# Observation of a symmetry-protected topological phase in external magnetic
fields

Zhihuang Luo${{}^{1}}^{*}$ Wenzhao Zhang2 Xinfang Nie3 & Dawei
Lu${{}^{3}}^{*}$

###### Abstract

Topological phases have greatly improved our understanding of modern
conception of phases of matter that go beyond the paradigm of symmetry
breaking and are not described by local order parameters. Instead,
characterization of topological phases requires the robust topological
invariants, whereas measuring these global quantities presents an outstanding
challenge for experiments, especially in interacting systems. Here we report
the real-space observation of a symmetry-protected topological phase with
interacting nuclear spins, and probe the interaction-induced transition
between two topologically distinct phases, both of which are classified by
many-body Chern numbers obtained from the dynamical response to the external
magnetic fields. The resulting value of Chern number
($\bar{\mathcal{C}h}=1.958\pm 0.070$) demonstrates the robust ground state
degeneracy, and also determines the number of nontrivial edge states. Our
findings enable direct characterization of topological features of quantum
many-body states through gradually decreasing the strength of the introduced
external fields.

Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing & School
of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai
519082, China.

Department of Physics, Ningbo University, Ningbo 315211, China.

Shenzhen Institute for Quantum Science and Engineering and Department of
Physics, Southern University of Science and Technology, Shenzhen 518055,
China.

Symmetry plays a fundamental role in classifying different phases of matter
and their associated phase transitions, where the symmetry is spontaneously
broken [1]. However, the discovery of topological phases goes beyond this
symmetry breaking paradigm, whose characterization requires the topological
invariants, such as the Chern numbers, that are robust against local
perturbations [2, 3]. A prominent example is the integer quantum Hall effect
[4], demonstrating that the Hall conductance in response to an external
electric field is quantized by integer Chern numbers. Recent progress of
topological insulators and superconductors [5, 6] has led to the broad notion
of symmetry-protected topological (SPT) phases [7, 8], which are short-range
entangled quantum states with the gapless or degenerate edge modes protected
by a symmetry. The edge states of SPT phases have been observed in a variety
of physical platforms [9, 10, 11, 12, 13, 14, 15, 16], and the momentum-
resolved Berry curvature and further relevant Chern number, as the central
measure of topology, have also been used to determine the band structures of
noninteracting SPT phases at the single-particle level [17, 18, 19, 20, 21].
Whereas in quantum many-body systems with interactions, where there may exist
a large number of SPT phases [8], the underlying Chern number is not easily
accessible. Because the interactions complicate the energy levels in real
space, and make it difficult to fully determine the band eigenstates based on
the previous methods of momentum space [17, 18, 19, 20, 21].

Going beyond the noninteracting systems, we report the real-space measurement
of many-body Chern numbers, and directly observe an interacting SPT phase. The
experiment is performed on the sample of trans-crotonic acid, which consists
of a staggered one-dimensional (1D) spin-$1/2$ chain of interacting carbon
nuclei. We use this system to realize a 1D Su-Schrieffer-Heeger (SSH) model
with an SPT phase [22, 23, 24]. By introducing the external magnetic fields to
the SSH model, the Berry curvatures are allow to be defined in the parameter
space of these fields, and can be extracted from the linear response of spin
magnetizations to the quench velocity [25, 26, 27, 28]. The relevant Chern
number obtained by the integral of Berry curvature determines the number of
degenerate ground states, and also the number of nontrivial edge states
according to the bulk-boundary correspondence. We characterize the two
topologically distinct phases, with the features of nonzero Chern number for
the topological phase and zero value for the trivial one, as the introduced
external magnetic field decreases to zero. Furthermore, we probe the
interaction-induced transition from a topological phase to a trivial one,
under a fixed small external field. Our measurements demonstrate clear
signatures of the SPT phase, and establish a general approach for exploring
the topological features of interacting quantum many-body systems.

The SSH model is a paradigmatic example with nontrivial topology, originally
describing electrons hopping on a 1D chain with staggered hopping amplitudes.
We consider the spin Hamiltonian of 1D SSH model derived via a Jordan-Wigner
transformation [29]

$\mathcal{H}_{\text{SSH}}^{\text{S}}=-J_{1}\sum_{i=1}^{N}(\sigma_{2i-1}^{x}\sigma_{2i}^{x}+\sigma_{2i-1}^{y}\sigma_{2i}^{y})-J_{2}\sum_{i=1}^{N-1}(\sigma_{2i}^{x}\sigma_{2i+1}^{x}+\sigma_{2i}^{y}\sigma_{2i+1}^{y}),$
(1)

which consists of $N$ unit cells with each unit cell hosting two spins. The
$\sigma_{i}^{x}$ $(\sigma_{i}^{y})$ stand for Pauli matrices. Depending on the
ratio of $J_{1}$ and $J_{2}$, the chain exhibits two topologically distinct
phases: a topological one with four-fold degenerate ground states if
$J_{1}<J_{2}$, and a trivial one with a single ground state if $J_{1}>J_{2}$.

We experimentally realized the 1D SSH model of the form of equation (1) using
the 13C-labeled trans-crotonic acid dissolved in d6-acetone, whose molecular
structure is depicted in Fig. 1a. It consists of four carbon-13 nuclear spins
(gray balls) and six hydrogen-1 nuclear spins (green balls). The oxygen-16
nuclei (red balls, 99.76% abundance) have no nuclear magnetic resonance (NMR)
signals due to its spin number $I=0$. All 1H nuclear spins were decoupled by
the WALTZ-16 decoupling sequence throughout the experiments [30]. The internal
spin interactions of four carbon-13 nuclei ($I=1/2$) read as
$\mathcal{H}_{\text{int}}=\frac{\pi}{2}\sum_{i=1}^{3}J_{i,i+1}\sigma_{i}^{z}\sigma_{i+1}^{z}$.
Here we only consider the nearest-neighbor couplings of $J_{12}=41.6$ Hz,
$J_{23}=69.7$ Hz, and $J_{34}=72.2$ Hz, which are dominant compared to others.
The chemical shifts and undesired $J$-couplings were refocused by spin-echo
sandwiches [29]. One can apply a pair of selective $\pi$ pulses shown in Fig.
1b to create the effective $J$-couplings,
$J_{ij}^{\text{eff}}=J_{ij}\frac{\tau_{2}-\tau_{1}}{\tau_{1}+\tau_{2}}$, which
are tunable by changing the intervals of $\tau_{1}$ and $\tau_{2}$. The
$\sigma_{i}^{x}\sigma_{i+1}^{x}$ ($\sigma_{i}^{y}\sigma_{i+1}^{y}$) terms can
be generated via rotating the $\sigma_{i}^{z}\sigma_{i+1}^{z}$ terms by a pair
of $\pi/2$ pulses along the $y$ ($x$) axis. Therefore, we realized the
effective Hamiltonian of $\mathcal{H}_{\text{SSH}}^{\text{S,
eff}}=-\frac{\pi}{2}\sum_{i=1}^{3}J_{i,i+1}^{\text{eff}}(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y})$
[29], corresponding to the spin Hamiltonian of 1D SSH model, for example,
which can be set to the topological configuration with
$J_{12}^{\text{eff}},J_{34}^{\text{eff}}\approx
0\text{Hz},J_{23}^{\text{eff}}=69.7\text{Hz}$, and the trivial configuration
with
$J_{12}^{\text{eff}},J_{34}^{\text{eff}}=41.6\text{Hz},J_{23}^{\text{eff}}\approx
0\text{Hz}$, respectively.

To capture the central characteristics of two topologically distinct phases,
we introduced an external magnetic field
$\vec{h}=(h_{r}\text{sin}\theta\text{cos}\phi,h_{r}\text{sin}\theta\text{sin}\phi,h_{r}\text{cos}\theta)$
into the SSH model, as shown in Fig. 1c, where the origin point (small black
ball) represents the pure Hamiltonian of SSH model, i.e.,
$\mathcal{H}(h_{r}=0)=\mathcal{H}_{\text{SSH}}^{\text{S}}$. The Hamiltonian in
the external magnetic field is
$\mathcal{H}_{\text{ext}}=-\sum_{i=1}^{2N}\vec{h}\cdot\vec{\sigma_{i}}$, which
was readily achieved by applying a radio-frequency (RF) pulse [29]. Now we can
define the Chern number over a closed manifold surface of parameter space of
$\vec{h}$ as

$\mathcal{C}h(h_{r})=\frac{1}{2\pi}\oint_{|\vec{h}|=h_{r}}dS_{\theta\phi}\mathcal{F}_{\theta\phi}.$
(2)

Here
$\mathcal{F}_{\theta\phi}\equiv\partial_{\theta}\mathcal{A}_{\phi}-\partial_{\phi}\mathcal{A}_{\theta}$
is the Berry curvature, and $\mathcal{A}_{\theta}\equiv
i\langle\psi_{g}|\partial_{\theta}|\psi_{g}\rangle$ is the Berry connection.
The $|\psi_{g}\rangle$ denotes the ground state of system. The topology of
Hamiltonian (1) can thus be characterized by
$\mathcal{C}h(0)=\text{lim}_{h_{r}\rightarrow 0}\mathcal{C}h(h_{r})$. We have
$\mathcal{C}h(0)\neq 0$ and $\mathcal{C}h(0)=0$ for the topological and
trivial phases respectively.

The ability to measure the Berry curvature is crucial for our experiment.
Based on a method proposed in [25], we measured the responses of the
generalized force along the $\phi$-direction,
$\mathcal{M}_{\phi}=-\langle\psi(t)|\partial_{\phi}\mathcal{H}|\psi(t)\rangle$.
The $|\psi(t)\rangle$ is an instantaneous state when the external magnetic
field evolves with a constant velocity $v_{\theta}=\pi/t_{f}$ along the quench
path, as shown in the red curve of Fig. 1c, where $\phi=\pi/2$. The
$\mathcal{M}_{\phi}$ at $\phi=\pi/2$ represents the sum of local spin
magnetizations in time, because
$\mathcal{M}_{\phi}|_{\phi=\pi/2}=-h_{r}\text{sin}\theta(t)\sum_{j=1}^{4}\langle\psi(t)|\sigma_{j}^{x}|\psi(t)\rangle$,
with $\theta(t)=v_{\theta}t$. Then the Berry curvature can be extracted from
the linear response of $\mathcal{M}_{\phi}$, according to

$\mathcal{M}_{\phi}=\text{const}+\mathcal{F}_{\theta\phi}v_{\theta}+\mathcal{O}(v_{\theta}^{2}).$
(3)

If the $v_{\theta}$ is sufficiently slow, the $\mathcal{O}(v_{\theta}^{2})$
can be negligible. The constant term gives the value of $\mathcal{M}_{\phi}$
in the adiabatic limit. Note that the Hamiltonian (1) has the rotation
invariance, meaning that the Berry curvature is independent of $\phi$ [29].
The Chern number is hence simplified into the integral with respect to
$\theta$ from $0$ to $\pi$.

For the measurement of Berry curvature, we begin with the initial ground
states at the north pole of spherical parameter manifold of an external
magnetic field. The initial states with different configurations of SSH model
were adiabatically prepared along the three paths illustrated in Fig. 1d. The
evolution time of adiabatic preparation obeys
$T=\mathcal{O}(1/\Delta_{\text{min}}^{2})$, where $\Delta_{\text{min}}$
denotes the minimum energy gap between the ground state and the first excited
state. The auxiliary sweeps of $h_{x}$ were added to avoid the closed gap. The
details on the preparation of topological phase, for example, can be found in
supplementary materials. We performed the measurements of the spin-spin
correlations along the $x$ direction,
$C_{ij}^{x}=\langle\sigma_{i}^{x}\sigma_{j}^{x}\rangle-\langle\sigma_{i}^{x}\rangle\langle\sigma_{j}^{x}\rangle$,
for different initial states using five independent readout experiments [29].
Figures 1e and 1f show the intercell spin correlation with $C_{23}^{x}=0.926$
for the topological phase, and intracell spin correlations with
$C_{12}^{x}=0.960,C_{34}^{x}=1.042$ for the trivial phase, respectively, where
the measured signals have been normalized using the initial ground states. In
the topological phase, two edge spins without correlations can be in spin up
or down states independently, which gives rise to the four-fold ground state
degeneracy.

We now turn to observe two topologically distinct phases with the measured
Chern number under different external magnetic fields. The system was set to
the topological and trivial configurations of the SSH model. The phase
diagrams of two configurations in the presence of external magnetic fields
were plotted in Figs. 2c and 3c, showing the calculated Chern number as a
function of external magnetic field $h_{r}$ and effective coupling
$J_{23}^{\text{eff}}$ or $J_{12}^{\text{eff}}$. For large $h_{r}$, both
figures exhibit the orange regions with $\mathcal{C}h=4$, corresponding to the
spin-polarized (SP) state induced by the external magnetic fields [29]. We
experimentally explored the regions along the dotted white lines, with fixed
$J_{23}^{\text{eff}}=69.7$ Hz (Fig. 2c) and $J_{12}^{\text{eff}}=41.6$ Hz
(Fig. 3c). The main results of Chern number are shown in Figs. 2d and 3d,
where each $\mathcal{C}h(h_{r})$ was obtained by the integrals of the Berry
curvatures, which were further extracted from the linear responses of the spin
magnetizations to the quench velocity using equation (3). The quench time
$t_{f}=350$ ms taken in our experiments was slow enough, such that the high-
order errors remain small [29]. Figures 2a and 2b show the measured spin
magnetizations and resulting Berry curvatures for $h_{r}=10$ Hz in the
topological configuration. Figures 3a and 3b show the similar results in the
trivial configuration. Other relevant data for measuring the Chern numbers can
be found in supplementary materials.

The Chern number reveals the singularities of the system, counting the total
number of degeneracy points enclosed in the spherical parameter manifold of
$\vec{h}$, which may be viewed as the sources of magnetic monopoles from the
Gauss’ law [29]. In Figs. 2c and 3c, we obtained $\bar{\mathcal{C}h}=1.958\pm
0.070$ for the topological phase, and $\bar{\mathcal{C}h}=-0.067\pm 0.063$ for
the trivial phase, respectively, as the $h_{r}$ trends to zero. Note that each
two-fold degenerate ground state forms a degeneracy point. The emergence of
two degeneracy points (small red balls) at $h_{r}=0$ determines the four-edge
states, featuring an SPT phase, according to the bulk-edge correspondence
(Fig. 2e). But the absence of a degeneracy point at $h_{r}=0$ indicates that
it is a trivial phase (Fig. 3e). We also demonstrated the different robustness
of degeneracy points against the small perturbations (see supplementary
materials). These signatures clearly distinguish the two topologically
different phases of the SSH model.

We further probed the interaction-induced transition between two topologically
distinct phases, where the external magnetic field strength was set to the
small value of $h_{r}=50$ Hz. The calculated phase diagram exhibits three
different regions of light green, light blue and orange, corresponding to the
topological phase with Chern number two, the trivial phase with Chern number
zero, and the SP phase with Chern number four (Fig. 4a). The area of SP phase
depends on the field strength $h_{r}$, and in the absence of interactions, its
Chern number grows linearly with the increase of system size, due to the
introduced external magnetic field itself (see supplementary materials). We
experimentally explored the region along the dotted white line with fixed
$J_{23}^{\text{eff}}=69.7$ Hz. By adjusting $J_{12}^{\text{eff}}$ and
$J_{34}^{\text{eff}}$, we observed a sharp transition from the nontrivial
Chern number ($\bar{\mathcal{C}h}=1.986\pm 0.048$) to the vanishing value
($\bar{\mathcal{C}h}=-0.076\pm 0.040$) (Fig. 4b), as expected from numerical
calculations, indicating the change in topology.

In conclusion, we have presented the experimental realization of a 1D SSH
model with interacting nuclear spins, and implemented a general method for the
real-space measurements of many-body Chern numbers through the linear
responses to the external magnetic fields. As the introduced field strength
decreases to zero, we have observed an interacting SPT phase with the nonzero
Chern number, that determines the number of the degenerate ground states and
also the nontrivial edge states according to the bulk-edge correspondence, and
a trivial phase with zero Chern number, which has only a single ground state.
In fact, it is not necessary to measure the Chern number in the zero magnetic
field limit, because which is invariant in weak fields for many cases [29].
The method for experimentally extracting the Chern number is also robust
against the noises of applied fields, regardless of the shapes of the closed
surface [29]. Using the measured Chern number, we have further probed the
interaction-induced transition between two topologically distinct phases under
a fixed small external field.

Our results provide the direct and central characterization of an interacting
SPT phase in real space. The realization techniques of the SSH model (e.g.,
refocusing scheme and quenching control) can be extended to larger nuclear
spin systems, such as the 13C-labeled polyacetylene, for further exploring the
more interesting phenomena, though the scalability of NMR platform in quantum
computing might be limited. Besides, the dynamical method of measuring the
Chern number can be generalized to other platforms and momentum spaces, and
may provide a promising route for topological characterization of many-body
states in higher dimensional systems, such as the topologically ordered states
with long-range quantum entanglement.

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This work was supported by the National Natural Science Foundation (Grants No.
11805008, No. 12074206, No. 12104213, No. 11734002, No. 12075110, and No.
11875159), Guangdong Provincial Key Laboratory (Grant No. 2019B121203005),
Guangdong Innovative and Entrepreneurial Research Team Program (2019ZT08C044),
and Science, Technology and Innovation Commission of Shenzhen Municipality
(KQTD20190929173815000 and JCYJ20200109140803865).

Z. L. proposed the idea, designed and performed the experiment. Z. L. analyzed
the data and wrote the manuscript. X. N. prepared the sample. Z. L. and W. Z.
developed the theoretical framework. Z. L. and D. L. revised the manuscript
and supervised the work. All authors discussed the results and contributed to
the writing of the manuscript.

The authors declare that they have no competing interests.

Correspondence and requests for materials should be addressed to Z. L.
<EMAIL_ADDRESS>or D. L. (ludw@sustech.edu.cn).

Figure 1: Experimental realization of the SSH model and measured spin-spin
correlations for different phases. a, Molecular structure of 13C-labeled
trans-crotonic acid, where $\text{C}_{1},\text{C}_{2},\text{C}_{3}\text{ and
}\text{C}_{4}$ nuclear spins are used as a four-qubit quantum simulator to
realize the SSH model. All hydrogen nuclei (green balls) are decoupled
throughout the experiments. The nearest-neighbor couplings of $J_{12}=41.6$
Hz, $J_{23}=69.7$ Hz, and $J_{34}=72.2$ Hz are dominating. b, Refocusing
scheme to create an effective coupling $J_{ij}^{\text{eff}}$, by a pair of
selective $\pi$ pulses acting on $\text{C}_{i}$ and $\text{C}_{j}$. By
adjusting $\tau_{1}$ and $\tau_{2}$, we can implement two topologically
distinct configurations of the SSH model. c, SSH model in a spherical space of
external magnetic field $\vec{h}$. When $h_{r}=0$, the origin point (small
black ball) represents the pure Hamiltonian of SSH model, i.e.,
$\mathcal{H}(h_{r}=0)=\mathcal{H}_{\text{SSH}}^{\text{S}}$. The quench path
(red curve) starts at the north pole and ramps along the $\phi=\pi/2$ meridian
with a constant velocity $v_{\theta}$ until it reaches the south pole at the
final time $t_{f}$. d, Adiabatic paths (labeled by $1,2,3$) for preparing
different ground states, corresponding to a topological phase, a trivial phase
and an arbitrary phase for given $J_{12}^{\text{eff}}\approx
J_{34}^{\text{eff}},J_{23}^{\text{eff}}$, respectively. The colormap indicates
the energy gap between the ground state and the first excited state. e and f
show the observed spin-spin correlations along the $x$ direction of
topological and trivial phases.  Figure 2: Observation of the topological
phase. The system was set to topological configuration, i.e.,
$J_{12}^{\text{eff}},J_{34}^{\text{eff}}\approx 0\text{Hz}$ and
$J_{23}^{\text{eff}}=69.7\text{Hz}$. a, Measured spin magnetizations
$\langle\sigma_{j}^{x}\rangle$ for $j=1,2,3,4$, which were generated along the
quenching path shown in Fig. 1c. The initial ground state was adiabatically
prepared at the north pole of spherical manifold ($h_{r}=10$ Hz) through the
path 1 (cyan lines) illustrated in Fig. 1d, and each
$\langle\sigma_{j}^{x}\rangle$ was measured at 101 time steps during the
quenching evolution. The light shadings show the numerical results. b, Berry
curvature extracted from the linear response of spin magnetizations. c,
Calculated phase diagram showing the Chern number as a function of external
magnetic field $h_{r}$ and effective coupling $J_{23}^{\text{eff}}$. The
dotted white line at $J_{23}^{\text{eff}}=69.7$ Hz indicates the region
explored experimentally in d. d, Measured Chern numbers in external magnetic
fields, which were obtained from the integral of Berry curvature
$\mathcal{F}_{\theta\phi}$. As $h_{r}\rightarrow 0$, the non-zero Chern number
($1.958\pm 0.070$) characterizes the topological phase directly. e, Schematic
of measuring the Chern number that corresponds to the total number of
degeneracy points (small red balls) enclosed in the spherical manifold. The
degeneracy points at $h_{r}=0$ indicate the non-zero Chern number of
topological phase of the SSH model. Figure 3: Observation of the trivial
phase. The system was set to trivial configuration, i.e.,
$J_{12}^{\text{eff}},J_{34}^{\text{eff}}=41.6\text{Hz}$ and
$J_{23}^{\text{eff}}\approx 0\text{Hz}$. a, Measured spin magnetizations
$\langle\sigma_{j}^{x}\rangle$ for $j=1,2,3,4$, which were generated along the
quenching path shown in Fig. 1c. The experiment began with an initial ground
state preparing at the north pole of spherical manifold ($h_{r}=10$ Hz)
through the adiabatic evolution along the path 2 (green lines) illustrated in
Fig. 1d, and each $\langle\sigma_{j}^{x}\rangle$ was measured at 101 time
steps during the quenching evolution. The light shadings show the numerical
results. b, Berry curvature extracted from the linear response of spin
magnetizations. d, Calculated phase diagram showing the Chern number as a
function of external magnetic field $h_{r}$ and effective coupling
$J_{12}^{\text{eff}}$. The dotted white line at $J_{12}^{\text{eff}}=41.6$ Hz
indicates the region explored experimentally in d. d, Measured Chern numbers
in external magnetic fields, which were obtained from the integral of Berry
curvature $\mathcal{F}_{\theta\phi}$. As $h_{r}\rightarrow 0$, the near-zero
Chern number ($-0.067\pm 0.063$) characterizes the trivial phase directly. e,
Schematic of measuring the Chern number that corresponds to the total number
of degeneracy points (small red balls) enclosed in the spherical manifold.
There are no degeneracy points at $h_{r}=0$, which indicates that it is the
trivial phase of the SSH model. Figure 4: Probing the topological transition.
a, Calculated phase diagram showing the Chern number as a function of
effective couplings $J_{12}^{\text{eff}}$ (or $J_{34}^{\text{eff}}$,
$J_{34}^{\text{eff}}=J_{12}^{\text{eff}}$) and $J_{23}^{\text{eff}}$, with a
fixed external magnetic field $h_{r}=50$ Hz. The light green, light blue and
orange regions denote the topological, trivial and spin-polarized (SP) phases,
respectively. The dotted white line at $J_{23}^{\text{eff}}=69.7$ Hz indicates
the region explored experimentally in b. b, Measured Chern number as a
function of $J_{12}^{\text{eff}}$. The solid red line shows the numerical
simulation and experimental data are obtained by integrating the Berry
curvature (see supplementary materials).

SUPPLEMENTARY INFORMATION
for
Observation of a symmetry-protected topological phase in external magnetic
fields
Zhihuang Luo${{}^{1}}^{*}$, Wenzhao Zhang2, Xinfang Nie3, & Dawei
Lu${{}^{3}}^{*}$

Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing & School
of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai
519082, China.

Department of Physics, Ningbo University, Ningbo 315211, China.

Shenzhen Institute for Quantum Science and Engineering and Department of
Physics, Southern University of Science and Technology, Shenzhen 518055,
China.

###### Contents

  1. S1 SSH model in external magnetic fields
    1. S1.1 SSH model
    2. S1.2 Many-body Chern number
    3. S1.3 Robustness against the external magnetic field
  2. S2 Experimental realization
    1. S2.1 Setup and sample
    2. S2.2 Realization of the SSH model
  3. S3 Numerical simulation and analysis
    1. S3.1 Adiabatic preparation of many-body ground states
    2. S3.2 Optimization of the quenching time
    3. S3.3 Robustness analysis of degeneracy points
  4. S4 Measurement and discussion
    1. S4.1 Measurement of spin-spin correlations
    2. S4.2 Observation of the spin-polarized phase
    3. S4.3 Other data for measuring the Chern numbers
    4. S4.4 Experimental error analysis

##  S1 SSH model in external magnetic fields

###  S1.1 SSH model

We consider the Su-Schrieffer-Heeger (SSH) model in a one-dimensional (1D)
lattice with $N=L/2$ unit cells [22, 23], each unit cell consisting of 2
fermions, as illustrated in Fig. S1A. Its Hamiltonian reads

$\mathcal{H}_{\text{SSH}}^{\text{F}}=-J_{1}\sum_{i=1}^{N}c_{2i-1}^{{\dagger}}c_{2i}-J_{2}\sum_{i=1}^{N-1}c_{2i}^{{\dagger}}c_{2i+1}+h.c,$
(S1)

where $c_{i}^{{\dagger}}$ and $c_{i}$ represent the fermionic creation and
annihilation operators. The hopping amplitudes are staggered, meaning that the
intracell hoppings $J_{1}$ are different from the intercell hoppings $J_{2}$.
Depending on the ratio of $J_{1}$ and $J_{2}$, the SSH model exhibits two
topologically distinct phases, shown In Fig. S1B. If $J_{1}<J_{2}$, the system
is in the symmetry-protected topological (SPT) phase with four-fold degenerate
ground states (Fig. S1C); if $J_{1}>J_{2}$, the system is in the trivial phase
with a single ground state (Fig. S1D). The SSH model is a paradigmatic example
for an SPT phase. The existence of SPT phase is relevant to two important
symmetries: particle number conversation and chiral symmetry that protects the
topological phase.

Using the following Jordan-Wigner transformation

$c_{j}=\prod_{i=1}^{j-1}\sigma_{i}^{z}\sigma_{j}^{+},\quad
c_{j}^{{\dagger}}=\prod_{i=1}^{j-1}\sigma_{i}^{z}\sigma_{j}^{-},$ (S2)

the SSH Hamiltonian (S1) can be rewritten in the form of spin-1/2 operators as

$\mathcal{H}_{\text{SSH}}^{\text{S}}=-J_{1}\sum_{i=1}^{N}(\sigma_{2i-1}^{x}\sigma_{2i}^{x}+\sigma_{2i-1}^{y}\sigma_{2i}^{y})-J_{2}\sum_{i=1}^{N-1}(\sigma_{2i}^{x}\sigma_{2i+1}^{x}+\sigma_{2i}^{y}\sigma_{2i+1}^{y}),$
(S3)

where $\sigma_{i}^{\alpha}(\alpha=x,y,z)$ represent the Pauli matrices. This
is the spin Hamiltonian of SSH model considered in our experiment. Using the
corresponding many-body ground states of Eq. (S3), the numerical calculations
of the spin-spin correlations along $x$ direction,
$C_{ij}^{x}=\langle\sigma_{i}^{x}\sigma_{j}^{x}\rangle-\langle\sigma_{i}^{x}\rangle\langle\sigma_{j}^{x}\rangle$,
are performed on a chain of 12 sites. We obtain $C_{2i,2i+1}^{x}=1$ for the
topological phase, and $C_{2i-1,2i}^{x}=1$ for the trivial phase, showing the
inter- and intracell correlations between spins in Figs. S1E and S1F,
respectively. It could be found that for the topological phase, there are two
independent edge spins, different from the correlated bulk spins. The
appearance of edge states at the boundary spins gives rise to the four-fold
ground state degeneracy.

Figure S1: (A) Geometry of the SSH model in a one-dimensional lattice with
open boundary conditions. Each two sites linked by thick lines is grouped into
a unit cell (circled by a dotted line). The hopping amplitudes are staggered,
i.e., the inracell hoppings $J_{1}$ (thick lines) are different from the
intercell hoppings $J_{2}$ (doubled thin lines). (B) Phase diagram of the SSH
model. There are two distinct phases: topological one (green) when
$J_{1}<J_{2}$ and trivial one (blue) when $J_{1}>J_{2}$. The boundary of
$J_{1}=J_{2}$ separates these phases. (C and D) Full energy spectrum of the
spin chain of 12 sites in the topological and the trivial configurations for
different particle numbers. The topological configuration exhibits a fourfold
degeneracy involving five, six (twofold degenerate), and seven particles,
whereas the trivial chain exhibits a single gapped ground state with six
particles. (E and F) Calculated spin-spin correlations along the x direction
for the topological and trivial phases.

###  S1.2 Many-body Chern number

To describe the central properties of SPT phase using the topological
invariants, we shall work with the following generic Hamiltonian by
introducing an external magnetic field of
$\vec{h}=(h_{r}\text{sin}\theta\text{cos}\phi,h_{r}\text{sin}\theta\text{sin}\phi,h_{r}\text{cos}\theta)$
into the spin form of SSH model

$\mathcal{H}(\vec{h})=-\sum_{i=1}^{L}\vec{h}\cdot\vec{\sigma_{i}}+\mathcal{H}_{\text{SSH}}^{\text{S}}.$
(S4)

It allows for defining the Berry curvature over the spherical parameter
surface of $\vec{h}$, and the further relevant Chern number obtained by the
integral of the Berry curvature. The Chern number manifests the singularities
of system enclosed in the spherical manifold, and can thus be used to capture
the signature of topology. If the field strength decreases to zero, the
nonzero value of Chern number reveals the existence of a nontrivial phase in
Eq. (S3); on the contrary, it is a trivial phase. Note that the Hamiltonian
(S4) has $U(1)$ invariance, i.e.,

$\mathcal{H}(h_{r},\theta,\phi)=R_{\text{tot}}^{z}(\phi)\mathcal{H}(h_{r},\theta,0)R_{\text{tot}}^{z}(-\phi),$
(S5)

with $R_{\text{tot}}^{z}=\prod_{i=1}^{L}e^{-i\phi\sigma_{i}^{z}/2}$.
Therefore, the Berry curvature is independent of $\phi$, and the Chern number
can be simplified into the integral of Berry curvature with respect to the
$\theta$ from 0 to $\pi$. When $\theta=0$ and $\pi$,
$\sum_{i=1}^{L}\sigma_{i}^{z}$ is a conversation quantity, which enables the
points of ground state degeneracy locating on the $z$ axis of $\vec{h}$. The
degeneracy points act as the sources of magnetic monopoles, and contribute the
nonzero terms to the integral of Berry curvature, which can be seen more
intuitively from the full expression of Berry curvature

$\mathcal{F}_{\theta\phi}(h_{r})=i\sum_{n\neq
0}\frac{\langle\psi_{0}|\partial_{\theta}\mathcal{H}|\psi_{n}\rangle\langle\psi_{n}|\partial_{\phi}\mathcal{H}|\psi_{0}\rangle-\langle\psi_{0}|\partial_{\phi}\mathcal{H}|\psi_{n}\rangle\langle\psi_{n}|\partial_{\theta}\mathcal{H}|\psi_{0}\rangle}{(E_{n}-E_{0})^{2}},$
(S6)

using the definitions of $\mathcal{F}_{\theta\phi}$ and $\mathcal{A}_{\theta}$
in the main text. Here $E_{n}$ and $|\psi_{n}\rangle$ denote the
$n_{\text{th}}$ eigenvalue and its corresponding eigenstate of Hamiltonian
(S4). It clearly shows that the $E_{n}=E_{0}$ for $n\neq 0$ are the singular
points of the above expression, corresponding to the ground state degeneracy.
As the analogy with the Gauss’s law, the Chern number in the integral form of
Eq. (S6) determines the number of degeneracy points, and also the number of
edge states according to the bulk-edge correspondence.

###  S1.3 Robustness against the external magnetic field

By the exact diagonalization of Hamiltonians (S4) with $L=6,8,10,12$ and the
integrals of Eq. (S6), we obtained the same numerical value of Chern number
two for the topological phase when the $h_{r}$ trends to zero, as shown in
Fig. S2A. It determines the four-fold degenerate edge states of SPT phase. For
the trivial phase with a single ground state, the numerical value of Chern
number is zero (Fig. S2B). It could be seen that the Chern number is invariant
in weak fields, as it is in the zero magnetic field limit. Therefore, we can
only measure the many-body Chern number in a weak field, which is feasible for
many systems, and more efficient experimentally for extracting the topological
features. The method for the measurement of Chern number is also robust
against the noises of applied fields, because the value of Chern number is
related to the distribution of magnetic monopoles, irrespectively of the
shapes of a closed surface (see Figures. S3A and S3B).

Figure S2: Calculated Chern numbers as a function of the field strength
$h_{r}$ for the topological configuration (A) and the trivial configuration
(B), respectively. As the $h_{r}$ trends to zero, the Chern numbers are
invariant with the value of two for the topological phase, and zero for the
trivial phase, regardless of the system sizes.  Figure S3: The introduced
external magnetic field is spherical (A) and arbitrary (B) closed surfaces.
The small red balls enclosed in the closed surfaces represent the degeneracy
points of Hamiltonian (S3) ($J_{1}<J_{2}$) without the external field, which
act as the sources of magnetic monopoles.

##  S2 Experimental realization

###  S2.1 Setup and sample

The experiment was carried out on a Bruker DRX 600 MHz spectrometer at
temperature 300 K. The spectrometer is constructed from a large
superconducting magnet, the radio frequency (RF) section, gradient field
coils, and the sample held in a glass tube, as shown in Fig. S4. The
superconducting magnet provides a static magnetic field along the $z$ axis,
with the magnitude of $B_{0}=14.1$ Tesla, which is extremely homogeneous
within at least 1 part in $10^{9}$. The RF section consists of the transmitter
and receiver, being that part of the spectrometer used for generating and
detecting the RF signals. The gradient field along the $z$ axis generated by
the gradient field coils can be used to average out the terms of zero quantum
coherence of density matrix. When the sample is placed in the magnetic field
of $B_{0}$, the NMR signal is produced by excitation of the nuclei (such as
${}^{1}\text{H},^{13}\text{C}$) of sample molecule with resonant RF fields,
and is detected with RF receivers. At the beginning of the experiment, the
nuclear spins are thermalized to the equilibrium state by waiting a long
period, which requires several minutes for well-prepared liquid samples. Under
the external control, RF pulses are then applied to implement the desired
unitary transformation on the initial state of spins. The final state is
measured from the peaks of frequency spectrum obtained by the Fourier
transformation of NMR signal that is generated via the free induction decay.

Figure S4:  Schematic of an NMR apparatus, including a large superconducting
magnet, the RF transmitter, receiver $\&$ amplifier, the control console and
recorder, the gradient field coils, and the sample held in a glass tube. The
superconducting magnet provides a static magnetic field $B_{0}=14.1$ Tesla
along the $z$ axis.

The sample employed in our experiments is the molecule of 13C-labeled trans-
crotonic acid dissolved in d6-acetone, shown in Fig. S5A. It consists of four
carbon nuclei with spin-1/2, which are used as our physical system. The
natural Hamiltonian of this system in the rotating frame reads

$\mathcal{H}_{\text{NMR}}=\sum_{i=1}^{4}\frac{\omega_{i}}{2}\sigma_{i}^{z}+\sum_{i<j,=1}^{4}\frac{\pi
J_{ij}}{2}\sigma_{i}^{z}\sigma_{j}^{z},$ (S7)

where $\omega_{i}$ represents the chemical shift of spin $i$, and $J_{ij}$ the
coupling constant between spins $i$ and $j$. The relevant Hamiltonian
parameters can be found in Fig. S5B. It shows that the nearest neighbor
couplings are dominating compared to others. The other interactions with the 1
H spins were decoupled by the WALTZ-16 decoupling sequence throughout the
experiments.

Figure S5C illustrates the 13C spectra of the thermal equilibrium state
$\rho_{\text{eq}}$ and the pseudo-pure state (PPS) $\rho_{0000}$, which was
prepared from the $\rho_{\text{eq}}$ by three line-selective shaped pulses.
The first shaped pulse with length of 80 ms was applied to selectively excite
the energy levels and make the populations equal of all levels except for
$0000$-level. It can be written as the product of a sequence of line-selective
pulses, i.e.,

$U_{1}=\prod_{n=1}^{14}e^{-i\theta_{n}\hat{I}_{x}^{(n_{1},n_{2})}},$ (S8)

where $\hat{I}_{x}^{(n_{1},n_{2})}$ is the single-transition operator between
energy levels $n_{1}$ and $n_{2}$. The flipping angles
$\\{\theta_{n}\\}_{n=1,2,\cdots,14}$ were optimized by numerical search
procedure, with appropriate values that satisfies the following condition:
$\text{diag}[U_{1}\rho_{\text{eq}}U_{1}^{{\dagger}}]=\text{diag}[\rho_{0000}]$.
The other shaped pulse with length of 24 ms were applied to remove zero
quantum coherence that cannot be averaged out by $z$-direction gradient
fields. They consists of the controlled-NOT gates between qubits $n$ and $m$,
denoted as $\text{CNOT}_{nm}$. For example, the second and third shaped pulses
of $U_{2}=\text{CNOT}_{12}$ and $U_{3}=\text{CNOT}_{34}$ were chosen in our
experiment. The intensity ratios for the rightmost peak in PPS $\rho_{0000}$
and the reference in the equilibrium spectrum are $0.704,0.782,0.807,1.605$
for $\text{C}_{1},\text{C}_{2},\text{C}_{3},\text{C}_{4}$, respectively. The
signal loss are caused by the gradient fields and the relaxation effect with
different spin-lattice relaxation time $T_{1}$ for different carbon nuclei.
The $T_{1}$ of $C_{4}$ is longer than other carbon nuclei.

Figure S5:  (A) Molecular structure of 13C-labeled trans-crotonic acid, where
$\text{C}_{1},\text{C}_{2},\text{C}_{3}\text{ and }\text{C}_{4}$ nuclear spins
are used as a four-qubits quantum simulator. All hydrogen nuclei (green balls)
are decoupled throughout the experiments. (B) Its Hamiltonian parameters,
where diagonal and off-diagonal elements represent the chemical shifts and
J-coupling constants (in Hz), respectively. The $T_{1}$ and $T_{2}$ are spin-
lattice relaxation times and spin-spin relaxation times (in seconds). (C)
Experimental carbon spectra of thermal equilibrium state $\rho_{eq}$ (blue)
and pseudo-pure state $\rho_{0000}$ (red).

###  S2.2 Realization of the SSH model

Our realization of the SSH model is performed on the four nuclear spins of the
trans-crotonic acid. In Eq. (S7), the terms of the chemical shifts
$\omega_{i}$ and undesired couplings $J_{ij}$ for $j\neq i$ can be refocused
by spin echo sandwiches, as shown in Fig. S6. We get

$U_{\text{free}}(\tau/2)R_{i}^{x}(\pi)U_{\text{free}}(\tau/2)=\text{exp}\left[-i\left(\sum_{j\neq
i}^{4}\frac{\omega_{j}}{2}\sigma_{j}^{z}+\sum_{j,k\neq i}^{4}\frac{\pi
J_{jk}}{2}\sigma_{j}^{z}\sigma_{k}^{z}\right)\tau\right]R_{i}^{x}(\pi),$ (S9)

where
$U_{\text{free}}(\tau/2)\equiv\text{exp}(-i\mathcal{H}_{\text{NMR}}\tau/2)$
represents the free evolution of the NMR Hamiltonian (S7), and
$R_{i}^{x}(\pi)\equiv\text{exp}(-i\pi\sigma_{i}^{x}/2)$ stands for a selective
$\pi$ pulse acting on spin $i$ along the $x$ axis. From the right-hand side of
Eq. (S9), it could be found that the $\omega_{i}$ and $J_{ij}$ for $j\neq i$
terms are removed. Using the spin echo sandwiches, we keep the dominate
couplings of $J_{12}=41.6$ Hz, $J_{23}=69.7$ Hz, and $J_{34}=72.2$ Hz,
compared to others ($J_{13}=1.4$ Hz, $J_{14}=7.0$ Hz, $J_{24}=1.2$ Hz). So the
corresponding spin interactions we reached here is

$\mathcal{H}_{\text{int}}=\frac{\pi}{2}\sum_{i=1}^{3}J_{i,i+1}\sigma_{i}^{z}\sigma_{i+1}^{z},$
(S10)

only with the nearest neighbor couplings. One can further apply the pulse
sequence shown in Fig. 1B of main text to create the effective couplings
$J_{ij}^{\text{eff}}$, where using a pair of selective $\pi$ pulses acting on
$\text{C}_{i}$ and $\text{C}_{j}$, we have

$U_{ij}(\tau_{2})R_{i}^{x}(\pi)U_{ij}(\tau_{1})R_{i}^{x}(\pi)=\text{exp}\left[-i\frac{\pi
J_{ij}^{\text{eff}}}{2}\sigma_{i}^{z}\sigma_{j}^{z}(\tau_{1}+\tau_{2})\right]$
(S11)

with $J_{ij}^{\text{eff}}=J_{ij}\frac{\tau_{2}-\tau_{1}}{\tau_{1}+\tau_{2}}$.
The $U_{ij}$ represents the evolution under the $J_{ij}$ coupling interaction.
We can now adjust the strength of $J_{ij}^{\text{eff}}$ by changing the
intervals of $\tau_{1}$ and $\tau_{2}$. If $\tau_{1}>\tau_{2}$, the
$J_{ij}^{\text{eff}}$ term has the negative sign; if $\tau_{1}=\tau_{2}$, the
$J_{ij}^{\text{eff}}$ term is turned off, corresponding to the same sequence
with the spin echo sandwich.

The $\sigma_{i}^{x}\sigma_{i+1}^{x}$ or $\sigma_{i}^{y}\sigma_{i+1}^{y}$ terms
can be generated via rotating the $\sigma_{i}^{z}\sigma_{i+1}^{z}$ terms by a
pair of $\pi/2$ hard pulses along the $y$ or $x$ axis. Therefore, combined
with the above techniques, we realize the effective Hamiltonian of SSH model

$\mathcal{H}_{\text{SSH}}^{\text{S,
eff}}=-\frac{\pi}{2}\sum_{i=1}^{3}J_{i,i+1}^{\text{eff}}(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}).$
(S12)

It can be set to the topological configuration with
$J_{12}^{\text{eff}},J_{34}^{\text{eff}}\approx
0\text{Hz},J_{23}^{\text{eff}}=69.7\text{Hz}$, and the trivial configuration
with
$J_{12}^{\text{eff}},J_{34}^{\text{eff}}=41.6\text{Hz},J_{23}^{\text{eff}}\approx
0\text{Hz}$.

The external Hamiltonian of an introduced magnetic field
$\mathcal{H}_{\text{ext}}$ can be realized by applying a hard RF pulse that
acts on all carbon nuclei, whose corresponding Hamiltonian is

$\mathcal{H}_{\text{RF}}=-\pi
B_{\text{RF}}\left[\text{cos}\phi_{\text{RF}}\sum_{i=1}^{4}\sigma_{i}^{x}-\text{sin}\phi_{\text{RF}}\sum_{i=1}^{4}\sigma_{i}^{y}\right],$
(S13)

where $B_{\text{RF}}$ and $\phi_{\text{RF}}$ are the tunable amplitude and
phase of RF pulse respectively. Under the rotation of a pair of $\pi/2$ hard
pulses along $y$ axis, the $\sum_{i=1}^{4}\sigma_{i}^{x}$ in Eq. (S13) becomes
$\sum_{i=1}^{4}\sigma_{i}^{z}$. Finally, we obtain the target Hamiltonian

$\mathcal{H}_{T}=-\pi
B_{\text{RF}}\left[-\text{sin}\phi_{\text{RF}}\sum_{i=1}^{4}\sigma_{i}^{y}+\text{cos}\phi_{\text{RF}}\sum_{i=1}^{4}\sigma_{i}^{z}\right]-\mathcal{H}_{\text{SSH}}^{\text{S,
eff}},$ (S14)

which can be mapped to the spin Hamiltonian (S4) of the SSH model in the
external magnetic field by setting $h_{r}=\pi B_{\text{RF}}$,
$\theta=2\pi-\phi_{\text{RF}}$, and $\phi=\pi/2$.

Figure S6: A spin echo sandwich of duration $\tau$. It consists of two
intervals of free evolution, both of duration $\tau/2$, “sandwiching” a line-
selective pulse of flip angle $\pi$.

##  S3 Numerical simulation and analysis

###  S3.1 Adiabatic preparation of many-body ground states

The initial ground states with different configuration of SSH model were
prepared adiabatically along the three paths shown in Fig. 1D of main text.
Taking the Path 1 for example, the adiabatic sweeps of $h_{x}$s and
$J_{23}^{\text{eff}}$ for preparing the topological phase are shown in Fig.
S7, where the auxiliary sweeps of the external field along x direction $h_{x}$
are added to avoid the closed gap when scanning the strength of
$J_{23}^{\text{eff}}$. We start with the initial state $|0000\rangle$, being
the ground state of $\mathcal{H}_{z}=-h_{z}\sum_{i=1}^{4}\sigma_{i}^{z}$, and
then sweeps the control parameters successively as follows: i)
$h_{x}:0\rightarrow 41.6$Hz; ii) $J_{23}^{\text{eff}}:0\rightarrow 69.7$Hz;
iii) $h_{x}:41.6\text{Hz}\rightarrow 0$. To ensure that the system always
stays in the instantaneous ground state, the control parameters need to change
slowly enough to satisfy the adiabatic condition [31]

$\left|\frac{\langle\psi_{g}|\dot{\psi}_{e}\rangle}{E_{e}-E_{g}}\right|=\frac{d|s(t)|}{dt}\left|\frac{\langle\psi_{g}|\partial_{s}\mathcal{H}(s)|\psi_{e}\rangle}{(E_{e}-E_{g})^{2}}\right|\ll
1.$ (S15)

Here the $|\psi_{g}\rangle$ and $|\psi_{e}\rangle$ represent the instantaneous
ground and first excited states, and $s(t)$ denotes the shorthand symbol of
control parameters of $h_{x}$ and $J_{23}^{\text{eff}}$. The above condition
determines the optimal sweep of control parameters of $h_{x}$ and
$J_{23}^{\text{eff}}$, shown in the solid red lines of Fig. S7. In
experiments, the sweeps were discretized 32 steps (green circles), and the
total time is $T=69.3$ ms. The right longitudinal axis represents the fidelity
between the theoretical and simulated ground states, that is,
$|\langle\psi_{g}^{\text{A}}(t_{i})|\psi_{g}^{\text{Th}}(t_{i})\rangle|$. It
can be found that the numerical value of fidelity for each time step is above
0.99, and final fidelity reaches to 0.998. That indicates it is credible that
the system always stay in the ground state in the adiabatic evolution.

Figure S7:  Adiabatic sweeps of $h_{x}$s and $J_{23}^{\text{eff}}$ along the
Path 1 (shown in Fig. 1(D) of main text) for preparing the ground state,
corresponding to a topological phase. The red solid lines were calculated
according to the adiabatic condition (see Eq. (S15)). The green circles are
the interpolations along the adiabatic sweeps. The blue triangles represent
the fidelities of
$|\langle\psi_{g}^{\text{A}}(t_{i})|\psi_{g}^{\text{Th}}(t_{i})\rangle|$
between the theoretical and simulated ground states during the discretized
time steps. When the evolution time T = 69.3 ms, the final fidelity is 0.998,
which indicates the adiabatic preparation of ground state is receivable.

###  S3.2 Optimization of the quenching time

In order to extract the Berry curvatures with the high accuracy from the
linear responses, the quench velocity $v_{\theta}$ should be slow enough such
that the high-order terms of $v_{\theta}$ are negligible. We simulated the
dynamical responses of the generalized force at different quench time
$t_{f}=\pi/v_{\theta}$, with fixed $h_{r}=50$ Hz. The $t_{f}$ is the time it
takes to ramp from the north pole of spherical manifold of $\vec{h}$ to its
south pole. The numerical Chern numbers are shown in Fig. S8, where the
$t_{f}$ becomes longer, the dynamical quench is more adiabatic and the phase
diagram approaches to the theoretical one (the final picture in Fig. S8)
calculated by the definition of Chern number. When $t_{f}=320$ ms, it is a
good simulation with the theoretical calculation. Therefore, the quench time
$t_{f}=350$ ms taken in our experiments was indeed slow enough, such that the
high-order errors remain small.

Figure S8: Chern number numerically extracted from the linear response at
different quench times $t_{f}$, where $t_{f}$ is the time it takes to ramp
from the north pole of spherical manifold to its south pole. In these
simulations, the external magnetic field was fixed to $h_{r}=50$ Hz. The
longer $t_{f}$ is more adiabatic and hence better in the simulation results.
The final picture (labelled by “Adiabatic”) is plotted by theoretical
calculation.

###  S3.3 Robustness analysis of degeneracy points

In Subsection (S1.2), we have explained the meaning of Chern number, which
reveals the singularities of systems and corresponds to the degeneracy points
enclosed in the spherical parameter manifold of $\vec{h}$. The nonzero value
of Chern number emerges in the integral form of Eq. (S6) if and only if the
ground states is degenerate. For the small systems with four qubits, the
energy-level diagrams of the topological and trivial configurations in the
presence of small perturbations are readily plotted by numerical calculations,
and we can then obtain the locations of points of degenerate ground states,
shown in Figs S9A and S9B. The small perturbations are introduced here for
visualizing the number of degeneracy points more intuitively. We further
analyze the robustness of topological and trivial phases by the numerical
calculations of the distance $dh_{z}$ between the degeneracy points. In the
presence of intracell hopping perturbation $J_{1}^{{}^{\prime}}$, the
topological degeneracies between the second and the third points in Fig. S9A
separate exponentially (see the green circles with the fitting curve of
$dh_{z}=0.39e^{0.28J_{1}^{{}^{\prime}}}-0.40$ in Fig. S9C), whereas in the
presence of intercell hopping perturbation $J_{2}^{{}^{\prime}}$, the trivial
degeneracies between the first and the second points in Fig. S9B separate
linearly (see the red circles with the fitting curve of
$dh_{z}=1.57J_{2}^{{}^{\prime}}$ in Fig. S9C). The results show that
topological degeneracies are more robust than trivial ones under small
perturbations.

Figure S9: Energy-level diagram and degeneracy points of topological (A) and
trivial (B) configurations in the presence of small perturbations. The
degeneracy points enclosed in the spherical field act as the sources of Chern
number and are analogous to magnetic monopoles. (C) Robustness of degeneracy
points against the perturbations of intracell hopping $J_{1}^{{}^{\prime}}$ or
intercell hopping $J_{2}^{{}^{\prime}}$. The green (red) circles represent the
degeneracy distance $dh_{z}$ between the second and the third (the first and
the second) points in the topological (trivial) energy-level diagram. The
fitting results show that topological degeneracies (separate exponentially)
are more robust than trivial ones (separate linearly) under small
perturbations.

##  S4 Measurement and discussion

###  S4.1 Measurement of spin-spin correlations

We performed the measurements of the spin-spin correlations for different
initial states using five independent readout experiments. The NMR signals
obtained through the quadrature detection are given by [32],

$S(t)=\text{Tr}\left[\rho_{g}R^{{\dagger}}e^{i{H}_{\text{NMR}}t}\sum_{j=1}^{4}(\sigma_{j}^{x}+i\sigma_{j}^{y})e^{-i{H}_{\text{NMR}}t}R\right],$
(S16)

where $\rho_{g}$ represents the initial ground states prepared by adiabatic
evolutions (see Subsection S3.1), and $R$ denotes the five readout pulses
listed in Table S1, where $I$ is the unit operator. The corresponding readout
spin operators can be used to reconstruct the spin-spin correlations of
$C_{ij}^{x}=\langle\sigma_{i}^{x}\sigma_{j}^{x}\rangle-\langle\sigma_{i}^{x}\rangle\langle\sigma_{j}^{x}\rangle$
and
$C_{ij}^{z}=\langle\sigma_{i}^{z}\sigma_{j}^{z}\rangle-\langle\sigma_{i}^{z}\rangle\langle\sigma_{j}^{z}\rangle$
for $i,j=1,2,3,4$. Figures S10A and S10B show the measured $C_{ij}^{z}$ of the
topological and trivial phases. Combined with the results of $C_{ij}^{x}$ in
Figs. 1E and 1F of main text, it further confirms the intercell spin
correlations with $C_{23}=-0.969$ for the topological phase, and the intracell
spin correlations with $C_{12}^{z}=-0.863,C_{34}^{z}=-0.984$ for the trivial
phase. Different from the two topologically distinct phases, the measured
$C_{ij}^{z}$ (Fig. S10C, left) and $C_{ij}^{x}$ (Fig. S10C, right) of spin-
polarized (SP) phase demonstrate that there exist no correlation between
spins.

Table S1: Readout pulses and their corresponding readout spin operators for reconstructing the spin-spin correlations. Readout pulses | Readout spin operators
---|---
$I$ | $\sigma_{1}^{x},\sigma_{2}^{x},\sigma_{3}^{x},\sigma_{4}^{x}$
$R_{1}^{y}(\pi/2)$ | $\sigma_{1}^{z},\sigma_{1}^{z}\sigma_{2}^{z},\sigma_{1}^{z}\sigma_{3}^{z},\sigma_{1}^{z}\sigma_{4}^{z},\sigma_{1}^{x}\sigma_{2}^{x},\sigma_{1}^{x}\sigma_{3}^{x},\sigma_{1}^{x}\sigma_{4}^{x}$
$R_{2}^{y}(\pi/2)$ | $\sigma_{2}^{z},\sigma_{2}^{z}\sigma_{3}^{z},\sigma_{2}^{z}\sigma_{4}^{z},\sigma_{2}^{x}\sigma_{3}^{x},\sigma_{2}^{x}\sigma_{4}^{x}$
$R_{3}^{y}(\pi/2)$ | $\sigma_{3}^{z},\sigma_{3}^{z}\sigma_{4}^{z},\sigma_{3}^{x}\sigma_{4}^{x}$
$R_{4}^{y}(\pi/2)$ | $\sigma_{4}^{z}$
Figure S10: Measured spin-spin correlations along the $z$ direction of (A)
topological and (B) trivial phases. (C) show the measured spin-spin
correlations along the $z$ (left) and $x$ (right) directions of spin-polarized
phase.

###  S4.2 Observation of the spin-polarized phase

We now discuss the influence of the introduced magnetic fields, which itself
lead to the external region of SP phase in the absence of the intracell- and
intercell hopping interactions of $J_{12}^{\text{eff}}$ and
$J_{23}^{\text{eff}}$, as shown in Fig. S11A. The pink region of SP phase
depends on the field strength $h_{r}$, and gradually decreases to zero as
$h_{r}\rightarrow 0$. The SP phase featuring the Chern number four can be
understood from the energy-level diagram (Fig. S11B). There are four points of
degenerate ground states in the presence of small perturbations of
$J_{12}^{\text{eff}}$ and $J_{23}^{\text{eff}}$, otherwise these points are
superimposed without the perturbations. We know that there is a one-to-one
correspondence between the number of degeneracy points and the Chern number
(Fig. S11C). When the system was set to the SP configuration (i.e.,
$J_{12}^{\text{eff}},J_{23}^{\text{eff}},J_{34}^{\text{eff}}=0$ Hz), we
measured the Chern number under different external magnetic fields shown in
Fig. S11D, and obtained the average Chern number $\mathcal{C}h=3.996\pm
0.065$. There is no obvious change of Chern number as $h_{r}\rightarrow 0$,
because the degeneracy points appear near $h_{r}=0$. To gain further insight
of the SP phase with the “nontrivial” Chern number, we analyze the scaling
behavior. Figure S11D shows the calculated Chern number dependence of system
size, where the Chern number grows linearly with the increase of cell number
for the SP phase which originates from the induced external magnetic fields.

Figure S11: (A) Calculated Chern number as a function of effective couplings
$J_{12}^{\text{eff}}$ and $J_{23}^{\text{eff}}$ for different external
magnetic fields $h_{r}$. The pink region of SP phase results from the external
magnetic field and gradually decreases as $h_{r}\rightarrow 0$. (B) Energy-
level diagram and degeneracy points of the SP configuration (i.e.,
$J_{12}^{\text{eff}},J_{23}^{\text{eff}},J_{34}^{\text{eff}}=0$ Hz) in the
presence of small perturbations. (C) Schematic of measuring the Chern number
that corresponds to the total number of degeneracy points (small red balls)
enclosed in spherical parameter space. The four degeneracy points at $h_{r}=0$
totally originate from the introduced external magnetic field itself. (D)
Measured Chern number in the SP configuration under different external
magnetic fields, which was obtained from the integral of Berry curvature
$\mathcal{F}_{\theta\phi}$ of Fig. S14B. There is no obvious change of Chern
number ($3.996\pm 0.065$), as $h_{r}\rightarrow 0$. (E) Calculated Chern
number as a function of cell number. In the SP phases, the Chern number grows
linearly with the increase of cell number.

###  S4.3 Other data for measuring the Chern numbers

The experimental Chern number of each $h_{r}$ was obtained by the integral of
the Berry curvatures, which were extracted from the linear responses of the
spin magnetizations. For the case of topological configuration, the measured
spin magnetizations $\langle\sigma_{j}^{x}\rangle$ for $j=1,2,3,4$ and the
resulting Berry curvatures $\mathcal{F}_{\theta\phi}$ under different external
magnetic fields are shown in Figs S12A and S12B, where each
$\langle\sigma_{j}^{x}\rangle$ was measured 101 times along the quench path.
The similar results for the cases of trivial and spin-polarized configurations
are shown in Figs. S13 and S14, respectively. Figure S15 shows the
experimental data of $\langle\sigma_{j}^{x}\rangle$ and
$\mathcal{F}_{\theta\phi}$ under different effective couplings
$J_{12}^{\text{eff}}$, with fixed $h_{r}=50$ Hz and $J_{23}^{\text{eff}}=69.7$
Hz, for probing the topological transition in the main text.

Figure S12: (A) Measured spin magnetizations of $\langle\sigma_{j}^{x}\rangle$
for $j=1,2,3,4$ under different external magnetic fields, in the case of
topological configuration, i.e., $J_{12}^{\text{eff}},J_{34}^{\text{eff}}=0$
Hz, $J_{23}^{\text{eff}}=69.7$ Hz. Each $\langle\sigma_{j}^{x}\rangle$ was
measured at 101 time steps during the quenching evolution. The solid lines
show the numerical results. (B) Extracted Berry curvatures from the linear
responses of the spin magnetizations in (A). The error bars are given by the
fitting uncertainties of experimental spectra, which refer to the readout
errors. Figure S13: (A) Measured spin magnetizations of
$\langle\sigma_{j}^{x}\rangle$ for $j=1,2,3,4$ under different external
magnetic fields, in the case of trivial configuration, i.e.,
$J_{12}^{\text{eff}},J_{34}^{\text{eff}}=41.6$ Hz, $J_{23}^{\text{eff}}=0$ Hz.
Each $\langle\sigma_{j}^{x}\rangle$ was measured at 101 time steps during the
quenching evolution. The solid lines show the numerical results. (B) Extracted
Berry curvatures from the linear responses of the spin magnetizations in (A).
The error bars are given by the fitting uncertainties of experimental spectra,
which refer to the readout errors. Figure S14: (A) Measured spin
magnetizations of $\langle\sigma_{j}^{x}\rangle$ for $j=1,2,3,4$, in the case
of spin-polarized configuration, i.e.,
$J_{12}^{\text{eff}},J_{34}^{\text{eff}},J_{23}^{\text{eff}}=0$ Hz. The
initial states for different external magnetic fields $h_{r}$ began with
$|0000\rangle$, corresponding to the ground states at the north pole of
parameter spheres, and each $\langle\sigma_{j}^{x}\rangle$ was measured at 101
time steps during the quenching evolution. The solid lines show the numerical
results. (B) Extracted Berry curvatures from the linear responses of the spin
magnetizations in (A). The error bars are given by the fitting uncertainties
of experimental spectra, which refer to the readout errors. Figure S15: (A)
Measured spin magnetizations of $\langle\sigma_{j}^{x}\rangle$ for $j=1,2,3,4$
under different effective couplings $J_{12}^{\text{eff}}$, with fixed
$h_{r}=50$ Hz and $J_{23}^{\text{eff}}=69.7$ Hz. Each
$\langle\sigma_{j}^{x}\rangle$ was measured at 101 time steps during the
quenching evolution. The solid lines show the numerical results. (B) Extracted
Berry curvatures from the linear responses of the spin magnetizations in (A).
The error bars are given by the fitting uncertainties of experimental spectra,
which refer to the readout errors.

###  S4.4 Experimental error analysis

We make an analysis to the data set of the case of topological configuration.
The standard deviations of the experimental and simulated spin magnetizations
were calculated via
$\sigma_{\text{exp(sim)}}=\sqrt{\sum_{i=1}^{M}(\langle\sigma_{j}^{x}\rangle_{\text{exp(sim)}}-\langle\sigma_{j}^{x}\rangle_{\text{th}})^{2}/M}$,
where $M=101$ represents the measured times. The experimental and simulated
results for different external magnetic fields are listed in Table. S2. It can
be seen that the sources of experimental errors mainly include the
imperfection of preparing the initial states (denoted as
$\sigma_{\text{sim}}^{i}$), the inhomogeneity of quenching the RF fields
(denoted as $\sigma_{\text{sim}}^{q}$), and the readout errors (denoted as
$\sigma_{\text{sim}}^{\text{r}}$). Here the simulated standard deviations are
investigated separately by only one error source, while the other procedures
are assumed to be ideal. For the initial state preparation, the major errors
are caused by the imperfection of the line-selective pulses in PPS
$\rho_{0000}$ ($\approx 1.21\%$) and the adiabatic evolution of ground states
at the north pole of spherical parameter manifold ($\approx 0.78\%$). The
external magnetic field is realized by a hard RF pulse, which is inhomogeneous
acting on all carbon nuclei with different amplitudes of $B_{\text{RF}}$ in
Eq. (S13). We simulated the quench dynamics based on a simple inhomogeneity
model, which assume that the output amplitude discrepancy of RF field
$B_{\text{RF}}$ is uniformly distributed between $\pm 3\%$. The average of
simulated standard deviations of $\sigma_{\text{sim}}^{\text{q}}$ is
approximately $1.50\%$. All measured values of $\langle\sigma_{j}^{x}\rangle$
in experiments are readout through the spectrum fitting. So the readout errors
can be evaluated using the fitting uncertainties, which are determined by the
absolute intensity of the spectral ground noise (white noise). It is worth
noting that another source of error comes from the decoherence effect,
however, which is small and can be ignored in our experiments, because the
measured signals have been normalized using the initial ground states.

Table S2: Standard deviations of the experimental and simulated spin magnetizations under different external magnetic fields in the case of topological configuration. The $\sigma_{\text{sim}}^{\text{i}},\sigma_{\text{sim}}^{\text{q}}$ and $\sigma_{\text{sim}}^{\text{r}}$ represent the standard deviations calculated by numerical simulations with only one error source, that is, the imperfection of preparing the initial states, the inhomogeneity of quenching the RF fields, and the readout errors, respectively. $h_{r}(\text{Hz})$ | 190 | 170 | 150 | 130 | 110 | 90 | 70 | 50 | 30 | 10
---|---|---|---|---|---|---|---|---|---|---
$\sigma_{\text{exp}}$ |  3.83% |  4.85% |  4.22% |  5.26% |  8.17% |  3.20% |  3.73% |  5.22% |  3.46% |  2.60%
$\sigma_{\text{sim}}^{\text{i}}$ |  1.79% |  2.10% |  2.37% |  2.41% |  2.46% |  1.96% |  1.91% |  1.78% |  1.68% |  1.59%
$\sigma_{\text{sim}}^{\text{q}}$ |  1.72% |  1.63% |  2.19% |  1.28% |  1.08% |  1.26% |  1.52% |  1.21% |  1.76% |  1.39%
$\sigma_{\text{sim}}^{\text{r}}$ |  1.41% |  1.64% |  1.50% |  2.00% |  3.19% |  1.04% |  0.92% |  1.18% |  1.60% |  1.19%