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effects of an electron - electron interaction in electronic systems have become a focus of the condensed matter physics 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 phenomenological landau parameters .
the system is metallic even with the interaction since the ground state is adiabatically connected to the free fermi sea where the excitations are given by an electron - hole _
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 recently and there are huge number of studies by several techniques as the bosonization and the conformal field theory . in the paper , we are trying to investigate the breakdown of the tl liquid behavior in a simple fermionic system . in a system with periodic potential ,
that is , on a lattice as a model hamiltonian , the allowed kinetic energy is restricted by a finite band width .
therefore the strong electron - electron interaction may bring an opening of the energy gap in electronic systems which is known as a mott transition .
the opening of the energy gap implies a metal - insulator transition which has become a focus again recently in connection to new materials as metal - oxides , organic materials and the high-@xmath0 superconductors .
as far as the conductivity of the electronic system is concerned , the mott transition can be understood as a freezing of the charge degree of freedom .
there can be spin related phenomena in a small energy scale , however , the mott transition is a phenomenon of the order of the electron - electron interaction .
noting this fact , we have focused on a charge degree of freedom and chosen a model hamiltonian of spinless fermions on a lattice    @xmath1    where the interaction can be long range .
( we set @xmath2 in the following . ) when the interaction is non - zero only for the nearest - neighbor ( nn ) sites , it is mapped to the spin @xmath3 antiferromagnetic xxz model by the jordan - wigner transformation . in this case , haldane investigated the model in detail by using the bethe ansatz solution of the xxz model@xcite .
generic ground states of the nn model are the tl liquid without the energy gap except at a half filling where the model has a metal - insulator transition at @xmath4 .
it is identified as an antiferromagnetic ising gap in term of the xxz spin model .    in this paper , we investigate the model , when the interaction is long range , with various filling factors of the fermion numbers . as shown later , it has a rich structure as a _
let us first consider a strong coupling limit @xmath5 of the model when the filling factor @xmath6 is rational where @xmath7 is the number of fermions and @xmath8 is the number of sites .
we use a periodic boundary condition in the following .
let us assume the interaction satisfies the downward convex condition , that is , @xmath9 ( @xmath10 ) to avoid a formation of charge clustering .
( see later . )
a possible form of the interaction which we use in the paper is @xmath11which reduced to a simple power law decaying interaction @xmath12 when @xmath13 in a sufficiently large system .
the nearest neighbor interaction can be recovered also by taking @xmath14 and @xmath15 . when the interaction is sufficiently large , the ground state of the system for the rational filling @xmath16 with mutually prime integers @xmath17 and @xmath18 was known for any @xmath17 and @xmath18 .
it is given by a one dimensional wigner crystal with period @xmath18@xcite .
although the ground state charge ordering ( crystal structure ) is complicated if @xmath17 and @xmath18 are large integers , it is explicitly given@xcite . in fig.[fig : chrg ] , shape of the charge ordering is shown for @xmath19 with @xmath20 and @xmath21 as an example . as is expected for the formation 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@xcite .
it is written as @xmath22 f_{[\frac k\rho]-1}^\infty -   ( [ \frac k\rho]-1 )   f_{[\frac k\rho]}^\infty { \rm \ if\ }   [ \frac k\rho ] = \frac k\rho   \\   ( [ \frac k\rho]+1 )   f_{[\frac k\rho]}^\infty -   [ \frac k\rho ]   f_{[\frac k\rho]+1 } ^\infty { \rm \ otherwise }   \end{array}\right .
\nonumber   \\
e_-(k ) & = &    -[\frac k\rho ] f_{[\frac k\rho]+1}^\infty +   ( [ \frac k\rho]+1 )   f_{[\frac k\rho ] } ^\infty \nonumber   % \label{eq : chmpm}\end{aligned}\]]where @xmath23 $ ] is a gauss symbol which denotes a least integer which is not larger that @xmath24 and @xmath25 is an order @xmath26 contribution mainly from the kinetic energy which is a smooth function of @xmath27 . in figs .
[ fig : chem ] , the chemical potentials are evaluated for two different interactions .
it is a devil s staircase which was first discussed by bak et al . in a context of the long range ising model@xcite .
the discontinuity of the chemical potential , @xmath28 , which is a key quantity to judge whether the system is metallic ( @xmath29 ) or not ( @xmath30 ) is evaluated as @xmath31this is generically non - negative for the potential which satisfies the convex downward condition .
( if this condition is not satisfied , @xmath32 can be negative for some filling factor which causes an instability .
it is identified as a charge clustering . )
it implies that the energy gap opens for any rational fillings if the interaction is strong enough .
further , as can be seen from eq .
( [ eq : gap ] ) , the energy gap @xmath33 only depends on @xmath18 .
its dependence is given by a power law @xmath34 .
from the consideration in the strong coupling limit , to have a mott insulator phase in the system with filling factor @xmath19 , finite interaction range over @xmath18 sites is crucial .
on the other hand , when the interaction is weak ( @xmath35 ) , the finite band width due to the lattice effect ( periodic potential ) is not so important . in this case
, one can approximate the system as a continuous model with a long range @xmath36 interaction and with the periodic boundary condition@xcite .
when @xmath37 , the continuum model is a sutherland model which has been studied extensively@xcite . if @xmath37 , the weak coupling model can be discussed using an information from the exact solutions . however , the strong coupling case , of course , neither the intermediate coupling case can not be discussed by the exact solutions . the ground state of the sutherland model is a tl liquid without an energy gap independent of the filling where the ground state is given by the jastraw wavefunction .
noting that there is an energy gap in the strong coupling , it suggests that there is a finite value of the interaction where the energy gap opens .
one possibility is that there is alway non - zero energy gap , that is , the critical value where the energy gap opens , @xmath38 .
however , 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 following ,
we give numerical results to confirm these consideration and study the critical region .
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 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 investigate 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 divided 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 @xmath39 which can be comparable with @xmath32 .    in fig .
[ fig : disp1o2 ] and fig .
[ fig : disp1o3 ] , lower parts of energy spectra are shown for systems with @xmath40 and @xmath41 , respectively
. when the interaction is sufficiently weak , one observes a behavior of the gapless tl liquid as shown in fig .
[ fig : disp1o2](a ) and fig .
[ fig : disp1o3](a ) . on the other hand , opening of the energy gap near @xmath42 , @xmath43 ( @xmath44 , @xmath45 for @xmath19 )
is clearly shown when the interaction is sufficiently strong ( see fig .
[ fig : disp1o2](b ) and fig .
[ fig : disp1o3](b ) ) .
as is known from the strong coupling analysis , there is a discrete symmetry breaking ( translational symmetry ) in the strong coupling phase . for the @xmath19 case ,
this is a @xmath46 symmetry breaking .
correspondingly there is almost @xmath18 degenerate ( ground ) states in a finite system .
they have different total momentum @xmath47 ( @xmath48 ) respectively . at these momentum sectors ,
the lowest energy state is given by one of the @xmath18 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 @xmath49 ) , is related to an energy barrier between the degenerate @xmath18 ground states barrier as shown in eq .
( [ eq : gap - exp ] ) . the physical energy gap which we concern is the second lowest one as seen in the figures . in the fig .
[ fig : soft1o3 ] , the lowest energy gap at @xmath50 for @xmath41 is plotted as a function of the system size for several values of the interaction strength . for @xmath51 ,
the gap size obeys a exponential law as @xmath52 which is a signature of the ( discrete ) symmetry breaking and @xmath53 is the order of the symmetry breaking potential .
the discrete symmetry breaking is confirmed numerically for @xmath51 in fig .  [
fig : soft1o3 ] . 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)@xcite . in fig .
[ fig : ab ] , the spectral flows of three different momentum sectors ( @xmath54 and @xmath55 ) for the @xmath41 system is shown where the @xmath56 symmetry breaking is expected .
it is clearly shown that the @xmath57 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 .
next let us investigate the critical region .
for a nearest neighbor model at half filling , the behavior of the gap opening is known from the bethe ansatz solution and is given by @xmath58 which is essentially singular at the critical point@xcite .
we have numerically investigate the behavior of the gap opening for several filing factors in our long range interaction model . in fig .
[ fig : gap1o2 ] ,  [ fig : gap1o3 ] , and  [ fig : gap1o4 ] , the excitation gap is plotted as a function of the interaction strength .    in the massless
tl liquid phase and the mott insulator phase with @xmath59 symmetry breaking , @xmath60 converges to the true energy gap in the thermodynamic limit .
therefore we extrapolate @xmath60 , the energy gap between the lowest energy at @xmath49 and @xmath61 to the infinite size limit by fitting them as a polynomial of the @xmath62 .
an example of the fitting is shown in fig .
[ fig : fit ] for @xmath41 case with several values of the interaction strength .
the numerical results shown in fig .
[ fig : gap1o2 ] ,  [ fig : gap1o3 ] , and  [ fig : gap1o4 ] can be well fitted by the following essentially singular form as a function of the interaction strength @xmath63it implies that the transition is a generalized kosterlitz - thouless ( kt ) type .
therefore the transition is of infinite order as a quantum phase transition .
this kind of the singular behavior is expected in the conformal field theory for the off - critical behavior near the conformally invariant critical phase . in the @xmath64 invariant case ,
the exponent @xmath65 is evaluated by the renormalization group analysis to be @xmath66@xcite . at the critical point of the model
has an apparent @xmath46 symmetry but may have higher symmetry .
then one of the possible guesses for the exponent is @xmath67 .
numerically it is difficult to determine @xmath68 and @xmath69 with sufficient accuracies due to the singular behavior of the energy gap .
however one may use @xmath70 as a lower bound of the @xmath71 since it is plausible to expect the wigner crystal at the strong coupling melts near @xmath72 .
this estimate agrees rather well with numerical results obtained ( figs .
[ fig : gap1o2 ] ,  [ fig : gap1o3 ] , and  [ fig : gap1o4 ] ) . in figs .
[ fig : phase ] , we have shown possible ground state phase diagram using the above estimate . in the figure , the interaction strength is plotted in a nonlinear scale .
it shows that there is alway insulator phase for any rational filling .
since the @xmath73 curve is the devil s staircase , @xmath74 shows a fractal structure .
correspondingly , the ground state phase diagram of the spinless fermions with long range interaction has a clear _
fractal structure_@xcite .
the author thanks y. morita , m. kohmoto , m. takahshi , k. maki and d. lidsky for helpful discussions .
this work was supported in part by grant - in - aid from the ministry of education , science and culture of japan .
spin model with inverse square exchange with anisotropy also has a similar fractal structure@xcite by the similar argument given in the manuscript . since _
only a part _ of the eigenstates is obtained analytically@xcite , the fractal nature of the system is not captured by the analytical solutions@xcite .
the metal - insulator transition in one dimensional fermionic systems with long - range interaction is investigated . 
 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 . 
 non - analytic 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 . 
 fractal nature of the ground state phase diagram is also revealed .    1.0 cm