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the electron crystal has created considerable interest since its possible existence was first pointed out by wigner  @xcite .
the three - dimensional wigner crystal and its two - dimensional counterpart have been extensively studied , and there exist beautiful experimental realizations of the latter using electrons trapped on the surface of liquid helium  @xcite .
more recently , wigner crystallization in one dimension has received renewed interest  @xcite ; for recent reviews see refs .   and .
the realization of a one - dimensional system requires the dominance of the confining potential over internal energies , in particular , the inter - particle interactions . upon increasing density ( and , thus ,
the interaction energy ) , or weakening the confining potential , the crystal deviates from its strictly one - dimensional structure .
it has been shown that at a critical density , a transition to a zigzag crystal takes place  @xcite . though not for electrons , this zigzag transition has indeed been observed using @xmath0mg@xmath1 ions in a quadrupole storage ring  @xcite .    here
we investigate the structural properties of the classical quasi - one - dimensional wigner crystal beyond the zigzag regime . while previous investigations  @xcite concentrated on regular structures , we are interested in the formation of defects . from symmetry considerations
the assumption of regular crystals is plausible at low densities when the number of rows is small , however , its validity is not at all obvious once the lateral extent of the crystal increases at higher densities .
in fact , one expects a nonuniform charge density in the direction transverse to the wire axis .
in particular , considering the electrostatics problem of charges confined by a parabolic potential , @xmath2 , the density profile should obey @xmath3 , where @xmath4 is the width of the system  @xcite .
therefore , the assumption of perfect rows with equal linear densities should eventually break down .
the formation of defects is of particular interest because they will have a direct impact on the transport properties of the system : while regular rows are locked , defects are expected to be mobile .
we consider classical particles in two dimensions interacting via long - range coulomb interaction .
the system is assumed to be infinite in the @xmath5-direction and confined in the transverse @xmath6-direction by a parabolic confining potential @xmath7 .
the energy of the system then reads @xmath8 where @xmath9 is the dielectric constant of the material and @xmath10 is the frequency of harmonic oscillations in the confining potential .    at low densities ,
the system is one - dimensional , and the particles minimize their mutual coulomb repulsion by occupying equidistant positions along the wire , forming a structure with short - range crystalline order  the so - called one - dimensional wigner crystal  @xcite . upon increasing the density , the inter - electron distance diminishes , and the resulting stronger electron repulsion eventually overcomes the confining potential @xmath7 , transforming the classical one - dimensional wigner crystal into a staggered ( zigzag ) chain . from the comparison of the coulomb interaction energy @xmath11 with the confining potential an important characteristic length scale emerges .
indeed , the transition from the one - dimensional wigner crystal to the zigzag chain is expected to take place when distances between electrons are of the order of the scale @xmath12 such that @xmath13 . within our model , _
i.e. _ , for a parabolic confining potential and coulomb interactions , the characteristic length scale @xmath12 is given as @xmath14 it is convenient for the following discussion to measure lengths in units of @xmath12 . to that purpose
we introduce a dimensionless density @xmath15 where @xmath16 is the linear density of the system .
rescaling lengths , the energy can be written as @xmath17,\end{aligned}\ ] ] where @xmath18 .
as a first step , we minimize the energy assuming regular rows , aiming to find approximate values for the density range in which a configuration with a given number of rows is stable . assuming staggering in the @xmath5-direction between neighboring rows and inversion symmetry of the @xmath6-positions of the rows with respect to the wire axis , the number of minimization parameters is @xmath19 ( @xmath20 ) for even ( odd ) number of rows @xmath21 , and the minimization is straightforward . within these constraints ,
the minimization of the energy with respect to the electron configuration reveals  @xcite that a one - dimensional crystal is stable for densities @xmath22 , whereas a zigzag chain forms at intermediate densities @xmath23 .
more rows appear as the density further increases .
the number of rows as a function of @xmath24 is shown in table [ table_1 ] .
.[table_1]number of rows in the crystal as a function of the dimensionless density @xmath24 , assuming regular structures . [ cols="^,^",options="header " , ]
eventually , one expects that more complicated structures will appear .
the simple configuration we studied is in competition with structures where defects appear away from the edges , such as structures of the type @xmath25 $ ] , for example .
detailed calculations within these simplified models reveal that such structures are indeed competitors for the ground state , but up to 13 rows such a minimum is not realized .
to approach the transition between regular and defected rows , we use a different trick . an unbiased search for the global minimum
is numerically costly because a simple minimization may get stuck in a metastable minimum .
however , if the initial guess of the electron configuration is sufficiently close to the global minimum , a simple minimization will converge .
having identified the structure of defects in region i , one may feed such structures into a simple minimization at lower densities .
the results of such a procedure have been included in figs .
[ fig - nu_def ] and [ fig - y ] .
there , structures with defect densities down to @xmath26 were obtained with the full minimization , whereas structures close to the phase boundary with lower defect structures were obtained with the method described here .
the results suggest that the defect density indeed vanishes at the transition which points to a second order phase transition .
we study quasi - one - dimensional systems of classical particles interacting via long - range coulomb interactions and confined by a parabolic potential in the transverse direction .
the ground state configurations are multi - row wigner crystals where the number of rows is controlled by the density ( or the strength of the confining potential ) .
we find that defects that accommodate the density variation in the transverse direction appear once the number of rows exceeds four .    defected structures have less particles in the outer than in the inner rows .
the full numerical minimization for five rows reveals that two distinct types of defected phases exist . upon increasing density ,
the regular structure at low - densities is replaced by a structure with symmetric defects , _
i.e. _ , where the center of the defect on the two outer rows is located at the same @xmath5-position . as the number of particles that can be simulated sets a lower limit on the defect density that can obtained , the full minimization allows one only to provide an upper limit for the density at the transition from regular to defected structures .
we extend our calculations to lower densities by using structures with the type of defect described above as the input for a simple minimization .
the results indicate that the defect density vanishes at @xmath27 and that the transition is of second order . to obtain symmetric defects ,
the longitudinal distortion of the inner rows , namely an increased density at the center of the defect , is crucial . any analytical description of the transition would have to take into account this distortion .    upon further increasing density
, the defect density rapidly increases . at a critical density , @xmath28 , structures with a different type of defect corresponding to a sinusoidal distortion of the rows with a phase shift of half a period between the two outer rows
become become the ground state .
this second regime is characterized by a defect density that barely varies with density and extends up to the transition to six rows .
the transition between the two defected phases with different symmetries is first order .
simplified models neglecting the corrugation of the rows only capture this second defected phase .
thus , these models do not allow one to further investigate the nature of the phase transition from regular to defected structures .
however , as this second phase occupies most of the density interval , they may be used to study the stability of regular structures upon increasing the number of rows .
we find that beyond nine rows , no stable regular structures exist .
taking into account that the simplified model overestimates the energy of defected structures , we expect that stable regular structures may disappear even earlier .
we would like to acknowledge stimulating discussions with k.  a. matveev , yu .
v.  nazarov , and a.  melikyan .
part of the calculations were performed at the ohio supercomputer center thanks to a grant of computing time .
this work was supported by the u.s .
department of energy , office of science , under contract no .
de - fg02 - 07er46424 .
the method we use essentially follows the steps outlined in ref .  .
it is based on the poisson summation formula relating summations over direct and reciprocal space , @xmath29 where the fourier transform of @xmath30 is defined as @xmath31 let us consider the function @xmath32 . by completing the square and carrying out the fourier transform integration , we obtain the fundamental equation @xmath33 where the reciprocal lattice vectors are given by @xmath34 with @xmath35 . the following definition of the incomplete @xmath36 function is extensively used and therefore given here for reference : @xmath37 the system we are considering contains a basic cell of length @xmath38 with @xmath39 electrons .
the spatial extent in the @xmath6-direction is limited by the confining potential . in the @xmath5-direction
, we impose periodic replications of the basic cell to avoid edge effects .
the interaction energy per cell of the system can be written @xmath40=\frac{1}{2}\sum_{i\ne j } \sum_{n}\frac{q_iq_j}{|{{\bf r}}_{ij}+nl\hat{{\bf x}}| } + \frac{1}{2}\sum_{j}q_j^2\sum_{n\ne 0}\frac{1}{|nl\hat{\bf x}|},\ ] ] where @xmath41 is the charge of particle @xmath42 , @xmath43 , and the index @xmath44 runs over replicas of the unit cell .
the artificial separation of the terms is for our convenience .
we then introduce the notation @xmath45 for @xmath46 and @xmath47 .
using eq .
( [ gam ] ) , we obtain the following representation , @xmath48 here we will introduce an artificial separation constant @xmath49 which will control the splitting of the summation between direct and reciprocal space .
we then have @xmath50 , where @xmath51 and @xmath52 to evaluate @xmath53 , we use eq .
( [ eq - poisson ] ) yielding @xmath54 while for @xmath55 the integration yields incomplete bessel functions  , @xmath56 the @xmath57 term ( denoted @xmath58 in the following ) is divergent and has to be treated separately . using the substitution @xmath59 and expanding the second exponential , one finds @xmath60 the divergent contribution comes from @xmath61 , namely @xmath62 the rest of the sum can be evaluated to @xmath63 thus , @xmath64 and @xmath65 splitting up @xmath66 in the same way , we find @xmath67 and @xmath68    at this stage we put everything together , @xmath69=\frac12\sum_{i\neq
j}q_iq_j\phi({\bf r}_{ij})+\frac12q_j^2\phi_0 $ ] , and combining various terms we obtain the result for the interaction energy per cell of the system ,    @xmath70&= & \frac{1}{2}\sum_{i , j}q_iq_j \left\{{\sum_n}'\frac{\mathrm{erfc}(\alpha|{\bf r}_{ij}+nl\hat{\bf x}|)}{|{\bf r}_{ij}+nl\hat{\bf x}| } + \frac{1}{l}\sum_{g\ne 0}e^{igx_{ij}}k_0\left(\frac{g^2}{4\alpha^2},\alpha^2y_{ij}^2\right)\right\}\\ & & -\frac{1}{2l}\sum_{i\ne j}q_iq_j\left[\gamma+\ln{\alpha^2y^2_{ij}}+\gamma(0,\alpha^2y^2_{ij})\right ] -\frac{\alpha}{\sqrt{\pi}}\sum_jq_j^2 -\frac{1}{2l}\left(\sum_jq_j\right)^2\left[\gamma-\ln 4\alpha^2+\lim_{g\rightarrow0}\ln g^2\right],\notag\end{aligned}\ ] ]    where the notation @xmath71 implies that for @xmath72 there is no self - interaction term in the summation . for a charge neutral system ,
the last term vanishes . for a system of electrons , as the one under consideration
, a uniform positive neutralizing background will exactly cancel the divergent term @xmath73 .
we define a dimensionless separation constant through @xmath74 and introduce dimensionless coordinates .
subsequently , the dimensionless interaction energy per electron in the simulation box can be cast as follows @xmath75=\frac{1}{2}\sum_{i , j}f[\{{\bf r}_{ij}\ } ] -\frac{n\widetilde\alpha}{\sqrt{\pi } } -n^2\left[\gamma-\ln{\left(\frac{2\widetilde\alpha}{l}\right)}\right],\notag\\\end{aligned}\ ] ] with    @xmath76&=&{\sum_n } ' \frac{\mathrm{erfc}\left(\widetilde\alpha\sqrt{(x_{ij}+n)^2+y_{ij}^2}\right)}{\sqrt{(x_{ij}+n)^2+y_{ij}^2 } } + 2\sum_{q=1}^{+\infty}\cos\left(2\pi qx_{ij}\right ) k_0\left(\frac{\pi^2q^2}{\widetilde\alpha^2},\widetilde\alpha^2y_{ij}^2\right ) -\ln\widetilde\alpha^2y_{ij}^2-\gamma(0,\widetilde\alpha^2y_{ij}^2).\end{aligned}\ ] ]
we study the structural properties of a quasi - one - dimensional classical wigner crystal , confined in the transverse direction by a parabolic potential . 
 with increasing density , the one - dimensional crystal first splits into a zigzag crystal before progressively more rows appear . while up to four rows the ground state possesses a regular structure , five - row crystals exhibit defects in a certain density regime . 
 we identify two phases with different types of defects . 
 furthermore , using a simplified model , we show that beyond nine rows no stable regular structures exist .