Quantum holonomy in Lieb-Liniger model

Abstract

We examine a parametric cycle in the N-body Lieb-Liniger model that starts
from the free system and goes through Tonks-Girardeau and super-Tonks-Girardeau
regimes and comes back to the free system. We show the existence of exotic
quantum holonomy, whose detailed workings are analysed with the specific sample
of two- and three body systems. The classification of eigenstates based on
clustering structure naturally emerges from the analysis.

Full text

arXiv:1304.5041v2 [cond-mat.quant-gas] 6 Jul 2013
N.YONEZAWA, A.TANAKA, T.CHEON OCU-PHYS 384, PREPRINT
Quantum holonomy in Lieb-Liniger model
Nobuhiro Yonezawa,1, ∗Atushi Tanaka,2, †and Taksu Cheon3, ‡
1Osaka City University Advanced Mathematical Institute (OCAMI), Sumiyoshi-ku, Osaka 558-8585, Japan
2Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
3Laboratory of Physics, Kochi University of Technology, Tosa Yamada, Kochi 782-8502, Japan
We examine a parametric cycle in the N-body Lieb-Liniger model that starts from the free system
and goes through Tonks-Girardeau and super-Tonks-Girardeau regimes and comes back to the free
system. We show the existence of exotic quantum holonomy, whose detailed workings are analyzed
with the specific sample of two- and three-body systems. The classification of eigenstates based on
clustering structure naturally emerges from the analysis.
PACS numbers: 02.30.Ik, 03.65.Vf, 67.85.-d
I. INTRODUCTION
Among the solvable models of quantum mechan-
ics, the Lieb-Liniger system [1] belongs to the selec-
tive class of models that are genuinely many-body. It
is a system made up of identical bosons interacting
through two-body contact force. It was later shown
that the one-dimensional system of identical fermions
with two-body contact interactions can be rigorously
mapped to the Lieb-Liniger system with strong and
weak coupling regimes interchanged [2, 3]. Several
further extensions of the model with anyon statistics
has been found [4–7], and they are also known to be
mathematically equivalent to the original model. The
thermodynamics of the Lieb-Liniger model has been
studied extensively [8–11].
What has made the Lieb-Liniger model a focus
of renewed recent attention is its experimental re-
alization in the form of Tonks-Girardeau gas [12–
14]. It has been shown that the coupling strength of
the Lieb-Liniger system can be experimentally con-
trolled through the Feshbach resonance mechanism
[15]. In recent experiments by Haller and collabora-
tors [16, 17], a smooth change of the coupling strength
from large negative values to large positive values,
where one finds the super-Tonks-Girardeau system
[18], has been realized.
∗Email:yonezawa@sci.osaka-cu.ac.jp
†Email:tanaka-atushi@tmu.ac.jp
‡Email:taksu.cheon@kochi-tech.ac.jp
The continuous transition from a strongly repul-
sive to strongly attractive regimes of Lieb-Liniger
model inspires us to propose following parametric cy-
cle C. We start with the noninteracting limit, in-
crease the coupling strength adiabatically, reaching
the strongly attractive regime crossing the ±∞ cou-
pling limit, then decrease the absolute value of nega-
tive coupling strength until it reaches the noninteract-
ing limit again. In this paper, we show that the ini-
tial energy eigenstates of the cycle are different from
the final eigenstates, although the initial and the final
Hamiltonians are identical.
This phenomenon, the so-called exotic quantum
holonomy, in which quantum eigenvalues and eigen-
states do not come back to the original ones af-
ter a cyclic parameter variation [19], belongs to a
wider class of quantum holonomy that comprises both
the celebrated Berry phase [20] and the Wilczek-Zee
holonomy [21] which appears in systems with degen-
erate eigenvalues. The exotic quantum holonomy in
the δ-function potential system was considered in [22].
Here we report a finding of the quantum holonomy in
many-body systems interacting through the δ-function
potential.
The plan of this paper is as follows. In Sec. II, we
derive the spectral equation for Lieb-Liniger model
in two different forms to demonstrate the presence of
quantum holonomies with respect to C. In Sec. III, we
show that the backward cycle is not always possible
due to the clustering of particles. This leads to the
concept of minimal states, which we utilize to classify
the spectrum of the system in Sec. IV. In Sec. V, we
page 1

N.YONEZAWA, A.TANAKA, T.CHEON OCU-PHYS 384, PREPRINT
provide another view of the quantum anholonomy by
focusing on the two-body system through the com-
plexification the coupling strength. Section VI con-
tains our conclusion.
II. ADIABATIC CYCLE CFOR
LIEB-LINIGER MODEL
Let us consider Nbosons confined in a one-
dimensional space. The system is described by the
Hamiltonian
H=−1
2
N
X
j=1
∂2
∂x2
j
+g
N
X
j=1
j−1
X
l=1
δ(xj−xl),(1)
where the unit is chosen such that ~and the mass
of a particle can be set to unity. The parameter g
is the interaction strength. We impose the periodic
boundary condition to the position space. For sim-
plicity, L= 2πis assumed, where Lis the period in
the position space. It is straightforward to extend our
analysis to an arbitrary L, as long as 0 < L < ∞.
We look at the dependence of eigenenergies and
eigenvectors on the coupling strength g. In partic-
ular, we focus on the cycle C, which consists of three
stages C(s)(s= 1,2,3). In the first stage C(1),gis
prepared to be 0 and is adiabatically increased to ∞.
Next, in stage C(2),gis suddenly flipped from ∞to
−∞. In the final stage C(3),gis again adiabatically
increased to 0, which is the initial value of g. We de-
note the initial and final points of Cas g= 0 and
g= 0−, respectively, to distinguish them.
The eigenvalue problem of H, Eq. (1), can be solved
by the Bethe ansatz, where an eigenfunction is com-
posed of Nplane waves specified by a set of quasimo-
menta, also called rapidity kj, which satisfy
exp (i2πkj) = Y
l6=j
kjl + ig
kjl −ig,(2)
where kjl =kj−kl[1]. We examine how kj(g)’s, which
are chosen to be smooth as gis varied, are changed by
the cycle C. The function kj(g) completely character-
izes the parametric evolution of eigenenergies, as well
as the “adiabatic” evolution of eigenvectors along C.
The analysis is decomposed into the three stages C(s)
(s= 1,2,3).
At the initial point g= 0 of the first stage C(1),kj(0)
takes an integer value. Without loss of generality, we
can choose the order of kj(g)’s so as to satisfy k1(g)<
k2(g)<··· < kN(g) for small positive g[23]. This
ensures k1(0) ≤k2(0) ≤ · · · ≤ kN(0).
We introduce two quantized quantities which is con-
served during the parametric evolution of kj(g) along
C(1). Such “topological invariants” provide a way to
evaluate the change of kj(g) induced by stage C(1).
First, during the interval 0 ≤g < ∞, we have an
integer
Ij(g)≡kj(g)−1
πX
j6=l
arctan g
kjl (g).(3)
This is a consequence of Eq. (2) and (t+ i)/(t−i) =
−e−2iarctan t,which is applicable as long as t−16= 0.
We use the principal branch of arctan throughout this
paper. This is justified for Eq. (3) because g/kjl (g)
does not cross its standard branch cuts, which em-
anate from ±ito ±i∞[24]. The continuity and dis-
creteness of Ij(g) in 0 ≤g < ∞imply that Ij(g) takes
a constant value, which can be determined from the
value of kj(g) at the initial point of C, i.e.,
Ij(g) = kj(0).(4)
Second, Eq. (2) and another formula for arctan
(t+ i)/(t−i) = e2i arctan(t−1),which holds for t6= 0,
implies that, in the interval 0 < g ≤ ∞,
Jj(g)≡kj(g) + 1
πX
l6=j
arctan kjl (g)
g(5)
is a half-integer for even Nand an integer for odd
N[1]. Following a similar argument for Ij(g) above,
we obtain the value of the invariant Jj(g) for 0 < g ≤
∞:
Jj(g) = kj(∞).(6)
Now we evaluate the change of kj(g) during C(1)
using these invariants. From Eqs. (3) and (5), we
obtain
kj(∞)−kj(0) = 1
2X
l6=j
sgnℜkjl (g)
g,(7)
where we used the identity
arctan(t) + arctan(1/t) = π
2sgn[ℜ(t)].(8)
page 2

N.YONEZAWA, A.TANAKA, T.CHEON OCU-PHYS 384, PREPRINT
We note that the right-hand side of Eq. (7) makes
sense only for 0 < g < ∞. Here, kjl (g) is positive for
j > l and negative for j < l, since we have assumed
the order of kj(g) at the initial point of C(1), and the
sign of kjl (g) does not change for g > 0 [23]. This
implies
X
l6=j
sgnℜkjl (g)
g=
j−1
X
l=1
−
N
X
l=j+1
.(9)
Accordingly, we obtain
kj(∞)−kj(0) = j−N+ 1
2.(10)
Next we examine the second stage C(2) , where gsud-
denly changes from ∞to −∞. Note that all kj(∞)’s
are finite because of Eq. (10). Since a finite root of
the Bethe equation, Eq. (2), at g=∞is also its root
at g=−∞, we employ a smooth extension of kj(g)
along C(2), i.e.,
kj(−∞) = kj(∞).(11)
Details of the justification of our choice are explained
in Appendix A.
We further extend kj(g)’s for the final stage C(3).
First, we impose that kj(g)’s satisfy Jj(g) = kj(∞)
within the interval −∞ ≤ g < 0. This implies that
kj(g)’s also satisfy Eq. (2). We provide an argument
that such kj(g)’s exist for −∞ ≤ g < 0, and are real-
valued in Appendix A. We accordingly conclude that
Jj(g) is independent of gwithin the interval −∞ ≤
g < 0, because kj(g)’s take real and finite values there.
Second, we examine Ij(g) [Eq. (3)] for −∞ < g ≤0.
In contrast to the analysis of Jj(g) above, we need to
inspect kj(0−), which is the final value of kj(g) in
C(3) and is different from the initial value kj(0). We
carry this out by extending kj(g)’s from the interval
−∞ ≤ g < 0. We explain the details of our argument
in Appendix B and only show the result that Ij(g)
agrees with kj(0−) within the interval −∞ < g ≤0.
The change of kj(g) in the path C(3) is given by
kj(0−)−kj(−∞) = −1
2X
l6=j
sgnℜkjl (g)
g.(12)
We can ensure that
k1(g)< k2(g)<···< kN(g) (13)
because it holds at g=−∞ (see Appendix B). Re-
calling the fact that gis negative here, we obtain
kj(0−)−kj(−∞) = j−N+ 1
2.(14)
Combining above three arguments, we obtain a non-
trivial change of kj(g) due to Cin the form
kj(0−)−kj(0) = 2j−(N+ 1).(15)
Note that the total momentum remains unchanged
during the cycle C. The final energy and state after
the adiabatic cycle, however, are different from the
initial ones, showing that Cinduces the eigenenergy
and eigenspace anholonomies [22]. We also remark
that k1< k2<··· < kNholds at the end of C.
This implies that we can repeat the adiabatic cycle
Carbitrarily, and the repetition of Cwill induce the
further instances of the eigenenergy and eigenspace
anholonomies.
We can summarize our results in terms of a mapping
between two sets of quasimomenta of free bosons, i.e.,
kj(0)’s and kj(0−)’s . It is sufficient to consider the
case that initial condition nj≡kj(0) satisfies n1≤
n2≤ · · · ≤ nN. With the notation n′
j≡kj(0−),
the mapping (n1, n2,...,nN)7→ (n′
1, n′
2,...,n′
N) =
F(n1, n2,...,nN), which is given by
F(n1, n2,...,nN)
= (n1−N+ 1, n2−N+ 3,...,nN+N−1),(16)
expresses the quantum holonomy induced by the cycle
C.
III. INVERSE CYCLE
We now examine the inverse of the cycle C. In con-
trast to the forward cycle C, the parametric variation
along the inverse C−1is not always possible. This is
because the clustering of particles at g=−∞ induces
the divergence of eigenenergy [1]. Such a clustering
invalidates the use of the Hamiltonian, Eq. (1). We
call an eigenstate of free boson at g= 0 a minimal
state if the the parametric variation along C−1is im-
possible. The precise condition for appearance of the
minimal state is the subject of this section.
page 3

N.YONEZAWA, A.TANAKA, T.CHEON OCU-PHYS 384, PREPRINT
Formally, C−1corresponds to the inverse of the
mapping F[Eq. (16)] on the sets of quasimomenta
at g= 0:
F−1(n1, n2,...,nN)
= (n1+N−1, n2+N−3, . . . , nN−N+ 1),(17)
where we impose the ordering condition n1≤n2≤
· · · ≤ nN. When the distance between nj’s are far
enough, F−1preserves the ordering. This is the case
that C−1can be realized, and the resultant energy and
quantum state are the solution of the eigenvalue prob-
lem of H[Eq. (1)] at g= 0. On the other hand, when
a pair of nj’s is too close, F−1breaks the ordering,
which implies the emergence of the clustering of par-
ticles during the inverse cycle. There are two possible
cases. The first case is where a pair of quasimomenta,
say, njand nj+1 , are degenerate, i.e., nj=nj+1 .
By applying F−1, the resultant quasimomenta satisfy
nj> nj+1. In fact, the eigenenergy diverges −∞ as
g→ −∞ during C−1. The second case, nj=nj+1 + 1,
also leads the clustering of particles.
The argument above is sufficient to determine the
condition for the minimal states. When there is, at
least, a pair of two quasimomenta at g= 0 that satis-
fies
|nj−nj+1| ≤ 1,(18)
states specified by njand nj+1 are minimal states.
IV. CLASSIFICATION OF SPECTRA
Because of the existence of quantum holonomy,
some states are reachable by the repetitions of para-
metric cycles Cand C−1starting from one particu-
lar eigenstate, while other states are not. This offers
the classification of whole eigenstates into families of
states connected by quantum holonomy. Such a family
can be specified by a minimal state introduced above,
because an arbitrary eigenstate with a finite energy
can become minimal by a finite repetition of C−1.
From one minimal state, we can find other minimal
states using the symmetries of the Hamiltonian (1).
Suppose that a minimal state is specified by quasi-
momenta (n1, n2,...,nN). The translational symme-
try implies that (n1+ 1, n2+ 1,...,nN+ 1) is also
-2
Hg=±¥L
-1
Hg=-1L
0
Hg=0L
1
Hg=1L
2
Hg=±¥L
x
1
2
3
E
FIG. 1. (Color online) Parametric evolution of eigenener-
gies of the two-body Lieb-Liniger model, where the xand
yaxes indicate (4/π) arctan gand √E, respectively. The
unit is chosen such that ~and the mass of a particle are
set to unity. The period of the position space is chosen to
be 2π. The thick (black) and thin (blue) lines correspond
to the families specified by the minimal states (0,0) and
(0,1), respectively. See Eqs. (21) and (22). Note that the
eigenenergies are continuous at g=±∞.
a minimal state, whose total momentum is larger
by Nthan the original one. For an arbitrary in-
teger ℓ, (n1+ℓ, n2+ℓ,...,nN+ℓ) is also a min-
imal state. The reflection symmetry implies that
(−nN,...,−n2,−n1) is also a minimal state, which
may or may not be different from the original state.
Hence, it is sufficient to find all minimal states
whose total momenta satisfy the condition
−N
2<X
j
nj≤N
2,(19)
to enumerate all minimal states using the translational
symmetry, offering a way to classify the spectra of
the Lieb-Liniger model completely. We illustrate this
classification for few-body cases.
We start the analysis of N= 2 case with two mini-
mal states,
(0,0) and (0,1).(20)
We obtain two families of eigenstates at g= 0 from
these two minimal states, by repeating C,
(0,0) 7→ (−1,1) 7→ (−2,2) 7→ · · · ,(21)
page 4

N.YONEZAWA, A.TANAKA, T.CHEON OCU-PHYS 384, PREPRINT
-2
Hg=±¥L
-1
Hg=-1L
0
Hg=0L
1
Hg=1L
2
Hg=±¥L
x
1
2
3
4
5
6
E
FIG. 2. (Color online) Eigenenergies of N= 3 fami-
lies, where the xand yaxes are the same as in Fig. 1.
The total momentum of all families shown here is zero.
The thick (black) line corresponds to the family [Eq. (24)]
specified by the minimal state (0,0,0). The thin (blue)
line corresponds to the (−1,0,1) family. These two fam-
ilies are trimers. The dotted (red), dashed (brown), and
dash-dotted (gray) lines are dimer families specified by
minimal states (−1,−1,2), (−2,−2,4), and (−3,−3,6),
respectively. Although there are level crossings, the adia-
batic theorem ensures that the adiabatic time evolution is
confined within a family [25]. The choice of the unit is the
same as in Fig. 1.
and
(0,1) 7→ (−1,2) 7→ (−2,3) 7→ · · · ,(22)
respectively. The eigenenergies of these families are
depicted in Fig. 1. By shifting the total momentum
from the two minimal states [Eq. (20)], we obtain an
infinite number of minimal states (ℓ, ℓ) and (ℓ, ℓ + 1)
with an arbitrary integer ℓ. The (ℓ, ℓ)- and (ℓ, ℓ + 1)-
families have the set of quasimomenta at g= 0 given
by {(ℓ−m, ℓ +m)}∞
m=0 and {(ℓ−m, ℓ + 1 + m)}∞
m=0,
respectively. This exhausts the minimal states and
families for N= 2.
The N= 3 case is far more complex than the N= 2
case. First, we consider the case that the total mo-
mentum is zero, where an infinite number of minimal
states can be found. We depict some of them in Fig. 2.
There are two minimal states,
(0,0,0) and (−1,0,1),(23)
-2
Hg=±¥L
-1
Hg=-1L
0
Hg=0L
1
Hg=1L
2
Hg=±¥L
x
1
2
3
4
5
6
7
8
E
FIG. 3. (Color online) Parametric evolution of eigenen-
ergies of the N= 4 case, where the xand yaxes are the
same as in Fig. 1. The thick (black) line corresponds to
the family (0,0,0,0) 7→ (−3,−1,1,3) 7→ (−6,−2,2,6) . . . .
The thin (blue) and dotted (red) lines correspond to
(−1,0,0,1) and (−1,−1,1,1) families, respectively. The
choice of the unit is the same as in Fig. 1.
which are called trimers [26], because the clustering of
all three particles occurs in the limit g→ −∞. The
family of eigenstates at g= 0 specified by the minimal
state (0,0,0) is
(0,0,0) 7→ (−2,0,2) 7→ (−4,0,4) 7→ .... (24)
Besides, there are an infinite number of minimal
states,
{(−ℓ, −ℓ, 2ℓ)}ℓ>0and {(−2ℓ, ℓ, ℓ)}ℓ>0,(25)
where the latter set can be induced through the use
of the reflection symmetry. These minimal states are
called dimers [26], because the clustering of two parti-
cles occurs in the limit g→ −∞. Second, we consider
the case Pjnj= 1. We have a trimer,
(0,0,1),(26)
and an infinite number of dimers
{(−ℓ, −ℓ, 2ℓ+ 1)}ℓ>0,{(−ℓ, −ℓ+ 1,2ℓ)}ℓ>0,
{(−2ℓ+ 1, ℓ, ℓ)}ℓ>0,{(−2ℓ, ℓ, ℓ + 1)}ℓ>0.(27)
Note that all minimal states that satisfy Pjnj=−1
can be obtained from the minimal state with Pjnj=
page 5

N.YONEZAWA, A.TANAKA, T.CHEON OCU-PHYS 384, PREPRINT
1 through the use of the reflection symmetry. We
obtain all other minimal states from above using the
translational and reflection symmetry.
It is possible to enumerate minimal states and as-
sociated spectral families in a similar way for larger
N. We simply close this section by showing several
families of the N= 4 system in Fig. 3.
V. EXCEPTIONAL POINTS
So far we have focused on the quantum holon-
omy induced by the real cycle C. In this sec-
tion, we examine the relationship between the ex-
otic quantum holonomy and non-Hermitian degener-
acy points, which are also known as Kato’s excep-
tional points [27, 28], using the complexification of the
coupling parameter g. When we adiabatically vary g
along a cycle that encloses an exceptional point, the
permutation of eigenenergies as well as eigenspaces
occurs. This resembles the exotic quantum holonomy.
Indeed, in Ref. [29] it is argued that, through an anal-
ysis of a quantum kicked top, the quantum holonomy
has a correspondence with the exceptional points. In
other words, it is conjectured that the eigenenergy and
eigenspace anholonomy can be understood as a result
of the metamorphosis of eigenenergies and eigenstates
induced by the encirclements around the exceptional
points. In the following, we offer another example of
this conjecture using the two-body Lieb-Liniger model
by deforming Cin the complexified gspace.
Due to the complexification of g, the Lieb-Liniger
Hamiltonian (1) becomes non-Hermitian, which de-
scribes a one-dimensional dissipative Bose system [30].
We obtain eigenenergies with complex-valued cou-
pling parameter gthrough numerical computation.
We here focus on the (0,0) family [Eq. (21)]. Let
En(g) denote the eigenenergy of the state whose quasi-
momenta take (−n, n) at g= 0. We depict En(g) for
n= 0,1,2 in Fig 4. We find that these eigenener-
gies compose a Riemann surface. Its Riemann sheets
En(g) are connected by the exceptional points and
associated branch cuts (cf. Ref. [31]).
Under the present choice of the branch cuts, all
exceptional points of the (0,0) family appear in the
E0(g) sheet. A pair of eigenenergies En(g) (n > 0)
-2
-1
0
1
-4
-2
0
ReHgL
ImHgL
HaL
-2
-1
0
1
-4
-2
0
ReHgL
ImHgL
HbL
-2
-1
0
1
-4
-2
0
ReHgL
ImHgL
HcL
FIG. 4. (Color online) Contour plots of ℜEn(c): (a) n= 0;
(b) n= 1; (c) n= 2. Lighter (darker) color indicates
larger (smaller) value of ℜE. Thin lines are the contours
of ℜEn(c). The exceptional points are indicated by solid
circles. Bold lines indicate the branch cuts. While all
complex exceptional points appear in E0(c), each En(c)
(n= 1,2) has a single exceptional point. (d) Schematic
explanation of complex cycles that enclose exceptional
points. We depict Cby a thick (red) line. Dashed (blue)
and dotted (black) curves indicate C1and C2, respectively.
See the main text. The choice of the unit is the same as
in Fig. 1.
and E0(g) has a pair of degenerate points gnand g∗
n,
where we choose ℑgn<0. We find that all degenerate
points are of degree two. Hence, the pair of eigenener-
gies for an exceptional point exhibits square-root-type
singularity. The encirclement around the exceptional
point gnin the complex gplane induces the permu-
tation of E0(g) and En(g). We numerically confirm
these properties of gnwith n= 1,2,...,10. We find
that ℜgnand ℑgndecrease monotonically as nin-
creases. We also obtain similar results for the (0,1)
family.
We note that our numerical finding can be explained
by a perturbation expansion for g=−∞ with a small
parameter g−1, as for the exceptional points that are
far from the real axis [32]. We will explain the details
in a forthcoming publication [33].
page 6

N.YONEZAWA, A.TANAKA, T.CHEON OCU-PHYS 384, PREPRINT
Let us consider the cycle that is a concatenation
of Cand C1in Fig 4 (d). Because this cycle en-
circles the exceptional point g1, the cyclic permuta-
tion (E0, E1) occurs. On the other hand, the cycle
composed of Cand C2induces the cyclic permutation
among (E0, E1, E2). As the cycle involves more deeper
exceptional points, the accuracy of the the resultant
permutation become better to approximate a shift to
eigenenergies (E0, E1,...)7→ (E1, E2,...), which is
realized by the quantum holonomy along the cycle C.
In this sense, we may say that the spectrum of Lieb-
Liniger model feels the exceptional points that reside
in the complex parameter space to induce the quan-
tum holonomy along C.
VI. CONCLUSION
We have shown in this work that an eigenstate of
the free Lieb-Liniger system g= +0 is transformed to
another eigenstate with higher energy in the process of
eigenspace anholonomy involving the parametric cycle
g: +0 →+∞:−∞ → −0. Experimental testing
should be within the range of current techniques [16,
17]. On the way to prove the existence of quantum
holonomy, we have demonstrated that the eigenstates
of the Lieb-Liniger model can be classified according
to their clustering property. The two- and three-boson
systems have been analyzed in detail.
Our result can be interpreted in terms of geometry.
Consisting of real numbers and ±∞, the parameter
space of coupling strength is homeomorphic to S1.
Therefore, our anholonomy is affected by the topol-
ogy of S1. The presence of two kinds of invariants,
Ij(g) [Eq. (3)] and Jj(g) [Eq. (5)], for the parametric
evolution of kj(g) reflects the fact that at least two
charts are required for S1. Converting one of spec-
trum condition to the other by using the formula of
arctan (8) corresponds to coordinate transformation.
The cycle of the winding number m,Cm, increases
kj(0) by m[2j−(N+ 1)].
The topological nature of the the quantum holon-
omy implies that it is stable against, at least, small
perturbations [34]. This also suggests that an experi-
mental realization of the quantum anholonomy is pos-
sible in one-dimensional bosonic systems.
ACKNOWLEDGEMENT
This research was supported by the Japan Ministry
of Education, Culture, Sports, Science and Technol-
ogy under the Grant numbers 22540396 and 24540412.
Appendix A: Extension of kj(g)’s to −∞ ≤ g < 0
We examine kj(g)’s that satisfy
kj(∞) = kj(g) + 1
πX
l6=j
arctan kjl (g)
g(A1)
in the interval −∞ ≤ g < 0 in this appendix. Our
argument consists of two parts. First, we provide an
argument that kj(g)’s are real and finite for −∞ ≤
g < 0. Second, we explain that such kj(g)’s are the
smooth extension of the ones defined in the first stage
C(1).
We have already examined kj(∞), which appears in
the left-hand side of Eq. (A1), in the main text. In
particular, kj(∞)’s are real and finite. Also, kj(∞)’s
are not degenerate, i.e.,
kj(∞)−kl(∞)>0,(A2)
for j > l, which is ensured by Eq. (10).
We introduce an assumption that plays the crucial
role in the following argument. We assume that there
uniquely exists {kj(g)}N
j=1 that satisfies Eq. (A1). We
note that this assumption indeed holds, as for g >
0 [8].
We show that kj(g)’s are real numbers by reductio
ad absurdum. Namely, we suppose that kj(g) is not
real and satisfies Eq. (A1) for a given j. Accordingly,
its complex conjugate kj(g)∗also satisfies Eq. (A1),
because Eq. (A1) is invariant under the complex con-
jugate. Since kj(g)∗is different from kj(g) and there
uniquely exists {kj(g)}N
j=1, there exists j′such that
kj′(g) = kj(g)∗and j′6=j. We compare kj(∞) and
{kj′(∞)}∗, which are real numbers. Using Eq. (A1),
we find kj(∞)− {kj′(∞)}∗= 0, which contradicts
Eq. (A2).
A corollary of the above proposition, i.e., kj(g) are
real for −∞ ≤ g < 0, is the continuity of kj(g)’s in
the stages C(2) as well as C(3), as mentioned in the
main text [see Eq. (11)]. In this sense, kj(g)’s that
page 7

N.YONEZAWA, A.TANAKA, T.CHEON OCU-PHYS 384, PREPRINT
satisfy Eq. (A1) are the smooth extension of kj(g) for
0< g ≤ ∞. We prove this corollary. Since kj(g)’s are
real numbers, we have
|arctan [kjl (g)/g]|< π/2,(A3)
which implies that kj(g)’s are finite, i.e.,
|kj(g)| ≤ |kj(∞)|+1
πX
l6=j
π
2<∞,(A4)
where we use Eq. (A1). Hence, we find
lim
g→−∞ kjl (g)/g = 0.(A5)
Taking the limit of Eq. (A1) as g→ −∞, we obtain
lim
g→−∞ kj(g) = kj(∞) + lim
g→−∞
1
πX
l6=j
arctan kjl (g)
g
=kj(∞).
(A6)
Hence, we conclude that kj(g) is continuous at g=
−∞. A similar argument above tells us that kj(g) is
also continuous at g=∞.
Appendix B: Extension of kj(g)’s from g < 0to
g= 0
We have explained the smooth extension of kj(g)’s
through the flip of gfrom ∞to −∞ in Appendix A.
Here we extend further kj(g)’s from g < 0 to g= 0 to
complete the analysis of the stage C(3) . We carry this
out by showing kj(0−) = Ij(g).
To prepare this, we show that kj(g)6=kj′(g) holds
for g < 0, and an arbitrary pair of (j, j ′). We prove
this by contradiction. Suppose that there exists g(<
0), where kj(g) = kj′(g) (j6=j′). Then Eq. (A1)
implies that kj(∞) = kj′(∞), which is inconsistent
with Eq. (A2).
Next we show that kj(g)6=kj′(g) (j6=j′) also
holds in the limit g→0−. We show this by using re-
ductio ad absurdum. Suppose kj(0−) = kj′(0−). We
can assume j > j ′without loss of generality. Hence,
Eq. (A1) under the limit g→0−implies
kj(∞)−kj′(∞) = lim
g→0−
2
πarctan kjj′(g)
g.(B1)
Thus we conclude kj(∞)−kj′(∞)≤0, since
arctan [kjj′(g)/g]<0 holds as long as g < 0. This
conclusion contradicts with Eq. (A2) with j > j′. We
thus show kj(0−)6=kj′(0−).
We examine Ij(g) [Eq. (3)] in the limit g→0−.
Since kjl (0−)6= 0 holds, as shown above, we find
limg→0−g/kjl (g) = 0. Hence, we obtain Ij(0−) =
kj(0−). Because Ij(g) is independent of gfor −∞ <
g < ∞, we conclude Ij(g) = kj(0−) for g≤0. We
note that this result and Eq. (4) imply continuity of
kj(g) at g= 0±. As for the proof of the continuity at
g= 0+, we refer to Ref. [23].
Finally, we remark that the present argument and
Eq. (A2) imply that the ordering of kj(g) satisfies
k1(g)< k2(g)<···< kN(g) for g≤0.
[1] E. H. Lieb and W. Liniger, Phys. Rev. 130, 1605
(1963).
[2] M. D. Girardeau, J. Math. Phys. 1, 516 (1960).
[3] T. Cheon and T. Shigehara, Phys. Rev. Lett. 82, 2536
(1999).
[4] E. Gutkin, Ann. Phys. (N.Y.) 176, 22 (1987).
[5] A. Kundu, Phys. Rev. Lett. 83, 1275 (1999).
[6] N. Oelkers, M. Batchelor, M. Bortz, and X. Guan, J.
Phys. A: Math. Gen. 39, 1073 (2006).
[7] M. T. Batchelor, X. W. Guan, and A. Kundu, J. Phys.
A: Math. Theor. 41, 352002 (2008).
[8] C. N. Yang and C. P. Yang, J. Math. Phys. 10, 1115
(1969).
[9] C. P. Yang, Phys. Rev. A 2, 154 (1970).
[10] M. Wadati, J. Phys. Soc. Jap. 71, 2657 (2002).
[11] T. Ichikawa, I. Tsutsui, and N. Yonezawa, Phys. Rev.
A86, 015602 (2012).
[12] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).
[13] T. Kinoshita, T. Wenger, and D. S. Weiss, Science
305, 1125 (2004).
[14] B. Paredes1, A. Widera, V. Murg, O. Mandel, S.
F¨olling, I. Cirac, G. V. Shlyapnikov, T. W. H¨ansch
and I. Bloch, Nature 429, 277 (2004).
[15] G. E. Astrakharchik, J. Boronat, J. Casulleras, and
S. Giorgini, Phys. Rev. Lett. 95, 190407 (2005).
[16] E. Haller, M. Gustavsson, M. J. Mark, J. G. Danzl,
page 8

N.YONEZAWA, A.TANAKA, T.CHEON OCU-PHYS 384, PREPRINT
and R. Hart, Science 325 (2009).
[17] E. Haller, M. J. Mark, R. Hart, J. G. Danzl, L. Re-
ichs¨ollner, V. Melezhik, P. Schmelcher, and H.-C.,
N¨agerl, Phys. Rev. Lett. 104, 153203 (2010).
[18] M. T. Batchelor, M. Bortz, X. W. Guan, and N. Oelk-
ers, J. Stat. Mech.: Th. Exp. 2005, L10001 (2005).
[19] T. Cheon and A. Tanaka, Europhys. Lett. 85, 20001
(2009).
[20] M. Berry, Proc. Roy. Soc. A 392, 45 (1984).
[21] F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111
(1984).
[22] T. Cheon, Phys. Lett. A 248, 285 (1998).
[23] T. Dorlas, Comm. Math. Phys. 154, 347 (1993).
[24] See, e.g., F. W. J. Olver, D. W. Lozier, R. F. Boisvert
and C. W. Clark, NIST Handbook of Mathematical
Functions (Cambridge University Press, Cambridge,
2010), Sec. 4.
[25] T. Kato, J. Phys. Soc. Japan 5, 435 (1950).
[26] J. G. Muga and R. F. Snider, Phys. Rev. A 57, 3317
(1998).
[27] T. Kato, Perturbation Theory for Linear Operators,
(Springer-Verlag, Berlin, 1980)
[28] W. D. Heiss, Czech. J. Phys. 54, 1091 (2004).
[29] S. W. Kim, T. Cheon and A. Tanaka, Phys. Lett. A
374, 1958 (2010).
[30] S. D¨urr, J. J. Garc´ıa-Ripoll, N. Syassen, D. M. Bauer,
M. Lettner, J. I. Cirac, and G. Rempe, Phys. Rev. A
79, 023614 (2009).
[31] W. D. Heiss and W.-H. Steeb, J. Math. Phys. 32,
3003 (1991).
[32] A. G. Ushveridze, J. Phys. A. 21, 955 (1988).
[33] A. Tanaka, N. Yonezawa and T. Cheon,
arXiv:1305.1693, to be published in J. Phys.
A.
[34] A. Tanaka and M. Miyamoto, Phys. Rev. Lett. 98,
160407 (2007).
page 9