-\Delta^{2}-\Delta^{\prime
2}}{\Delta^{\prime}}\cos\left(\frac{k_{0}}{2}\right).$ (45)

$\Delta$$\Delta^{\prime}$ Figure 1: Phase diagram for Eq. (1) with $V=0$. Gray
area denotes gapless phases whereas white areas denote the gapped Kitaev
chain. The axis perpendicular to $(\Delta,\Delta^{\prime})$ is to add the non-
vanishing interaction $V$.

###  IV.2 A representative study for the interacting cases: $\Delta=1.0$

By fixing $\Delta=1.0$ and selecting different $\Delta^{\prime}$, we can
obtain a rough idea of the whole phase diagram for the interacting model Eq.
(1). First we choose a fixed system size $L=121$, which we find is large
enough to accurately illustrate the phase diagram. Note that, in the main
text, the bond dimension of all MPSs is fixed at $\chi=64$, which we find is
sufficient to obtain converged physical measurements. A larger bond dimension
is tested and verified in Appendix C. Furthermore, we select
$\Delta^{\prime}=0.0,0.5$, which belong to gapped superconductor phases at
$V=0$; and $\Delta^{\prime}=1.0,1.5$, which are gapless at $V=0$ as we
discussed above in the non-interacting limit. These four cases are illustrated
by red dots in FIG. 1. For $V\rightarrow\pm\infty$, obviously Eq. (11) will
lead to antiferromagnetic$-z$ (ferromagnetic$-z$) states, respectively.
Therefore, we restrict ourselves to scan the parameter range
${-15.0}\leqslant{V}\leqslant{15.0}$, which turns out to be sufficient.

Once obtaining a converged ground state from a randomly initialized MPS, first
we plot the middle-bond EE $S_{m}$ and variance $v$ in FIG. 2 as the functions
of $V$, in which we can identify the gapped ground states with a vanishing
variance $v$. EE and variance maximize at the same time implying that the
system is approaching a critical point. For $V>0$, we notice that that there
is a robust critical point at $V\approx{4.0}$, which seems independent of the
choice of $\Delta^{\prime}$. While for $V<0$, non-vanishing $\Delta^{\prime}$
can expand one critical point for $\Delta^{\prime}=0.0$ (see FIG. 2(a)) to a
finite gapless regime as shown in FIG. 2(b, c, d). Larger $\Delta^{\prime}$
induces a wider gapless phase. For example, $\Delta^{\prime}=1.5$ can induce a
critical phase for $-5.0\lessapprox{V}\lessapprox{2.0}$. Within this gapless
phase, the variance $v$ also fluctuates and shows distinctions between $V>0$
and $V<0$ regions, which implies that the gapless phases can be further
distinguished. By taking both FIG. 3(d) and FIG 2(d) into consideration, we
can conclude that the left phase boundary at $V\approx{-5.0}$ of this
$\Delta^{\prime}$ induced gapless phase is of a quantum Berezinsky-Kosterlitz-
Thouless (BKT) Kosterlitz and Thouless (1973) type, characterized by a central
charge $c=1$. The right phase boundary at $V\approx 2.0$ is of a first-order
type. Different types of quantum phases and their transitions can also be
further illustrated by the distributions of bipartite entanglement spectrum in
the ground state. We use $\Delta^{\prime}=0.5,1.5$ as two examples, which are
shown in FIG. 4.

In FIG. 5, we plot two examples of EE and bond energy for
$\Delta=1.0,\Delta^{\prime}=1.5$. FIG. 5(a, c) denote $V=-3.0$, which is deep
in the gapless phase. By fitting from Cardy’s formula Eq. (39), the central
charge reads $c\approx{0.574}$ for the odd lattice $L=121$, where we can see
that the oscillations near the center of the 1d chain is incommensurate. In
comparison, in an even lattice of $L=120$, the central charge
$c\approx{0.917}$ turns out to be quite different. We address this issue in
Appendix B and attribute the underestimation of central charge on odd lattices
to the exact ground state degeneracy, due to anti-commutation relation between
inversion and parity symmetries. Even lattices can be regarded as a
perturbation which opens a small finite-size gap between the two degenerate
ground states on an odd lattice, which leads to more accurate estimations of
the central charge. In FIG. 5(b, d), $V=1.8$ lies in the gapless phase but in
proximity to the first order transition into the Kitaev chain. Compared to
$V=-3.0$ case in FIG. 5(a, c), here the oscillations in terms of both EE and
bond energy are quite different, with a larger period persisting into the
bulk. We found that fitting to Cardy’s formula failed to produce a sensible
central charge.

In FIG. 6, we plot the order parameters defined in Eq. (32, 34), from which we
can see that, for $\Delta^{\prime}=1.5$, in the gapped phase
$2.0\lessapprox{V}\lessapprox{4.0}$, the non-vanishing $\langle{M}_{x}\rangle$
implies the spontaneous breaking of parity symmetry $\mathcal{P}$, pointing to
a Kitaev chain. Meanwhile for $V\rightarrow\pm\infty$, non-vanishing
$\langle{M}_{z}\rangle$ implies breaking of inversion symmetry $\mathcal{I}$.
Because of their incompatible unbroken symmetries, the phase transition
between these two symmetry breaking phases is beyond the Landau paradigm of
spontaneous symmetry breaking. It requires a more detailed study, which we
present in the next section.

Figure 2: Middle-bond entanglement entropy $S_{m}$ (blue circle) and variance
$v$ (purple cross) in the case of $\Delta=1.0$. $L=121$. (a, b, c, d) denote
$\Delta^{\prime}=(0.0,0.5,1.0,1.5)$, respectively.  Figure 3: First (blue
circle markers) and second (purple triangle markers) derivatives of the ground
state energy density $\epsilon_{0}$. $\Delta=1.0$. $L=121$. (a, b, c, d)
denote $\Delta^{\prime}=(0.0,0.5,1.0,1.5)$, respectively.  Figure 4: Middle-
bond bipartite entanglement spectrum defined by Eq. (36). $\Delta=1.0$.
$L=121$. (a) $\Delta^{\prime}=0.5$. (b) $\Delta^{\prime}=1.5$.  Figure 5: Bond
energy $E_{b}$ (insets) and bipartite EE $S_{A}$.
$\Delta=1.0,\Delta^{\prime}=1.5$. (a, b) $L=121$. (c, d) $L=120$. (a, c)
$V=-3.0$. (b, d) $V=1.8$. Hollow circles represent the original data while
filled circles represent the extracted uniform EE as defined in Eq. (38).
Figure 6: Magnetic order parameters $M_{z}$ and $M_{x}$ in the case of
$\Delta=1.0$. $L=121$. For $V>0$, $M_{z}=M_{\text{AFM}-z}$. For $V<0$,
$M_{z}=M_{\text{FM}-z}$. (a, b, c, d) denote
$\Delta^{\prime}=(0.0,0.5,1.0,1.5)$, respectively.

###  IV.3 Overview of the whole phase diagram

We also obtained data for $\Delta=0.5,1.5,2.0$ in a range of different
$\Delta^{\prime}$. The general structure of the phase diagrams as a function
of $V$ are similar to $\Delta=1.0$ case. In particular, a larger $\Delta$ will
drive the critical point $V_{c}$ between the Kitaev chain ($\langle
M_{x}\rangle\neq 0$) and inversion-breaking superconductor ($\langle
M_{z}\rangle\neq 0$) to a larger value. based on these numerical results, we
can qualitatively draw the schematic phase diagrams as shown in FIG. 7: (a)
for a fixed $\Delta=1.0$ and (b) a three-dimensional phase diagram as a
function of $(V,\Delta,\Delta^{\prime})$.

$V$$\Delta^{\prime}$TSC (M-$x$)AFM-$z$FM-$z$(a)$V$$\Delta^{\prime}$$\Delta$TSC
(M-$x$)AFM-$z$FM-$z$GaplessGapless(b) Figure 7: (a) Planar schematic phase
diagram $(V,\Delta^{\prime})$ with $\Delta=1.0$. Gray areas denote gapless
regimes. Red points denote DQCPs. (b) Three-dimensional schematic phase
diagram $(V,\Delta,\Delta^{\prime})$ for the model. Red faces denote
continuous second-order phase transitions. Blue face denotes discontinuous
first-order phase transitions. Yellow face denotes BKT-type phase transitions.
Closed faces form gapless regimes.

##  V Characterizing the critical behaviors

###  V.1 Finite-size analysis of the critical points

After having a basic understanding of the phase diagram, we take a closer look
at the critical point separating the parity-breaking Kitaev chain and the
inversion-breaking superconductor. As we have mentioned, since the two gapped
phases are not related to each other by spontaneous symmetry breaking, this
unconventional critical point is beyond the Ginzburg-Landau-Wilson paradigm.

In this subsection, we focus on the case with $\Delta=1.0$. In the first
place, we compute the Binder cumulant Binder (1981a, b)

$U_{L}=\frac{\langle{M}^{4}\rangle}{\langle{M}^{2}\rangle^{2}}$ (46)

around the unconventional critical point $V\approx{4.0}$. It follows the
finite-size scaling ansatz $U_{L}=g_{U}\left(|\delta|L^{1/\nu}\right)$, in
which $g_{U}$ does not scale with $L$. $\nu$ is the critical exponent for the
correlation length $\xi=|\delta|^{-\nu}$, in which $\delta$ is the reduced
interacting parameter defined as $\delta\equiv{V-V_{c}}$. Since the function
$g_{U}$ is independent of the finite lattice sizes at the critical point,
numerical data for $U_{L}$ given by different lattice sizes will intersect at
the same point.

In FIG. 8 we plot two Binder cumulants $U_{z}$ and $U_{x}$ corresponding to
$M_{\text{AFM}-z}$ and $M_{x}$, respectively. Furthermore, if we compute the
derivative of the Binder cumulant, we can extract the correlation-length
critical exponent $\nu$ since

$\frac{dU}{dV}\propto{L}^{1/\nu}$ (47)

and it also reaches its maximum at the critical point Nahum  _et al._ (2015).

Figure 8: Binder cumulants $U_{z}$ (monotonically increasing solid markers)
and $U_{x}$ (monotonically decreasing hollow markers) around the critical
point in the case of $\Delta=1.0$. (a, b, c, d) denote
$\Delta^{\prime}=(0.0,0.5,1.0,1.5)$, respectively. Dashed lines mark the
intersection point(s) for $U_{z}$ and $U_{x}$.  Figure 9: Derivatives of the
Binder cumulants $dU_{z}/dV$ (solid markers) and $dU_{x}/dV$ (hollow markers)
around the critical point. $\Delta=1.0$. (a, b, c, d) denote
$\Delta^{\prime}=(0.0,0.5,1.0,1.5)$, respectively. Dashed lines mark the peaks
for $dU_{z}/dV$ and $dU_{x}/dV$. Insets show the finite-size logarithm
fittings at the critical point(s) marked by the dashed lines.  Figure 10:
Magnetic order parameters $M_{\text{AFM}-z}$ (monotonically increasing solid
data markers) and $M_{x}$ (monotonically decreasing hollow data markers)
around the critical point. $\Delta=1.0$. (a, b, c, d) denote
$\Delta^{\prime}=(0.0,0.5,1.0,1.5)$, respectively. Insets show the logarithm
finite-size fittings at the critical point(s).

For $\Delta^{\prime}=0.0$ as shown in FIG. 8(a), 9(a), the two critical points
determined by $U_{z}$ and $U_{x}$ coincide with each other, leading to a
single critical point between the two gapped symmetry-breaking phases. This
unique critical point is an analog of the deconfined quantum critical point
(DQCP) in 2D Senthil  _et al._ (2004b, a), as we will discuss in more detail
in the next subsection. As we gradually increase $\Delta^{\prime}>0$, we find
that this single critical point starts to split into two, hosting a stable
gapless phase in between. The two phase boundaries adjacent to it are
determined by the scaling of two order parameters $M_{x}$ and $M_{z}$.
Furthermore, from the finite-size analysis in FIG. 10, we can infer that both
order parameters will vanish within this narrow gapless phase in the
thermodynamic limit, suggesting that it preserves both inversion and parity
symmetries. This gapless nature of this phase is also inferred by a large
central charge within it, which will be discussed in details later.

After nailing down the critical points, we turn to the finite-size scaling
study for the corresponding magnetizations as

$\langle{M}\rangle=L^{-\beta/\nu}g_{M}\left(|\delta|L^{1/\nu}\right),$ (48)

where $\beta$ is the critical exponent for the magnetic order parameter.
Therefore, exactly at the critical point we have
$\langle{M}\rangle\propto{L^{-\beta/\nu}}$, which can be used to extract the
related critical exponents. This procedure is shown in FIG. 10 and insets
there. All these results are summarized in TABLE 1, where we can see that
$M_{z,x}$ share the same critical exponents up to numerical errors even when
the single critical point splits into two phase boundaries. This suggests an
emergent symmetry relating the two order parameters. When the two order
parameters become critical at the same point, this single critical point is a
1d DQCP as we will discuss soon.

For a larger $\Delta$, we find that the single DQCP seems to persist for the
full range of $\Delta^{\prime}$, instead of splitting into two phase
boundaries with a gapless phase in between. We list the critical points and
exponents for $\Delta=1.5$ in TABLE 2. Finally, we want to mention that in
addition to the numerical error summarized in the Table, the finite step size
$\delta V=0.02$ of data points can also lead to errors of the critical
exponents, which is hard to evaluate.

Table 1: Critical point(s) and critical exponents for $\Delta=1.0$. $\Delta^{\prime}$ | $\left[V_{c}^{z},V_{c}^{x}\right]$ | $\nu_{z}$ | $\beta_{z}/\nu_{z}$ | $\nu_{x}$ | $\beta_{x}/\nu_{x}$
---|---|---|---|---|---
0.0 | [4.00, 4.00] | 1.01(0) | 0.24(4) | 1.01(1) | 0.24(4)
0.5 | [4.01, 4.00] | 1.02(7) | 0.23(9) | 1.02(9) | 0.23(4)
1.0 | [4.03, 4.02] | 1.08(5) | 0.24(9) | 1.08(5) | 0.24(5)
1.5 | [4.10, 4.03] | 1.20(1) | 0.22(3) | 1.21(1) | 0.22(8)
Table 2: Critical point(s) and critical exponents for $\Delta=1.5$. $\Delta^{\prime}$ | $\left[V_{c}^{z},V_{c}^{x}\right]$ | $\nu_{z}$ | $\beta_{z}/\nu_{z}$ | $\nu_{x}$ | $\beta_{x}/\nu_{x}$
---|---|---|---|---|---
0.0 | [5.00, 5.00] | 0.89(1) | 0.21(5) | 0.89(3) | 0.21(5)
0.5 | [5.00, 5.00] | 0.89(7) | 0.23(2) | 0.91(2) | 0.20(3)
1.0 | [5.02, 5.02] | 0.93(4) | 0.21(2) | 0.93(3) | 0.23(7)
1.5 | [5.04, 5.04] | 0.99(7) | 0.23(5) | 1.00(3) | 0.24(2)

###  V.2 Realizing DQCPs by fusing two dualed order parameters and the
possible breakdown of the Tomonaga-Luttinger liquid

We find that the parameter $\Delta$ determines the properties of the critical
point(s) for $V>0$. By fixing $\Delta^{\prime}=0.5$, we focus on two
representative cases of $\Delta=0.5,1.5$. The finite-size scaling analysis
similar to the previous section is shown in FIG. 11 and FIG. 12. The critical
exponents obtained from them are summarized in TABLE 3.

For $\Delta<1.0$, the two order parameters $M_{z,x}$ approach criticality at
different $V$, leaving a narrow gapless phase between them where both order
parameters $M_{z,x}$ vanish. Such a gapless phase is allowed by the LSM
theorem. The peaks in FIG. 11(a) are also less sharp, which means the the
corresponding critical exponent $\nu$ is larger. However, for $\Delta>1.0$,
$M_{z,x}$ approach criticality at one single critical point $V_{c}$. The
narrow gapless phase shrinks into a single critical point. The critical peaks
in FIG. 11(b) are also sharper and lead to a smaller $\nu$. In this sense, we
say the two phase boundaries of vanishing $M_{z,x}$ fuse together and realize
a DQCP.

Figure 11: Derivatives of the Binder cumulants $dU_{z}/dV$ (solid markers) and
$dU_{x}/dV$ (hollow markers) around the critical point(s). Insets show the
logarithm finite-size fittings at the critical point(s).
$\Delta^{\prime}=1.5$. (a) $\Delta=0.5$. (b) $\Delta=1.5$.  Figure 12:
Magnetic order parameters $M_{\text{AFM}-z}$ (monotonically increasing solid
markers) and $M_{x}$ (monotonically decreasing hollow markers) around the
critical point. $\Delta^{\prime}=1.5$. (a) $\Delta=0.5$. (b) $\Delta=1.5$.

To get more understanding for this process, we take a look at the effective
field action of a U(1) Tomonaga-Luttinger liquid (TLL) theory expressed in
terms of two variables $\phi$ and $\theta$:

$H_{LL}[\theta,\phi]=\frac{u}{2\pi}\int{dx}\left[\frac{1}{g}\left(\partial_{x}\theta\right)^{2}+g\left(\partial_{x}\phi\right)^{2}\right],$
(49)

where $u$ is the renormalized velocity and $g$ is the Luttinger parameter.
Here $\Pi=\partial_{x}\theta/\pi$ is the canonical conjugate of phase variable
$\phi$.

In the dictionary of Abelian bosonization, the spin operators in a
spin-$\frac{1}{2}$ chain can be expressed as

$\displaystyle S^{z}(x)\pm\hskip 1.0pt\mathrm{i}\hskip 1.0ptS^{x}(x)\sim
e^{\mp\hskip 1.0pt\mathrm{i}\hskip 1.0pt\theta}\big{[}(-1)^{x}\pm\cos
2\phi\big{]},$ (50) $\displaystyle
S^{y}(x)\sim-\frac{\nabla\phi}{\pi}+(-1)^{x}\cos 2\phi.$ (51)

Under the symmetry operations, the boson fields transform as

$\displaystyle\mathcal{I}:~{}~{}$
$\displaystyle\theta(x)\rightarrow\pi-\theta(-x),~{}\phi(x)\rightarrow\frac{\pi}{2}-\phi(-x),$
(52) $\displaystyle\mathcal{P}_{f}:~{}~{}$
$\displaystyle\theta(x)\rightarrow-\theta(x),~{}~{}\phi(x)\rightarrow\frac{\pi}{2}-\phi(x)$
(53) $\displaystyle\tilde{\mathcal{C}}:~{}~{}$
$\displaystyle\theta(x)\rightarrow\pi-\theta(x),~{}\phi(x)\rightarrow\phi(x),$
(54) $\displaystyle\tilde{T}_{x}:~{}~{}$
$\displaystyle\theta(x)\rightarrow\theta(x+1),~{}\phi(x)\rightarrow\frac{\pi}{2}-\phi(x+1).~{}$
(55)

where we have set the lattice constant as unity. Note that
$\tilde{\mathcal{C}}$ and $\tilde{T}_{x}$ are anti-unitary symmetries while
inversion $\mathcal{I}$ and Ising $\mathcal{P}_{f}$ symmetries are unitary.

The $J_{x}=J_{z}$ (or equivalently $2(t+\Delta)=V$) limit of our model (11)
corresponds to the XXZ spin-$\frac{1}{2}$ chain, characterized by Giamarchi
and Press (2004)

$\displaystyle J_{y}/J_{x,z}=2(t-\Delta)/V=-\cos(\pi g/2),$ (56)
$\displaystyle u(2-g)=\sqrt{V^{2}-4(t-\Delta)^{2}}=4\sqrt{t\Delta}$ (57)

for $V\geq 2|t-\Delta|$ in effective action (49). In particular, the
Heisenberg limit $V=2t,~{}\Delta=0$ with $g=2$ and $u=\pi V$, characterizes
the single critical point separating the Luttinger liquid phase at
$V>2(t-\Delta)$ and the Ising antiferromagnetic phase at $V<2(t-\Delta)$.

The leading-order back-scattering terms introduced by $|J_{x}-J_{z}|$, $J_{y}$
and $\Gamma$ couplings in (11) are

$\displaystyle\mathcal{H}_{b.s.}=g_{\theta}\cos(2\theta)+g_{y}\cos(4\phi)+g_{\Gamma}\nabla\phi\nabla\theta\cos\theta+\cdots$
(58)

In particular the $g_{\Gamma}$ term from the $\Gamma$ (or $\Delta^{\prime}$)
coupling in the lattice model has a scaling dimension of
$\text{dim}(g_{\Gamma})=2+\frac{g}{4}$ and is hence irrelevant, suggesting the
stability of the single critical point for a small $\Delta^{\prime}$ in the
$\Delta=0$ limit. While the dimension of $g_{y}$ term is $\text{dim}(\cos
4\phi)=4/g>2$, the $g_{\theta}$ term has a scaling dimension of
$\text{dim}(\cos 2\theta)=g<2$. Since the only relevant term is
$\cos(2\theta)$, when $g_{\theta}$ changes sign, the ground state goes through
a transition from the $AFM_{z}$ phase to the $AFM_{x}$ phase.

In this Luttinger liquid phase of model (11), the antiferromagnetic order
parameters for the $AFM_{z,x}$ phases in the bosonized language read

$\displaystyle M_{z}=\sum_{j}(-1)^{j}S^{z}_{j}\sim\cos\theta,$ (59)
$\displaystyle M_{x}=\sum_{j}(-1)^{j}S^{x}_{j}\sim\sin\theta,$ (60)

and they share the same scaling dimension

$\displaystyle\text{dim}[M_{x}]=\text{dim}[M_{z}]=\frac{g}{4}.$ (61)

We notice that a similar unification of two order parameters ($FM_{z}$ and
VBS) at a DQCP in 1D is recently discussed by Ref. Jiang and Motrunich (2019).
While in the context of Ref. Jiang and Motrunich (2019) the relation between
the two order parameters is only clear in the bosonized dual picture, here in
our example of 1D DQCP a standard bosonization treatment already reveals the
emergent symmetry between the two distinct order parameters $M_{x}$ and
$M_{z}$, one ($M_{z}$) breaking inversion symmetry $\mathcal{I}$ while the
other ($M_{x}$) breaks fermion parity $\mathcal{P}_{f}$.

The nature of this DQCP can be revealed by looking into the dual domain wall
variables of e.g. the $AFM_{z}$ phase. Here we follow the strategy of Ref.
Jiang and Motrunich (2019) to identify the projective symmetry action on the
domain wall variables $\\{\vec{\mu}_{j+1/2}|j\in\mathbb{Z}\\}$ on the spin
chain (11):

$\displaystyle\mu_{j+1/2}^{x}=\sigma_{j}^{z}\sigma^{z}_{j+1},$
$\displaystyle\mu^{z}_{j-1/2}\rho_{j}^{z}\mu^{z}_{j+1/2}=\sigma_{j}^{x},$
$\displaystyle\rho_{j}^{x}=\sigma_{j}^{z}$ (62)

which are constraint by the Gauss’ law

$\displaystyle\mu^{x}_{j+1/2}=\rho^{x}_{j}\rho^{x}_{j+1}$ (63)

Here $\rho_{j}^{z}$ is the link variable for the $Z_{2}$ gauge field, while
$\mu^{z}_{j+1/2}$ creates a $Z_{2}$ gauge charge (i.e. domain wall of $M_{z}$
order parameter) on the dual site $j+\frac{1}{2}$. In terms of the dual
variables, the symmetry operations can be written as

$\displaystyle\mathcal{P}_{f}=\prod_{j}\rho_{j}^{x}=\prod_{j=\text{odd}}\mu^{x}_{j+1/2}:~{}\mu^{z}_{j+1/2}\rightarrow(-1)^{j}\mu^{z}_{j+1/2};~{}~{}~{}~{}~{}$
(64)
$\displaystyle\tilde{T_{x}}=T_{x}\cdot\mathcal{K}:~{}\mu^{z}_{j+1/2}\rightarrow\mu^{z}_{j+3/2};$
(65)
$\displaystyle\tilde{\mathcal{C}}=(\prod_{j}\rho^{z}_{j})\cdot\mathcal{K}:~{}\mu^{z}_{j+1/2}\rightarrow\mu^{z}_{j+1/2};$
(66)
$\displaystyle\mathcal{I}=(\prod_{j}\rho^{z}_{j})\cdot\mathcal{O}_{I}:~{}~{}\mu^{z}_{j+1/2}\rightarrow\mu^{z}_{-j-1/2}.~{}~{}~{}$
(67)

Most importantly, the Ising/parity symmetry $\mathcal{P}_{f}$ and inversion
$\mathcal{I}$ anticommutes on the domain wall variable $\mu^{z}_{j+1/2}$ even
on a periodic spin chain of an even length:

$\displaystyle\mathcal{P}_{f}\cdot\mathcal{I}\circ\mu^{z}_{j+1/2}=-\mathcal{I}\cdot\mathcal{P}_{f}\circ\mu^{z}_{j+1/2}.$
(68)

This projective symmetry action on the domain wall variable is captured by a
nontrivial projective symmetry group Wen (2002), i.e. a nontrivial group
cohomology $\omega\in\mathcal{H}^{2}[G,Z_{2}]$ where $G$ is the symmetry group
of the spin chain Chen (2017). This means destruction of the $AFM_{z}$ phase
by condensing domain wall $\mu^{z}_{j+1/2}$ will inevitably breaks the
symmetry, leading to e.g. an $AFM_{x}$ phase which spontaneously breaks the
Ising symmetry. This is in parallel with 2D DQCPs where defects of one ordered
phase carry nontrivial quantum numbers of another symmetry, and condensation
of this defect will necessarily break another symmetry while restoring the
originally broken symmetry.

Regarding the critical exponents of this critical point, $\nu$ can be related
to the scaling dimension of the relevant perturbation $\cos(2\theta)$ by

$\nu=\frac{1}{\mathfrak{d}-g}$ (69)

in this TLL theory Jiang and Motrunich (2019). Here $\mathfrak{d}=2$ is the
space-time dimension. With the value of $\beta/\nu$ at hand and the general
scaling relation $2\beta=\nu(\mathfrak{d}-2+\eta)$ Francesco  _et al._ (1997),
we can immediately obtain the critical exponent $\eta$, namely the _anomalous
dimension_ for the two point correlation function
$\Gamma(n)=|n|^{2-\mathfrak{d}-\eta}$. The scaling dimension for the order
parameter reads $\text{dim}[M]=\eta/2=\beta/\nu$.

On one hand, from TABLE 3 fixed with $\Delta^{\prime}=1.5$ we can see that,
only for $\Delta=1.5$, these two order parameters vanish at the same critical
point. Numerical results of the critical exponents appears to be the same
within numerical errors. Moreover, they are consistent with Eq. (69) and (61),
implying that it is indeed described by a $c=1$ TLL theory in the long-
wavelength limit. More data in TABLE 2 suggests that the Luttinger parameter
$g$ can vary within a finite range if $g<2$. The scaling dimensions of
$M_{x,z}$ are also the same at the critical point. The emergent larger
$U(1)_{\theta}$ symmetry unifies these two order parameters together and can
rotate from one to the other.

On the other hand, if $\Delta=0.5$, the critical points for these two order
parameters split, leading to a stable gapless phase between the two ordered
phases. Numerical results of the critical exponents contradict Eq. (69) and
(61), which implies that this TLL theory is not valid any longer. The analysis
given by Abelian bosonization in Eq. (61) seems to break down for a small
$\Delta/t$. However, interestingly, notice that the scaling dimensions of
$M_{x,z}$ for the two phase boundaries where $M_{x,z}$ vanish respectively are
still identical within numerical error, but become smaller than their values
at the DQCP. Currently we do not have a good theoretical understanding of this
gapless phase or how it emerges from the TLL at larger $\Delta/t$, and we
leave these interesting questions for future works.

Table 3: Critical point(s) and critical exponents for $\Delta^{\prime}=1.5$. $\Delta$ | $\left[V_{c}^{z},V_{c}^{x}\right]$ | $\nu_{z}$ | $\beta_{z}/\nu_{z}$ | $\nu_{x}$ | $\beta_{x}/\nu_{x}$
---|---|---|---|---|---
0.5 | [3.34, 2.88] | 1.47(1) | 0.19(7) | 1.51(3) | 0.20(6)
1.0 | [4.10, 4.03] | 1.20(1) | 0.22(3) | 1.21(1) | 0.22(8)
1.5 | [5.04, 5.04] | 0.99(7) | 0.23(5) | 1.00(3) | 0.24(2)

###  V.3 Finite-size analysis of the central charge

As discussed in Appendix B in detail, in a 1d chain of an odd length, there
will be two exactly degenerate ground states with opposite fermion parities,
due to the anti-commutation relation between inversion and parity symmetry
operations. This leads an underestimated to entanglement entropy, and hence an
underestimated central charge (see e.g. FIG. 5(a)) by fitting the Cardy’s
formula Eq. (39) numerically.

To resolve this issue, we use an even system size, which splits the exact
degeneracy in the spectrum, as illustrated in Appendix E. Even lattices allow
us to extract the entanglement entropy and the central charge more reliably.
Here we consider the same parameter range as in Sec. V.2. The results are
shown in FIG. 13. At the DQCP in FIG. 13(b), $\Delta=\Delta^{\prime}=1.5$, the
critical exponents are given by $\nu=1$ and $g=1$ in TABLE 3. The
corresponding central charge is approaching unity as the system size
increases, consistent with a Tomonaga-Luttinger liquid (TLL). After the single
DQCP splits into two phase boundaries, within the stable gapless phase, the
central charge $c\rightarrow 1.3$ as we compute up to $L=1600$ in FIG. 13(a).
This suggests that the stable gapless phase sandwiched by the two gapped
symmetry-breaking phases cannot be described by a TLL with $c=1$.

In FIG. 13(c), we make the scaling analysis deep in the gapless phase at a
negative $V=-3.0$. This stable gapless phase lies between the FMz phase and
the Kitaev chain features a central charge of $c\rightarrow 1$, again pointing
to a TLL.

Figure 13: Finite-size scaling of fitted central charge $c$.
$\Delta^{\prime}=1.5$ on even lattices. $\chi=256$. (a) $\Delta=0.5,V=3.00$.
(b) $\Delta=1.5,V=5.04$. (c) $\Delta=1.0,V=-3.0$.

##  VI Concluding remarks

In this paper, we studied the phase diagram and quantum phase transitions in a
1d interacting fermion model with a Lieb-Schultz-Mattis (LSM) type anomaly. In
the presence of a site-centered inversion symmetry, any gapped symmetric
ground state must be a Kitaev chain with a Majorana zero mode on each open
end. Via the Jordan-Wigner transformation, it is equivalent to a spin-$1/2$
model whose gapped ground states must break either the inversion or an Ising
symmetry, which corresponds to the fermion parity in the fermion model. Such a
LSM system provides a rich playground to identify unconventional quantum phase
transitions between different ordered phases, not related to each other by
spontaneous symmetry breakings, hence beyond the Ginzburg-Landau-Wilson
paradigm.

To understand the phase diagram of a generic fermion model with symmetric
nearest-neighbor couplings, we first solve the non-interacting limit $V=0$. In
the interacting cases with $V\neq{0}$, we implement the variational MPS method
to numerically study the model. The phase diagram and phase boundaries are
obtained using variance, entanglement entropy and ground state energy.
Furthermore, we focus on the unconventional quantum phase transition between
the inversion-breaking phase and the parity-breaking phase (i.e. the Kitaev
chain), and carry out a detailed finite-size scaling analysis to extract
critical exponents. This is combined with Abelian bosonization and projective
symmetry group analysis, to understand the nature of this “deconfined”quantum
critical point (DQCP), where both inversion-breaking and parity-breaking order
parameters vanish simultaneously. We find that the numerically measured
critical exponents are captured by the Luttinger parameter in a $U(1)$
Luttinger liquid. We have also identified a stable gapless phase, which is
symmetric and separates the two symmetry-breaking phases. While the nature of
this gapless phase and how it emerges from the DQCP remains unknown, we leave
this interesting question for future works.

###### Acknowledgements.

WZ acknowledges helpful discussions with Xiao-Qi Sun, Shuo Yang and Shuai A.
Chen. This work is supported by National Science Foundation under award number
NSF DMR-1653769 (WZ,YML) and Partnerships for Research and Education in
Materials Grant No. NSF DMR-1828019 (DNS).

##  Appendix A Symmetry implementations on the fermion chain vs. the spin
chain

Here we address in detail the symmetry operations in the fermion language vs.
the spin language.

In the fermion model,

$\displaystyle H$
$\displaystyle=\sum_{j}(-)^{j}\left[t\left(c_{j}^{\dagger}c_{j+1}+h.c.\right)+\left(\Delta
c_{j}^{\dagger}c_{j+1}^{\dagger}+h.c.\right)\right]$ (70)
$\displaystyle+\left(\text{i}\Delta^{\prime}c_{j}^{\dagger}c_{j+1}^{\dagger}+h.c.\right)+V\left(n_{j}-\frac{1}{2}\right)\left(n_{j+1}-\frac{1}{2}\right)$

By the Jordan-Wigner transformation in Eq. (10), the above fermion model is
transformed into a spin chain

$H_{\text{spin}}=\sum_{j}\sum_{\alpha=x,y,z}J_{\alpha}S_{j}^{\alpha}S_{j+1}^{\alpha}+(-)^{j}\Gamma\left(S_{j}^{x}S_{j+1}^{y}+S_{j}^{y}S^{x}_{j+1}\right)$
(71)

where the exchange couplings are given by

$\displaystyle J_{x}=2(t+\Delta),~{}~{}J_{y}=2(t-\Delta),~{}~{}J_{z}=V,$
$\displaystyle\Gamma=-2\Delta^{\prime}.$ (72)

Three symmetries are clearly present in the spin model:

$\displaystyle\mathcal{P}_{f}=\prod_{j}Z_{j}:~{}(S_{j}^{x},S_{j}^{y},S_{j}^{z})\rightarrow(-S_{j}^{x},-S_{j}^{y},S^{z}_{j}),~{}~{}~{}$
(73)
$\displaystyle\tilde{\mathcal{C}}=(\prod_{j}X_{j})\cdot\mathcal{K}:~{}(S_{j}^{x},S_{j}^{y},S_{j}^{z})\rightarrow(S_{j}^{x},S_{j}^{y},-S^{z}_{j}),~{}~{}~{}$
(74)
$\displaystyle\tilde{T}_{x}=T_{x}\cdot\mathcal{K}:~{}(S_{j}^{x},S_{j}^{y},S_{j}^{z})\rightarrow(S_{j+1}^{x},-S_{j+1}^{y},S^{z}_{j+1}).~{}~{}$
(75)

They are nothing but the magnetic translation (4), fermion parity (i.e. Ising
symmetry) (13) and anti-unitary particle-hole symmetry (6) discussed in the
fermion context.

Although the Hamiltonian remains local in both the fermion and the spin
representations, due to the Jordan-Wigner string, a locality-preserving
symmetry operation in one representation may appear to be non-local in the
other representation. One example is the inversion symmetry (2) discussed in
this work. Below we write down two possible sets of inversion symmetry
operations: the first one is non-local in the spin language; the second one
preserves locality in the spin language but looks non-local in the fermion
language. Since all numerical simulations are carried out in the spin
representation, in the main text we will stick to the 2nd set of inversion
symmetry summarized in Appendix A.2.

###  A.1 Non-local inversion symmetry in the spin representation

We first consider the following inversion symmetry

$\displaystyle c_{j}\overset{\mathcal{I}}{\longrightarrow}\hskip
1.0pt\mathrm{i}\hskip 1.0ptc_{-j}^{\dagger}$ (76)

in the fermion chain. On an open spin chain of length $L=2N+1$, the associated
inversion symmetry generator in the spin model (30) is

$\displaystyle\mathcal{I}=e^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{\pi}{4}\sum_{j}(-1)^{j+N+1}\sigma^{z}_{j}}\cdot e^{\hskip
1.0pt\mathrm{i}\hskip
1.0pt\frac{\pi}{4}\mathcal{P}_{f}}(\prod_{r}\sigma^{x}_{r})\cdot\mathcal{O}_{I}.$
(77)

where $\mathcal{O}_{I}$ is the spatial inversion operator.

In terms of the spin language, the parity operator
$\mathcal{P}_{f}=\prod_{l=-N}^{N}\left(1-2c_{l}^{\dagger}c_{l}\right)=\prod_{l=-N}^{N}\left(-\sigma_{l}^{z}\right)$.
Note that
$\sigma_{j}^{+}\sigma_{j}^{z}=-\sigma_{j}^{+},\sigma_{j}^{z}\sigma_{j}^{+}=\sigma_{j}^{+}$.
Thus we have $\left\\{\mathcal{P}_{f},\sigma_{j}^{+}\right\\}=0$. Similarly,
$\left\\{\mathcal{P}_{f},\sigma_{j}^{-}\right\\}=0$. Since we have the inverse
Jordan-Wigner transformation
$\sigma_{j}^{z}=2c_{j}^{\dagger}c_{j}-1,\sigma_{j}^{-}=\prod_{l=-N}^{j-1}\left(-\sigma_{l}^{z}\right)c_{j},\sigma_{j}^{+}=c_{j}^{\dagger}\prod_{l=-N}^{j-1}\left(-\sigma_{l}^{z}\right)$.
Thus we can find that, under the inversion symmetry $\mathcal{I}$,

$\displaystyle\sigma_{j}^{z}$ $\displaystyle\rightarrow-\sigma_{-j}^{z},$ (78)
$\displaystyle\sigma_{j}^{-}$
$\displaystyle\rightarrow\text{i}(-)^{N+j}\sigma_{-j}^{+}\mathcal{P}_{f}=\text{i}(-)^{N+j+1}\mathcal{P}_{f}\sigma_{-j}^{+},$
$\displaystyle\sigma_{j}^{+}$
$\displaystyle\rightarrow\text{i}(-)^{N+j+1}\mathcal{P}_{f}\sigma_{-j}^{-}.$

Therefore, due to the Jordan-Wigner string, the above inversion symmetry
$\mathcal{I}$ is not a locality-preserving unitary:

$\displaystyle
S_{j}^{x}\overset{\mathcal{I}}{\longrightarrow}(-1)^{j+N+1}\mathcal{P}_{f}\cdot
S_{-j}^{y},$ $\displaystyle
S_{j}^{y}\overset{\mathcal{I}}{\longrightarrow}(-1)^{j+N+1}\mathcal{P}_{f}\cdot
S_{-j}^{x},$ $\displaystyle
S_{j}^{z}\overset{\mathcal{I}}{\longrightarrow}-S^{z}_{-j}.$ (79)

Interestingly, a (non-local) string order parameter is required to preserve
this non-local inversion symmetry, unlike the usual local order parameter for
the case of a locality-preserving symmetry. This is discussed in more detail
later, in Appendix D.

###  A.2 Locality-preserving inversion symmetry in the spin representation

Alternatively, there is also a locality-preserving inversion symmetry
preserved in the spin chain (71):

$\displaystyle\mathcal{I}=(\prod_{j}X_{j})\cdot\mathcal{O}_{I}$ (80)

under which the spins transform as

$\displaystyle(S_{j}^{x},S_{j}^{y},S_{j}^{z})\overset{\mathcal{I}}{\longrightarrow}(S_{-j}^{x},-S_{-j}^{y},-S_{-j}^{z}).$
(81)

This symmetry, however does not have a local form in the fermion language:

$\displaystyle
c_{j}\overset{\mathcal{I}}{\longrightarrow}-\mathcal{P}_{f}c_{-j}^{\dagger}$
(82)

We shall stick to this locality-preserving inversion symmetry in the main
text.

##  Appendix B Zero-mode, entanglement entropy and finite-size analysis in the
XY model on even and odd open chains

In this section, we use the pedagogical 1D XY model to illustrate the issue of
zero-mode and its effects on the EE on even and odd lattices under OBC, which
is similar to our model when it comes to the lower EE and underestimation of
the central charge. Different DMRG methods could also make some subtle
difference. The Hamiltonian is

$H_{XY}=\sum_{j=0}^{L-2}\left[\left(\frac{1+\gamma}{2}\right)\sigma_{j}^{x}\sigma_{j+1}^{x}+\left(\frac{1-\gamma}{2}\right)\sigma_{j}^{y}\sigma_{j+1}^{y}\right],$
(83)

where $\gamma$ is a free parameter. $\gamma=0$ is the critical point, at which
the system becomes gapless. Eq. (83) is equivalent to the free 1D
superconductor model

$H_{f\text{-}XY}=\sum_{j=0}^{L-2}\left[\left(c_{j}^{\dagger}c_{j+1}-c_{j}^{\dagger}c_{j+1}\right)+\gamma\left(c_{j}^{\dagger}c_{j+1}^{\dagger}-c_{j}^{\dagger}c_{j+1}^{\dagger}\right)\right].$
(84)

By a unitary transformation, the single quasi-particle spectrum of Eq. (84)
can be computed exactly van Hemmen (1980) as
$H_{f\text{-}XY}=\sum_{k=0}^{L-1}\lambda_{k}\left(d_{k}^{\dagger}d_{k}-\frac{1}{2}\right)$.
The many-body excitation spectrum is given by various filling combinations of
the single quasi-particle spectrum. For $\gamma=0$, on a finite odd lattice
$L$, we find that there is always an _exact_ zero-mode $\lambda_{0}=0$. While
on a even lattice, there is a finite-size non-zero but very small gap
$\lambda_{0}\neq{0}$. That is, on a finite odd lattice with OBC, the two
ground states $|\Psi_{0}\rangle=|0\rangle$ and
$|\Psi_{1}\rangle=d_{0}^{\dagger}|0\rangle$ are precisely degenerated. They
belong to different topological sectors characterized by the fermion parity
$\mathcal{P}_{f}$.

If the ground state of Eq. (84) is a Slater determinant, the reduced density
matrix of a subsystem $A$ containing $M$ sites can be written as
$\rho_{A}=e^{-\mathcal{H}_{A}}/\mathcal{Z}$ and its bipartite entanglement
spectrum $\\{\omega\\}$ can be analytically extracted from the correlation
matrix Peschel (2003); Peschel and Eisler (2009); Calabrese (2016).
$\mathcal{Z}$ is the partition function. Therefore the corresponding EE reads

$\displaystyle S_{A}$
$\displaystyle=-\text{tr}\left(\rho_{A}\ln\rho_{A}\right)$ (85)
$\displaystyle=-\sum_{l=0}^{M-1}\left(\frac{\omega_{l}}{2}\right)\tanh\left(\frac{\omega_{l}}{2}\right)+\sum_{l=0}^{M-1}\ln\left[2\cosh\left(\frac{\omega_{l}}{2}\right)\right].$

We find that bond energy and EE in $|\Psi_{0}\rangle$ and $|\Psi_{1}\rangle$
are the same as shown in FIG. 14. However, EE in the superposed state
$|\Psi\rangle=\alpha|\Psi_{0}\rangle+\beta|\Psi_{1}\rangle$ cannot be
analytically computed since the superposition of two Slater determinants may
not be written as another Slater determinant. If we simulate the XY chain
using a randomly initialized MPS, we can converge to the minimally entangled
state, which turns out to the superposition of $|\Psi_{0,1}\rangle$ and
results in a lower EE as well as a underestimation of the central charge. They
are illustrated in FIG. 15 and we think this is the reason for the
incommensurability observed in other odd spin chains Deschner and Sørensen
(2013). If we add boundary perturbations such as
$H_{1}=h\left(\sigma_{0}^{z}+\sigma_{L-1}^{z}\right)$ in the early sweeping
stage to select the MPS within a fixed parity sector, numerically we indeed
can obtain the results in FIG. 14 for odd lattices.

Next we carry out some finite-size analysis in terms of even and odd lattices
for the XY model. From FIG. 16 we can see that the variance $v$ is sharper on
odd lattices. We consider the Neel order parameter

$M_{x}=\frac{1}{L}\sum_{j=0}^{L-1}(-)^{j}\sigma_{j}^{x}$ (86)

and the corresponding Binder culmulant $U_{x}$. From FIG. 17 and FIG. 18 we
can see that on even and odd lattices, the XY model exhibits almost similar
critical properties up to some numerical errors. However, their EE can be
dramatically different.

Figure 14: Bond energy $E_{b}$ (insets) and bipartite EE $S_{A}$ in the XY
spin chain under OBC computed from the exact solution. $L=121$. Hollow circles
represent the original data while filled circles represent the extracted
uniform EE as defined in Eq. (38). (a) $|\Psi_{0}\rangle$ without zero-mode.
(b) $|\Psi_{1}\rangle$ with zero-mode. Both fitted central charges are
$c\approx{0.962}$. Figure 15: Bond energy $E_{b}$ (insets) and bipartite EE
$S_{A}$ in the open XY chain obtained by MPS with $\chi=64$. Hollow circles
represent the original data while filled circles represent the extracted
uniform EE as defined in Eq. (38). (a) $L=120$. Fitted central charge
$c\approx{0.960}$. (b) $L=121$. Fitted central charge $c\approx{0.375}$.
Figure 16: Variance of the open XY chain. $\chi=64$. Insets show the logarithm
fittings of $v$ at the critical point. (a) Even lattices. (b) Odd lattices.
Figure 17: (a) Binder cumulant’s derivative and (b) magnetization in XY model
on odd chains. $\chi=64$. Fitted critical exponents as given by insets:
$\nu_{x}=1.11(1),\beta_{x}/\nu_{x}=0.24(4)$. Figure 18: (a) Binder cumulant’s
derivative and (b) magnetization in XY model on even chains. $\chi=64$. Fitted
critical exponents as given by insets:
$\nu_{x}=1.02(9),\beta_{x}/\nu_{x}=0.24(4)$.

##  Appendix C Larger bond-dimension test

To test the convergence of our numerical computation, we also repeat the
simulation with the same parameters as in FIG. 9 and FIG. 10 up to the bond-
dimension $\chi=128$. They are re-plotted as FIG. 19 and FIG. 20. The critical
exponents obtained with $\chi=128$ are summarized in TABLE 4, in which we find
the numbers are almost identical to those shown in TABLE 1. In this sense, we
claim that our numerical simulation has already well converged with $\chi=64$
and the error-bar estimation is faithful.

Figure 19: Derivatives of the Binder cumulants $dU_{z}/dV$ (solid markers) and $dU_{x}/dV$ (hollow markers) around the critical point. $\chi=128$. $\Delta=1.0$. (a, b, c, d) denote $\Delta^{\prime}=(0.0,0.5,1.0,1.5)$, respectively.  Figure 20: Magnetic order parameters $M_{\text{AFM}-z}$ (monotonically increasing solid data markers) and $M_{x}$ (monotonically decreasing hollow data markers) around the critical point. $\chi=128$. $\Delta=1.5$. (a, b, c, d) denote $\Delta^{\prime}=(0.0,0.5,1.0,1.5)$, respectively. Table 4: Critical point(s) and critical exponents for $\Delta=1.0$. $\chi=128$. $\Delta^{\prime}$ | $\left[V_{c}^{z},V_{c}^{x}\right]$ | $\nu_{z}$ | $\beta_{z}/\nu_{z}$ | $\nu_{x}$ | $\beta_{x}/\nu_{x}$
---|---|---|---|---|---
0.0 | [4.00, 4.00] | 1.01(2) | 0.24(4) | 1.01(2) | 0.24(4)
0.5 | [4.01, 4.00] | 1.02(8) | 0.23(9) | 1.03(0) | 0.23(3)
1.0 | [4.03, 4.02] | 1.08(5) | 0.24(9) | 1.08(5) | 0.24(5)
1.5 | [4.10, 4.03] | 1.20(7) | 0.22(3) | 1.21(2) | 0.22(8)

##  Appendix D A string order parameter that preserves non-local inversion
symmetry (77)

We can construct another kind of non-local string order parameter according to
the rule given by Eq. (78). It is invariant under the inversion symmetry
$\mathcal{I}$ and looks like

$\displaystyle{O}_{\text{Str-}x}\equiv\sum_{j=-N}^{N}\frac{1+\text{i}(-)^{N+j+1}Q}{2}\sigma_{j}^{x}$
(87)
$\displaystyle=\sum_{j=-N}^{N}\frac{\sigma_{j}^{x}+(-)^{N+j+1}\sigma_{j}^{y}\prod_{l=-N,l\neq{j}}^{N}\sigma_{l}^{z}}{2}.$

To write ${O}_{\text{Str-}x}$ in a MPO form, we have to encode the operator
string into the productions of matrices living on _each_ site. It turns out
that the corresponding MPO has a dimension of $D=4N+2$ and can be written as

$V_{\text{Str-}x}^{[j]}=\frac{1}{2}\begin{pmatrix}\mathbbm{1}&\dots&0\\\
\vdots&\ddots&\vdots\\\ \sigma^{x}&\dots&\mathbbm{1}\end{pmatrix}.$ (88)

On different sites, $V_{\text{Str-}x}^{[j]}$ has different forms. For $j=-N$,
$\left(2V_{\text{Str-}x}^{[-N]}\right)_{D-1,D-2}=-\sigma^{y};\left(2V_{\text{Str-}x}^{[-N]}\right)_{D-1,D-2-l}=\sigma^{z},l=1,\dots,2N$.
For $j=N$,
$\left(2V_{\text{Str-}x}^{[N]}\right)_{1,0}=-\sigma^{y};\left(2V_{\text{Str-}x}^{[N]}\right)_{1+l,0}=\sigma^{z},l=1,\dots,2N$.
For $-N<j<N$,

$\displaystyle\left(2V_{\text{Str-}x}^{[j]}\right)_{D-1-(N+j)-l,D-2-(N+j)-l}$
(89) $\displaystyle=\left\\{\begin{aligned} &\sigma^{z},~{}l\neq(N+j);\\\
&(-)^{N+j+1}\sigma^{y},~{}l=(N+j)\end{aligned}\right.$

for $l=0,\dots,2N$.

Figure 21: String order parameter and $z$-magnetic order parameter.
$\Delta=1.0$. $L=121,\chi=64$. (a, b, c, d) denote
$\Delta^{\prime}=(0.0,0.5,1.0,1.5)$, respectively.

From FIG. 21 we can see that in the gapped TSC phases, the non-vanishing
string order parameter $\langle{O}_{\text{Str-}x}\rangle$ does imply the
spontaneous symmetry breaking of the associated parity symmetry
$\mathcal{P}_{f}$, which is consistent with the results given by the local
order parameter $M_{x}$.

##  Appendix E Numerical results on on even lattices as a perturbative
analysis

On a finite lattice under OBC, although the symmetry $\mathcal{I}$ is only
precisely well-defined on odd lattices as we discussed in the main text, we
can regard the even ones as a kind of perturbation on the boundary by removing
one site.

###  E.1 Critical exponents at the critical point(s)

In TABLE 5 we perform the finite-size analysis on even lattices following the
same parameters in TABLE 1, which show that although specific numbers are
different but they are close and follow the same trend.

Table 5: Critical exponents at $\Delta=1.0$ on even lattices $\Delta^{\prime}$ | $\left[V_{c}^{z},V_{c}^{x}\right]$ | $\nu_{z}$ | $\beta_{z}/\nu_{z}$ | $\nu_{x}$ | $\beta_{x}/\nu_{x}$
---|---|---|---|---|---
0.0 | [4.00, 4.00] | 1.00(7) | 0.24(4) | 1.00(7) | 0.24(4)
0.5 | [4.01, 4.00] | 1.02(3) | 0.24(1) | 1.02(3) | 0.23(7)
1.0 | [4.04, 4.01] | 1.06(8) | 0.23(6) | 1.07(1) | 0.23(2)
1.5 | [4.13, 4.01] | 1.14(9) | 0.19(3) | 1.16(3) | 0.20(9)

###  E.2 Central charge in the gapless phase

Away from the phase boundaries namely deeply in the gapless phase induced by
non-vanishing $\Delta^{\prime}$, we believe that Cardy’s formula could work
well. In FIG. 22 we show that the fitted central charge as well as the
variance on an even lattice $L=120$. Furthermore, we also present several
representative examples using larger lattices $L=400$ and $L=800$ with the
bond dimension $\chi=256$ to make more accurate estimation of the central
charge as shown in TAB. 6.

Figure 22: Fitted central charge $c$ and variance $v$ in the negative-$V$ gapless phase for $\Delta=1.0$ and $\Delta^{\prime}=1.5$ on an even lattice $L=120$. $\chi=64$. Table 6:  Fitted central charge $c$ in the negative-$V$ gapless phase for $\Delta=1.0,\Delta^{\prime}=1.5$ on other larger even lattices. $\chi=256$ for $L=400,800$. $\chi=512$ for $L=1000$. $V$ | -4.0 | -3.0 | -2.0 | -1.0 | 0.0 | 1.0
---|---|---|---|---|---|---
$L=400$ | 0.949 | 0.961 | 0.982 | 0.988 | 0.988 | 0.974
$L=800$ | 0.964 | 0.971 | 0.987 | 0.991 | 0.991 | 0.979
$L=1000$ | 0.970 | 0.975 | 0.992 | 0.996 | 0.995 | 0.989

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