# Persistent Current of SU(N) Fermions

Wayne J. Chetcuti  Dipartimento di Fisica e Astronomia, Via S. Sofia 64, 95127
Catania, Italy  INFN-Sezione di Catania, Via S. Sofia 64, 95127 Catania, Italy
Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE
Tobias Haug  Centre for Quantum Technologies, National University of
Singapore, 3 Science Drive 2, Singapore 117543, Singapore  Leong-Chuan Kwek
Centre for Quantum Technologies, National University of Singapore, 3 Science
Drive 2, Singapore 117543, Singapore  MajuLab, CNRS-UNS-NUS-NTU International
Joint Research Unit, UMI 3654, Singapore  Luigi Amico  On leave from
Dipartimento di Fisica e Astronomia ’Ettore Majorana’, Università di Catania,
Italy  Quantum Research Centre, Technology Innovation Institute, Abu Dhabi,
UAE  Centre for Quantum Technologies, National University of Singapore, 3
Science Drive 2, Singapore 117543, Singapore  LANEF ’Chaire d’excellence’,
Universitè Grenoble-Alpes & CNRS, F-38000 Grenoble, France

###### Abstract

We study the persistent current in a system of SU(N) fermions with repulsive
interaction, confined in a ring-shaped potential and pierced by an effective
magnetic flux. Several surprising effects emerge. As a combined result of spin
correlations, (effective) magnetic flux and interaction, spinons can be
created in the ground state such that the elementary flux quantum can change
its nature. The persistent current landscape is affected dramatically by these
changes. In particular, it displays a universal behaviour. Despite its
mesoscopic character, the persistent current is able to detect a quantum phase
transition (from metallic to Mott phases). Most of, if not all, our results
could be experimentally probed within the state-of-the-art quantum technology,
with neutral matter-wave circuits providing a particularly relevant platform
for our work.

Quantum technology intertwines basic research in quantum physics and
technology to an unprecedented degree: different quantum systems, manipulated
and controlled from the macroscopic spatial scale down to individual or atomic
level, can be platforms for quantum devices and simulators with refined
capabilities; on the other hand, the acquired technology prompts new studies
of fundamental aspects of quantum science with an enhanced precision and
sensitivity. Amongst the various quantum systems relevant for quantum
technologies, ultracold atomic systems play an important role due to their
excellent coherent properties and enhanced control and flexibility of the
operating conditions Acín  _et al._ (2018). Atomtronics is an emerging
research area in quantum technology exploiting cold atoms matter-wave circuits
with a variety of different architectures Amico  _et al._ (2021). Being
characterized by distinctive physical principles, atomic circuits can define a
quantum technology with specific features. In particular, one of the peculiar
knobs that can be exploited in atomtronics is the statistics of the particles
forming the quantum fluid flowing in the circuit. Most of the studies carried
out so far have been devoted to atomtronic circuits of ultracold bosons,
whilst ones comprised of interacting ultracold fermions require extensive
investigations.

In this paper, we focus on quantum fluids comprising of interacting
multicomponent spin SU($N$) fermions. Strongly interacting fermions with $N$
spin components, as provided by alkaline-earth and ytterbium cold atomic
gases, are highly non-trivial multicomponent quantum systems Cazalilla  _et
al._ (2009); Gorshkov  _et al._ (2010). Such systems extend beyond the physics
of interacting spin-$\frac{1}{2}$ electrons found in condensed matter systems
Pagano  _et al._ (2015); Cappellini  _et al._ (2014). They are very relevant
both for high-precision measurement Ludlow  _et al._ (2015); Marti  _et al._
(2018); Poli  _et al._ (2013) and to enlarge the scope of cold atoms quantum
simulators of many-body systems Livi  _et al._ (2016); Kolkowitz  _et al._
(2017); Scazza  _et al._ (2014); Hofrichter  _et al._ (2016). Additionally,
atom-atom interactions can be made independent on the nuclear spin. This
feature effectively enlarges the symmetry of the systems to the SU($N$) one.
Such a feature makes cold alkaline-earth atoms, especially with lattice
confinements, an ideal platform to study exotic quantum matter, including
higher spin magnetism, spin liquids and topological matter Capponi  _et al._
(2016); Cazalilla and Rey (2014); Guan  _et al._ (2013) and, beyond condensed
matter physics, in QCD Wiese (2014).

Here, we consider $N_{p}$ SU($N$) fermions with repulsive interaction, trapped
in a ring-shaped circuit of mesoscopic size $L$ Imry (2002) and pierced by an
effective magnetic field. We study the persistent current response to this
applied field, which provides a standard avenue to probe the coherence of the
system Gefen  _et al._ (1984).

Different regimes depending on the filling fraction $\nu=N_{p}/L$ are
explored. i) For incommensurate $\nu$, the persistent current is non-vanishing
for any value of the interaction. Monitoring the numerical results for the
spectrum of the system with the exact Bethe ansatz analysis Sutherland (1968),
we find that as the effective magnetic flux increases, spinon excitations can
be created in the ground state. Such a remarkable phenomenon occurs as a
specific ‘screening’ of the external flux, which being continuously adjustable
quantity, can be compensated by spinons excitations (quantized in nature) only
partially. This in turn results in an imbalance and causes the persistent
current to display characteristic oscillations with a period of ${1}/{N_{p}}$
shorter than the bare flux quantum. For two-spin component fermions in the
large interaction regime, such a phenomenon was studied in Kusmartsev (1991);
Yu and Fowler (1992). We shall see that such a process depends on $N_{p}$,
number of spin components and interaction in a non-trivial way. ii) In
contrast with the SU($2$) case Lieb and Wu (1968), SU($N$) fermions with $N>2$
undergo a Mott quantum phase transition for a finite value of interaction
$U=U_{c}$ at integer fillings. Accordingly, a metallic behaviour crosses over
to a regime in which the current is exponentially suppressed. This regime is
also monitored by Bethe ansatz Sutherland (1975) and corroborated by exact
diagonalization and Density Matrix Renormalization Group (DMRG) White (1992);
Stoudenmire and R. White . We shall see that, despite the persistent current
being mesoscopic in nature, the onset of the Mott transition is marked by a
clear finite size scaling. The onset to the gapped phase progressively hinders
the spinon creation phenomenon.

Methods – A system of $N_{p}$ SU($N$) fermions residing in a ring-shaped
lattice composed of $L$ sites threaded with a magnetic flux $\phi$ can be
modeled using the Hubbard model Capponi  _et al._ (2016)

$\mathcal{H}_{\textrm{SU}(N)}=-t\sum_{j=1}^{L}\sum\limits_{\alpha=1}^{N}\big{(}e^{\imath\frac{2\pi\phi}{L}}c^{\dagger}_{\alpha,j}c_{\alpha,j+1}+\textrm{h.c.}\big{)}+\frac{U}{2}\sum\limits_{j}n_{j}(n_{j}-1)$
(1)

where $c_{\alpha,j}^{\dagger}$ ($c_{\alpha,j}$) creates (annihilates) a
fermion with colour $\alpha$,
$n_{j}=\sum_{\alpha}c_{\alpha,j}^{\dagger}c_{\alpha,j}$ is the local particle
number operator for site $j$. The parameters $t$ and $U>0$ account for the
hopping strength and on-site repulsive interaction respectively. The effective
magnetic field is realized through Peierls substitution $t\rightarrow
te^{\imath\frac{2\pi\phi}{L}}$. For $N=2$, the Hubbard model describing
spin-$\frac{1}{2}$ fermions is obtained. In this case, the Hamiltonian (1) is
integrable by Bethe Ansatz (BA) for any $U/t$ and $\nu$ Lieb and Wu (1968).
For $N>2$, the BA integrability is preserved in the continuous limit of
vanishing lattice spacing, (1) turning into the Gaudin-Yang-Sutherland model
describing SU($N$) fermions with delta interaction Sutherland (1968); Decamp
_et al._ (2016); such a regime is achieved by (1) in the dilute limit of small
$\nu$. Another integrable regime of (1) is obtained for $n_{j}=1\forall j$ and
large repulsive values of $U\gg t$ for which the system is governed by the
Lai-Sutherland model Sutherland (1975); Capponi  _et al._ (2016).

In our approach, the exact solution plays a very important role in classifying
the eigenstates of the system. According to the general theory of BA solvable
models, the spectrum of the model is obtained through the solution of coupled
transcendental equations, which are paramterized by a specific set of numbers
called the quantum numbers Takahashi (1999). Indeed, they label all the
excitations. For the specific case of the integrable SU($N$) Hubbard models,
the many-body excitations are labeled by quantum numbers: $I_{a},\;a=1\dots
N_{p}$ and $J_{\beta_{j}},\;\beta_{j}=1\dots M_{j}\hskip
5.69054pt\mathrm{for}\hskip 5.69054ptj=1\dots N-1$ with $I_{a}$ and
$J_{\beta_{j}}$ with being integers or half-odd integers, and $M_{j}$ refers
to the number of particles with a given component Essler  _et al._ (2005);
Takahashi (1999); Sutherland (1968) (see Supplementary material). It is well
known that, at zero flux $\phi$, the ground state is found to be characterized
by $I_{a}=I_{1},I_{1}+1,I_{1}+2,\ldots I_{1}+N_{p}$ and
$J_{\beta_{1}}=J_{\beta_{1}},J_{\beta_{1}}+1,J_{\beta_{1}}+2,\ldots
J_{\beta_{1}}+M_{1}$. Instead, sequences of
$I_{a}=I_{1}-1,{\atop\vee},I_{1}+1,I_{1}+2,\ldots I_{1}+N_{p}$ and
$J_{\beta_{j}}=J_{\beta_{j}}-1,{\atop\vee},J_{\beta_{j}}+1,J_{\beta_{j}}+2,\ldots
J_{\beta_{j}}+M_{j}$ with ‘holes’ correspond to excitations; in particular
holes in $\\{J_{\beta}\\}$ characterize the so-called spinon excitations
Andrei (1995).  For SU($N$) fermions, there can be $N-1$ different types of
such spinon states Frahm and Schadschneider (1995); Lee and Schlottmann
(1989). For non-vanishing $\phi$, we shall see the quantum numbers
configurations $\\{I_{a},J_{\beta}\\}$ can change. For intermediate
interactions and intermediate fillings, the model $(\ref{eq:1})$ is not
integrable and approximated methods are needed to access its spectrum. Indeed,
Hubbard models for SU($2$) and SU($N$) fermions enjoy very different physics.
For incommensurate fillings, a metallic behaviour is found with characteristic
oscillations of the spin-spin and charge correlation functions that, for $N>2$
can be coupled to each other. At integer $\nu$, fermions may be in a Mott
phase. Such a phase is suppressed only exponentially for $N=2$ Lieb and Wu
(1968); in striking contrast, for $N>2$ the system displays a Mott transition
for a finite value of $U/t$ Capponi  _et al._ (2016); Xu  _et al._ (2018).

Figure 1: Persistent current $I(\phi)$ at incommensurate filling for SU($3$)
fermions with different interaction strengths $U$ in the dilute filling regime
of the Hubbard model. The exact diagonalization $L=30,N_{p}=3$ is monitored
with the BA of the Sutherland-Gaudin-Yang model. The red, black and green dots
in the main figure depict the Bethe ansatz results for the persistent current
for $U$ = 0.1, 1.0 and 10,000 respectively. These dots are meant to be a guide
to the eye, to aid in perceiving the fractionalization of the persistent
current with increasing interaction Insets show how the BA energies need to be
characterized by $X\neq 0$, to be the actual ground state. At $U=0$, the
ground state energy is a periodic sequence of parabolas meeting at degeneracy
points $\phi_{d}$ ($\phi_{d}=1/2$ for the case in this figure). The values of
the flux at which spinons are created $\phi_{s}$ have been included as an
example in the top inset, which for an interaction $U=1$, range from 0.37 to
0.63. Figure 2: Figures of merit for spinon creation in the ground state of
SU($N$) fermions. We consider the minimum value of $U$ required for spinons to
be created in the ground state for a given value of $\phi$; all the values of
$\phi$ where a spinon is created are recorded. The displayed curves are
calculated by monitoring all the distances $|\phi_{s}-\phi_{d}|$ at which the
state with no spinons crosses states with any spinon states, where $\phi_{s}$
is the flux at which spinons are created and $\phi_{d}$ is the degeneracy
point (see Fig. 1).  a) Spinon creation flux distance $|\phi_{s}-\phi_{d}|$
against interaction $U$. The inset contains the data in the intermediate $U$
regime. b)Spinon creation flux distance against the interaction, rescaled by
$N$ and $N_{p}$ respectively, in the limit of low $UN_{p}$. In this limit,
spinon production is found to be a universal function of $N_{p}/N$ c) Spinon
creation flux distance $|\phi_{s}-\phi_{d}|$ against interaction per particle
$U/N_{p}$ in the low $U/N_{p}$ regime. One can observe the enhancement of
spinon production with increasing $N$. All the presented results are obtained
by BA of Gaudin-Yang-Sutherland model for $L=40$, with $N_{p}=1$(circles),
$2$(squares), $3$(crosses) per spin component, with $N=2,3,4,5$. Thus, the
dilute limit of the Hubbard model (1) is covered.

At mesoscopic size, the properties discussed above are displayed as specific
traits Imry (2002). In this regime, even though the application of the
magnetic flux does not change the nature of the possible excitations, we shall
see that the latter may be indeed promoted to ground states. Our diagnostic
tool is the persistent current, providing access to the particles’ phase
coherence Gefen  _et al._ (1984); Shastry and Sutherland (1990). At zero
temperature, the persistent current of the system is given by
$I(\phi)=-\frac{\partial E_{0}}{\partial\phi}$ where $E_{0}$ is the ground
state energy. The persistent current is of mesoscopic nature in that it
corresponds to $1/L$ corrections of the ground state energy Gefen  _et al._
(1984).

For a quantum system in a ring, the angular momentum is quantized (see Moulder
_et al._ (2012); Wright  _et al._ (2013) for recent experiments). Accordingly,
$I(\phi)$ displays a characteristic sawtooth behaviour, with a periodicity
that Leggett proved to be fixed by the effective flux quantum $\phi_{0}$ of
the system Byers and Yang (1961); Onsager (1961); Leggett (1991). Furthermore,
the persistent current is parity dependent: for systems with even (odd) number
of spinless particles, the energy is decreased (increased) by the application
of the external flux; therefore, the persistent current displays a
paramagnetic (diamagnetic) behaviour. Leggett predictions are independent of
disorder. In particular, the periodicity of the persistent current reflects
the structure of the ground-state. For example in the case of a BCS ground-
state, the period of the persistent current is halved due to the formation of
Cooper pairs Byers and Yang (1961); Pecci  _et al._ (2021). Similarly for
bosonic systems, the persistent current fractionalized with a period of
$1/N_{p}$, indicating the formation of a bound state of $N_{p}$ particles Polo
_et al._ (2020). The specific parity effects also hold for SU($N$) systems and
SU(2) fermions Byers and Yang (1961); Onsager (1961) (see Supplementary
material).

In our approach, we combine exact diagonalization or DMRG analysis with,
whenever possible, BA results. Specifically: in the integrable regimes of
dilute systems (described by Gaudin-Yang-Sutherland model), and a filling of
one-particle per site $\&$ large interactions (captured by the Sutherland
model), the BA results (through the Bethe quantum numbers introduced above)
are exploited as bookkeeping to monitor the eigenstates provided by the
numerical results. This way, the nature and physical content of the system’s
ground state can be established as functions of the parameters. We shall see
that the actual lowest energy of the system can only be obtained with Bethe
quantum numbers corresponding to spinon excitations. In the non-integrable
regimes, we rely on numerical analysis. Here, only systems with an equal
number of particles per species are considered. In the following, the energy
scale is given by $t=1$.

Persistent current of SU($N$) fermions at incommensurate fillings – Our
analysis begins in the low $\nu$ regimes (continuous limit) wherein we can
rely on exact results based on the Gaudin-Yang-Sutherland model BA. The
numerical analysis shows that, by increasing $\phi$, specific energy level
crossings occur in the ground state of the system. The BA analysis (see
Supplemental material) allows us to recognize such level crossings as ground
state transitions between no-spinon states and spinon states. Specific
$1/{N_{p}}$ periodic oscillations occur in the ground state energy as $\phi$
is varied; therefore, a curve with $N_{p}$ cusps/parabolic-wise segments per
flux quantum emerges. Such a feature was evidenced for two-spin component
fermions in the large interaction regime Kusmartsev (1991); Yu and Fowler
(1992). Here, we find that spinon creation defines a phenomenon occurring for
any value of $U$; additionally, we shall see that the spinon creation
mechanism displays a non-trivial dependence on the number of spin components
$N$. Indeed, the different $N-1$ spinon configurations are found to play a
relevant role for the phenomenon. The quantity
$X=\sum_{j}^{N-1}\sum_{\beta_{j}}^{M_{j}}J_{\beta_{j}}$ can be exploited to
characterize the properties of the specific spinon excitations that are
created in the ground state.

Specifically, for small and intermediate $U$, while the system’s ground state
with no spinons is found to be non-degenerate, the one with spinons can be
made of degenerate multiplets corresponding to Bethe states with distinct
configurations of the quantum numbers $J_{\beta_{j}}$ (see inset of Fig. 1).
By further increasing $U$, the spinon states organize themselves in multiplets
of increasing degeneracy on a wider interval of the flux. At large but finite
$U$, the exact BA analysis shows that the spectrum can be reproduced by a
suitable continuous limit of a SU($N$) $t-J_{eff}$ model with
$J_{eff}={{4}E_{\infty}/{(UL)}}$, where $E_{\infty}$ is the energy of the
Gaudin-Yang-Sutherland model at infinite interaction (see Supplementary
material). We remark that the specific features of the SU($N$) fermions enter
the entire energy spectrum of the system through the SU($N$) quantum numbers
$\\{I_{a},J_{\beta_{j}}\\}$. In the limit of infinite $U$, the persistent
current is analytically obtained as (derivations in Supplementary material)

$I(\phi)=-2\bigg{(}\frac{2\pi}{L}\bigg{)}^{2}\sum\limits_{a}^{N_{p}}\bigg{[}I_{a}+\frac{X}{N_{p}}+\phi\bigg{]}$
(2)

Equation (2) shows that, in this regime, the persistent current displays
$1/N_{p}$ reduced periodicity; such a phenomenon is observed for $UL/N_{p}\gg
1$, for any number of spin components $N$. Therefore, in this regime, the bare
flux quantum of the system is evenly shared among all the particles. We note
that, in the infinite $U$ regime, the ground state reaches the highest
degeneracy (see inset of Fig.1).

As a global view of spinon creation in the ground state, we monitor, for
different values of $U$, $N$, and $N_{p}$, the values of the flux $\phi_{s}$
at which the ground state energy in the system is no longer given by a state
with no spinons – Fig 2. Such values provide the number of spinons that can be
present in the ground state at a given $U$. At moderate $U$, spinon production
is found to be a universal function of the $N_{p}/N$– see Fig. 2b; for systems
with lower $N_{p}$, spinons are generated at a lower value of interaction. For
large $U$, spinon production is dictated by $N_{p}$, with a fine structure
that is determined by $N$: Systems with higher $N_{p}$ produce spinons at a
lower value of $U$; for fixed $N_{p}$, systems with the lower value of
$N_{p}/N$ generate spinons at a lower $U$– see inset of Fig 2a. Such a
phenomenon depends on the specific degeneracies of the system discussed
previously, that facilitate spinon creation by increasing $N$. This feature
emerges also by analysing the dependence of the phenomenon on the interaction
per particle $U/N_{p}$ – Fig. 2c. We observe that $N$ enhances the spinon
production . While the number of spinons decreases with $N_{p}$ for $N=2$,
such a trend appears to be reversed for $N>2$. For intermediate values of $U$,
discontinuities arise in the curves in cases where $N_{p}/N>1$ (see
Supplementary material). These discontinuities correspond to jumps $\Delta X$
in the spinon character $X$. By comparing systems with the same $N_{p}$ but
different $N$, we note that the discontinuities tend to be smoothed out by
increasing $N$ and $L$ (see Supplementary material). The value of $\Delta X$
results to be parity dependent.

Commensurate fillings regime – At integer filling fractions $\nu=1$, the
system enters a Mott phase for $U>U_{\text{c}}$ (thermodynamic limit). In this
phase, a spectral gap opens. For small $U$, the current is a nearly perfect
sawtooth. For our mesoscopic system, we observe that $I(\phi)$ is smoothed
out, indicating the onset of the Mott phase transition by increasing $U$ (see
Fig. 3a). Such a behavior is found to hold for all $N$. The gap indicating the
onset of the Mott phase transition is studied in Fig. 3c (see Supplementary
material). For $N=2$ such gap opens at $U=0$; for $N>2$ the spectral gap opens
at a finite value of $U$. Both the current amplitude
$I_{\text{max}}=\max_{\phi}(I)$ and $\Delta E_{\min}$ are suppressed
exponentially for large $U$ – Fig. 3b and Fig. 3c. $\Delta E_{min}$ is around
the same specific value $U$ for larger system sizes ($L\geq 8$), which depends
on $N$ ($U\approx 2$ for $N=3$, $U\approx 3$ for $N=4$). We carry out a finite
size scaling analysis Barber (1983) of the current $I$ for values of $U$
around the Mott instability. In Fig. 4a, the persistent currents display a
crossing point at a particular value $U^{*}\approx 2.9$ (see also Schlottmann
(1992); Manmana  _et al._ (2011); Assaraf  _et al._ (1999); Buchta  _et al._
(2007)); a clear data collapse is obtained in Fig. 4b.

The onset to a gapped phase affects the spinon creation process substantially.
For $N=2$ ($U^{*}=0$), spinon states have energies larger than the ground
state energy for any value of $U$. In contrast for $N>2$, spinons can be
created for $U<U^{*}$ (see inset of Fig. 3a); for $U>U^{*}$ spinon energies
result to be well separated from the ground state energy. We note that the
Lai-Sutherland BA results can reproduce the qualitative features of the low
lying states of the model even for intermediate $U$ obtained by numerics; as
expected, for large $U$, BA and numerics match exactly (see Supplementary
material).

Figure 3: SU($N$) persistent current $I(\phi)$ at integer filling. a)
$I(\phi)$ for $N=3$, $L=9$ against flux $\phi$. Insets display BA results of
the Sutherland-Lai model for different spinon configurations $X$ for $N=3$ and
$N_{p}=L=6$ compared with exact diagonalization. The ground state energy at
small $\phi$ can only be reached with spinon Bethe quantum numbers
configuration. b) Maximal current $I_{\text{max}}=\max_{\phi}(I)$ for $N=3$
(lower curves) and $N=4$ (upper curves, shifted by factor 20) plotted against
$U$ c) Minimal energy gap $E_{\text{min}}$ against $U$ for $N=3$ (lower
curves) and $N=4$ (upper curves, shifted by 0.3). All curves with $L>9$ were
calculated with DMRG.

a) b)

Figure 4: Finite size scaling of the persistent current for $N=3$. a) Finite
size critical crossing of $I_{\text{max}}$ at $U^{*}=2.9$ b) Data collapse.
$L=6,9$ were calculated with exact diagonalization; larger $L$ were obtained
with DMRG. The critical indices are $\eta\approx 0.2$ and $\zeta\approx 0.7$

Conclusions– In this work, the coherence of a quantum gas of SU($N$)
interacting fermions as quantified by the persistent current, is studied. The
analysis is carried out both for incommensurate and commensurate filling $\nu$
regimes. We highlight the nature of the ground state of the system by
corroborating the numerical analysis (exact diagonalization and DMRG) with
Bethe ansatz, which allows the access to the specific physical nature of the
system’s states. For both incommensurate and commensurate $\nu$, the ground
state can have spinon nature. Such a remarkable phenomenon implies that the
spin correlations can lead to a re-definition of the system’s effective flux
quantum and, for incommensurate $\nu$ cases, yields the $1/N_{p}$ fractional
periodicity for the persistent current observed at large $U$ (see insets of
Fig.1). The reduction of the effective flux quantum indicates that a form of
‘attraction from repulsion’ can occur in the system, with this feature being
consistent with Chakravarty and Kivelson (2001). Despite the similarities,
such a phenomenon follows a very different route from the flux quantum
fractionalization occurring for electrons with pairing force interaction (that
could compared to our study for $N=2$ only) Byers and Yang (1961); Onsager
(1961) and for bosons with attractive interaction (occurring as consequence of
quantum bright solitons formation) Naldesi  _et al._ (2019, 2020): For SU($N$)
fermions the persistent current and the aforementioned redefinition of the
flux quantum reflects the coupling between the spin and matter degrees of
freedom.  The ground state spinon creation displays a marked dependence on the
number of spin components $N$ with distinctions between the $N=2$ and $N>2$
cases (see Fig. 2). At moderate $U$, spinon production is found to be a
universal function of $N_{p}/N$ – see Fig. 2b.  For integer $\nu$, spinon
creation is suppressed by increasing $U$. The sawtooth shape of the current is
smoothed out (see Fig. 3). This feature arises since the Mott gap hinders both
the motion of the particles and the creation of spinons in the ground state.
Remarkably, a clear finite size scaling behaviour is observed for $N>2$,
albeit the persistent current is a mesoscopic quantity (see Fig. 4). Such a
result provides an operative route for the detection of the Mott phase
transition in SU($N$) systems, a notoriously challenging problem in the field.

We believe that systems in physical conditions and parameter ranges as
discussed here, can be realized experimentally on several physical platforms,
including cold atom quantum technology Pagano  _et al._ (2015); Cappellini
_et al._ (2014); DeSalvo  _et al._ (2010); Stellmer  _et al._ (2013) with the
twist provided by atomtronics Amico  _et al._ (2021). Recently, the persistent
current of SU(2) fermions was experimentally realized Cai  _et al._ (2021).
The momentum distribution through the time of flight expansion of cold atom
systems has been demonstrated to provide a precise probe for persistent
currents Amico  _et al._ (2021). Most of the features that we observe in the
persistent current are expected to emerge in time of flight images of the
system.  Finite temperature is expected to reduce the visibility of the time
of flight images. Nonetheless, based on the predictions for $N=2$, the main
features of the persistent current are expected to be feasibly detectable
Amico  _et al._ (2005). The temperature can activate transitions between
different angular momentum states, thereby causing a decay in the amplitude of
the persistent current Polo  _et al._ (2019). Recently, the dependence of
persistent currents on finite temperature effects has been investigated for
SU(2) fermions Patu and Averin (2021). However, the actual decay of the
persistent current of SU($N$) fermions needs an in-depth study.

In particular, here we mention that it was previously demonstrated how the
flux fractionalization could allow to approach the Heisenberg quantum limit
for rotation sensing Naldesi  _et al._ (2020). Our study indicates how SU($N$)
systems can provide the platform for high precision sensors.

Acknowledgements We thank Anna Minguzzi for discussions. The Grenoble LANEF
framework ANR-10-LABX-51-01 are acknowledged for their support with mutualized
infrastructure.

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Supplementary material

Persistent Current of SU(N) Fermions.

In the following sections, we provide supporting details of the theory
discussed in the main manuscript.

The derivation of the persistent current for SU($N$) fermions is sketched out
in the limit of infinite interaction $U$. The analytics are carried out for
the two integrable limits of the Hubbard model: incommensurate low filling
fractions and integer fillings. A specific analysis is devoted to the energy
and consequently the persistent current at large but finite $U$. The
persistent current undergoes a non-trivial change of the bare flux quantum.
This feature occurs because of the presence of spinons in the ground state of
the system. Spinons of different types correspond to specific Bethe quantum
numbers configurations. The Bethe quantum number configurations needed for the
given value of $X$ are provided. We then proceed to discuss
$|\phi_{s}-\phi_{d}|$ providing the figure of merit for the generation of
spinons. Spinon generation is inhibited for commensurate fillings due to a
spectral gap that opens up for a finite value of $U$. Lastly, the parity
effect for incommensurate systems is considered.

###  .1 Derivation of the Persistent Current in the limit of infinite U for
SU(N) Fermions

The derivation of the persistent current for SU($N$) fermions in the limit of
infinite interaction $U$, is sketched out for the two integrable limits of the
SU($N$) Hubbard model.

A system of interacting fermions with SU($N$) spin symmetry residing on a
chain of length $L$ threaded by an effective magnetic flux $\phi$, is
described by the Gaudin-Yang-Sutherland model Sutherland (1968); Decamp  _et
al._ (2016),

$\mathcal{H}=\sum\limits_{m=1}^{N}\sum\limits_{i=1}^{N_{m}}\bigg{(}-\imath\frac{\partial}{\partial
x_{i,m}}-\frac{2\pi}{L}\phi\bigg{)}^{2}+4U\sum\limits_{i<j,m,n}\delta(x_{i,m}-x_{j,n})$
(3)

where $N_{m}$ is the number of electrons with colour $\alpha$ of SU($N$)
symmetry with $m=1,\dots N$. The model is integrable by Bethe ansatz and is
given by the following set of equations.

$e^{\imath(k_{j}L-\phi)}=\prod\limits_{\alpha=1}^{M_{1}}\frac{4\bigg{(}k_{j}-\lambda_{\alpha}^{(1)}\bigg{)}+\imath
U}{4\bigg{(}k_{j}-\lambda_{\alpha}^{(1)}\bigg{)}-\imath U}\hskip
14.22636ptj=1,\ldots,N_{p}$ (4)
$\prod\limits_{\beta\neq\alpha}^{M_{r}}\frac{2\bigg{(}\lambda_{\alpha}^{(r)}-\lambda_{\beta}^{(r)}\bigg{)}+\imath
U}{2\bigg{(}\lambda_{\alpha}^{(r)}-\lambda_{\beta}^{(r)}\bigg{)}-\imath
U}=\prod\limits_{\beta=1}^{M_{r-1}}\frac{4\bigg{(}\lambda_{\alpha}^{(r)}-\lambda_{\beta}^{(r-1)}\bigg{)}+\imath
U}{4\bigg{(}\lambda_{\alpha}^{(r)}-\lambda_{\beta}^{(r-1)}\bigg{)}-\imath
U}\cdot\prod\limits_{\beta=1}^{M_{r+1}}\frac{4\bigg{(}\lambda_{\alpha}^{(r)}-\lambda_{\beta}^{(r+1)}\bigg{)}+\imath
U}{4\bigg{(}\lambda_{\alpha}^{(r)}-\lambda_{\beta}^{(r+1)}\bigg{)}-\imath
U}\hskip 14.22636pt\alpha=1,\ldots,M_{r}$ (5)

for $r=1,\ldots,N-1$ where $M_{0}=N_{p}$, $M_{N}=0$ and
$\lambda_{\beta}^{(0)}=k_{\beta}$. $N_{p}$ denotes the number of particles,
$M_{r}$ corresponds to the colour with $k_{j}$ and $\lambda_{\alpha}^{(r)}$
being the charge and spin momenta respectively. The energy corresponding to
the state for every solution of these equations is
$E=\sum\limits^{N_{p}}_{j}k_{j}^{2}$.

Taking the SU(3) case as an example, one obtains a set consisting of three
non-linear equations

$e^{\imath
k_{j}L}=\prod\limits_{\alpha=1}^{M_{1}}\frac{4(k_{j}-\lambda_{\alpha}^{(1)})+\imath
U}{4(k_{j}-\lambda_{\alpha}^{(1)})-\imath U}$ (6)
$\prod\limits_{\beta\neq\alpha}^{M_{1}}\frac{2(\lambda_{\alpha}^{(1)}-\lambda_{\beta}^{(1)})+\imath
U}{2(\lambda_{\alpha}^{(1)}-\lambda_{\beta}^{(1)})-\imath
U}=\prod\limits_{\beta=1}^{M_{0}=N_{p}}\frac{4(\lambda_{\alpha}^{(1)}-k_{\beta})+\imath
U}{4(\lambda_{\alpha}^{(1)}-k_{\beta})-\imath
U}\prod\limits_{\beta=1}^{M_{2}}\frac{4(\lambda_{\alpha}^{(1)}-\lambda_{\beta}^{(2)})+\imath
U}{4(\lambda_{\alpha}^{(1)}-\lambda_{\beta}^{(2)})-\imath U}$ (7)
$\prod\limits_{\beta\neq\alpha}^{M_{2}}\frac{2(\lambda_{\alpha}^{(2)}-\lambda_{\beta}^{(2)})+\imath
U}{2(\lambda_{\alpha}^{(2)}-\lambda_{\beta}^{(2)})-\imath
U}=\prod\limits_{\beta=1}^{M_{1}}\frac{4(\lambda_{\alpha}^{(2)}-\lambda_{\beta}^{(1)})+\imath
U}{4(\lambda_{\alpha}^{(2)}-\lambda_{\beta}^{(1)})-\imath U}$ (8)

which can be re-written in logarithmic form as

$k_{j}L+2\sum\limits_{\alpha=1}^{M_{1}}\arctan\Bigg{(}\frac{4(k_{j}-\lambda_{\alpha}^{(1)})}{U}\Bigg{)}=2\pi(I_{j}+\phi)\hskip
14.22636ptj=1,\ldots,N_{p}$ (9)

$2\sum\limits_{\beta=1}^{N_{p}}\arctan\Bigg{(}\frac{4(\lambda_{\alpha}^{(1)}-k_{\beta})}{U}\Bigg{)}+2\sum\limits_{a=1}^{M_{2}}\arctan\Bigg{(}\frac{4(\lambda_{\alpha}^{(1)}-l_{a})}{U}\Bigg{)}-2\sum\limits_{\beta=1}^{M_{1}}\arctan\Bigg{(}\frac{2(\lambda_{\alpha}^{(1)}-\lambda_{\beta}^{(1)})}{U}\Bigg{)}=2\pi
J_{\alpha}\hskip 14.22636pt\alpha=1,\ldots,M_{1}$ (10)

$2\sum\limits_{\beta=1}^{M_{1}}\arctan\Bigg{(}\frac{4(l_{a}-\lambda_{\beta}^{(1)})}{U}\Bigg{)}-2\sum\limits_{b=1}^{M_{2}}\arctan\Bigg{(}\frac{2(l_{a}-l_{b})}{U}\Bigg{)}=2\pi
L_{a}\hskip 14.22636pta=1,\ldots,M_{2}$ (11)

where $\lambda_{\beta}^{(2)}$ was changed to $l_{a}$ for the sake of
convenience with $I_{j}$, $J_{\alpha}$ and $L_{a}$ being the Bethe quantum
numbers, the first being associated with charge momenta and the other two for
spin momenta. Carrying out a summation over $\alpha$ and over $a$ for
Equations (10) and (11) respectively,

$2\sum\limits_{\alpha=1}^{M_{1}}\sum\limits_{\beta=1}^{N_{p}}\arctan\Bigg{(}\frac{4(\lambda_{\alpha}^{(1)}-k_{\beta})}{U}\Bigg{)}+2\sum\limits_{\alpha=1}^{M_{1}}\sum\limits_{a=1}^{M_{2}}\arctan\Bigg{(}\frac{4(\lambda_{\alpha}^{(1)}-l_{a})}{U}\Bigg{)}-2\sum\limits_{\alpha=1}^{M_{1}}\sum\limits_{\beta=1}^{M_{1}}\arctan\Bigg{(}\frac{2(\lambda_{\alpha}^{(1)}-\lambda_{\beta}^{(1)})}{U}\Bigg{)}=2\pi\sum\limits_{\alpha=1}^{M_{1}}J_{\alpha}$
(12)

$2\sum\limits_{a=1}^{M_{2}}\sum\limits_{\beta=1}^{M_{1}}\arctan\Bigg{(}\frac{4(l_{a}-\lambda_{\beta}^{(1)})}{U}\Bigg{)}-2\sum\limits_{a=1}^{M_{2}}\sum\limits_{b=1}^{M_{2}}\arctan\Bigg{(}\frac{2(l_{a}-l_{b})}{U}\Bigg{)}=2\pi\sum\limits_{a=1}^{M_{2}}L_{a}\
$ (13)

and noting that the last term on the left hand side in both of the above
equations goes to zero, leads one to the following expression

$2\sum\limits_{\alpha=1}^{M_{1}}\sum\limits_{\beta=1}^{N_{p}}\arctan\Bigg{(}\frac{4(\lambda_{\alpha}^{(1)}-k_{\beta})}{U}\Bigg{)}=2\pi\Bigg{(}\sum\limits_{\alpha=1}^{M_{1}}J_{\alpha}+\sum\limits_{a=1}^{M_{2}}L_{a}\Bigg{)}$
(14)

In the limit $\frac{U}{N_{p}}\rightarrow\infty$, the $k_{j}$ terms can be
neglected since they are significantly smaller in magnitude compared to the
spin momenta. Consequently,

$2\sum\limits_{\alpha=1}^{M_{1}}\sum\limits_{\beta=1}^{N_{p}}\arctan\Bigg{(}\frac{4\lambda_{\alpha}^{(1)}}{U}\Bigg{)}=2\pi\Bigg{(}\sum\limits_{\alpha=1}^{M_{1}}J_{\alpha}+\sum\limits_{a=1}^{M_{1}}L_{a}\Bigg{)}\implies
2\sum\limits_{\alpha=1}^{M_{1}}\arctan\Bigg{(}\frac{4\lambda_{\alpha}^{(1)}}{U}\Bigg{)}=\frac{2\pi}{N_{p}}\Bigg{(}\sum\limits_{\alpha=1}^{M_{1}}J_{\alpha}+\sum\limits_{a=1}^{M_{1}}L_{a}\Bigg{)}$
(15)

which upon substitution in Equation (9) yields

$k_{j}L=2\pi\Bigg{[}I_{j}+\frac{1}{N_{p}}\Bigg{(}\sum\limits_{\alpha=1}^{M_{1}}J_{\alpha}+\sum\limits_{a=1}^{M_{2}}L_{a}\Bigg{)}+\phi\Bigg{]}$
(16)

Squaring the above expression,

$k_{j}^{2}=\bigg{(}\frac{2\pi}{L}\bigg{)}^{2}\Bigg{[}I_{j}^{2}+2I_{j}\Bigg{(}\frac{X}{N_{p}}+\phi\Bigg{)}+\Bigg{(}\frac{X}{N_{p}}\Bigg{)}^{2}+2\phi\frac{X}{N_{p}}+\phi^{2}\Bigg{]}$
(17)

the ground state energy of the system is given by

$E_{0}=\sum\limits_{j}^{N_{p}}k_{j}^{2}=\bigg{(}\frac{2\pi}{L}\bigg{)}^{2}\Bigg{[}\sum\limits_{j}^{N_{p}}I_{j}^{2}+2\sum\limits_{j}^{N_{p}}I_{j}\Bigg{(}\frac{X}{N_{p}}+\phi\Bigg{)}+\bigg{(}\frac{X}{N_{p}}\bigg{)}^{2}+N_{p}\Bigg{(}2\phi\frac{X}{N_{p}}+\phi^{2}\Bigg{)}\Bigg{]}$
(18)

assuming the $I_{j}$ quantum numbers are a consecutive integer/half-integer
set, where
$X=\Bigg{(}\sum\limits_{\alpha=1}^{M_{1}}J_{\alpha}+\sum\limits_{a=1}^{M_{2}}L_{a}\Bigg{)}$.
At zero temperature the persistent current of the system is defined as,

$I(\phi)=-\frac{\partial E_{0}}{\partial\phi}$ (19)

Therefore, the persistent current in the limit of infinite $U$ turns out to
be,

$I(\phi)=-2\bigg{(}\frac{2\pi}{L}\bigg{)}^{2}\sum\limits_{j}^{N_{p}}\bigg{[}I_{j}+\frac{X}{N_{p}}+\phi\bigg{]}$
(20)

In the case of SU($N$) fermions, one would still have the same expression for
the persistent current. The only difference is that
$X=\sum\limits_{j}^{N-1}\sum\limits^{\alpha_{j}}J_{\alpha_{j}}$.

The other integrable limit of the SU($N$) Hubbard model, is for commensurate
filling fractions in the presence of a lattice. The model is described by the
Lai-Sutherland model Sutherland (1975); Capponi  _et al._ (2016) with the
energy of the system being given by $E=-2\sum\limits_{j}^{N_{p}}\cos k_{j}$.
The Bethe ansatz equations for this model are similar to the ones outlined in
Equations (4) and (5). However, in this case in Equation (4) there is $\sin
k_{j}$ instead of $k_{j}$ on the right hand side and for Equation (5) one
substitutes $\lambda_{\beta}^{(0)}=\sin k_{\beta}$ when required. By following
the same procedure one arrives to Equation (16). Substituting this expression
in the energy of the system, one arrives to

$E_{0}(\phi)=-E_{m}\cos\Bigg{[}\frac{2\pi}{L}\Bigg{(}D+\frac{X}{N_{p}}+\phi\Bigg{)}\Bigg{]}$
(21)

and in turn the persistent current is of the following form,

$I(\phi)=-E_{m}\bigg{(}\frac{2\pi}{L}\bigg{)}\sin\Bigg{[}\frac{2\pi}{L}\Bigg{(}D+\frac{X}{N_{p}}+\phi\Bigg{)}\Bigg{]}$
(22)

where
$E_{m}=2\frac{\sin\big{(}\frac{N_{p}\pi}{L}\big{)}}{\sin\big{(}\frac{\pi}{L}\big{)}}$
where $D=\frac{I_{max}+I_{min}}{2}$, which comes about due to the $I_{j}$
being consecutive for the ground state configuration. The above expression is
a generalization of the ground state energy for SU(2) fermions obtained in
Kusmartsev (1991); Yu and Fowler (1992). In particular, at infinite $U$ for
the same number of particles the pre-factor $E_{m}$ is the same for SU($N$) as
it was for SU(2). This in turn implies that the ground state energy for
fermions carrying different SU($N$) spin, say SU(2) and SU(3), will coincide
if their phase shift is the same. The same also holds true for expression
(18).

##  I Corrections to the infinite $U$ limit: derivation of the Energy Spin
correction

In this section we generalize the energy spin correction, obtained for SU(2)
fermions in Yu and Fowler (1992), for SU($N$) fermions. At infinite $U$ the
system is highly degenerate, meaning that there are multiple ways of choosing
the spin rapidity $J_{\alpha}$ distribution Kusmartsev (1991); Yu and Fowler
(1992). In order to find out the lowest energy state at finite $U$ when the
degeneracy is lifted, leading order $\frac{1}{U}$ corrections have to be
introduced for the Bethe ansatz equations at infinite $U$. When $U$ is at
infinity, the charge momenta $k_{j}$ are of order unity whilst the spin
momenta $\lambda_{\beta}$ are of order $U$. With this picture in mind, we
expand the arctangent function in Equation (9) to leading order in
$\frac{k_{j}}{U}$. Defining the scaled variables $x_{\alpha}$ as,

$x_{\alpha}=\displaystyle{\lim_{U\to\infty}}\bigg{(}\frac{2\lambda_{\alpha}}{U}\bigg{)}$
(23)

through Taylor expansion one finds that,

$f(x+h)=\arctan(2x_{\alpha})-2\frac{k_{j}}{U}\frac{1}{x^{2}_{\alpha}+\frac{1}{4}}$
(24)

Therefore, for large but finite $U$, the $k_{j}$ have leading $\frac{1}{U}$
corrections,

$\delta
k_{j}=-2\frac{k_{j}}{UL}\sum\limits_{\alpha}^{M}\frac{1}{x^{2}_{\alpha}+\frac{1}{4}}$
(25)

where the $x_{\alpha}$ has to satisfy the remaining Bethe equations which in
the SU(3) case for example are Equations (11) and (12). The total ground state
energy reads,

$E=\sum\limits_{j}^{N_{p}}(k_{j}+\delta
k_{j})^{2}=\sum\limits_{j=1}^{N_{p}}\bigg{(}k_{j}^{2}+2k_{j}\delta
k_{j}+(\delta k_{j})^{2}\bigg{)}$ (26)

Therefore, the leading order $\frac{1}{U}$ correction is given by

$+2k_{j}\delta
k_{j}=-\frac{4}{UL}\sum\limits_{j}^{N_{p}}k_{j}^{2}\sum\limits_{\alpha}^{M}\frac{1}{x^{2}_{\alpha}+\frac{1}{4}}=J_{eff}\sum\limits_{\alpha}^{M}\frac{1}{x^{2}_{\alpha}+\frac{1}{4}}$
(27)

In the presence of a lattice, the energy correction is of a similar form

$+2\delta
k_{j}\sin(k_{j})=J_{eff}\sum\limits_{\alpha}^{M}\frac{1}{x^{2}_{\alpha}+\frac{1}{4}}$
(28)

but
$J_{eff}=-\frac{4}{UL}\Bigg{(}\sum\limits_{j=1}^{N_{p}}\sin^{2}k_{j}\Bigg{)}$
in this case, whereby $k_{j}$ in Equation (27) was replaced by $\sin k_{j}$.
The leading order $\frac{1}{U}$ correction to the Bethe ansatz equations for
SU($N$) fermions has the same expression as the one obtained for SU(2) in Yu
and Fowler (1992). This was to be expected since Equation (9), which is the
primary equation relating the charge and spin rapidities, is the same for all
SU($N$).

##  II Bethe Ansatz Spinon Configurations

To obtain the minimum energy for a given value of the flux $\phi$, one
requires that the summation over the spin rapidities satisfies the degeneracy
point equation having the form Kusmartsev (1991); Yu and Fowler (1992)

$\frac{2w-1}{2N_{p}}\leq\phi+D\leq\frac{2w+1}{2N_{p}}\hskip
14.22636pt\textrm{where}\hskip 14.22636ptX=-w$ (29)

with $w$ only being allowed to have integer or half-integer values due to the
nature of the spin rapidities.

Consider the case of three fermions with SU(3) spin. There are three sets of
quantum numbers: one pertaining to the charge momenta $I_{j}$ and the other
two belonging to the spin momenta denoted as $J_{\alpha_{1}}$ and
$J_{\alpha_{2}}$. The ground state configuration for such a system is given as
$I_{j}=\\{-1,0,1\\}$, $J_{\alpha_{1}}=\\{-0.5,0.5\\}$ and
$J_{\alpha_{2}}=\\{0\\}$. The correction of the spin quantum numbers for all
the values of the flux per Equation (27) is as follows

Magnetic flux | $J_{\alpha_{1}}$ | $J_{\alpha_{2}}$ | $X$
---|---|---|---
$0.0-0.1$ | $\\{-0.5,0.5\\}$ | $\\{0\\}$ | $0$
$0.2-0.5$ | $\\{-1.5,0.5\\}$ | $\\{0\\}$ | $-1$
$0.6-0.8$ | $\\{-0.5,1.5\\}$ | $\\{0\\}$ | $+1$
$0.9-1.0$ | $\\{-0.5,0.5\\}$ | $\\{0\\}$ | $0$
Table 1: Spin quantum number configurations with the flux for $N_{p}=3$ with
SU(3) spin with $M_{1}=2$ and $M_{2}=1$.

As can be seen from Table (1), in cases where $X=0$, the spin quantum number
configuration is different from the ground state one and ‘holes’ are
introduced such that the spin quantum number configurations are no longer
consecutive, with the $I_{j}$ set remaining unchanged. There are two notable
points worthy of mention. The first is that one could have chosen a different
way to arrange the set of quantum numbers. An alternative arrangement is given
by Table (2). The target value $X$ is reached via a different configuration,
which in turn leads to a degenerate state. Such a phenomenon is a
characteristic property of SU($N$) systems that is not present for SU(2). As
$N$ increases, the number of degenerate states that are present in the system
increases due to the various Bethe quantum number configurations that one can
adopt.

Magnetic flux | $J_{\alpha_{1}}$ | $J_{\alpha_{2}}$ | $X$
---|---|---|---
$0.0-0.1$ | $\\{-0.5,0.5\\}$ | $\\{0\\}$ | $0$
$0.2-0.5$ | $\\{-0.5,0.5\\}$ | $\\{-1\\}$ | $-1$
$0.6-0.8$ | $\\{-0.5,0.5\\}$ | $\\{+1\\}$ | $+1$
$0.9-1.0$ | $\\{-0.5,0.5\\}$ | $\\{0\\}$ | $0$
Table 2: Alternative spin quantum number configurations with the flux for
$N_{p}=3$ with SU(3) spin with $M_{1}=2$ and $M_{2}=1$.

The other point concerns the value of $X$ for $\phi=0.6-0.8$ and
$\phi=0.9-1.0$. According to Equation (29), $X$ should be equal to $-2$ and
$-3$ respectively. The reason behind this is due to the fact that the
degeneracy equation has to be applied within a specific flux range that
depends on the parity of the system: for a flux in the interval of $-0.5$ to
$0.5$ for $N_{p}=N(2n+1)$ and in the range of $\phi=0.0$ to $1.0$ in the case
of $N_{p}=N(2n)$. The ground state energy of the system is given by a series
of parabolas in the absence of an effective magnetic flux. These parabolas
each have a well defined angular momentum $l$. They intersect at the degneracy
points, which is parity dependent, and are shifted with respect to each other
by a Galilean translation Bloch (1973). Consequently, when the magnetic flux
piercing the system falls outside the range outlined previously, one needs to
change the $I_{j}$ quantum numbers in order to offset the increase in angular
momentum $l$ that one obtains on going to the next energy parabola.

For positive $\phi$ one requires that the $I_{j}$ quantum numbers need to all
be shifted by one to the left. For example in the case considered above for
$\phi>0.5$, the $I_{j}$ go from $\\{-1,0,1\\}$ to $\\{-2,-1,0\\}$ for
$0.5<\phi<1.5$. On going to the next parabola, they would need to be shifted
again by one to the left. In the case of negative $\phi$, the shift occurs to
the right.

Note that there are other combinations of the quantum numbers, not outlined in
Tables 1 amd 2, whose total sum reaches the target value of $X$. However,
these configurations do not give the lowest value for the energy as the ones
mentioned, even though the value of $X$ is the same. At infinite $U$, the
system is solely dependent on the value of $X$ and not on the arrangement of
the spin quantum number configuration. Consequently, the system is highly
degenerate. This is also observed in the SU(2) case. However, as mentioned in
the derivation of the energy correction, the degeneracy is lifted on going to
large but finite $U$ and one is left with only one combination that gives the
lowest energy in the case of SU(2) systems. On the other hand, for SU($N$)
systems whilst this degeneracy is also lifted, they also benefit from an extra
‘source’ of degeneracy due to the different configurations of the Bethe
quantum numbers as shown in Tables 1 and 2.

##  III Spinon creation in the ground state for SU(N) Fermions

The SU($N$) Hubbard model is not integrable in all limits, unlike its SU(2)
counterpart. The Hamiltonian is integrable by Bethe ansatz for incommensurate
and commensurate filling fractions. In this section, we take a look at spinon
creation for SU($N$) fermions in these two regimes.

Figure 5: Spinon creation in incommensurate SU($N$) fermionic systems. The
case of $N=3$ is considered for $N_{p}=3$ fermions residing on a ring composed
of $L=30$ sites. The above figures show how the Bethe ansatz energies need to
be characterized by spinon quantum numbers in order to have the actual ground
state for various values of the interaction $U$. All curves are calculated
with the Bethe ansatz of the Gaudin-Yang-Sutherland model and exact
diagonalization.

For a system with incommensurate filling fractions, spinons are created with
increasing $U$ as can be observed from Fig. 5. Level crossings occur between
the ground state with no spinons and levels with spinon character $X$, with
the value of $X$ obtained as outlined in the previous section. The creation of
spinons starts out around the degeneracy point $\phi_{d}$ (see Fig. 5b), which
is parity dependent. The degeneracy point $\phi_{d}$ is 0 for $N_{p}=N(2n)$
and 0.5 for $N_{p}=N(2n+1)$ systems. Comparing Fig. 5a and Fig. 5d, we observe
that the elementary flux quantum $\phi_{0}$ has been renormalized in the
latter case and that $1/N_{p}$ periodicity is achieved, resulting in $N_{p}$
cusps/parabolic-wise segments that corresponds to 3 in this case.

Figure 6: Spinon creation in commensurate SU($N$) fermionic systems. The
systems taken in consideration are SU(2) with $N_{p}=6$ (left column), SU(3)
with $N_{p}=6$ (middle column) and SU(4) with $N_{p}=4$ (right column). The
different Bethe ansatz energies of the Lai-Sutherland model characterized by
different spinon configurations needed to make up the ground state of the
system are considered for different values of interaction with $U=1$ (top
row), $U=5$ (middle row) and $U=100$ (last row). All the presented results are
obtained with Bethe ansatz of the Gaudin-Yang-Sutherland model for $N_{p}=L$.
The Bethe ansatz states with no spinons, having two different colours (orange
and red), are used to indicate that the $I_{j}$ quantum numbers are shifted
due to being in different energy parabolas. In the case of $U=1$, the Bethe
ansatz did not converge for certain values of the flux. This does not have an
impact on what we are trying to discuss here and so they were left out.

Figure 7: SU($N$) energy gap at integer filling. Minimal energy gap
$E_{\textrm{min}}=\textrm{min}_{\phi}(\Delta E)$ for different $N$ against $U$
at comparable system sizes ($N=2$ and $N=4$ with $L=8$ and $N=3$ with $L=9$).
All curves were obtained by exact diagonalization.

In the case of commensurate filling fractions, spinon creation is drastically
impacted by a spectral gap that opens around the transition to the Mott phase
(see Fig. 7). The energy gap is determined as the minimal gap for any flux
$\Delta E=\textrm{min}_{\phi}(\Delta E)$. For the special case $N=2$, the gap
opens at $U=0$, whereas for any other $N$ it opens at non-zero $U$ indicating
the onset to the Mott phase transition. Indeed if spinon creation in a system
with SU(2) fermions (Figs. 6a),d),g)) is compared to systems with SU(3) (Figs.
6b),e),h)) and SU(4) (Figs. 6c),f),i)) spin components, we note that no
spinons are created in the SU(2) case for any value of $U$. On the other hand
for SU($N$) systems, spinon creation is present in the system for values of
$U$ below the threshold value of where the transition happens $U^{*}$, which
was calculated to be around 2.9. An interesting feature that pops up, is that
after passing $U^{*}$, one no longer needs to change the $I_{j}$ quantum
numbers on going from one energy parabola to the other, as can be seen by
comparing (Figs. 6c),f)).

Figure 8: Figure of merit for spinon creation in the ground state of SU($N$)
fermions against the interaction per particle $U/N_{p}$. We consider the
minimum value of $U$ required for spinons to be created in the ground state
for a given value of $\phi$; all the values of $\phi$ where a spinon is
created are recorded. The displayed curves are calculated by monitoring all
the distances $|\phi_{s}-\phi_{d}|$ at which the state with no spinons crosses
states with any spinon configurations where $\phi_{s}$ is the flux at which
spinons are created and $\phi_{d}$ is the degeneracy point. The inset contains
the data in the intermediate regime. The discontinuities observed in the
intermediate $U/N_{p}$ regime when $N_{p}/N>1$, are more pronounced for larger
values of $N_{p}/N$ for a system with the same $N_{p}$ but different $N$. All
the presented results are obtained by Bethe ansatz of Gaudin-Yang-Sutherland
model for$L=40$, with $N_{p}=1$(circles), 2(squares), 3(crosses) per spin
component, with $N$ = 2, 3, 4, 5.

A good way to visualize the impact of SU($N$) fermions on spinon creation in
the ground state is by having a figure of merit. By considering all the values
of the flux $\phi$ at which spinons can be created and noting down the value
of the interaction $U$, at which there is a level crossing between the ground
state and an excited state, one can obtain this figure of merit. The curves
displayed in Fig. 8 are obtained by monitoring the spinon flux distance
$|\phi_{s}-\phi_{d}|$ at which there are crossings between states with spinons
and those without, with $\phi_{s}$ being the flux at which spinons are created
and $\phi_{d}$ being the degeneracy point. Figure depicts the spinon creation
flux distance against the interaction per particle $U/N_{p}$. The inset
denotes the data in the intermediate $U/N_{p}$ regime. Due to the specific
$N-1$ types of excitations that are inherently present in SU($N$) fermions for
$N>2$, spinon creation is facilitated with $N$.  This can be clearly seen from
Figs. 9a),b). In the case where $N_{p}/N>1$, discontinuities arise in the
intermediate $U$ regime as can be seen from the insets of Figs. 9a),b). The
discontinuites arise due to jumps $\Delta X$ in the spinon character $X$ and
are absent when $N_{p}/N=1$ (inset of Fig. 9a). Additionally, when comparing
systems containing the same $N_{p}$, the discontinuities tend to smoothen out
with increasing $N$ and $L$. This can be clearly seen from Figs. 9b),c)

Figure 9: Comparison of spinon creation in SU(2) fermions and SU($N$)
fermions. a) Spinon creation flux distance $|\phi_{s}-\phi_{d}|$ against
interaction $U$ is considered for a ring of $L=40$ sites with $N_{p}=4$
fermions with $N=2$ and $N=4$ spin components, where $\phi_{s}$ is the flux at
which spinons are created and $\phi_{d}$ is the degeneracy point. The
intermediate $U$ regime (inset) highlights the discontinuity present in the
SU(2) case. b) Spinon creation flux distance against interaction for a ring of
$L=40$ sites with $N_{p}=6$ fermions with $N=2$ and $N=3$ spin components. The
inset depicts the disconituities for intermediate $U$ in both systems. c)
Spinon creation is for a system with $N_{p}=6$ particles with $N=3$ with
various system sizes, $L=20$, $L=30$ and $L=40$. All the presented results are
obtained with Bethe ansatz of the Gaudin-Yang-Sutherland model.

##  IV Parity effect

Specific parity effects are observed for SU($N$) fermions. Both for
commensurate and incommensurate fillings, the persistent current is found
diamagnetic (paramagnetic) for ring systems containing $N_{p}=(2n+1)N$
($N_{p}=(2n)N$) fermions, with $n$ being an integer. The nature of the current
can be deduced by looking at the ground state energy of the system, whereby if
the system has a minimum (maximum) at zero flux, then it is diamagnetic
(paramagnetic) - Fig. 10. Such phenomena generalize the $4n$/$4n+2$ of
spin-$\frac{1}{2}$ fermions Waintal  _et al._ (2008). Indeed, for both non-
integer and integer fillings fractions, we demonstrate how results of Byers-
Yang, Onsager and Leggett on the landscape of the system persistent current
can be generalized to SU($N$) fermions Byers and Yang (1961); Onsager (1961);
Leggett (1991).

Figure 10: Parity effect for SU($N$) fermions. Ground state energy
$E_{0}(\phi)$ is plotted against the flux $\phi$ for different $N$ ranging
from 3 (circles) to 5 (diamonds). Since the energy is suppressed (increased)
by the effective magnetic field, systems with even (odd) number of particle
per spin component are paramagnetic (diamagnetic). All the presented results
are obtained by Bethe ansatz of Gaudin-Yang-Sutherland model for $L=30$, with
$N_{p}$ taken to be 1 particle and 2 particles per species for each $N$
corresponding to $n=0,1$ respectively.

a) b)
c) d)
e) f)
g) h)

Figure 11: SU($N$) persistent current and the corresponding ground state
energy at incommensurate filling for different interaction strengths $U$.
a),b) Ground state energy and Persistent current for $N_{p}=3$ for $U=0.1$.
c),d) Ground state energy and persistent current for $N_{p}=3$ for $U=10,000$.
e),f) Ground state energy and Persistent current for $N_{p}=6$ for $U=0.1$.
g),h) Ground state energy and persistent current for $N_{p}=6$ for $U=10,000$.
All curves are calculated with Bethe ansatz for the Gaudin-Yang-Sutherland
model with $L=20$.

The behaviour of this parity effect holds for small and intermediate $U$ but
it is washed out at infinite $U$ for incommensurate fillings or above a finite
threshold of interaction for integer fillings. Indeed, the character of the
current is diamagnetic, since the fractionalization of the bare flux quantum
causes the ground state energy to always be a minimum at zero flux. In the
cases where the current already had a diamagnetic nature at small values of
$U$, its nature remains unchanged. The washing out of the persistent current
can be clearly observed from Fig. 11 whereby comparison of SU(3) systems with
$N_{p}=3$ and $N_{p}=6$ clearly show the stark difference in the nature of the
current for the latter case between the different regimes of $U$.