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landau s theory of fermi liquids has proven tremendously successful at describing a wide range of systems of interacting fermions .
in particular , the theory legitimizes the free electron model by casting fermionic systems in terms of weakly - interacting quasiparticles .
systems of electrons in 1d provide an intriguing example of departure from the landau fermi liquid paradigm , exhibiting non - fermi - liquid behavior for any finite strength of the electron - electron interaction.@xcite perhaps the simplest model of electrons in one dimension is the one - dimensional ( 1d ) homogeneous electron gas ( heg ) , which comprises electrons on a uniform positively - charged background.@xcite    the strong correlation occurring in 1d ensures that the excitations are not electron - like quasiparticles , but are instead collective in nature .
an appropriate description of the low - energy spectrum of the 1d heg comes from the theory of tomonaga and luttinger.@xcite there are several experimental signatures of the tomonaga - luttinger ( tl ) liquid which distinguish it from the normal fermi liquid ; these are largely accessible to transport and tunneling experiments . for example , the conductivity of a 1d channel as a function of the temperature is expected to vary logarithmically in the presence of weak disorder for the fermi liquid , and as a power law for the tl liquid.@xcite analogous relations hold for the differential conductivity and the optical conductivity .
also associated with the lack of quasiparticles in the tl liquid is spin - charge separation , whereby spin and charge excitations propagate at different characteristic velocities.@xcite    one - dimensional models are easy to envisage , but experimental observation of 1d behavior is potentially problematic .
low - dimensional systems are never entirely independent of their 3d environment , leading to effects which have the potential to obscure the 1d behavior .
furthermore , the presence of impurities has been shown to alter drastically the behavior of a tl liquid.@xcite however , even in manifestly 3d systems , behavior unambiguously characteristic of electrons in 1d arises surprisingly frequently .
features associated with the luttinger model have been observed in organic conductors ( e.g. , tetrathiafulvalene - tetracyanoquinodimethane and the bechgaard salts),@xcite transition metal oxides,@xcite carbon nanotubes,@xcite edge states in quantum hall liquids,@xcite semiconductor heterostructures,@xcite confined atomic gases,@xcite and atomic nanowires.@xcite theoretical work on electrons in 1d thus has a large region of potential applicability .
the exactly - solvable luttinger model describes electrons moving in one dimension with short - range interactions and linear dispersion .
studies with long - range interactions have found that the exponents and excitation velocities are nontrivially altered.@xcite one thus expects to be able to describe the 1d heg within the luttinger model framework , but the exact behavior of the parameters of the model is largely unclear .
the interactions that we study here are long - ranged , possessing a @xmath1 coulombic tail .
this is most applicable to systems where screening is a small effect , such as isolated metallic carbon nanotubes and semiconductor structures where there is negligible coupling to the substrate .
the 1d heg has been studied with a variety of theoretical and computational approaches .
the principal distinction between various studies is the choice of electron - electron interaction .
the bare coulomb interaction , @xmath1 , which describes an infinitely - thin wire , is perhaps conceptually the simplest choice , although it is largely avoided in the literature@xcite in its original form due to the divergence at @xmath2 . instead
, many previous authors have removed the singularity while retaining the long - range behavior by investigating interaction potentials of the form @xmath3 , where @xmath4 is a parameter related to the width of the wire .
this interaction has been studied analytically@xcite and numerically.@xcite    otherwise , one can derive an effective 1d interaction by factorizing the wave function into longitudinal and transverse parts and assuming that the transverse component is the ( 2d ) single - particle ground state of the confining potential .
the 1d interaction is then the matrix element of the 3d coulomb interaction with respect to the transverse eigenfunctions.@xcite an example of this is the harmonic wire , in which the transverse confinement is provided by a parabolic potential , leading to a gaussian density profile in the transverse plane .
the harmonic wire has been studied with quantum monte carlo ( qmc),@xcite variants of the singwi - tosi - land - sjlander approach,@xcite and the fermi hypernetted - chain approximation.@xcite    we have studied both the infinitely - thin wire and the harmonic wire using qmc . in this article
we report qmc calculations of the momentum density ( md ) , energy , pair - correlation function ( pcf ) , and static structure factor ( ssf ) of the infinitely - thin wire at a variety of densities and system sizes .
the md results in particular show the non - fermi - liquid character of the system and allow us to recover one of the tl parameters .
the total energy data that we provide are exact and may be regarded as a benchmark for future work .
we also present calculations of the md for the harmonic wire , again extracting one of the tl parameters .
the rest of this paper is structured as follows : the models for which we perform our calculations are described in sec .
[ sec : models ] . in sec.[qmc_details ] we outline qmc methods and provide the details of our approach .
we report the ground state energies of both models in sec.[sec : energies ] and describe the pcfs in sec .  [
sec : pcf ] . in sec.[sec : ssf ] we give the ssfs that we find for the infinitely - thin wire and in sec .
[ sec : md ] we give the mds for both models .
we describe the procedure for estimating a parameter of the tl model in sec.[sec : ll_params ] . finally , we draw our conclusions in sec.[sec : conclusions ] .
we use hartree atomic units ( @xmath5 ) throughout this article .
the hamiltonians for both of the models we have studied may be written as @xmath6 where @xmath7 is the madelung energy ( the interaction of a particle with its own background and periodic images ) , @xmath8 is the distance between electron @xmath9 and electron @xmath10 , and @xmath11 is the ewald interaction ; this is the interaction of an electron at @xmath12 with another electron at @xmath13 , all of electron @xmath10 s periodic images , and @xmath14-th of the uniform positive background .
the two models that we have studied differ in the @xmath11 and @xmath7 terms .
the ewald interaction for the infinitely - thin wire may be written @xmath15 which is calculated in practice using an accurate approximation based on the euler - maclaurin summation formula ; see eq .
( 4.8 ) of ref . for details .
the interaction of eq .  ( [ eq : inf_thin_int ] ) diverges as @xmath16 when @xmath17 . in higher dimensions ,
the divergence in the interaction energy is canceled by an equal and opposite divergence in the kinetic energy , so that nodes do not necessarily occur where two antiparallel spins occupy the same position.@xcite in the infinitely - thin 1d system , the curvature of the wave function is unable to compensate for the divergence in the interaction potential , so the trial wave function has nodes at all of the coalescence points for both parallel and antiparallel spin pairs .
the result is that the ground state energy is independent of the spin - polarization and depends only on the density . in other words
, the lieb - mattis theorem@xcite does not apply and the paramagnetic and ferromagnetic states are degenerate for the interaction of eq.([eq : inf_thin_int ] ) .
we have examined only the fully spin - polarized case for the infinitely - thin wire .
the second model we have studied describes electrons in a 2d confinement potential given by @xmath18 where @xmath19 is the width parameter and @xmath20 is the magnitude of the projection of the electron position onto the plane perpendicular to the axis of the wire .
the ewald - like interaction for this model may be written as@xcite @xmath21\cos(gnx_{ij } ) \ ; , \label{eq : ewald_like_sum}\end{aligned}\ ] ] where @xmath22 .
equation ( [ eq : ewald_like_sum ] ) possesses a long - range coulomb tail and is finite at @xmath23 . a derivation of eq .
( [ eq : ewald_like_sum ] ) is given in appendix [ app : ewald ] .
for the harmonic wire we have probed different polarizations , @xmath24 .
we computed expectation values using the variational and diffusion monte carlo ( vmc and dmc , respectively ) methods as implemented in the casino program.@xcite for the infinitely - thin wire , we combined vmc and dmc results to form extrapolated estimates@xcite where applicable , whereas for the harmonic wire we used vmc alone .    in the vmc method the expectation value of the hamiltonian with respect to a trial wave function is calculated using a stochastic integration technique.@xcite trial wave functions usually contain a number of free parameters ; we optimized the free parameters in our wave function by unreweighted variance minimization@xcite and linear - least - squares energy minimization.@xcite dmc is a stochastic projector technique for solving the many - body schrdinger equation and generates configurations distributed according to the product of the trial wave function and its ground state component.@xcite dmc calculations of expectation values of operators that commute with the hamiltonian are in principle exact for systems in which the wave function nodes are known ; this is the case for both the infinitely - thin and harmonic wires .
we used a slater - jastrow wave function for both systems , where the jastrow factor comprised two - body terms consisting of smoothly truncated polynomials and a sum of cosines with periodicity commensurate with that of the simulation cell.@xcite the orbitals in the slater determinant were plane waves with wave vectors up to @xmath25 for the paramagnetic systems and @xmath26 for the ferromagnetic systems .
the orbitals were evaluated at quasiparticle coordinates related to the actual coordinates by a backflow transformation.@xcite backflow provides an efficient way of describing three - body correlations in the 1d heg , but leaves the exact nodal surface unchanged .
one method for assessing the wave function quality is to examine the fraction of the correlation energy retrieved , @xmath27 , where @xmath28 is the hartree - fock energy , and @xmath29 and @xmath30 are the dmc and vmc energies , respectively .
we tested several types of wave function for the infinitely - thin wire with @xmath31 a.u .
, @xmath32 , and @xmath33 ; our vmc calculations retrieved @xmath34 of the correlation energy when we used a two - body jastrow factor and backflow transformations [ the error bars were @xmath35 a.u . ] , which is the type of wave function we use throughout this paper . while it is indeed the case that dmc is formally exact for the 1d heg , the quality of the trial wave function is important for the statistical efficiency of the dmc method and the accuracy with which expectation values of operators that do not commute with the hamiltonian may be computed
.      the dmc energy did not change beyond statistical error upon varying the number of walkers between 640 and 2000 , so we used @xmath37 walkers in our calculations and assumed that population control bias is negligible .
the dependence of the energy upon the dmc timestep @xmath38 was also investigated ; fig .
[ fig : dtdmc ] shows that for small @xmath38 the energy is constant .
we performed our calculations at a single timestep given by @xmath39 .
this fairly conservative choice was made to ensure that time step bias is entirely negligible .
the rms distance diffused by each electron in a single step was thus slightly less than @xmath40 .
for the infinitely - thin wire , we used simulation cells containing 37 , 55 , 73 , and 99 particles subject to periodic boundary conditions for our calculations of the energy , pcf , and ssf .
our md calculations for the infinitely - thin wire also used a much larger cell with @xmath41 . for the harmonic wire , we used cells with @xmath42 , @xmath43 , and occasionally @xmath44 for the @xmath33 systems and cells with @xmath45 and @xmath46 for the @xmath47 systems .
previous work encountered difficulties in sampling different spin configurations of the harmonic wire for @xmath48 due to the presence of `` pseudonodes '' at the antiparallel coalescence points,@xcite although these problems were largely overcome by the use of lattice - regularized diffusion monte carlo ( lrdmc ) in ref .. the problem occurred because for strong , repulsive interactions the wave function can become small when two antiparallel spins approach one another . combined with a small time step this can lead to simulations where opposite spins exchange positions infrequently and the space of spin configurations is explored very inefficiently .
use of a small time step is a necessary part of the algorithm of projector methods like dmc .
we have avoided ergodicity problems by using vmc to study the harmonic wire ; in the vmc method there is no restriction other than ergodicity on the transition probability density and one may propose moves however one wishes provided that the acceptance probability is modified accordingly .
we use electron - by - electron sampling with the transition probability density given by a gaussian centered on the initial electron position .
the vmc `` time step '' in fact bears no relation to real time and is simply the variance of the transition probability density . in practice ,
the unmodified time steps ( chosen to achieve a @xmath49 acceptance ratio ) used in vmc are usually large enough to eliminate ergodicity problems , although we found some cases where it was necessary to enforce a lower limit on the width of the transition probability density .
table [ table : spinexchange ] shows the frequency with which electrons changed positions in our simulations for both high and low density systems with strong and weak confinement .
.frequency with which electrons paths cross in our vmc simulations of the harmonic wire .
the quantity @xmath50 is the proportion of proposed single - electron moves that result in a change in the ordering of the particles .
a typical calculation comprises between @xmath51 and @xmath52 proposed single - electron moves .
the data shown are for @xmath45 .
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we present calculations of the energy , pair correlation function ( pcf ) , static structure factor ( ssf ) , and momentum density ( md ) for the one - dimensional electron gas using the quantum monte carlo method . 
 we are able to resolve peaks in the ssf at even - integer - multiples of the fermi wave vector , which grow as the coupling is increased . 
 our md results show an increase in the effective fermi wave vector as the interaction strength is raised in the paramagnetic harmonic wire ; this appears to be a result of the vanishing difference between the wave functions of the paramagnetic and ferromagnetic systems . 
 we have extracted the luttinger liquid exponent from our mds by fitting to data around @xmath0 , finding good agreement between the exponent of the ferromagnetic infinitely - thin wire and the ferromagnetic harmonic wire .