# Synthetic Hall tube of interacting fermions

Xiaofan Zhou  State Key Laboratory of Quantum Optics and Quantum Optics
Devices, Institute of Laser spectroscopy, Shanxi University, Taiyuan 030006,
China  Gang Chen<EMAIL_ADDRESS>State Key Laboratory of Quantum Optics
and Quantum Optics Devices, Institute of Laser spectroscopy, Shanxi
University, Taiyuan 030006, China  Collaborative Innovation Center of Extreme
Optics, Shanxi University, Taiyuan, Shanxi 030006, China  Collaborative
Innovation Center of Light Manipulations and Applications, Shandong Normal
University, Jinan 250358, China  Suotang Jia  State Key Laboratory of Quantum
Optics and Quantum Optics Devices, Institute of Laser spectroscopy, Shanxi
University, Taiyuan 030006, China  Collaborative Innovation Center of Extreme
Optics, Shanxi University, Taiyuan, Shanxi 030006, China

###### Abstract

Motivated by a recent experiment [J. H. Han, et. al., Phys. Rev. Lett. 122,
065303 (2019)], we investigate many-body physics of interacting fermions in a
synthetic Hall tube, using state-of-the-art density-matrix renormalization-
group numerical method. Since the inter-leg couplings of this synthetic Hall
tube generate an interesting spin-tensor Zeeman field, exotic topological and
magnetic properties occur. Especially, four new quantum phases, such as
nontopological spin-vector and -tensor paramagnetic insulators, and
topological and nontopological spin-mixed paramagnetic insulators, are
predicted by calculating entanglement spectrum, entanglement entropies, energy
gaps, and local magnetic orders with 3 spin-vectors and 5 spin-tensors.
Moreover, the topologically magnetic phase transitions induced by the
interaction as well as the inter-leg couplings are also revealed. Our results
pave a new way to explore many-body (topological) states induced by both the
spiral spin-vector and -tensor Zeeman fields.

##  I Introduction

Since the discovery of the quantum Hall effect Hall , the exploration of novel
topological states of matter has been attracted great attention in both theory
and experiment, since they provide important applications in designing novel
quantum devices and processing quantum information. The Hofstadter-Harper
Hamiltonian is one of the fundamental models that are used to investigate
topological states Hofstadter1976 . The experimental realization of such
Hamiltonian in cold atomic gases opens up the avenue of simulating topological
states Goldman2016 ; Tai2017 ; Cooper2019 . In cold atoms system, the internal
degrees of freedom of atoms, such as the hyperfine spins Mancini2015 ;
Stuhl2015  and clock states Livi2016 ; zhou2017current , can be treated as
synthetic dimension to simulate the $D$+1 dimensional quantum physics using
$D$ dimensional lattices Boada2012 , e.g., four dimensional quantum Hall
effect Price2015  and chiral edge current of Hall ribbons Mancini2015 ;
Stuhl2015 .

Using three hyperfine states as synthetic lattice dimension and coupling them
through synthetic gauge fields by two-photon Raman process SDprl14 , spin-1
spin-orbit coupling Natu2015 ; Barbarino2015 ; spin-1SOC2016 ; SOC-1BEC1 ;
Pixley2017  and spin-tensor-momentum coupling Luo2017 ; Li2020  have also been
implemented. When the links between the hyperfine states are cyclic with a
gauge flux $\phi=2\pi/3$, the optical lattice can form a synthetic Hall tube
Barbarino2018 ; Luo2020 , which is a simple Hofstadter-Harper Hamiltonian
Hofstadter1976 . The synthetic Hall tube supports a generalized inversion
symmetry-protected topological insulator Nourse2016 , which is similar as the
integer quantum Hall state. Since the time-reversal, particle-hole, and chiral
symmetries are broken, this topological insulator belongs to the symmetry
class A (unitary) of the Altland-Zirnbauer classification Altland1997 ;
Schnyder2008 ; Ludwig2016 ; Chiu2016 . As varying one of the inter-leg
coupling strength, there exists a topological phase transition with a closing
band gap at critical point Barbarino2018 . In a recent experiment, this
interesting synthetic Hall tube has been realized successfully in the
alkaline-earth fermions Han2019 .

Apart from the single-particle quantum regulation in cold atoms experiments,
the interactions between the internal states can be controlled via Feshbach
resonances Chin2010 , and more importantly, generate rich many-body phenomena
Raghu2008 ; Bloch2008 ; Mueller2017 ; Rachel2018 ; Junemann2017 ; zhou2017SPT
; Leseleuc2019 . However, the interacting synthetic Hall tube has not been
fully investigated. In this paper, we investigate many-body properties of such
system, based on state-of-the-art density-matrix renormalization-group (DMRG)
numerical method dmrg1 ; dmrg2 . Since the inter-leg couplings of this
synthetic Hall tube generate an interesting spin-tensor Zeeman field, it is
very necessary to explore magnetic properties of the system, apart from the
interaction-driven topological transition. Due to the coexistence of the
spiral spin-vector and -tensor Zeeman fields in the synthetic Hall tube, local
magnetic orders with 3 spin-vectors and 5 spin-tensors should be introduced
zhou2020 . In terms of the calculated entanglement spectrum, entanglement
entropies, energy gaps, and local magnetic orders, we find four new quantum
phases such as nontopological spin-vector and -tensor paramagnetic insulators,
and topological and nontopological spin-mixed paramagnetic insulators.
Moreover, the topologically magnetic phase transitions induced by the
interaction as well as the inter-leg couplings are also revealed. Our results
pave a new way to explore many-body (topological) states induced by both the
spiral spin-vector and -tensor Zeeman fields.

Figure 1: (a) Schematics of the system setup with three Raman lasers
$R_{1,2,3}^{\sigma,\pi,\sigma}$, which are represented respectively by the
yellow, blue, and red arrows. $R_{1}^{\sigma}$ has an angle $\eta$ from
$x$-axis. A magnetic field $B$ along $z$-axis is applied to lift the spin
degeneracy of the ground state $|^{1}S_{0}\rangle$. (b) Three hyperfine spin
states in ground state $|^{1}S_{0}\rangle$ of alkaline-earth(-like) atoms
173Yb are coupled by three two-photon Raman transitions. (c) Synthetic Hall
tube with a uniform flux $\phi$ on each side plaquette and interaction $U$
between these hyperfine spin states.

##  II Model and Hamiltonian

Similar as Ref. Han2019 , here we consider the alkaline-earth fermions 173Yb
trapped in an effective one-dimensional optical lattice (along $x$-direction),
as shown in Fig. 1(a). Three hyperfine spin states of the ground state
$|^{1}S_{0}\rangle$, $|1\rangle=|F=5/2,m_{F}=-5/2\rangle$,
$|2\rangle=|F=5/2,m_{F}=-3/2\rangle$ and $|3\rangle=|F=5/2,m_{F}=-1/2\rangle$,
are chosen as three legs, as shown in Fig. 1(b). Three linearly-polarized
Raman laser beams $R_{1,2,3}^{\sigma,\pi,\sigma}$ are used to make three two-
photon Raman transitions between the states $|^{1}S_{0},F=5/2\rangle$ and
$|^{3}P_{1},F=7/2\rangle$. The couplings $|1\rangle\leftrightarrow|2\rangle$
and $|2\rangle\leftrightarrow|3\rangle$ are the $\pi-\sigma$ transitions
($\Delta m_{F}=1$), while the coupling $|1\rangle\leftrightarrow|3\rangle$ is
the $\sigma-\sigma$ transition ($\Delta m_{F}=2$), as shown in Fig. 1(b).
Thus, the three-component atomic tunneling along the lattice and three-leg
couplings with a complex phase factor form a synthetic tube with a uniform
flux per plaquette, as shown in Fig. 1(c).

When the effective one-dimensional optical lattice is deep enough and the Rabi
frequency of the two-photon Raman transitions is not too large, we use the
single-band approximation to derive the tight-binding model Hamiltonian
Han2019

$\hat{H}=\hat{H}_{\text{hop}}+\hat{H}_{\Omega}+\hat{H}_{\text{int}},$ (1)

where the tunneling Hamiltonian

$\hat{H}_{\text{hop}}=\sum_{j,\sigma}(-t\hat{c}_{j+1,\sigma}^{\dagger}\hat{c}_{j,\sigma}+\text{H.c.}),$
(2)

the inter-leg coupling Hamiltonian

$\hat{H}_{\Omega}=\frac{1}{2}\sum_{j,\sigma\neq\sigma^{\prime}}(\Omega_{\sigma\sigma^{\prime}}e^{i\phi
j}\hat{c}_{j,\sigma}^{\dagger}\hat{c}_{j,\sigma^{\prime}}+\text{H.c.}),$ (3)

and the interaction Hamiltonian

$\hat{H}_{\text{int}}=U\\!\\!\\!\sum_{j,\sigma\neq\sigma^{\prime}}\\!\\!\hat{n}_{j,\sigma}\hat{n}_{j,\sigma^{\prime}}.$
(4)

In Eq. (1), $\hat{c}_{j,\sigma}$ ($\hat{c}_{j,\sigma}^{\dagger}$) is the
annihilation (creation) operator for a fermion at the real lattice site
$j=1,\cdots,L$ with spin $\sigma=\left(1,2,3\right)$ and the lattice length
$L$, $\hat{n}_{j,\sigma}\equiv\hat{c}_{j,\sigma}^{\dagger}\hat{c}_{j,\sigma}$
is the number operator. $t$ is the tunneling rate,
$\Omega_{\sigma\sigma^{\prime}}$ is the Rabi frequency of the two-photon Raman
transition between the spin states $\left|\sigma\right\rangle$ and
$\left|\sigma^{\prime}\right\rangle$ and is set to $\Omega_{12}=\Omega_{23}$
for simplicity, the $j$-dependent complex phase factor $e^{i\phi j}$ results
from the momentum imparted by the two-photon Raman transitions, the flux
$\phi=k_{R}d_{x}(1-\cos\eta)$ with $k_{R}$ being the recoil momentum of the
Raman lasers and $d_{x}$ being the lattice constant, $U$ is the interaction
strength, and H.c. is the Hermitian conjugate.

The Hamiltonian (1) has a distinct advantage that all parameters can be tuned
independently. For example, $t$ can be tuned by varying the depth of the
optical lattice, $\Omega_{\sigma\sigma^{\prime}}$ can be controlled by
adjusting the magnitudes of the Raman laser beams, $\phi$ can be manipulated
by controlling the angle $\eta$, and $U$ can be tuned via the external
magnetic field through orbital Feshbach resonance ren1 ; ofrexp1 ; ofrexp2  or
via the transverse trapping frequencies through the confinement induced
resonance Zhang2016 ; Bergeman2003 . In the following, we mainly consider the
case of the half filling, i.e., $n=N/L=1$ with $N$ being the total number of
atoms, since the system exhibits a synthetic Hall tube in such condition. We
also address the repulsive interaction $U>0$ and set $t=1$ as a unit.

In the absence of interaction ($U=0$), when $\phi=2\pi/3$ and
$\Omega_{-}<\Omega_{31}<\Omega_{+}$ with $\Omega_{\pm}=\pm
3t+\sqrt{\Omega_{12}^{2}+9t^{2}}$, this synthetic Hall tube supports a
topological insulator protected by generalized inversion symmetry Nourse2016 ;
Barbarino2018 ; Han2019 . Since the time-reversal, particle-hole, and chiral
symmetries are broken, the topological insulator belongs to the unitary
symmetry class A (unitary) of the Altland-Zirnbauer classification and is
characterized by a $\mathbb{Z}$ invariant Altland1997 ; Schnyder2008 ;
Ludwig2016 ; Chiu2016 . More interestingly, the inter-leg couplings generate
spatially periodic spin-vector and -tensor Zeeman fields with the following
Hamiltonian

$\begin{split}\hat{H}_{\Omega}&=\sum_{j}\Omega_{12}\left[\cos(\phi
j)S_{j}^{x}-\sin(\phi j)S_{j}^{y}\right]\\\ &+\Omega_{31}\left[\cos(\phi
j)\left(N_{j}^{xx}-N_{j}^{yy}\right)+\sin(\phi
j)N_{j}^{xy}\right],\end{split}$ (5)

where
$\mathbf{S}_{j}=\sum_{\sigma\sigma^{\prime}}{b}_{j\sigma}^{{\dagger}}\mathbf{F}_{\sigma\sigma^{\prime}}{b}_{j\sigma^{\prime}}$
with $\mathbf{F}_{\sigma\sigma^{\prime}}$ being the spin operators of the
total angular momentum $F=1$,
$N^{\alpha\beta}=\\{S^{\alpha},S^{\beta}\\}/2-\delta_{\alpha\beta}\mathbf{S}^{2}/3$
with being the anticommutation relation and $\alpha(\beta)=(x,y,z)$,
$\Omega_{12}$ and $\Omega_{31}$ are called the spin-vector and -tensor Zeeman
fields respectively, and $2\pi/\phi$ is the spiral period of the Zeeman field.
When $\Omega_{31}=0$, the synthetic Hall tube reduces to the spin-1 spin-orbit
coupled optical lattice only with the spin-vector Zeeman field Natu2015 ;
Barbarino2015 ; spin-1SOC2016 ; SOC-1BEC1 , which has a trivial topology.
Notice that $\Omega_{12}$ can also be treated as the spin-tensor Zeeman field
since the Hamiltonian (5) has the rotational symmetry.

In the presence of weak interaction, the topological insulator with the
$\mathbb{Z}$ invariant still exists since the generalized inversion symmetry
remains Morimoto2015 . When further increasing the interaction strength, the
topology of the system becomes trivial. On the other hand, the interaction
Hamiltonian (4) can also be rewritten as a magnetic form

$\hat{H}_{\text{int}}=\frac{U}{2}\sum_{j}{S_{j}^{z}}^{2}(\hat{n}_{j}-1).$ (6)

It shows clearly that at the half filling $\left(n=1\right)$, the interaction
has a little contribution on the magnetism of the system, i.e., the magnetic
properties of the synthetic Hall tube are mainly determined by
$\hat{H}_{\Omega}$.

Based on above qualitative analysis, it can be found that the synthetic Hall
tube exhibits exotic topological and magnetic properties arising from the
competition between the tunneling, spin-vector and -tensor Zeeman fields, and
interaction. In order to quantitatively reveal them, we will perform state-of-
the-art DMRG numerical method, for which we retain $400$ truncated states per
DMRG block and perform $30$ sweeps with a maximum truncation error $\sim
10^{-10}$.

Figure 2: (a1, b1, c1) Schematic diagrams of the spin-vector density arrows
$\vec{S_{j}}$ and the spin-tensor density ellipsoids $T_{j}$. The blue arrow
denotes the spin-vector $\vec{S}$, while the red ellipsoid reflects the spin-
tensor $T$, in which the black arrows are the ellipsoid’s axis orientations.
(a2, b2, c2) Spatial distributions of
$\left[l_{S}(j),\theta_{S}(j),\phi_{S}(j)\right]$ for the vector-density
arrows $\vec{S_{j}}$. (a3, b3, c3) Distributions of the axis lengths
$l_{T}^{n}\left(j\right)$ ($n=a,b,c$) of the spin-tensor density ellipsoids
$T_{j}$. (a4, b4, c4) Distributions of the orientational Euler angles
$\theta_{T}$, $\phi_{T}$, $\phi_{T}^{\prime}$ of the spin-tensor density
ellipsoids $T_{j}$. We set $\Omega_{31}/t=0$ for (a1)-(a4),
$\Omega_{31}/t=12.3$ for (b1)-(b4), and $\Omega_{31}/t=19$ for(c1)-(c4). In
all subfigures, we have $\Omega_{12}/t=12.3$, $U/t=0$, $L=32$, and $n=1$.

##  III Order parameters

The many-body topological properties can be well described by the degeneracy
in entanglement spectrum, entanglement entropy, chemical potential spectrum,
and excited energy gap. The entanglement spectrum is defined as Li2008

$\xi_{i}=-\ln(\rho_{i}),$ (7)

with $\rho_{i}$ being the eigenvalue of the reduced density matrix
$\hat{\rho}_{A}=\mathrm{Tr_{B}|\psi\rangle\langle\psi|}$, where $|\psi\rangle$
is the ground-state wavefunction and $A,B$ correspond to the left or the right
half of the one-dimensional chain. The system is topological if the
entanglement spectrum is degenerate since the entanglement spectrum resembles
the energy spectrum of edge excitations and vice versa Zhao2015 ; Yoshida2014
; Turner2011 ; Pollmann2010 ; Fidkowski2010 ; Flammia2009 ; Li2008 . The
quantum criticality of the interaction-driven topological phase transition can
be governed by the von Neumann entropy Flammia2009 ; Hastings2010 ; Daley2012
; Abanin2012 ; jiang2012 ; Islam2015

$S_{\mathrm{vN}}=-\mathrm{Tr_{A}[\hat{\rho}_{A}\log\hat{\rho}_{A}]}.$ (8)

The divergence of the von Neumann entropy at the critical point not only
indicates a continuous transition but also yields a central charge, which
reflects the universality class of phase transition. The von Neumann entropy
of a subchain of length $l$ is given by

$S_{\mathrm{vN}}=\frac{C}{6}\ln\left[\sin\frac{\pi l}{L}\right]+\text{const},$
(9)

in which the slope at large distance gives the central charge $C$ of the
conformal field theory underlying the critical behavior centralcharge1 ;
centralcharge2 .

The appearance of edge states is usually considered to be a hallmark of
topological properties for the bulk system. The topological insulator of the
synthetic Hall tube has two gapless edge states inside the gap between the
lowest and the upper branches in the chemical potential spectrum Barbarino2018
, which is essentially the energy required to add an atom to a system of $N$
atoms and can be defined as

$\mu=E_{g}^{o}(N)-E_{g}^{o}(N-1).$ (10)

Here, $E_{g}^{o}(N)$ is the ground-state energy of $N$ atoms under open
boundary condition. The topological ground state of the synthetic Hall tube is
nondegenerate and separated from the first excited state by a finite gap,
which closes and reopens in the process of topological phase transition
Han2019 . The excited energy gap is defined as

$\Delta_{e}=E_{e}^{p}(N)-E_{g}^{p}(N),$ (11)

where $E_{e}^{p}(N)$ [$E_{g}^{p}(N)$] is the first-excited (ground) state
energy of $N$ atoms under periodic boundary condition.

Due to the coexistence of the spin-vector and -tensor Zeeman fields, the
magnetism of the synthetic Hall tube should be described by whole spin-1 local
magnetic orders (8 spin-moments with 3 spin-vectors and 5 spin-tensors) and
their correlations zhou2020 . The local spin-vector

$\vec{S}_{j}=(\langle S_{j}^{x}\rangle,\langle S_{j}^{y}\rangle,\langle
S_{j}^{z}\rangle)^{T},$ (12)

while the local spin-tensor fluctuation matrix $T_{j}\ $has tensor moments

$T_{j}^{\alpha\beta}=\langle\\{S_{j}^{\alpha},S_{j}^{\beta}\\}\rangle/2-\langle
S_{j}^{\alpha}\rangle\langle S_{j}^{\beta}\rangle.$ (13)

Geometrically, $\vec{S}_{j}$ is characterized by an arrow and $T_{j}$ is
governed by an ellipsoid (with principle axis lengths
$l_{T}^{n}\left(j\right)$ ($n=a,b,c$) and orientations
$\vec{v}_{T}^{n}\left(j\right)$ given by the square-roots of the eigenvalues
and eigenvectors of $T_{j}^{\alpha\beta}$ Bharath ). Since the magnetic
properties are mainly determined by $\hat{H}_{\Omega}$, all the insulators are
spiral paramagnetic phases without any long-range correlations. As a result,
eight independent geometric parameters, including the length $l_{S}$ and
spherical coordinates $\theta_{S}$, $\phi_{S}$ of the arrow, the two axis
lengths $l_{T}^{a,b}$ with the third axis length
$l_{T}^{c}=\sqrt{{2-(l_{S})}^{2}-{(l_{T}^{a})}^{2}-{(l_{T}^{b})}^{2}}$, and
the orientational Euler angles $\theta_{T}$, $\phi_{T}$, $\phi_{T}^{\prime}$
of the ellipsoid, are chosen to quantitatively characterize and geometrically
visualize the magnetic orders.

##  IV Quantum phases

###  IV.1 Noninteracting case ($U=0)$

We first address the case of the noninteracting case ($U=0$). For
$\Omega_{-}<\Omega_{31}<\Omega_{+}$, the ground state is a topological
insulator and vice versa. As a result, we can discuss the magnetisms of the
topological and nontopological insulators as varying the spin-tensor Zeeman
field $\Omega_{31}$ for a fixed spin-vector Zeeman field $\Omega_{12}/t=12.3$.
When $\Omega_{31}=0$, the system is the same as the spin-1 spin-orbit coupled
optical lattice only with the spin-vector Zeeman field Natu2015 ;
Barbarino2015 ; spin-1SOC2016 ; SOC-1BEC1 . In this case, the spin-vector
arrow has a unit length $l_{S}=1$ and spirals in the $x-y$ plane (i.e.,
$\theta_{S}$ is a constant and $\phi_{S}$ changes cyclically), as shown in
Figs. 2(a1) and 2(a2). The spin-tensor ellipsoid almost is a plate with large
$l_{T}^{b,c}$ and small $l_{T}^{a}$ [see Fig. 2(a3)], and also spirals with a
cyclical variation $\phi_{T}$ and constants $\theta_{T}$, $\phi_{T}^{\prime}$
[see Fig. 2(a4)], since the spin-tensor ellipsoid depends crucially on the
three spin-vector operators $S^{\alpha}$ [see Eq. (13)]. This paramagnetic
insulator dominated only by the spin-vector is called nontopological spin-
vector paramagnetic insulator (NTSV). For $\Omega_{31}<\Omega_{-}$ (i.e., a
small spin-tensor Zeeman field), the ground state is still the NTSV.

For the topological regime with $\Omega_{-}<\Omega_{31}<\Omega_{+}$, the spin-
vector arrows also spiral in the $x-y$ plane but have short lengths $l_{S}<1$,
as shown in Figs. 2(b1) and 2(b2). In this case, the spin-vector cannot fully
describe the magnetic properties and the spin-tensor should be considered. The
spin-tensor ellipsoids have finite $l_{T}^{a,b,c}$ [see Fig. 2(b3)], and also
spiral in the $x-y$ plane with a cyclical variation $\phi_{T}$ and constants
$\theta_{T}$, $\phi_{T}^{\prime}$ [see Fig. 2(b4)]. Different from the NTSV,
this spiral spin-tensor ellipsoids only depend on the spin-tensor Zeeman
field. This topological insulator is called topological spin-mixed
paramagnetic insulator (TSM). For $\Omega_{31}>\Omega_{+}$ (i.e., a large
spin-tensor Zeeman field), the spin-vector arrow vanishes (i.e., $l_{S}=0$),
as shown in Figs. 2(c1) and 2(c2). In this case, the magnetic orders are fully
dominated by the spin-tensor ellipsoid. The ellipsoids have $l_{T}^{b,c}\sim
1$ and $l_{T}^{a}\rightarrow 0$ [see Fig. 2(c3)], and also spiral in the $x-y$
plane with a cyclical variation $\phi_{T}$ and constants $\theta_{T}$,
$\phi_{T}^{\prime}$ [see Fig. 2(c4)]. This paramagnetic insulator without the
spin-vector is called nontopological spin-tensor paramagnetic insulator
(NTST).

Figure 3: (a) The length of the spin-vector arrow $l_{S}$ and the axis lengths
of the spin-tensor ellipsoid $l_{T}^{a,b,c}$ as functions of the spin-tensor
Zeeman field $\Omega_{31}/t$ with a spin-vector Zeeman field
$\Omega_{12}/t=12.3$. (b) Phase diagram in the $\Omega_{12}-\Omega_{31}$
plane. In all subfigure, we have $U=0$, $L=32$, and $n=1$.

The above analysis of Fig. 2 shows that there exist two topologically magnetic
phase transitions as increasing the spin-tensor Zeeman field $\Omega_{31}/t$.
One is the transition from the NTSV to the TSM at
$\Omega_{31}^{c1}=\Omega_{-}$. At this critical point, the spin-vector arrow
length $l_{S}$ drops rapidly, but the ellipsoid’s axis lengths $l_{T}^{a,b,c}$
increase rapidly [see Fig. 3(a)]. The other is the transition from the TSM to
the NTST at $\Omega_{31}^{c2}=\Omega_{+}$. At this critical point, the spin-
vector arrow length $l_{S}$ suddenly becomes zero, and the ellipsoid’s axis
lengths $l_{T}^{a,b}$ ($l_{T}^{c}$) increase (decrease) abruptly [see Fig.
3(a)]. Figure 3(b) shows the phase diagram in the $\Omega_{31}-\Omega_{12}$
plane. Both the phase transitions of NTSV$\leftrightarrow$TSM and
TSM$\leftrightarrow$NTST [see blue lines in Fig. 3(b)] are of second-order
with a closing excited energy gap $\Delta_{e}$ at the critical points
Barbarino2018 ; Han2019 .

###  IV.2 Interacting case ($U>0$)

Figure 4: (a) The lowest four levels in the entanglement spectrum $\xi_{i}$
($i=0,1,2,3$), (b) the von Neumann entropy $S_{\mathrm{vN}}$, (c) the chemical
potential spectrum $\mu$, (d) the excited energy gap $\Delta_{e}$, and (e) the
length of the spin-vector arrow $l_{S}$ and the axis lengths of the spin-
tensor ellipsoid $l_{T}^{a,b,c}$ as functions of the interaction strength
$U/t$. In (a, b, c, e), $\Omega_{31}/t=11$ and $L=32$, and open boundary
condition is used. In (d), $\Omega_{31}/t=11$ and $L=12$, and periodic
boundary condition is used. The inset of (b) shows that the von Neumann
entropy of a subchain of length $l$ as a function of $\sin(\pi l/N)$ for a
chain with $L=32$ at the critical point $U_{c}/t=1.85$. The solid line is the
linear fit: $S_{\mathrm{vN}}={\frac{C}{6}}\ln[\sin(\pi l/N)]+1.22$ with
$C\approx 2$. The central charge is six times the slope of the linear fit. (f)
The length of the spin-vector arrow $l_{S}$ and the axis lengths of the spin-
tensor ellipsoid $l_{T}^{a,b,c}$ as functions of the spin-tensor Zeeman field
$\Omega_{31}/t$ with the interaction strength $U/t=6$. In all subfigures, we
have $\Omega_{12}/t=12.3$ and $n=1$.

We now explore many-body properties induced by the repulsive interaction
($U>0$). We first address the topological properties driven by interaction,
when $\Omega_{12}/t=12.3$ and $\Omega_{31}/t=11$. For a weak interaction, the
entanglement spectrum $\xi_{i}$ is two-fold degeneracy, and no longer
degenerate beyond a critical interaction strength $U_{c}/t\sim 1.85$, as shown
in Fig. 4(a). Without any symmetry breaking in this processing, it is a
typical topological phase transition from a topological insulator to a
nontopological insulator. As demonstrated in Fig. 4(b), sharp features of the
von Neumann entropy $S_{\mathrm{vN}}$ emerge at the critical point. From inset
of Fig. 4(b), we estimate $C\sim 1.97$, which is close to the universality
class of the Luttinger liquid ($C=2$) and shows the continuous of the
topological phase transition. On the other hand, in the absence of
interaction, there are two gapless edge states inside the bulk band gap in the
chemical potential spectrum $\mu$. As increasing the interaction strength
beyond a critical value $U_{c}/t$, these edge states merge into the bulk band
and the system becomes a nontopological insulator, as shown in Fig. 4(c).
Moreover, the excited energy gap $\Delta_{e}$ closes at the same critical
value and then reopens, as shown in Fig. 4(d). This critical point is well
consistent with that derived from entanglements in Figs. 4(a) and 4(b).

We now explore the magnetic orders in the presence of interaction. Based on
the above graphics of Fig. 2, it can be found that the spin-vector arrows and
the spin-tensor ellipsoids exhibit the same spiral features, but show the
distinct lengths $l_{S}$ and $l_{T}^{a,b,c}$. This means that these lengths
are adequate to describe the magnetic properties. As a result, we only
calculate the lengths $l_{S}$ and $l_{T}^{a,b,c}$ and ignore the angles
$\theta_{S}$, $\phi_{S}$, $\theta_{T}$, $\phi_{T}$, and $\phi_{T}^{\prime}$
hereafter.

In Fig. 4(e), we plot the lengths $l_{S}$ and $l_{T}^{a,b,c}$ as functions of
the interaction strength $U/t$. This figure shows that at the critical point
$U_{c}/t$, the spin-vector arrow length $l_{S}$ suddenly increases to
$l_{S}\sim 1$, and the spin-tensor ellipsoid’s axis lengths $l_{T}^{a,b}$ drop
rapidly, which indicate that this topologically magnetic phase transition from
the TSM to the NTSV occurs. In Fig. 4(f), we plot the lengths $l_{S}$ and
$l_{T}^{a,b,c}$ as functions of the spin-tensor Zeeman field strength
$\Omega_{31}$ for a large interaction strength $U/t=6$. In this case, all the
insulators are nontopological since the entanglement spectrum $\xi_{i}$ is
nondegenerate. Interestingly, as increasing the spin-tensor Zeeman field
$\Omega_{31}$, the vector length $l_{S}$ and the ellipsoid’s axis lengths
$l_{T}^{a,b}$ firstly remain, then $l_{S}$ rapidly decreases to $l_{S}<1$ and
$l_{T}^{a,b}$ ($l_{T}^{c}$) increase (decrease) rapidly. The corresponding
phase is called the nontopological spin-mixed paramagnetic insulator (NTSM).
Further increasing the spin-tensor Zeeman field $\Omega_{31}$, $l_{S}$
suddenly drops to $l_{S}\sim 0$, and $l_{T}^{a,b}$ rapidly increase to
$l_{T}^{a,b}\sim 1$, i.e., the system enters into the NTST. Note that for a
small interaction, the fundamental properties are similar to that in Fig. 3(a)
and thus not plotted here.

Figure 5: Phase diagrams in the $\Omega_{31}-U$ plane for the different spin-
vector Zeeman fields $\Omega_{12}/t=12.3$ (a) and $\Omega_{12}/t=6.0$ (b). The
blue and black lines show the continuous phase transitions. The blue lines
denote the liquids with $C=2$. In all subfigure, we have $L=32$ and $n=1$.

Finally, with the help of the calculated entanglement spectrum, entanglement
entropies, energy gaps, and local magnetic orders, in Fig. 5 we map out phase
diagrams the $\Omega_{31}-U$ plane for the different spin-vector Zeeman fields
$\Omega_{12}/t=12.3$ (a) and $\Omega_{12}/t=6.0$ (b). This figure shows
clearly four different phases such as the TSM, the NTSM, the NTSV, and the
NTST, which are well controlled by both the spin-vector and -tensor Zeeman
fields as well as the repulsive interaction. Moreover, all the phase
transitions with a closing excited energy gap $\Delta_{e}$ are of second
order.

##  V Conclusions

Before ending up this paper, we briefly discuss how to observe these quantum
phases and phase transitions in cold atom experiments. The entanglement
entropy can be measured using quantum interference of many-body twins of
ultracold atoms in optical lattices Islam2015 . The excited energy gap closing
in the processing of topological phase transition can be observed via
momentum-resolved analysis of the quench dynamics Han2019 . The local magnetic
orders can be measured by isolating the sites of interest using additional
site-resolved potentials Greiner2011 ; Greiner2016 ; Greiner2017 ; Bloch2016 .
Thus, all the quantum phases and phase transitions can be observed in current
experimental setups.

In conclusion, we have studied the many-body physics of interacting synthetic
Hall tube by the state-of-the-art DMRG numerical method. We have found four
quantum phases, including the TSM, the NTSM, the NTSV, and the NTST, by means
of the calculated entanglement spectrum, entanglement entropies, energy gaps,
and local magnetic orders. These quantum phases depend crucially on the
interaction and the spiral spin-vector and -tensor Zeeman fields induced by
the inter-leg couplings. Our work paves a new way to explore many-body
(topological) states induced by both the spiral spin-vector and -tensor Zeeman
fields.

## Acknowledgments

This work is supported by the National Key R&D Program of China under Grant
No. 2017YFA0304203; the NSFC under Grant No. 11674200; the Fund for Shanxi
‘1331 Project’ Key Subjects Construction; and Research Project Supported by
Shanxi Scholarship Council of China.

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