Gap-opening transition and fractal ground-state phase diagram in one-dimensional fermions
with long-range interaction: Mott transition as a quantum phase transition of infinite order
Yasuhiro Hatsugai
*
Department of Applied Physics, University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo 113, Japan
~Received 20 January 1997; revised manuscript received 29 May 1997!
The metal-insulator transition in one-dimensional fermionic systems with long-range interaction is investi-
gated. We have focused on an excitation spectrum by the exact diagonalization technique in sectors with
different momentum quantum numbers. At rational fillings, we have demonstrated gap opening transitions
from the Tomonaga-Luttinger liquid to the Mott insulator associated with a discrete symmetry breaking by
changing the interaction strength. Finite interaction range is crucial to have the Mott transition at a rational
filling away from the half filling. It is consistent with the strong coupling picture where the Mott gap exists at
any rational fillings with sufficiently strong interaction. The critical regions as a quantum phase transition are
also investigated numerically. Nonanalytic behavior of the Mott gap is the characteristic in the weak coupling.
It is of the order of the interaction in the strong coupling. It implies that the metal-insulator transition of the
model is of the infinite order as a quantum phase transition at zero temperature. The fractal nature of the
ground-state phase diagram is also revealed. @S0163-1829~97!04143-X#
I. INTRODUCTION
Effects of an electron-electron interaction in electronic
systems have become a focus of the condensed-matter phys-
ics recently. In a three or higher dimensions, it is widely
believed that Landau’s fermi-liquid theory is valid and the
effect of the interaction is absorbed into several phenomeno-
logical Landau parameters. The system is metallic even with
the interaction if the ground state is adiabatically connected
to the free Fermi sea where the excitations are given by an
electron-hole gapless excitation across the Fermi surface. In
one dimension, however, the ground state is unstable against
a perturbation of the interaction and the ground state is given
by the so-called Tomonaga-Luttinger ~TL! liquid. Although
the Fermi-liquid theory is not valid in one dimension, the TL
liquid is also metallic and the excitations are gapless. This
TL liquid has been focused on again recently and there are a
huge number of studies by several techniques such as the
bosonization and the conformal field theory.
1–7
In the paper,
we are trying to investigate the breakdown of the TL liquid
behavior in a simple fermionic system.
5,4,12
In a system with
periodic potential, that is, on a lattice as a model Hamil-
tonian, the allowed kinetic energy is restricted by a finite
bandwidth. Therefore the strong electron-electron interaction
may bring an opening of the energy gap in electronic sys-
tems which is known as a Mott transition. The opening of the
energy gap implies a metal-insulator transition which has
become a focus again in connection to new materials as
metal-oxides, organic materials, and the high-T
C
supercon-
ductors.
As far as the conductivity of the electronic system is con-
cerned, the Mott transition can be understood as a freezing of
the charge degree of freedom. There can be spin related phe-
nomena in a small energy scale, however, the Mott transition
is a phenomenon of the order of the electron-electron inter-
action. Noting this fact, we have focused on a charge degree
of freedom and chosen a model Hamiltonian of spinless fer-
mions on a lattice,
H52t
(
j
c
j11
†
c
j
1H.c.1
(
iÞj
V
u
i2 j
u
n
i
n
j
~
V
k
>0
!
,
~1!
where the interaction can be long range. ~We set t5 1 in the
following.! When the interaction is nonzero only for the
nearest-neighbor ~NN! sites, it is mapped to the spin-1/2 an-
tiferromagnetic XXZ model by the Jordan-Wigner transfor-
mation. In this case, Haldane investigated the model in detail
by using the Bethe Ansatz solution of the XXZ model.
3
Ge-
neric ground states of the NN model are the TL liquid with-
out the energy gap except at a half filling where the model
has a metal-insulator transition at V
1
5 t. It is identified as an
antiferromagnetic Ising gap in terms of the XXZ spin model.
In this paper, we investigate the model, when the interac-
tion is long range, with various filling factors of the fermion
numbers. As shown later, it has a rich structure as a fractal.
II. STRONG COUPLING
Let us first consider a strong coupling limit V@ t of the
model when the filling factor
r
5 M/L is rational, where M is
the number of fermions, and L is the number of sites. We use
a periodic boundary condition in the following. Let us as-
sume the interaction satisfies the downward convex condi-
tion, that is, (i1 j)V
l
< jV
l2i
1iV
l1j
~l,j, l,i,j!L! to
avoid a formation of charge clustering. ~See later.! A pos-
sible form of the interaction which we use in the paper is
V
j
5 Vf
j
L
~
a
!
, ~2!
f
j
L
~
a
!
5
1
S
L
p
sin
p
j
L
D
a
~
a
>1
!
, ~3!
which reduced to a simple power law decaying interaction
V
j
5 V/j
a
when j! L in a sufficiently large system. The
nearest-neighbor interaction can be recovered also by taking
PHYSICAL REVIEW B 15 NOVEMBER 1997-IVOLUME 56, NUMBER 19
56
0163-1829/97/56~19!/12183~7!/$10.00 12 183 © 1997 The American Physical Society
a
→` and L→ `. When the interaction is sufficiently large,
the ground state of the system for the rational filling
r
5 p/q with mutually prime integers p and q was known for
any p and q. It is given by a one-dimensional Wigner crystal
with period q.
8
Although the ground-state charge ordering
~crystal structure! is complicated if p and q are large inte-
gers, it is explicitly given.
8
In Fig. 1, the shape of the charge
ordering is shown for
r
5 p/q with q5 113 and p
5 1,... ,q21 as an example. As is expected for the forma-
tion of the charge ordering, there are some commensurate
conditions to stabilize the system. This commensurability
condition brings a fractal structure into the system as shown
later. The chemical potential in the thermodynamic limit is
evaluated by a similar consideration applied for the long-
range Ising model.
9
It is written as
m
`
~
r
1 0
!
5 V
(
k5 1
`
e
1
~
k,
r
!
1 F
~
r
!
, ~4!
m
`
~
r
2 0
!
5 V
(
k5 1
`
e
2
~
k,
r
!
1 F
~
r
!
, ~5!
e
1
~
k
!
5
H
F
k
r
G
f
@
k/
r
#
2 1
`
2
S
F
k
r
G
2 1
D
f
@
k/
r
#
`
if
F
k
r
G
5
k
r
S
F
k
r
G
1 1
D
f
@
k/
r
#
`
2
F
k
r
G
f
@
k/
r
#
1 1
`
otherwise,
e
2
~
k
!
52
F
k
r
G
f
@
k/
r
#
11
`
1
S
F
k
r
G
11
D
f
@
k/
r
#
`
,
where
@
x
#
is a Gauss symbol which denotes a least integer
which is not larger that x and F(
r
) is an order t contribution
mainly from the kinetic energy which is a smooth function of
r
. In Fig. 2, the chemical potentials are evaluated for two
different interactions. It is a devil’s staircase which was first
discussed by Bak and Bruinsma in a context of the long-
range Ising model.
9
The discontinuity of the chemical poten-
tial, D
m
, which is a key quantity to judge whether the system
is metallic (D
m
5 0) or not (D
m
. 0) is evaluated as
D
m
`
S
r
5
p
q
D
5
m
`
~
r
1 0
!
2
m
`
~
r
2 0
!
5 Vg
~
r
!
,
g
S
r
5
p
q
D
5
(
s51
`
sq
~
f
sq11
`
1f
sq21
`
22 f
sq
`
!
. ~6!
This is generically non-negative for the potential which sat-
isfies the convex downward condition. ~If this condition is
not satisfied, D
m
can be negative for some filling factor
FIG. 1. Ground-state charge ordering of the one-dimensional
spinless fermions with long-range interaction in the strong cou-
pling. The black points are the positions of the particles.
r
5 p/q
with q5 113 and p5 1,... ,q21. The horizontal axis is the spacial
direction j, j5 1,... ,q21.
FIG. 2. Evaluation of the chemical potential in the strong cou-
pling limit. The smooth part F(
r
) is set to be zero for simplicity. ~a!
a
5 2, ~b!
a
5 1.5.
12 184 56
YASUHIRO HATSUGAI
which causes an instability. It is identified as a charge clus-
tering.! It implies that the energy gap opens for any rational
fillings if the interaction is strong enough. Further, as can be
seen from Eq. ~6!, the energy gap D
m
(p/q) only depends on
q. Its dependence is given by a power law D
m
`
@
r
5 (p/q)
#
;(const/q
b
). From the consideration in the strong
coupling limit, to have a Mott insulator phase in the system
with filling factor
r
5 p/q, a finite interaction range over q
sites is crucial.
III. WEAK COUPLING
On the other hand, when the interaction is weak (V! t),
the finite bandwidth due to the lattice effect ~periodic poten-
tial! is not so important. In this case, one can approximate
the system as a continuous model with a long range g/r
a
interaction and with the periodic boundary condition.
10
When
a
5 2, the continuum model is a Sutherland model
which has been studied extensively.
11,13
If
a
5 2, the weak
coupling model can be discussed using the information from
the exact solutions. However, in the strong coupling case, of
course, neither of the intermediate coupling cases can be
discussed by the exact solutions. The ground state of the
Sutherland model is a TL liquid without an energy gap inde-
pendent of the filling where the ground state is given by the
Jastraw wave function.
Noting that there is an energy gap in the strong coupling
at a rational filling
r
, it suggests that there is a finite value of
the interaction where the energy gap opens. One possibility
is that there is always a nonzero energy gap, that is, the
critical value where the energy gap opens, V
C
(
r
)5 0. How-
ever, even in the nearest-neighbor model where the energy
gap can be most stable, there is a gapless TL phase ~XY
phase in the XXZ model!. Therefore we can expect that there
are always finite regions of the gapless TL liquid phases in
any filling factors and inevitably the Mott transition.
In the weak coupling, the opening of the energy gap
causes an instability of the TL liquid which can be described
by the Umklapp operators generated in the higher order in
perturbation theory.
5,4,7
By this perturbational consideration,
to have the opening of the energy gap at a higher commen-
surability, strong repulsion is required. For models with short
range interaction ~the Hubbard model, etc.!, the condition for
the instability cannot be satisfied since the so-called TL pa-
rameter K can have the restricted value. It implies that the
finite range interaction is important to have the instability.
This is also consistent with the strong coupling picture.
In the next section, we give numerical results to confirm
these considerations ~strong coupling and weak coupling!
and special efforts are focused on the critical region ~inter-
mediate coupling! where the other technique cannot be ap-
plied.
IV. NUMERICAL RESULTS AND DISCUSSIONS
As we have discussed, the ground state of the spinless
fermions with long-range interaction has two phases for any
rational filling when the interaction strength is varied. The
FIG. 3. Low energy spectra of the spinless fermions with long-range interaction classified by the momentum for
r
5 1/2. (
a
5 2). The
different symbols are for different system sizes; n: L510, L: L514, h: L5 18, s: L5 22. ~a! V5 1. The thin lines are a guide for the
eyes. ~b! V54.
56
12 185GAP-OPENING TRANSITION AND FRACTAL GROUND-...
one is the TL liquid metallic phase and the other is the Mott
insulator phase. The transition between the two is a typical
quantum phase transition at zero temperature. In this section,
these phase transitions are demonstrated numerically. The
main focus of the numerical calculations here is to investi-
gate a critical behavior near the gap opening ~the transition
point!. Our main strategy is to investigate the system from an
insulator side.
We use the exact diagonalization technique for systems
with a periodic boundary condition. The Hilbert space is di-
vided into several sectors with different momentum quantum
numbers and diagonalized within them to obtain the lowest
few energies. For small systems, the full spectra are also
obtained. Due to the small system size available, it is not
efficient to calculate the chemical potential directly. Instead
of it, we calculate an excitation gap E
ex
which can be com-
parable with D
m
.
In Fig. 3 and Fig. 4, lower parts of energy spectra are
shown for systems with
r
5 1/2 and
r
5 1/3, respectively.
When the interaction is sufficiently weak, one observes a
behavior of the gapless TL liquid as shown in Fig. 3~a! and
Fig. 4~a!. On the other hand, opening of the energy gap near
k52nk
F
, n50,61,62,... ~52
p
m/q mod q, m
5 1,2,... ,q for
r
5 p/q! is clearly shown when the inter-
action is sufficiently strong @see Fig. 3~b! and Fig. 4~b!#.As
is known from the strong coupling analysis, there is a dis-
crete symmetry breaking ~translational symmetry! in the
strong coupling phase. For the
r
5 p/q case, this is a Z
q
symmetry breaking. Correspondingly there is almost q de-
generate ~ground! states in a finite system. They have differ-
ent total momentum 2nk
F
(n51,... ,q mod q), respec-
tively. At these momentum sectors, the lowest energy state is
given by one of the q degenerate ground states. Therefore the
lowest energy gap, the energy difference between the lowest
energy state at the momentum and the true ground state of
the finite system ~usually k5 0!, is related to an energy bar-
rier between the degenerate q ground-states barrier as shown
in Eq. ~7!. The physical energy gap that we are concerned
with is the second lowest one as seen in the figures. In the
Fig. 5, the lowest energy gap at k5 2k
F
for
r
5 1/3 is plotted
as a function of the system size for several values of the
interaction strength. For V5 32, the gap size obeys an expo-
nential law as
E
ex
~
k5 2k
F
,L
!
;e
2 cL
, ~7!
which is a signature of the ~discrete! symmetry breaking and
c is the order of the symmetry breaking potential. The dis-
crete symmetry breaking is confirmed numerically for V532
in Fig. 5. To confirm the discrete symmetry breaking, we
have also calculated a spectral flow, that is, the energy as a
function of the Aharonov-Bohm flux through the periodical
system ~ring!.
14
In Fig. 6, the spectral flows of three different
momentum sectors ~k5 0, 2
p
/3, and 4
p
/3! for the
r
5 1/3
system are shown where the Z
3
symmetry breaking is ex-
pected. It is clearly shown that the three low energy states are
entangled with each other. That is, these three states are
equivalent in the thermodynamic limit which is a signature
of the discrete symmetry breaking.
FIG. 4. Low energy spectrum of the spinless fermions with long-range interaction classified by the momentum for
r
5 1/3. ~
a
5 2, t
5 1.! The different symbols are for different system sizes; n: L5 9, L: L5 15, h: L5 21, s: L5 27. ~a! V5 4. The thin lines are a guide
for the eyes. ~b! V532.
12 186 56
YASUHIRO HATSUGAI
FIG. 5. The energy gap at k5 2k
F
is shown as a function of the
system sizes for
r
5 1/3 in log scale. (
a
5 2). The different lines are
for different values of the interaction strength; V5 4, 8, 16, and 32
from above.
FIG. 6. The spectral flows ~energies as a function of the
Aharonov-Bohm flux F through the ring system! of the three lowest
energy states with momentum k5 0, 2
p
/3, and 4
p
/3 for the system
with the filling factor
r
5 1/3 (L5 21). ~
a
5 2 and V532!. F
0
is the
flux quantum. When F5 LF
0
5 21F
0
, the system returns to the
original state by the ~small! gauge invariance.
FIG. 7. The excitation gap E
ex
as a function of V for
r
5 1/2.
(
a
5 2). The data are taken by extrapolating to the infinite size.
FIG. 8. The excitation gap E
ex
as a function of V for
r
5 1/3.
(
a
5 2). The data are taken by extrapolating to the infinite size.
56
12 187GAP-OPENING TRANSITION AND FRACTAL GROUND-...
