# Gauge fields on $B$-branes over $\mathbb{CP}^{n}$

Andrés Viña  Departamento de Física. Universidad de Oviedo. García Lorca 18.
33007 Oviedo. Spain<EMAIL_ADDRESS>
###### Abstract.

Considering the $B$-branes over the complex projective space
${\mathbb{P}}^{n}$ as the objects of the bounded derived category
$D^{b}({\mathbb{P}}^{n})$, we prove that the cardinal of the set of
holomorphic gauge fields on a given $B$-brane ${\mathscr{G}}^{\bullet}$ is
$\leq 1$. Moreover, the cardinal is $1$ iff each ${\mathscr{G}}^{p}$ is
isomorphic to a direct sum of copies of ${\mathscr{O}}_{{\mathbb{P}}^{n}}$.

###### Key words and phrases:

Holomorphic connections, derived category of ${\mathbb{P}}^{n}$, gauge fields
on $B$-branes

MSC 2020: 53C05, 18G10

##  1\. Introduction

From the mathematical point of view, a $B$-brane over a complex manifold $X$
is an object of the bounded derived category $D^{b}(X)$ [1, 2].

The definition of a holomorphic connection on a vector bundle proposed by
Deligne [4, page 6] can be generalized in a natural way to objects of
$D^{b}(X)$, giving rise to the concept of a holomorphic field on $B$-branes
[11]. In this context, the set ${\mathfrak{c}}({\mathscr{F}}^{\bullet})$ of
gauge fields on the $B$-brane ${\mathscr{F}}^{\bullet}$, if it is nonempty, is
an affine space whose vector space is ${\rm
Hom}_{D^{b}(X)}({\mathscr{F}}^{\bullet},\,\Omega^{1}_{X}{\mathscr{F}}^{\bullet})$
(see below, in Section 2).

The purpose of this note is to study the cardinal
$\\#\,{\mathfrak{c}}({\mathscr{G}}^{\bullet})$ of the set
${\mathfrak{c}}({\mathscr{G}}^{\bullet}),$ where ${\mathscr{G}}^{\bullet}$ is
a $B$-brane over the complex projective space ${\mathbb{P}}^{n}$. In Theorem
1, we prove that this cardinal is $\leq 1$, and Theorem 2 gives a necessary
and sufficient condition for this cardinal to be $1$.

In the proofs of these theorems plays an important role the fact that the set

(1.1)
${\mathcal{E}}=\\{{\mathscr{O}}(-n),\,{\mathscr{O}}(-n+1),\dots,{\mathscr{O}}(-1),\,{\mathscr{O}}\\}$

is a full exceptional family in $D^{b}({\mathbb{P}}^{n})$ [7, Sect 8.3]. Thus,
$D^{b}({\mathbb{P}}^{n})$ is equivalent to the smallest triangulated
subcategory that contains this exceptional family. This fact allows us to pass
from general branes to complexes whose terms are direct sums of elements of
the set (1.1). On these branes the functor ${\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}\,.\,,\,\Omega^{1}(\,.\,)\big{)}$ can be
expressed in terms the cohomology of ${\mathbb{P}}^{n}$. This reduction
permits us to prove the results stated in Theorems 1 and 2.

Some well-known results in cohomology will be developed in some detail, in
order to facilitate the understanding of the proofs of these theorems for
those mathematical physicists who are not fully familiar with sheaf
cohomology.

##  2\. The set of gauge fields

Given ${\mathscr{G}}^{\bullet}$ an object of $D^{b}({\mathbb{P}}^{n})$, we set
$\Omega^{1}({\mathscr{G}}^{\bullet}):=\Omega^{1}\otimes_{\mathscr{O}}{\mathscr{G}}^{\bullet}$,
where $\Omega^{1}$ is the sheaf of holomorphic $1$-forms on
${\mathbb{P}}^{n}$. By ${\mathscr{J}}({\mathscr{G}}^{\bullet})$ we denote the
corresponding first jet complex. That is,
${\mathscr{J}}({\mathscr{G}}^{\bullet})$ is the complex of abelian sheaves
$\Omega^{1}({\mathscr{G}}^{\bullet})\oplus{\mathscr{G}}^{\bullet}$ endowed
with the following ${\mathscr{O}}$-module structure

(2.1)
$f\ast(\alpha^{\bullet}\oplus\sigma^{\bullet})=(f\alpha^{\bullet}+\partial
f\otimes\sigma^{\bullet})\oplus f\sigma^{\bullet},$

where $f$ is an holomorphic function,
$\alpha^{\bullet}\in{\Omega^{1}}({\mathscr{G}}^{\bullet})$ and
$\sigma^{\bullet}\in{\mathscr{G}}^{\bullet}$.

One has the following natural exact sequence of complexes of
${\mathscr{O}}$-modules, which in general does not split in the category of
complexes of ${\mathscr{O}}$-modules,

(2.2)
$0\to\Omega^{1}({\mathscr{G}}^{\bullet})\to{\mathscr{J}}({\mathscr{G}}^{\bullet})\overset{\pi}{\rightarrow}{\mathscr{G}}^{\bullet}\to
0$

A holomorphic gauge field on the $B$-brane ${\mathscr{G}}^{\bullet}$ is a
right inverse in $D^{b}({\mathbb{P}}^{n})$ of the morphism $\pi$ [11]. The
existence of holomorphic gauge fields can be studied by considering certain
${\rm Ext}$ groups.

Ext functors. In general, given two bounded below complexes $A^{\bullet}$ and
$B^{\bullet}$ in an abelian category ${\mathfrak{C}}$, the complex ${\rm
Hom}^{\bullet}(A^{\bullet},\,B^{\bullet})$ is defined by (see [8, page 17])

(2.3) ${\rm Hom}^{m}(A^{\bullet},\,B^{\bullet})=\prod_{p\in{\mathbb{Z}}}{\rm
Hom}_{\mathfrak{C}}\big{(}A^{p},\,B^{p+m}\big{)},$

with the differential $\delta_{H}$.

(2.4)
$(\delta_{H}^{m}g)^{p}=\delta_{B}^{m+p}g^{p}+(-1)^{m+1}g^{p+1}\delta_{A}^{p}.$

If the category ${\mathfrak{C}}$ has enough injectives,

${\rm Ext}^{i}(A^{\bullet},\,B^{\bullet}):=H^{i}{\rm
Hom}^{\bullet}(A^{\bullet},\,I^{\bullet}),$

where $I^{\bullet}$ is a complex of injective objects, which is $q$-isomorphic
to $B^{\bullet}$.

Some properties of funtors ${\rm Ext}^{i}$ to be used here are the following:

  1. (a)

${\rm Ext}^{0}$ is the ${\rm Hom}$ functor of the bounded below derived
category $D^{+}(\mathfrak{C})$.

  2. (b)

If ${\mathfrak{C}}$ is the category of ${\mathscr{O}}$-modules and
${\mathscr{G}}^{\bullet}$ is a complex of locally free sheaves, then it is not
necessary to take an injective resolution in the definition of functors ${\rm
Ext}^{i}({\mathscr{G}}^{\bullet},\,.\,)$. That is, ${\rm
Ext}^{i}({\mathscr{G}}^{\bullet},\,{\mathscr{F}}^{\bullet})=H^{i}{\rm
Hom}^{\bullet}({\mathscr{G}}^{\bullet},\,{\mathscr{F}}^{\bullet})$ [6, Chap
III, 6.5.1].

  3. (c)

The family of functors ${\rm Ext}^{i}$ is a $\delta$-functor [6, page 205].

The fundamental properties of ${\rm Ext}$ functors can be found [5, 8, 9].

The Atiyah class. From (2.2) together with the above property (c), it follows
the Ext exact sequence

(2.5) $\displaystyle 0\to\,$ $\displaystyle{\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{G}}^{\bullet},\,\Omega^{1}({\mathscr{G}}^{\bullet})\big{)}\to{\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{G}}^{\bullet},\,{\mathscr{J}}({\mathscr{G}}^{\bullet})\big{)}\overset{\pi_{*}}{\longrightarrow}$
$\displaystyle\to\,$ $\displaystyle{\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{G}}^{\bullet},\,{\mathscr{G}}^{\bullet}\big{)}\overset{\Delta}{\rightarrow}{\rm
Ext}^{1}\big{(}{\mathscr{G}}^{\bullet},\,\Omega^{1}({\mathscr{G}}^{\bullet})\big{)}\to\dots$

The element $\Delta({\rm id})\in{\rm
Ext}^{1}\big{(}{\mathscr{G}}^{\bullet},\,\Omega^{1}({\mathscr{G}}^{\bullet})\big{)}$
is the Atiyah class [3] $a({\mathscr{G}}^{\bullet})$ of
${\mathscr{G}}^{\bullet}$.

From the exactness of (2.5), we deduce:

  1. (1)

The vanishing of $a({\mathscr{G}}^{\bullet})$ is a necessary and sufficient
condition for the existence of a holomorphic gauge field on the brane
${\mathscr{G}}^{\bullet}$.

  2. (2)

The set of holomorphic gauge fields on this brane, if it is nonempty, is an
affine space that has ${\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{G}}^{\bullet},\,\Omega^{1}({\mathscr{G}}^{\bullet})\big{)}$
as vector space.

###  2.1. Case ${\mathscr{O}}(k)$

If ${\mathscr{A}}$ is an ${\mathscr{O}}$-module and $k$ is an integer, we use
the habitual notation ${\mathscr{A}}(k)$ for the twisted
${\mathscr{O}}$-module ${\mathscr{A}}\otimes_{\mathscr{O}}{\mathscr{O}}(k)$.
The functor $F_{(k)}:=-\otimes_{\mathscr{O}}{\mathscr{O}}(k)$ is an
automorphism of the category of ${\mathscr{O}}$-modules, since $F_{(k)}$ and
$F_{(-k)}$ are inverse. Moreover, given two ${\mathscr{O}}$-modules
${\mathscr{A}}$, ${\mathscr{B}}$, then

(2.6) ${\rm Hom}_{\mathscr{O}}({\mathscr{A}}(k),\,{\mathscr{B}})\simeq{\rm
Hom}_{\mathscr{O}}({\mathscr{A}},\,{\mathscr{B}}(-k)).$

The brane ${\mathscr{G}}^{\bullet}$, such that
${\mathscr{G}}^{0}={\mathscr{O}}(k)$ and ${\mathscr{G}}^{i}=0$ for $i\neq 0$,
will be denoted by ${\mathscr{O}}(k)$. As ${\mathscr{O}}(k)$ is a locally free
${\mathscr{O}}$-module, by the above property (b)

(2.7) ${\rm Ext}^{i}({\mathscr{O}}(k),\,{\mathscr{F}}^{\bullet})=H^{i}{\rm
Hom}_{\mathscr{O}}\big{(}{\mathscr{O}}(k),\,{\mathscr{F}}^{\bullet}\big{)}=H^{i}{\rm
Hom}_{\mathscr{O}}\big{(}{\mathscr{O}},\,{\mathscr{F}}^{\bullet}(-k)\big{)},$

where (2.6) has been used.

In particular, if the complex ${\mathscr{F}}^{\bullet}$ reduces to
$\Omega^{1}(k)$, then the differential $\delta_{H}$ defined in (2.4) is
trivial and from (2.7) together with the property (a), one deduces

${\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{O}}(k),\,\Omega^{1}(k))={\rm
Hom}_{\mathscr{O}}({\mathscr{O}},\,\Omega^{1})=\Gamma({\mathbb{P}}^{n},\,\Omega^{1})=H^{0}({\mathbb{P}}^{n},\,\Omega^{1}).$

From the above item (2), together with
$H^{0}({\mathbb{P}}^{n},\,\Omega^{1})=0$ [10, page 4], it follows the
following proposition about the cardinal of the set of the holomorphic gauge
fields on ${\mathscr{O}}(k)$.

######  Proposition 1.

$\\#\\{\text{holomorphic gauge fields
on}\,\,{\mathscr{O}}(k)\\}\in\\{0,\,1\\}.$

Let us assume that $\psi\in{\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{O}}(k),\,{\mathscr{J}}({\mathscr{O}}(k))\big{)}$
is a holomorphic gauge field on ${\mathscr{O}}(k)$. As

$\displaystyle{\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{O}}(k),\,{\mathscr{J}}({\mathscr{O}}(k))\big{)}=H^{0}{\rm
Hom}^{\bullet}\big{(}{\mathscr{O}}(k),\,{\mathscr{J}}({\mathscr{O}}(k))\big{)}=$
$\displaystyle{\rm
Hom}^{0}\big{(}{\mathscr{O}}(k),\,{\mathscr{J}}({\mathscr{O}}(k))\big{)}={\rm
Hom}_{\mathscr{O}}\big{(}{\mathscr{O}}(k),\,\Omega^{1}(k)\oplus{\mathscr{O}}(k)\big{)}$

and $\psi$ is a right inverse of $\pi$ (see (2.2)) in
$D^{b}({\mathbb{P}}^{n})$, it can be written as $\psi=\nabla\oplus{\rm id}$,
with $\nabla:{\mathscr{O}}(k)\to\Omega^{1}(k)$. The property
$\psi(f\sigma)=f\ast\psi(\sigma)$ implies that
$\nabla(f\sigma)=df\sigma+f\nabla(\sigma)$, for $f\in{\mathscr{O}}$. Hence,
$\nabla$ can be considered as a connection, in the usual sense, on the line
bundle $L$ associated to ${\mathscr{O}}(k)$.

The connection form of $\nabla$ in a local frame $e$, defined on an open $U$,
is the holomorphic $1$-form $A$ determined by $\nabla(e)=A\otimes e$. The
curvature $F$ of this connection is a $2$-form defined over the whole
${\mathbb{P}}^{n}$, which on $U$ is equal to $\partial A$. Hence, $F$ is a
global holomorphic $2$-form. The first Chern class of $L$ is defined by the
form $\frac{i}{2\pi}F$. As $H^{0}({\mathbb{P}}^{n},\,\Omega^{2})=0$ [10, page
4], it turns out that the Chern class vanishes; that is, $k=0$.

On the other hand, for the case $k=0$, the morphism

$\psi:{\mathscr{O}}\to\Omega^{1}\oplus{\mathscr{O}},\;\;\sigma\mapsto\partial\sigma\oplus\sigma$

satisfies $\psi(f\sigma)=f\ast\psi(\sigma).$ It is a holomorphic gauge field
on ${\mathscr{O}}$. From the above arguments together with Proposition 1, it
follows the proposition.

######  Proposition 2.

The cardinal $\\#\,{\mathfrak{c}}({\mathscr{O}}(k))=\delta_{0k}.$

###  2.2. General case

As usual, we denote with ${\mathscr{F}}^{\bullet}[l]$, with
$l\in{\mathbb{Z}}$, the complex ${\mathscr{F}}^{\bullet}$ shifted $l$ to the
left. Given ${\mathscr{A}},{\mathscr{B}}$ elements of the generating set
(1.1), let us consider morphisms $h$ between
${\mathscr{A}}^{\prime}:={\mathscr{A}}[l]$ and
${\mathscr{B}}^{\prime}:={\mathscr{B}}[l^{\prime}]$. We denote by ${\rm
Cone}(h)={\mathscr{A}}^{\prime}[1]\oplus{\mathscr{B}}^{\prime}$ the mapping
cone of $h$ [5, page 154]. We define ${\mathcal{E}}^{(1)}$ the set obtained
adding to ${\mathcal{E}}$ the elements of the form ${\rm Cone}(h)$. Hence, an
element of ${\mathcal{E}}^{(1)}$ is a complex whose term at a position $p$ is
either $0$, or ${\mathscr{O}}(k)$, or a direct sum of
${\mathscr{O}}(k_{1})\oplus{\mathscr{O}}(k_{2})$, with $-n\leq
k,k_{1},k_{2}\leq 0$.

Repeating the process with the elements of ${\mathcal{E}}^{(1)}$, one obtains
${\mathcal{E}}^{(2)}$, etc. The objects of the triangulated subcategory
generated by the set ${\mathcal{E}}$ are elements which belong to some
${\mathcal{E}}^{(m)}$.

Therefore, an object of the triangulated subcategory of
$D^{b}({\mathbb{P}}^{n})$ generated by the set (1.1) is a complex
$({\mathscr{G}}^{\bullet},\,d^{\bullet})$, where ${\mathscr{G}}^{p}$ is a
sheaf of the form

(2.8) ${\mathscr{G}}^{p}=\bigoplus_{i\in S_{p}}{\mathscr{O}}(k_{pi}),$

with $-n\leq k_{pi}\leq 0$ and $i$ varying in a finite set $S_{p}$. (When $i$
“runs over the empty set”, the direct sum is taken to be $0$).

Since ${\mathscr{G}}^{\bullet}$ is a complex of locally free
${\mathscr{O}}$-modules

$\displaystyle{\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{G}}^{\bullet},\,\Omega^{1}{\mathscr{G}}^{\bullet}\big{)}=H^{0}{\rm
Hom}^{\bullet}\big{(}{\mathscr{G}}^{\bullet},\,\Omega^{1}{\mathscr{G}}^{\bullet}\big{)}=$
$\displaystyle\\{g\in{\rm
Hom}^{0}\big{(}{\mathscr{G}}^{\bullet},\,\Omega^{1}{\mathscr{G}}^{\bullet}\big{)}\,|\,\delta_{H}g=0\\},$

where $\delta_{H}$ is the operator defined in (2.4). Hence, according to
(2.3), it follows

${\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{G}}^{\bullet},\,\Omega^{1}{\mathscr{G}}^{\bullet}\big{)}\subset{\rm
Hom}^{0}\big{(}{\mathscr{G}}^{\bullet},\,\Omega^{1}{\mathscr{G}}^{\bullet}\big{)}=\prod_{p}{\rm
Hom}_{\mathscr{O}}({\mathscr{G}}^{p},\,\Omega^{1}{\mathscr{G}}^{p}).$

By the additivity of the functor ${\rm Hom}_{\mathscr{O}}(\,.\,,\,.\,)$, it
follows

${\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{G}}^{\bullet},\,\Omega^{1}{\mathscr{G}}^{\bullet}\big{)}\subset\prod_{p}\bigoplus_{i,j}{\rm
Hom}_{\mathscr{O}}\big{(}{\mathscr{O}}(k_{pi}),\,\Omega^{1}(k_{pj})\big{)},$

The summand ${\rm
Hom}_{\mathscr{O}}\big{(}{\mathscr{O}}(k_{pi}),\,\Omega^{1}(k_{pj})\big{)}$ is
equal to

${\rm
Hom}_{\mathscr{O}}\big{(}{\mathscr{O}},\,\Omega^{1}(k_{pj}-k_{pi})\big{)}=\Gamma({\mathbb{P}}^{n},\,\Omega^{1}(k_{pj}-k_{pi}))=0,$

since $H^{0}({\mathbb{P}}^{n},\,\Omega^{1}(k))=0$, for any $k$. Therefore,
${\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{G}}^{\bullet},\,\Omega^{1}{\mathscr{G}}^{\bullet}\big{)}=0.$

One has the following theorem, which generalizes Proposition 1.

######  Theorem 1.

If ${\mathscr{G}}^{\bullet}$ is a $B$-brane over ${\mathbb{P}}^{n}$, then
$\\#\,\mathfrak{c}({\mathscr{G}}^{\bullet})\leq 1.$

Let $\psi$ be a holomorphic gauge field on the above $B$-brane
${\mathscr{G}}^{\bullet}$. Then

$\displaystyle\psi\in{\rm
Hom}_{D^{b}({\mathbb{P}}^{n})}\big{(}{\mathscr{G}}^{\bullet},\,{\mathscr{J}}({\mathscr{G}}^{\bullet})\big{)}=H^{0}{\rm
Hom}^{\bullet}\big{(}{\mathscr{G}}^{\bullet},\,{\mathscr{J}}({\mathscr{G}}^{\bullet})\big{)}\subset$
$\displaystyle\prod_{p}{\rm
Hom}_{\mathscr{O}}\big{(}{\mathscr{G}}^{p},\,(\Omega^{1}{\mathscr{G}}^{p}\oplus{\mathscr{G}}^{p})\big{)}.$

Thus, $\psi$ determies a family
$\\{\psi^{p}:{\mathscr{G}}^{p}\to\Omega^{1}{\mathscr{G}}^{p}\oplus{\mathscr{G}}^{p}\\}$
of morphisms of ${\mathscr{O}}$-modules. As $\psi$ is a right inverse of
$\pi$, $\psi^{p}=\nabla^{p}\oplus{\rm id}^{p}$, where
$\nabla^{p}:{\mathscr{G}}^{p}\to\Omega^{1}{\mathscr{G}}^{p}$. Furthermore, as
in Section 2.1, $\nabla^{p}$ is a holomorphic connection on
${\mathscr{G}}^{p}$, for any $p$.

The trace ${\rm tr}(F^{p})$ of curvature of $\nabla^{p}$ is a holomorphic
$2$-form on ${\mathbb{P}}^{n}$; as $H^{0}({\mathbb{P}}^{n},\,\Omega^{2})=0$,
that trace vanishes. Hence, the first Chern class of vector bundle associated
to the locally free sheaf ${\mathscr{G}}^{p}$ vanishes.

On the other hand, the first Chern class of ${\mathscr{G}}^{p}$ is the sum

$\sum_{i}c_{1}({\mathscr{O}}(k_{pi})).$

This class is 0 iff $k_{pi}=0$ for all $i$, since the $k_{pi}\leq 0$.

Therefore, the existence of a holomorphic gauge field on the brane
${\mathscr{G}}^{\bullet}$ defined in (2.8) implies that
${\mathscr{G}}^{\bullet}$ is a sequence of direct sum of copies of
${\mathscr{O}}$

(2.9) $\dots\to\bigoplus_{i\in
S_{p}}{\mathscr{O}}\overset{d^{p}}{\longrightarrow}\bigoplus_{i\in
S_{p+1}}{\mathscr{O}}\to\dots$

Since ${\rm
Hom}_{\mathscr{O}}({\mathscr{O}},\,{\mathscr{O}})\simeq{\mathbb{C}}$, the map
$d^{p}$ is given by a constant complex matrix.

On the other hand, given the brane (2.9), if the set of indices $S_{p}$ has
$m_{p}$ elements, on $\oplus_{1}^{m_{p}}{\mathscr{O}}$ we define the map
$\varphi^{p}$,

$\varphi^{p}(\sigma_{1}\oplus\dots\oplus\sigma_{m_{p}})=\partial\sigma_{1}\oplus\dots\oplus\partial\sigma_{m_{p}}.$

It is a holomorphic connection on $\oplus_{1}^{m_{p}}{\mathscr{O}}$. Moreover,
the family $\\{\varphi^{p}\\}$ is compatible with the “constant” differentials
$d^{p}$. Thus, this family defines a holomorphic gauge field on the brane
defined by the complex (2.9). Therefore, we have the following theorem.

######  Theorem 2.

A $B$-brane on ${\mathbb{P}}^{n}$ admits a holomorphic gauge field iff it is
isomorphic to a brane of the form (2.9).

## References

  * [1] P. S. Aspinwall: $D$-branes on Calabi-Yau manifolds. In J.M. Maldacena (ed) Progress in String Theory, World Sci. Publ., 2005, 1-152.
  * [2] P. S. Aspinwall et al.: Dirichlet branes and mirror symmetry. Clay mathematics monographs vol 4. Amer. Math. Soc., Providence 2009.
  * [3] M. Atiyah: Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85, 181-207 (1957).
  * [4] P. Deligne: Equations différentialles à points singuliers réguliers. Lecture Notes 163. Springer-Verlag, Berlin 1970.
  * [5] S. I. Gelfand and Y. I. Manin: Methods of homological algebra. Springer, Berlin 2002.
  * [6] R. Hartshorne: Algebraic geometry. Springer-Verlag 1983.
  * [7] D. Huybrechts: Fourier-Mukai transforms in algebraic geometry. Oxford U. P., Oxford 2006.
  * [8] B. Iversen: Cohomology of sheaves. Springer-Verlag. Berlin 1986.
  * [9] M. Kashiwara, and P. Schapira: Sheaves on manifolds. Springer-Verlag 2002.
  * [10] Ch. Okonek, M. Schneider and H. Spindler: Vector Bundles on Complex Projective Spaces. Birkhäuser, Basel 1988.
  * [11] A. Viña: Yang-Mills fields on $B$-branes. arXiv:2206.10238 [math. AG].


# Spin-flip induced superfluidity in a ring of spinful hard-core bosons

K. K. Kesharpu<EMAIL_ADDRESS>E. A. Kochetov  Bogoliubov Laboratory
of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980,
Russia

###### Abstract

The $t-J$ Hamiltonian of the spinful hard-core bosonic ring in the _Nagaoka
limit_ is solved. The energy spectrum becomes quantized due to presence of
spin, where each energy level corresponds to a cyclic permutation state of the
spin chains. The ground state is true ferromagnetic when the ring contains
$N=2,3$ spinful hard-core bosons; for all other $N$ it is a mixture of the
ferromagnetic and non-ferromagnetic states. This behaviour is different from
the fermionic ring, where ground state is true ferromagnetic only for $N=3$.
It is shown that the intrinsic spin generated gauge fields are analogous to
the synthetic gauge fields generated by rotation of either the condensate or
the confining potential. It is argued that the low lying excited levels of the
spin flipped states intrinsically support the superfluidity. Possible ways to
experimentally verify these results are also discussed.

##  I Introduction

Rapid progress in the experimental techniques of trapping and manipulating
cold atoms has opened numerous possibilities for their use in quantum
simulation and quantum computing Georgescu  _et al._ (2014); Bruzewicz  _et
al._ (2019); Schäfer  _et al._ (2020); Bloch (2005). Spinor bosons—atoms with
integer spins—in optical lattice are one of the modern work horse to probe the
physics of the strongly correlated systems Bloch  _et al._ (2012, 2012, 2008);
Lewenstein  _et al._ (2007). It has several advantages over their condensed
matter counterparts e.g., precise knowledge of underlying microscopic model,
possibility to control the parameters of lattice Hamiltonian, and absence of
impurity in physical realizations. In this work we theoretically investigate
the physics of one of the simplest system, yet rich in physics, that can be
constructed using spinor bosons: the _1D_ ring lattice loaded with spinful
hard-core bosons (HCB) 111When two or more bosons can not occupy the same
position due to strong repulsive inter particle potential, they are known as
hard-core bosons Girardeau (1960)..

The spinless bosonic ring is a well studied problem both theoretically and
experimentally Morsch and Oberthaler (2006); Cazalilla  _et al._ (2011);
Manninen  _et al._ (2012); Bloch  _et al._ (2008); Lewenstein  _et al._
(2007). In Ref. Manninen  _et al._ (2012) the _Yarst_ states for fermionic and
bosonic ring were given. In Ref. Cazalilla  _et al._ (2011) several exactly
solvable models for one dimensional bosonic systems were reviewed. In Ref.
Wang and Zhu (1995) a model for neutral _spinless_ HCB on a ring was solved.
Ref. Boninsegni (2001) considered _spinful_ HCB on a $N\times N$ plaquette. On
this plaquette they solved the usual _t-J_ Hamiltonian by dividing the spinful
HCB into two different species of spinless HCB. Ref. Fledderjohann  _et al._
(2005) also considered spinful HCB on finite two-dimensional plaquette. They
investigated the effect of fermionic and bosonic statistics on the emergence
of ferromagnetic phase. In Ref. Bergkvist  _et al._ (2006) the phase diagram
of _1D_ chain, when even and odd number of spinful bosons are present at a
single site, were investigated. In Ref. Nie  _et al._ (2018, 2013) the ground
state properties of the spinful fermions and bosons in thermodynamic limits
were studied. Most of the work done till now are related either to the bosons
on one and two dimensional periodic lattices, and their behaviour in the
thermodynamic limit ($N\gg 1$), or _spinless_ HCB on a ring. Apart from some
comments about the energy levels of spinful bosonic ring in Ref. Manninen  _et
al._ (2012), a comprehensive study of the properties of the spinful hard-core
bosons on a ring away from the thermodynamic limit is still missing.

One of the interesting effect observed in the ring of bosons is the persistent
current, which is related to the superfluidity Cooper (2008); Fetter (2009);
Vilchynskyy  _et al._ (2013). In Ref. Cai  _et al._ (2010) the phase diagram
of superfluid and insulating phases was studied for spinless hard-core bosons.
Ref. Lin and Wu (2007) investigated the ground state and superfluidic
properties of the spinless hard-core bosons in one dimensional potential.
Recently, Ref. Polo  _et al._ (2020) estimated the values of persistent
current for two hard-core bosons in a ring lattice. From the experimental side
the persistent current was observed in spinor (not hard-core) condensates
Beattie  _et al._ (2013), and fermion rings Chin  _et al._ (2006); Cai  _et
al._ (2022). It is well known that ground state can never support the
persistent current Pitaevskiĭ and Stringari (2016); Lieb  _et al._ (2002);
Leggett (1999). Hence, one accesses the excited superfluidic states by
applying a velocity field, either by rotating the confined particles in the
ring lattice, or the ring lattice itself Lieb  _et al._ (2002); Cooper (2008);
Fetter (2009); Vilchynskyy  _et al._ (2013); Pitaevskiĭ and Stringari (2016).
The rotation of the lattice is analogous to the generation of synthetic gauge
fields, which in turn is related to the twisted boundary condition Lieb  _et
al._ (2002); Cooper (2008). We show that, for the case of hard-core spinful
bosons the twisted boundary condition is generated intrinsically without
application of any external velocity field.

_The main focus of our work is the investigation of the ground state
properties and the necessary condition for occurrence of superfluidity in
these systems._ This article is structured as follows. In Sec. II we solve the
Hamiltonian of the spinful HCB on a ring in _Nagaoka limit_. In this limit the
spin and charge degrees of freedom can be treated separately, hence, the
Hamiltonian is easily solvable. In Sec. III we investigate the dependence of
the ground state energies on the total spin and the structure of the total
spin chains. We discuss the necessary conditions for the emergence of
superfluidity in these systems. Here, we also suggest the experimental setups
to corroborate our theoretical predictions. Finally, In Sec. IV we summarise
the results.

##  II The Model

We take a ring of $L$ sites and $N$ spinful HCB with spin projections
$\sigma=\left\\{\uparrow,\downarrow\right\\}$ 222Although, the bosons don’t
have two spin projections $\pm 1/2$ like fermions, but two _hyperfine_ states
of some atoms can be considered as two spin projections of the bosons. These
spins are known as the _pseudo spins_ , and these atoms are known as the
_spinor bosons_ Stamper-Kurn and Ueda (2013); Kawaguchi and Ueda (2012).
Naturally, one can map the _pseudo spins_ to the spin-up state
$(\ket{\uparrow})$ and the spin-down state $(\ket{\downarrow})$.. Because of
the hard-core nature of the bosons, in the ring every site contains _at most
single boson_ ($N\leq L$). The _t-J_ Hamiltonian in the _Nagaoka limit_ with
periodic boundary conditions Nagaoka (1966, 1965) can be written as [see
supplementary materials]:

$H=-t\sum\limits_{i=1,\sigma}^{L-1}\tilde{b}_{i\sigma}^{\dagger}\tilde{b}_{i+1\sigma}-t\sum\limits_{\sigma}\tilde{b}_{L\sigma}^{\dagger}\tilde{b}_{1\sigma}+\mathrm{H.c}.$
(1)

Here, $t$ is the boson hopping factor from the _i_ -th site to the
neighbouring _i+1_ -th site. $\tilde{b}_{i\sigma}^{\dagger}$ and
$\tilde{b}_{i\sigma}$ are the HCB creation and annihilation operators with
spin projections $\sigma=\left\\{\uparrow,\downarrow\right\\}$. The operators
$\tilde{b}_{i\sigma}$ and $\tilde{b}_{i\sigma}^{\dagger}$ implies single
occupancy constraint:
$\sum\limits_{\sigma}b_{i\sigma}^{\dagger}b_{i\sigma}\leq 1$.

Due to the 1D nature of the ring, and the absence of the spin interaction, the
initial spin configuration of the spin chain is fixed during electron hopping.
Hence, we can separate the spin and charge degree of freedoms in the
Hamiltonian:

$H=-t\sum\limits_{i=1}^{L-1}{b}_{i}^{\dagger}{b}_{i+1}-t{b}_{L}^{\dagger}{b}_{1}\hat{P}+\mathrm{H.c}.$
(2)

Here, ${b}_{i}^{\dagger}({b}_{i})$ is a _spinless_ hard-core bosonic creation
(annihilation) operator. The spin content of the problem is encoded in the
spin permutation operator $\hat{P}$. It displaces the spin to the next non-
empty site:

$\hat{P}\ket{s_{1}^{z},s_{2}^{z},\dots,s^{z}_{N}}=\ket{s_{N}^{z},s_{1}^{z},\dots,s^{z}_{N-1}}.$

The boson hopping part of Eq. (2) is analogous to the _XY_ spin-chain
Hamiltonian Holstein and Primakoff (1940); Cazalilla  _et al._ (2011); Lieb
_et al._ (1961); Wang and Zhu (1995). Hence, one can use the _Jordan-Wigner
transformation_ to represent the Hamiltonian in Eq. (2) in terms of spinless
fermionic operators [see supplementary materials]:

$H=-t\sum\limits_{i=1}^{L-1}\hat{f}_{i}^{\dagger}\hat{f}_{i+1}+\>t\>\mathrm{e}^{\imath\pi
N}\hat{f}_{L}^{\dagger}\hat{f}_{1}\>\hat{P}+\text{H.c}.$ (3)

Here, $\hat{f}_{i}^{\dagger}$ ($\hat{f}_{i}$) is the spinless fermionic
creation (annihilation) operator. Defining the function,

$h(N)=\begin{cases}0,\quad\text{odd }N\\\ 1,\quad\text{even }N\end{cases},$
(4)

we can write Eq. (3) as:

$H=-t\sum\limits_{i=1}^{L-1}\hat{f}_{i}^{\dagger}\hat{f}_{i+1}-\>t\>\mathrm{e}^{\imath\pi
h(N)}\hat{f}_{L}^{\dagger}\hat{f}_{1}\>\hat{P}+\text{H.c}.$ (5)

The spin permutation and spinless fermionic operators are separately
diagonalized, because they are independent of each other. The eigenvalues
($\lambda_{\nu}$) and eigenfunction ($\psi_{\nu}$) of $\hat{P}$ are Ivantsov
_et al._ (2020):

$\displaystyle\lambda_{\nu}=\mathrm{e}^{\imath 2\pi p_{\nu}/N_{\nu}};$ (6a)
$\displaystyle\ket{\psi_{\nu}}=\frac{1}{\sqrt{N_{\nu}}}\sum\limits_{q=0}^{N_{\nu}-1}\mathrm{e}^{\imath
2\pi p_{\nu}\frac{q}{N_{\nu}}}\hat{P}^{q}\>\ket{\tilde{\psi}_{\nu}}.$ (6b)

Here, $\nu$ enumerates all possible disconnected spin blocks. A spin block
contains only connected spin chains. When two spin chains can be transformed
into each other by application of $\hat{P}$ operator they are connected,
otherwise they are disconnected. $N_{\nu}$ represents the total number of
connected spin chains in $\nu$-th spin block. $p_{\nu}$ enumerates the
connected spin chains in the $\nu$-th spin block; it takes the values
$p_{\nu}=0,1,\dots,N_{\nu}-1$. $\tilde{\psi}_{\nu}$ is the wave function of
one of the spin chain of $\nu$-th spin block.

For example, we have a chain of 4 sites and 3 particles. We take the spin-
chain $\ket{\uparrow\bullet\uparrow\downarrow}$ out of $2^{3}$ possible spin-
chains. It is connected to the $\ket{\uparrow\bullet\downarrow\uparrow}$ spin-
chain, as
$\hat{P}^{2}\ket{\uparrow\bullet\uparrow\downarrow}=\ket{\uparrow\bullet\downarrow\uparrow}$.
In this case both these configurations belong to the same $\nu$-th spin block.
This particular $\nu$-th block will have 3 possible configurations, hence
$N_{\nu}$ = 3, and $p_{\nu}=0,1,2$ 333The three configurations ($N_{\nu}$ = 3)
corresponding to three $p_{\nu}$ are: (i)
$p_{\nu}=0\equiv\ket{\uparrow\bullet\uparrow\downarrow}$, (ii)
$p_{\nu}=1\equiv\ket{\downarrow\bullet\uparrow\uparrow}$, (iii)
$p_{\nu}=2\equiv\ket{\uparrow\bullet\downarrow\uparrow}$.. The wave function
of three $p_{\nu}$ states can be found using Eq. (6b) [see App. C].
Consequently, every spin-chain in the $\nu$-th spin block has its own wave
function and spin momentum $p_{\nu}$. The number of disconnected blocks will
depend on the number of particles present in the ring ($N$) and the spin of
these particles ($s^{z}$). Due to these disconnected block of spin chains, the
total spin Hamiltonian corresponding to the $\hat{P}$ operator is a block
Hamiltonian with $\nu$ blocks. We find the Hamiltonian corresponding to the
$\nu$-th block by substituting $\lambda_{\nu}$ from Eq. (6b) into Eq. (5):

$H_{\nu}=-t\sum\limits_{i=1}^{L-1}\hat{f}_{i}^{\dagger}\hat{f}_{i+1}-\>t\>\mathrm{e}^{i2\pi\left[\frac{p_{\nu}}{N_{\nu}}+\frac{h(N)}{2}\right]}\hat{f}_{L}^{\dagger}\hat{f}_{1}+\text{H.c}.$
(7)

The total Hamiltonian of the whole system is a direct sum of these spin block
Hamiltonians: $H=\sum\limits_{\nu}\oplus H_{\nu}$.

Eq. (7) is nothing but the tight binding model with a penetrating magnetic
flux $\Phi_{\nu}\equiv
2\pi\left[\frac{p_{\nu}}{N_{\nu}}+\frac{h(N)}{2}\right]$ through the ring.
Using the gauge $f_{i}\mapsto\mathrm{e}^{\imath\Phi_{\nu}x_{i}/L}f_{i}$ one
maps the Hamiltonian in Eq. (7) onto twisted Hamiltonian:

$H_{\nu}=-t\sum\limits_{i}^{L-1}\mathrm{e}^{\imath\frac{\Phi_{\nu}}{L}}\hat{f}_{i}^{\dagger}\hat{f}_{i+1}-\>t\>\mathrm{e}^{\imath\frac{\Phi_{\nu}}{L}}\hat{f}_{L}^{\dagger}\hat{f}_{1}+\text{H.c}.$
(8)

Here, $x_{i}=1,2,\dots,L$, enumerates the $L$ sites. The locally induced phase
factor $\mathrm{e}^{\imath\Phi_{\nu}/L}$ is known as _Peierls phase_. The
explicit expression for the $k$-th mode energy of this tight binding
Hamiltoinian $H_{\nu}$ is [see supplementary materials]:

$\displaystyle
E_{\text{PBC}}(k,\nu,p_{\nu};N,L)=-2t\cos\frac{2\pi}{L}\left(k+\frac{p_{\nu}}{N_{\nu}}+\frac{h(N)}{2}\right).$
(9)

The total energy is found by summing over all $N$ low lying $k$-th mode
energies:

$\displaystyle E_{PBC,\>g}=$
$\displaystyle-2t\frac{\sin\left[(N+1+h(N))\pi/2L\right]}{\sin(\pi/L)}\cos\left[\frac{\frac{4\pi}{L}\left(\frac{p_{\nu}}{N_{\nu}}+\frac{h(N)}{2}\right)+\frac{(N-1+h(N))\pi}{L}}{2}\right]$
(10)
$\displaystyle-2t\frac{\sin\left[(N+1+h(N))\pi/2L\right]}{\sin(\pi/L)}\cos\left[\frac{\frac{4\pi}{L}\left(\frac{p_{\nu}}{N_{\nu}}+\frac{h(N)}{2}\right)-\frac{(N-1+h(N))\pi}{L}}{2}\right]$
$\displaystyle+2t\left[1+h(N)\right]\cos\left[\frac{2\pi}{L}\left(\frac{p_{\nu}}{N_{\nu}}+\frac{h(N)}{2}\right)\right].$

The Hamiltonian of the antiperiodic boundary condition
$\left(\tilde{b}_{L+1,\sigma}=-\tilde{b}_{1,\sigma}\right)$ is written by
introducing the extra phase $\mathrm{e}^{i\pi}$ in the second term of Eq. (1).
Repeating the aforementioned procedure, the _k_ -th mode energy levels and the
ground state energies can be calculated. One can directly find these
expressions from Eq. (9) and (10) by replacing $h(N)\to h(N+1)$. If a magnetic
field $\mathbf{B}$ is applied perpendicular to the ring, then an additional
flux $\Phi_{B}=2\pi\mathbf{B}A$—here, $A$ is the area of the ring—penetrates
through the ring. To find the total energy one repeats the above calculation
by substituting $\Phi_{\nu}\mapsto\Phi_{\nu}+\Phi_{B}$, and adds the total
spin ($S_{\nu}$) dependent _Zeeman energy_ , $Z=g\mu_{B}BS_{\nu}$ [see
supplementary materials]. Here, $g$ is the _Lande factor_ ; $\mu_{B}$ is the
_Bohr magneton_ ; $S_{\nu}$ is the total spin of the $\nu$-th block.

##  III Ground state properties and Superfluidity

Figure 1: (Color online) Energy levels for the periodic boundary condition
when $N=4,5$ bosons reside on a ring of $L=8$ sites. The _x_ -axis represents
the spin ($\mathrm{S}$), and the _y_ -axis represents the energy ($E/2t$).
$\nu$ represents the spin block. The spin chain configurations corresponding
to different $\mathrm{S}$ and $\nu$ are shown in Tab. 1 and 2. For $\nu=2$
(square, orange) and $\nu=3$ (down triangle, green) the $p_{\nu}=0$ state is
not available [see App. A]. For both $S=0$ and $S=1/2$ spin states, correspond
two spin blocks: $\nu=3$ (down triangle, green) and $\nu=4$ (up triangle,
red).

In a ring states corresponding to the cyclic permutation ($p_{\nu}$) of the
initial spin configurations become available, because particles can jump
directly from the _L_ -th site to the _1_ -st site, One can further group
these permutation states into irreducible representations of the cyclic
symmetry groups $C_{n}$. It should be noted that a single cyclic group can
contain more than one spin blocks ($\nu$). For example, for $N=4$ the $C_{4}$
group contains the $\nu=2,3$ spin blocks as shown in Tab. 1. The more general
relation between $N$, $p_{\nu}$, and $\nu$ can be found using _Burnside’s
Lemma_ Xavier  _et al._ (2020).

We show the detailed spin configurations for $N=4$ and $5$ in Tab. 1 and 2
respectively. The corresponding energy levels with $L=8$ are shown in Fig. 1.
Here, for $N\geq 4$ the ground state is a mixture of the ferromagnetic phase
($S=N/2$) and non-ferromagnetic phases ($S=N/2-2,N/2-3,\dots$). It should be
noted that the single spin flipped phase ($S=N/2-1$) is absent in the ground
state due to the unavailability of the $p_{\nu}=0$ state [see App. A].
Interestingly for $N=2,3$, the ground state is pure ferromagnetic, because the
spin flipped phases ($S=N/2-2,N/2-3,\dots$) are not available. This behaviour
is different from the fermionic ring, for which the ground state is pure
ferromagnetic only for $N=3$ Ivantsov  _et al._ (2020).

Physically the spinful hard-core bosonic ring can be realised by loading
spinor bosons Stamper-Kurn and Ueda (2013); Kawaguchi and Ueda (2012) in
optical tweezers Grier (2003); Neuman and Block (2004); de Goër de Herve  _et
al._ (2021) or paul traps Schneider  _et al._ (2012); Tomza  _et al._ (2019).
The two hyperfine states ($F$) of the spinor bosons can be considered as two
_pseudo-spin_ states. Recently numerous experiments have successfully
generated several 2D and 3D crystals with high fidelity Nogrette  _et al._
(2014); Barredo  _et al._ (2018, 2016); Endres  _et al._ (2016); Kim  _et al._
(2016); D’Onofrio  _et al._ (2021). Hence generating 1D rings should not be
difficult. One of the interesting fact to observe experimentally is the
dependence of the energy levels on underlying spin structures. For example one
can generate a ring of spinor bosons with the initial spin configuration
$\ket{\uparrow\uparrow\downarrow\downarrow}$ ($\nu=3$ in Tab. 1). Then the
system is excited to the higher energy levels ($p_{\nu}=1,2,3$) through
rotation of the confining potentials Cooper (2008); Fetter (2009). To return
to the ground state the system should radiate the energy proportional to the
$E(p_{\nu})-E(p_{\nu}=0)$, which can be easily measured. In the next step one
can prepare the system with the spin arrangement
$\ket{\uparrow\downarrow\uparrow\downarrow}$ ($\nu=4$ in Tab. 1). Analogously
the system will be excited to the higher energy levels, and the radiated
energy will be measured. In the former case the radiated energy will be higher
than the later case, because only single $p_{\nu}=1$ state is available. _It
will be the direct experimental evidence of the spin chain configuration
dependent quantization of the energy in the spinful hard-core bosonic rings._

As an example of the physics displayed by the spinful hard-core bosonic ring,
let us show that a slight change of the underlying spin structure of the HCB
on a ring might provide a necessary condition for a superfluidity to emerge.
According to the two-fluid picture, the superfluid contains both normal as
well as superfluid components. One therefore defines a quantity, the so called
_superfluid fraction ( $f_{s}$)_, to represent the _degree of superfluidity_.
There are different ways to calculate it Lieb  _et al._ (2002); Prokof’ev and
Svistunov (2000). We use the definition where $f_{s}$ is calculated through
the reaction of the system under a change in boundary conditions.
Mathematically, the change in boundary conditions is equivalent to imposing a
linear phase variation $\Theta x/L$ over length $L$ of the system Lieb  _et
al._ (2002). Hence, if $\Psi(x)$ is the wave function of the superfluid, then
$\Psi(x+L)=\mathrm{e}^{i\Theta}\Psi(x)$. It should be stressed that, the phase
variation should be linear in $x$ to conserve the symmetry of the system and
avoid a phase slip. Physically it means that the particles acquire a similar
phase $\Theta/L$ while tunneling to the neighbouring sites.

Figure 2: (Color online) Dependence of superfluid fraction ($f_{s}$) on number
of sites ($L$) while number of particles ($N$) is fixed. $f_{s}$ is calculated
using Eq. (11) with phase $\theta=2\pi/N$ for $N=5$ (blue, plus), $N=6$ (red,
down-triangle), $N=9$ (orange, circle), $N=10$ (violet, cross), $N=21$ (green,
square), and $N=26$ (brown, up-triangle).

Physically, the twisted phase $\Theta$ is imposed by rotating the system with
some angular velocity $\omega$ Lieb  _et al._ (2002); Roth and Burnett (2003a,
b). For a unit radius 1D ring, $\Theta$ is related to the superfluid velocity:
$v_{s}=\hbar\Theta/(mL)$ Roth and Burnett (2003b); Leggett (1999); Lieb  _et
al._ (2002). Experimentally, the twisted phase can be imposed through atom-
light interactions Dalibard  _et al._ (2011), rotating the confining potential
Cooper (2008), or rotating the confined atoms Fetter (2009). However, there is
another way to impose twisted boundary condition: _through the change in the
underlying spin configurations._ Indeed the phase factor $\Phi_{\nu}/L$ in Eq.
(8), which is dependent on the spin configuration through $\nu$, is equivalent
to the twisted phase. The persistent current appears when $\Phi_{\nu}\ll\pi$,
because only for this case that high energy excitation are absent in the
system Roth and Burnett (2003a). In this limit the superfluid density is Roth
and Burnett (2003b):

$f_{s}=\frac{L^{2}}{tN}\frac{E_{\Phi_{\nu}}-E_{0}}{\Phi_{\nu}^{2}}.$ (11)

Here, $E_{0}$ ($E_{\Phi_{\nu}}$) is the energy of the system in the absence
(presence) of the phase twist.

Eq. (11) is directly applicable when the ring contains _odd and large_ number
of particles. In this case the low lying excited energy levels ($p_{\nu}\ll
N_{\nu}$) satisfy the condition $\Phi_{\nu}^{\text{odd}}:=2\pi
p_{\nu}/N_{\nu}\ll\pi$. For even $N$ the twisted phase takes on the form
$\Phi_{\nu}^{\text{even}}:=2\pi p_{\nu}/N_{\nu}+\pi$. An extra phase factor of
$\pi$ accounts for a passage from odd $N$ to even $N$. Therefore to calculate
$f_{s}$ induced solely by a change in the spin structure at fixed even $N$, we
should replace
$\Phi_{\nu}^{\text{even}}\mapsto\Phi_{\nu}^{\text{even}}-\pi=\Phi^{\text{odd}}\ll\pi$.
In Fig. 2 we plotted the dependence of the superfluid fraction on the number
of sites in the ring for $N\geq 5$ and $p_{\nu}/N_{\nu}=1/N$. It can be
observed that, when $N\approx L$ (commensurate) superfluidity is absent
($f_{s}\approx 0$). However, the superfluid fraction increases as $L$
increases (incommensurate). The occurrence of the superfluidity for
incommensurate case ($N/L\ll 1$) is a manifestation of the fact that a $1D$
dilute gas of hard-core bosons is always superfluid Lieb  _et al._ (2002).

It should be mentioned that, the condition $\Phi_{\nu}\ll\pi$, is _necessary
but not sufficient_ for appearance of superfluidity, because it does not say
anything about the stability of the persistent currents Pitaevskiĭ and
Stringari (2016). We propose the following experiment to detect superfluidity
generated by a single-spin flip. One can prepare a 1D ring of spinor bosons
using optical tweezers, and containing $N\gg 1$. All the particles should be
in the same spin state, in other words the system is in ferromagnetic phase
($S=N/2$). Then spin of a single atom is flipped. Due to absence of the
$p_{\nu}=0$ state for spin $S=N/2-1$, the ground state will then be the first
excited state ($p_{\nu}=1$). It is equivalent to generating a small twisted
phase $\Phi=2\pi/N$. One can then experimentally find the _matter-wave
interference pattern_ and _structure factor_ to get information about
superfluidity Roth and Burnett (2003a).

##  IV Conclusions

We have shown that in the spinful hard-core bosonic ring energy spectrum is
quantized. Here each excited energy level corresponds to a cyclic permutation
state ($p_{\nu}$) of the spin chain. Interestingly, depending on the
underlying spin configuration of the spin chains, the spin blocks with
identical total spin ($S$) can have different size ($N_{\nu}$). For example,
although the total spin of the spin-chains
$\ket{\uparrow\uparrow\downarrow\downarrow}$ and
$\ket{\uparrow\downarrow\uparrow\downarrow}$ are same ($S=0$), however due to
different spin configurations the size of the corresponding spin blocks will
be four and two respectively [see Tab. 1]. To corroborate this fact, one can
perform experiments measuring the radiated energy from the first excited state
($p_{\nu}=1$) to the ground states. For
$\ket{\uparrow\uparrow\downarrow\downarrow}$ the radiated energy will always
be lower compared to $\ket{\uparrow\downarrow\uparrow\downarrow}$. We showed
that the ground state phase is true ferromagnetic only for $N=2,3$. It is
important, as for fermionic ring the ground state is true ferromagnetic only
for $N=3$. Usually the superfluid fraction is measured by measuring the energy
change due to imposed twisted phase $\Phi_{\nu}\ll\pi$. We showed that apart
from already existing methods for generating twisted phase—rotation of either
the condensate or the confining potential, and the light atom interaction—one
can use the spin generated intrinsic phases $2\pi p_{\nu}/N_{\nu}$ as a
twisted phase. This provides a new way to generate twisted phase in hard-core
bosonic rings. We argue that the low lying energy levels occurring due to
cyclic permutation of the spin chains ($p_{\nu}\ll N$), can support the
persistent current without any external excitation, when: (i) $N\gg 1$, (ii)
$N/L\ll 1$. In other words, superfluid emerges spontaneously during transition
from fully polarised state ($S=N/2$) to the spin flipped states. In this
article we also proposed several experiments to corroborate the above
mentioned results.

###### Acknowledgements.

This article is partly supported by the RFBR Grants No. 21-52-12027. K.K.K
would like to thank Pavel D. Grigoriev for useful discussion on the problem.

##  Appendix A Unavailability of $p_{\nu}=0$ state for $\mathrm{S}=(N/2)-1$

For the case of $p_{\nu}=0$ the spin wave function $\ket{\psi}$ in Eq. (6b) is
symmetric. It corresponds to the fully polarised case. Hence if we have $x$
number of $p_{\nu}=0$ state, then one of the $p_{\nu}=0$ state corresponds to
fully polarised case, and $x-1$ number of $p_{\nu}=0$ states corresponds to
the spin flipped case. For example, we take a ring with five spinful HCB
($N=5$). The spin configurations for $N=5$ is shown in Tab. 2. Here we observe
that for $S=5/2$ single $p_{\nu}=0$ state, for $S=3/2$ single $p_{\nu}=0$
state, and for $S=1/2$ two $p_{\nu}=0$ states are available. For $S=3/2$ no
$p_{\nu}=0$ is available, because the single $p_{\nu}=0$ corresponds to the
fully polarised (ferromagnetic) state. However, for $S=1/2$ because two
$p_{\nu}=0$ states are available, one $p_{\nu}=0$ state corresponds to the
fully polarised (ferromagnetic) state, and the other $p_{\nu}=0$ corresponds
to the non-ferromagnetic phase.

##  Appendix B Spin configurations for $N=4$ and $N=5$ _spinful HCB_ on a ring

In Tab. 1 we represent all possible spin configurations of four spinful HCB.
It should be noted that number of sites $L$ does not have any effect on the
spin configurations.

$S$ | $\nu$ | $N_{\nu}$
---|---|---
2 | 1 | $N_{1}=1$
|  | $p_{1}=0\equiv\ket{\uparrow\uparrow\uparrow\uparrow}$
1 | 2 | $N_{2}=4$
|  | $p_{2}=0\equiv\ket{\uparrow\uparrow\uparrow\downarrow}$
|  | $p_{2}=1\equiv\ket{\downarrow\uparrow\uparrow\uparrow}$
|  | $p_{2}=2\equiv\ket{\uparrow\downarrow\uparrow\uparrow}$
|  | $p_{2}=3\equiv\ket{\uparrow\uparrow\downarrow\uparrow}$
0 | 3 | $N_{3}=4$
|  | $p_{3}=0\equiv\ket{\uparrow\uparrow\downarrow\downarrow}$
|  | $p_{3}=1\equiv\ket{\downarrow\uparrow\uparrow\downarrow}$
|  | $p_{3}=2\equiv\ket{\downarrow\downarrow\uparrow\uparrow}$
|  | $p_{3}=3\equiv\ket{\uparrow\downarrow\downarrow\uparrow}$
| 4 | $N_{4}=2$
|  | $p_{4}=0\equiv\ket{\uparrow\downarrow\uparrow\downarrow}$
|  | $p_{4}=1\equiv\ket{\downarrow\uparrow\downarrow\uparrow}$
Table 1:  The spin configurations for $N=4$ bosons. Column <<S>> reprsents the
total spin of the chain. Column <<$\nu$> represents the enumerated spin
blocks. $N_{\nu}$ reprsents the total number of connected spin chains
contained in the $\nu$-th spin block. $p_{\nu}$ enumerates the connected spin
chains in the $\nu$-th block.

In Tab. 2 we represent all possible spin configurations of five spinful HCB.

$S$ | $\nu$ | $N_{\nu}$
---|---|---
$\frac{5}{2}$ | 1 | $N_{1}=1$
|  | $p_{1}=0\equiv\ket{\uparrow\uparrow\uparrow\uparrow\uparrow}$
$\frac{3}{2}$ | 2 | $N_{2}=5$
|  | $p_{2}=0\equiv\ket{\uparrow\uparrow\uparrow\uparrow\downarrow}$
|  | $p_{2}=1\equiv\ket{\downarrow\uparrow\uparrow\uparrow\uparrow}$
|  | $p_{2}=2\equiv\ket{\uparrow\downarrow\uparrow\uparrow\uparrow}$
|  | $p_{2}=3\equiv\ket{\uparrow\uparrow\downarrow\uparrow\uparrow}$
|  | $p_{2}=4\equiv\ket{\uparrow\uparrow\uparrow\downarrow\uparrow}$
$\frac{1}{2}$ | 3 | $N_{3}=5$
|  | $p_{3}=0\equiv\ket{\uparrow\uparrow\uparrow\downarrow\downarrow}$
|  | $p_{3}=1\equiv\ket{\downarrow\uparrow\uparrow\uparrow\downarrow}$
|  | $p_{3}=2\equiv\ket{\downarrow\downarrow\uparrow\uparrow\uparrow}$
|  | $p_{3}=3\equiv\ket{\uparrow\downarrow\downarrow\uparrow\uparrow}$
|  | $p_{3}=4\equiv\ket{\uparrow\uparrow\downarrow\downarrow\uparrow}$
| 4 | $N_{4}=5$
|  | $p_{4}=0\equiv\ket{\uparrow\uparrow\downarrow\uparrow\downarrow}$
|  | $p_{4}=1\equiv\ket{\downarrow\uparrow\uparrow\downarrow\uparrow}$
|  | $p_{4}=2\equiv\ket{\uparrow\downarrow\uparrow\uparrow\downarrow}$
|  | $p_{4}=3\equiv\ket{\downarrow\uparrow\downarrow\uparrow\uparrow}$
|  | $p_{4}=4\equiv\ket{\uparrow\downarrow\uparrow\downarrow\uparrow}$
Table 2:  The spin configurations for $N=5$ bosons. Column <<S>> reprsents the
total spin of the chain. Column <<$\nu$> represents the enumerated spin
blocks. $N_{\nu}$ reprsents the total number of connected spin chains
contained in the $\nu$-th spin block. $p_{\nu}$ enumerates the connected spin
chains in the $\nu$-th block.

##  Appendix C Matrix representation of spin wave function

For a spin chain configuration $\ket{\uparrow\bullet\uparrow\downarrow}$,
using Eq. (6b), the wave functions of the spin chain can be represented in a
compact formu sing matrix notation
$\ket{\psi(p_{\nu})}=(1/\sqrt{N_{\nu}})\mathrm{C}\ket{\tilde{\psi}_{\nu}}$.
Here $\ket{\psi(p_{\nu})}$ is a $N_{\nu}\times 1$ column matrix. Its
components represent the wave function corresponding to $p_{\nu}$-th value.
$\mathrm{C}$ is the $N_{\nu}\times N_{\nu}$ matrix. Its rows and columns are
indexed as $p_{\nu}=0,1,\dots N_{\nu}$. Hence $\mathrm{C}_{mn}$-th terms is
$\mathrm{e}^{\imath 2\pi m(n/N_{\nu})}$. The wave function
$\ket{\tilde{\psi}_{\nu}}$ is a $N_{\nu}\times 1$ column matrix of all
possible connected spin chain of the $\nu$-th block. If the $\nu$-th block
represents all the connected spin chain of
$\ket{\uparrow\bullet\uparrow\downarrow}$, the total wave function of the
$\nu$-th block is:

$\begin{bmatrix}\psi_{\nu}(p_{\nu}=0)\\\ \psi_{\nu}(p_{\nu}=1)\\\
\psi_{\nu}(p_{\nu}=2)\end{bmatrix}=\frac{1}{\sqrt{N_{\nu}}}\begin{bmatrix}1&1&1\\\
1&\mathrm{e}^{\imath 2\pi/3}&\mathrm{e}^{\imath 4\pi/3}\\\
1&\mathrm{e}^{\imath 4\pi/3}&\mathrm{e}^{\imath
8\pi/3}\end{bmatrix}\begin{bmatrix}\ket{\uparrow\bullet\uparrow\downarrow}\\\
\ket{\downarrow\bullet\uparrow\uparrow}\\\
\ket{\uparrow\bullet\downarrow\uparrow}\end{bmatrix}.$ (12)

Eq. (12) can be generalized to arbitrary $N$.

## References

  * Georgescu  _et al._ (2014) I. M. Georgescu, S. Ashhab, and F. Nori, Reviews of Modern Physics 86, 153 (2014).
  * Bruzewicz  _et al._ (2019) C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Applied Physics Reviews 6, 021314 (2019).
  * Schäfer  _et al._ (2020) F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Nature Reviews Physics 2, 411 (2020).
  * Bloch (2005) I. Bloch, Nature Physics 1, 23 (2005).
  * Bloch  _et al._ (2012) I. Bloch, J. Dalibard, and S. Nascimbène, Nature Physics 8, 267 (2012).
  * Bloch  _et al._ (2008) I. Bloch, J. Dalibard, and W. Zwerger, Reviews of Modern Physics 80, 885 (2008).
  * Lewenstein  _et al._ (2007) M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(De), and U. Sen, Advances in Physics 56, 243 (2007).
  * Note (1) When two or more bosons can not occupy the same position due to strong repulsive inter particle potential, they are known as hard-core bosons Girardeau (1960).
  * Morsch and Oberthaler (2006) O. Morsch and M. Oberthaler, Reviews of Modern Physics 78, 179 (2006).
  * Cazalilla  _et al._ (2011) M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, Reviews of Modern Physics 83, 1405 (2011).
  * Manninen  _et al._ (2012) M. Manninen, S. Viefers, and S. Reimann, Physica E: Low-dimensional Systems and Nanostructures 46, 119 (2012).
  * Wang and Zhu (1995) Z. D. Wang and J.-X. Zhu, Physical Review B 52, 5275 (1995).
  * Boninsegni (2001) M. Boninsegni, Physical Review Letters 87, 087201 (2001).
  * Fledderjohann  _et al._ (2005) A. Fledderjohann, A. Langari, E. Müller-Hartmann, and K.-H. Mütter, The European Physical Journal B - Condensed Matter and Complex Systems 43, 471 (2005).
  * Bergkvist  _et al._ (2006) S. Bergkvist, I. P. McCulloch, and A. Rosengren, Physical Review A 74, 053419 (2006).
  * Nie  _et al._ (2018) W. Nie, H. Katsura, and M. Oshikawa, Physical Review B 97, 125153 (2018).
  * Nie  _et al._ (2013) W. Nie, H. Katsura, and M. Oshikawa, Physical Review Letters 111, 100402 (2013).
  * Cooper (2008) N. Cooper, Advances in Physics 57, 539 (2008).
  * Fetter (2009) A. L. Fetter, Reviews of Modern Physics 81, 647 (2009).
  * Vilchynskyy  _et al._ (2013) S. I. Vilchynskyy, A. I. Yakimenko, K. O. Isaieva, and A. V. Chumachenko, Low Temperature Physics 39, 724 (2013).
  * Cai  _et al._ (2010) X. Cai, S. Chen, and Y. Wang, Physical Review A 81, 023626 (2010).
  * Lin and Wu (2007) Y. Lin and B. Wu, Physical Review A 75, 023613 (2007).
  * Polo  _et al._ (2020) J. Polo, P. Naldesi, A. Minguzzi, and L. Amico, Physical Review A 101, 043418 (2020).
  * Beattie  _et al._ (2013) S. Beattie, S. Moulder, R. J. Fletcher, and Z. Hadzibabic, Physical Review Letters 110, 025301 (2013).
  * Chin  _et al._ (2006) J. K. Chin, D. E. Miller, Y. Liu, C. Stan, W. Setiawan, C. Sanner, K. Xu, and W. Ketterle, Nature 443, 961 (2006).
  * Cai  _et al._ (2022) Y. Cai, D. G. Allman, P. Sabharwal, and K. C. Wright, Physical Review Letters 128, 150401 (2022).
  * Pitaevskiĭ and Stringari (2016) L. P. Pitaevskiĭ and S. Stringari,  _Bose-Einstein Condensation and Superfluidity_ , first edition ed., International Series of Monographs on Physics No. 164 (Oxford University Press, Oxford, United Kingdom, 2016).
  * Lieb  _et al._ (2002) E. H. Lieb, R. Seiringer, and J. Yngvason, Physical Review B 66, 134529 (2002).
  * Leggett (1999) A. J. Leggett, Reviews of Modern Physics 71, S318 (1999).
  * Note (2) Although, the bosons don’t have two spin projections $\pm 1/2$ like fermions, but two _hyperfine_ states of some atoms can be considered as two spin projections of the bosons. These spins are known as the _pseudo spins_ , and these atoms are known as the _spinor bosons_ Stamper-Kurn and Ueda (2013); Kawaguchi and Ueda (2012). Naturally, one can map the _pseudo spins_ to the spin-up state $(\ket{\uparrow})$ and the spin-down state $(\ket{\downarrow})$.
  * Nagaoka (1966) Y. Nagaoka, Physical Review 147, 392 (1966).
  * Nagaoka (1965) Y. Nagaoka, Solid State Communications 3, 409 (1965).
  * Holstein and Primakoff (1940) T. Holstein and H. Primakoff, Physical Review 58, 1098 (1940).
  * Lieb  _et al._ (1961) E. Lieb, T. Schultz, and D. Mattis, Annals of Physics 16, 407 (1961).
  * Ivantsov  _et al._ (2020) I. Ivantsov, H. B. Xavier, A. Ferraz, and E. Kochetov, Physical Review B 101, 195107 (2020).
  * Note (3) The three configurations ($N_{\nu}$ = 3) corresponding to three $p_{\nu}$ are: (i) $p_{\nu}=0\equiv\ket{\uparrow\bullet\uparrow\downarrow}$, (ii) $p_{\nu}=1\equiv\ket{\downarrow\bullet\uparrow\uparrow}$, (iii) $p_{\nu}=2\equiv\ket{\uparrow\bullet\downarrow\uparrow}$.
  * Xavier  _et al._ (2020) H. B. Xavier, E. Kochetov, and A. Ferraz, Physical Review B 101, 045112 (2020).
  * Stamper-Kurn and Ueda (2013) D. M. Stamper-Kurn and M. Ueda, Reviews of Modern Physics 85, 1191 (2013).
  * Kawaguchi and Ueda (2012) Y. Kawaguchi and M. Ueda, Physics Reports 520, 253 (2012).
  * Grier (2003) D. G. Grier, Nature 424, 810 (2003).
  * Neuman and Block (2004) K. C. Neuman and S. M. Block, Review of Scientific Instruments 75, 2787 (2004).
  * de Goër de Herve  _et al._ (2021) M. de Goër de Herve, Y. Guo, C. De Rossi, A. Kumar, T. Badr, R. Dubessy, L. Longchambon, and H. Perrin, Journal of Physics B: Atomic, Molecular and Optical Physics 54, 125302 (2021).
  * Schneider  _et al._ (2012) C. Schneider, D. Porras, and T. Schaetz, Reports on Progress in Physics 75, 024401 (2012).
  * Tomza  _et al._ (2019) M. Tomza, K. Jachymski, R. Gerritsma, A. Negretti, T. Calarco, Z. Idziaszek, and P. S. Julienne, Reviews of Modern Physics 91, 035001 (2019).
  * Nogrette  _et al._ (2014) F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L. Béguin, A. Vernier, T. Lahaye, and A. Browaeys, Physical Review X 4, 021034 (2014).
  * Barredo  _et al._ (2018) D. Barredo, V. Lienhard, S. de Léséleuc, T. Lahaye, and A. Browaeys, Nature 561, 79 (2018).
  * Barredo  _et al._ (2016) D. Barredo, S. de Léséleuc, V. Lienhard, T. Lahaye, and A. Browaeys, Science (New York, N.Y.) 354, 1021 (2016).
  * Endres  _et al._ (2016) M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz, A. Krajenbrink, C. Senko, V. Vuletic, M. Greiner, and M. D. Lukin, Science 354, 1024 (2016).
  * Kim  _et al._ (2016) H. Kim, W. Lee, H.-g. Lee, H. Jo, Y. Song, and J. Ahn, Nature Communications 7, 13317 (2016).
  * D’Onofrio  _et al._ (2021) M. D’Onofrio, Y. Xie, A. J. Rasmusson, E. Wolanski, J. Cui, and P. Richerme, Physical Review Letters 127, 020503 (2021).
  * Prokof’ev and Svistunov (2000) N. V. Prokof’ev and B. V. Svistunov, Physical Review B 61, 11282 (2000).
  * Roth and Burnett (2003a) R. Roth and K. Burnett, Physical Review A 68, 023604 (2003a).
  * Roth and Burnett (2003b) R. Roth and K. Burnett, Physical Review A 67, 031602 (2003b).
  * Dalibard  _et al._ (2011) J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, Reviews of Modern Physics 83, 1523 (2011).
  * Girardeau (1960) M. Girardeau, Journal of Mathematical Physics 1, 516 (1960).