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two - dimensional ( 2d ) systems play a central role in contemporary condensed matter physics . novel phenomena such as the quantum hall effect @xcite observed when a 2d electron gas at low temperature is subjected to a strong magnetic field , as well as practical developments based on quasi-2d systems , e.g. , high - mobility field - effect transistors @xcite or semiconductor quantum - well lasers @xcite have brought significant technological advances . such devices are based on the quasiequilibrium response of the internal electron or electron - hole plasmas to an external stimulation . hence it is essential to understand the fundamental quantum - statistical properties of such two - dimensional interacting plasmas at finite temperatures . one of the well - known differences between 2d and 3d nonrelativistic quantum mechanics is the presence of at least one bound state for any symmetric attractive potential in two dimensions . this bound state , with binding energy @xmath0 , brings a non - vanishing contribution of @xmath1 to the two - body part of the partition function even if the interaction is weak and the state is very shallow . for a dilute gas this contribution introduces a deviation from the ideal gas law that is larger than the correction due to the fermi or bose statistics of the particles . however , it is clear that 2d gases with a vanishing inter - particle interaction strength should be well described by free fermi or bose gas models . this contradiction will be resolved in the present paper . a related question is what happens to the partition function when additional bound states appear with increasing strength of interaction between the particles . in the 3d case the answer to this question is based on a careful consideration of states in the continuum , which are modified by the interaction , i.e. , taking scattering into account in the partition function calculation @xcite . the same approach should be applied in two dimensions , however , scattering theory in two dimensions is relatively undeveloped compared to the 3d case . for example , the relation between low - energy scattering and bound states , which has important consequences in the statistical mechanics , has been considered only very recently @xcite in two dimensions . in the present paper the connection between scattering and the statistical mechanics of a 2d plasma is studied .    in the next section we introduce the relation of the two - dimensional scattering phase shift to the partition function and show that a proper consideration of the scattering states removes discontinuities in the partition function in an analogous fashion to the 3d case . in sects . [ short - range ] and [ ehplasma ] we explore this in more detail using an analytical model with an attractive square well potential before turning to a more realistic model of the statically screened electron - hole plasma , which is the main focus of this paper . we also compare in both cases the influence on the second virial coefficient of the exchange interaction and the screened direct interaction . such considerations are crucial in understanding the nature of the strongly correlated electron - hole plasma in semiconductor quantum wells . the two - body interaction part of the partition function of 2d interacting boltzmann particles is given by @xmath2 where @xmath3,@xmath4 is the reduced mass , @xmath5 is the projection of the angular momentum onto the axis normal to the plane of 2d motion ( @xmath6 ) , @xmath7 is the 2d scattering phase shift @xcite dependent on the relative - motion momentum @xmath8 , @xmath9 are the bound - state energies ( index @xmath10 enumerates bound states with given @xmath11 ) , and the double sum in the first term ranges only over bound states . ( [ eq1 ] ) is the 2d analogue of the beth - uhlenbeck formula and can be derived in the same fashion as in the 3d case @xcite . often only the first term in eq . ( [ eq1 ] ) is considered when calculating an internal partition function , neglecting the phase shift term . for an attractive potential @xmath12 , as @xmath13 decreases bound state energies increase towards the continuum . as such a state reaches zero energy , a partition function that contains only the bound state sum will be discontinuous . these unphysical singularities would extend to all the thermal properties , such as pressure and specific heat . integrating by parts we can rewrite eq . ( [ eq1 ] ) as @xmath14 for nonzero temperature the integral term in the right - hand side of eq . ( [ eq2 ] ) is a smooth function of the interaction strength @xmath15 . if the phase shifts satisfy the condition : @xmath16 where @xmath17 is the number of bound states with given @xmath5 , the zero - energy part of the phase shift integral in eq . ( [ eq1 ] ) exactly cancels the zero - energy part of the bound state sum , removing the discontinuity in @xmath18 as a function of the interaction strength . this cancellation is similar to the well known behaviour in 3d where the partition function discontinuities are removed @xcite with the help of levinson s theorem @xcite . equation ( [ eq3 ] ) constitutes the 2d statement of levinson s theorem .    as a central theorem of scattering theory @xcite , levinson s theorem has been discussed for dirac particles , multichannel scattering , multi - particle single - channel scattering , one - dimensional scattering systems , impurities in aharonov - bohm rings , systems with non - uniform effective mass , and even for time - periodic potentials @xcite . however , its applicability to the 2d scattering problem has been considered only recently . in ref . @xcite the 2d statement of levinson s theorem , eq . ( [ eq3 ] ) , was proposed and verified empirically , while in ref . @xcite this theorem was more rigorously established for cutoff potentials using the green - function method . the above arguments , based on the partition function continuity requirement , provide an additional statistical mechanical justification of levinson s theorem in 2d . the two - body interaction part of the partition function can be used to calculate the second virial coefficient , @xmath19 , which characterizes the first correction to the ideal gas law in the low - density expansion of pressure @xmath20 @xmath21 is positive for repulsive potentials , causing an increase in the pressure over its ideal - gas value , and negative for attractive potentials , causing a decrease in the pressure . a calculation of the second virial coefficient is meaningful in the dilute gas regime , where the mean inter - particle spacing @xmath22 is larger than the thermal wavelength @xmath23 and the higher order terms in eq . ( [ eq4 ] ) are negligible . for a free , bose or fermi gas in two dimensions , @xmath24 @xcite , the plus sign applying to the fermi case , as the pauli principle introduces an effective repulsion between fermions , thereby increasing the pressure . there is extensive ongoing research in the statistical mechanics of anyons , 2d particles obeying fractional statistics @xcite , and the second virial coefficient of a free anyon gas lies in between the bosonic and fermionic value @xcite . in this paper we focus on the relative importance of interaction - induced bound and scattering states on the second virial coefficient . the 3d analysis of the second virial coefficient @xcite is easily reformulated for a 2d interacting gas @xcite . for a system of identical particles with spin @xmath25 the second virial coefficient is @xmath26 where the upper sign is for bosons and the lower sign for fermions . the exclusion principle modifies the sum over @xmath5 in eq . ( [ eq2 ] ) depending on the angular momentum parity , and the partition function ( using eq . ( [ eq3 ] ) ) is @xmath27 \nonumber \\ & & \nonumber\\ & ~+~&(2s+1){2\over{\pi q_t^2 } } \int_0^\infty q dq   \left[\sum_{m=-\infty}^{\infty }   \left(s+{1\pm(-1)^m \over 2}\right ) \delta_m(q ) \right ]   \exp(-q^2/q_t^2)~. \label{eq6 } \end{aligned}\ ] ]    the electron - hole plasma constitutes a mixture of two components and for a binary mixture of components @xmath28 and @xmath29 having second virial coefficients @xmath30 and @xmath31 and densities @xmath32 and @xmath33 , respectively , the second virial coefficient is @xcite @xmath34 in eq . ( [ eq7 ] ) @xmath35 and @xmath36 where @xmath37 and @xmath18 is given by eq . ( [ eq1 ] ) or eqs . ( [ eq2 ] ) and ( [ eq3 ] ) with the properly chosen reduced mass @xmath38 . our first example , the boltzmann gas with weak short - range attraction , is chosen to elucidate how the second virial coefficient at given temperature vanishes when the binding energy @xmath39 decreases , even though the bound - state part of the internal partition function @xmath40 approaches unity rather than vanishes . to trace the precise nature of the cancellation of @xmath41 it is convenient to use eq . ( [ eq1 ] ) for @xmath18 without the application of levinson s theorem . let us assume that the 2d particles interact via an attractive square - well potential of radius @xmath42 and depth @xmath43 . this simple model allows analytical treatment which provides insight into the generic behaviour of a gas of attracting particles . to evaluate the partition function we need to analyse both the bound and scattering states in this potential . the binding energies for any value of angular momentum @xmath5 can be easily found by matching the logarithmic derivative of the radial wave function at @xmath44 : @xmath45 where @xmath46 , @xmath47 , @xmath48 is the bessel function of the first kind and @xmath49 is the modified bessel function of the second kind . note that for @xmath50(i.e . smaller than the first root of @xmath51 ) there is only one bound state ( having @xmath52 ) and for @xmath53 this state is very shallow , e.g. for @xmath54 , @xmath55 , for @xmath56 , @xmath57 . for a shallow @xmath58 state the transcendental equation for binding energies eq . ( [ eq10 ] ) reduces to @xmath59 where @xmath60 ( @xmath61 is euler s constant ) .    for the unbound states with positive energy of the relative motion , @xmath62 , scattering phase shifts can be found in a similar fashion . for small values of the momentum , @xmath63 , all phase shifts for @xmath64 are small compared to @xmath65 ( @xmath25-wave scattering ) @xcite . the tangent of the @xmath25-wave scattering phase shift for @xmath66 ( i.e. @xmath67 ) is given by @xcite : @xmath68 substituting @xmath69 from eq . ( [ eq11 ] ) into eq . ( [ eq12 ] ) we get @xmath70 note that this expression does not contain parameters of the potential @xmath71 and @xmath72 explicitly , and it is valid for an arbitrary potential well with a shallow @xmath73 level @xcite .    since the integrand in the partition function , eq . ( [ eq1 ] ) , contains an exponential factor @xmath74 , the wavevectors @xmath75 which are larger than the thermal wave vector @xmath76 give negligible contribution to the value of the integral . therefore for the short - range interaction or for low temperature , satisfying condition @xmath77 , the scattering phase shifts need only be considered for @xmath78 . then all the terms in the phase shift sum in eq . ( [ eq1 ] ) can be neglected except for the term with @xmath52 . finding the derivative of @xmath65 from eq . ( [ eq13 ] ) we obtain for the two - body interaction part of the partition function @xmath79 the integral in eq . ( [ eq15 ] ) is the ramanujan integral @xcite which can be rewritten as @xcite @xmath80 where @xmath81 with @xmath82 . thus , the partition function acquires a very simple form @xmath83 a similar result has been obtained recently for contact - interacting particles @xcite .    to consider the small @xmath84 asymptotic of the function @xmath85 it is convenient to expand the integral eq . ( [ eq17 ] ) in descending powers of @xmath86 : @xmath87^{-3}\right)~. \label{eq19}\ ] ] from eq . ( [ eq19 ] ) one can see that @xmath18 and hence the second virial coefficient @xmath88 both vanish when @xmath89 , although one bound state always exists . so the lowest - order density correction to the 2d ideal gas law vanishes only slowly as @xmath90 as the binding energy is reduced . note that , when the potential supports several bound states , the contribution of any shallow bound state with @xmath58 is cancelled by the scattering phase shift integral in the `` logarithmic '' manner described above . for @xmath64 the cancellation has a power - law dependence in @xmath91 @xcite . this implies that higher - order levinson s theorems responsible for continuity of the partition function derivatives @xcite are different for @xmath58 and @xmath92 , whereas the zeroth - order levinson s theorem in two dimensions has the same form , eq . ( [ eq3 ] ) , for all @xmath11 .    for extremely weak interaction potential , such that @xmath93 , from eqs . ( [ eq11 ] ) and ( [ eq19 ] ) it follows that @xmath94 , which coincides with the perturbation theory result . in the other limit for large @xmath84 values ( @xmath95 ) , @xmath96 @xcite , therefore the exponential dependence of the partition function on the binding energy is recovered .    in fig . [ fig1 ] we plot the ratio of the total partition function @xmath97 to its bound - state part @xmath98 as a function of @xmath99 . we do this for both the full expression , eq . ( [ eq17 ] ) , and the first two terms in the asymptotic expansion , eq . ( [ eq19 ] ) . one can see that the asymptotic expression ( dashed line in fig . [ fig1 ] ) is accurate only for very small values of @xmath99 . over a wide range of @xmath91 both scattering and bound - state terms are important , e.g. when @xmath100 the scattering term produces a 20% correction to @xmath97 . when @xmath101 the bound - state contribution dominates completely . 16truecm    it is instructive to compare the contributions to the second virial coefficient of the direct and the exchange interactions . for example for a gas of spinless bosons the second virial coefficient is @xmath102 . for @xmath18 given by eq . ( [ eq18 ] ) , the direct interaction term ( @xmath103 ) is smaller than the exchange term ( @xmath104 ) for small binding energies . thus , as required for vanishing binding energy , the second virial coefficient is governed by the statistics of particles , despite the existence of a single bound state . in this section we study a more realistic model of an interacting gas than the one considered in sect . [ short - range ] . we consider a mixture of the two types of fermi particles , positively charged holes ( h ) and negatively charged electrons ( e ) . this model is important for understanding the thermodynamic and optical properties of the electron - hole plasma in semiconductor quantum wells . in the low - temperature , low - density limit most electron - hole pairs are bound into excitons . this limit has been studied extensively because of the recent proposals of exciton condensates in this system @xcite . the properties of the degenerate 2d electron - hole plasma are also well known @xcite . we consider the case when the temperature is comparable to the exciton binding energy so the occupation of continuum states ( and therefore screening ) is significant , although the carrier density is low enough to neglect the terms higher than @xmath105 in the virial expansion . the temperature and density conditions under investigation are close to those in the regime where excitonic gain in wide - gap semiconductors is anticipated @xcite . we assume for simplicity equal masses @xmath106 and spins @xmath107 for both species . then @xmath108 and the second virial coefficient for the mixture eq .  ( [ eq7 ] ) acquires the form : @xmath109 here @xmath110 and the charge neutrality condition @xmath111 is taken into account . the screened coulomb attraction between electrons and holes is modeled using the fourier representation of the interaction potential : @xmath112 where @xmath113 is the screening wave vector . hereafter we employ 3d excitonic rydberg units where length and energy are scaled , respectively , by the effective bohr radius @xmath114 and rydberg ry@xmath115 . for electron - electron and hole - hole repulsion the same potential with the opposite sign equation ( [ eq21 ] ) is the well - known thomas - fermi expression for the coulomb potential statically screened by a 2d electron gas . being the long - wavelength static limit of the random phase approximation , eq . ( [ eq21 ] ) is a simple model for the screened coulomb potential in two dimensions . nevertheless , this expression reflects the fact that the statically screened potential in two dimensions decreases at large distances slower than in the 3d case ( as a power law rather than exponentially ) . despite numerous more realistic corrections @xcite eq .  ( [ eq21 ] ) remains the most widely used approximation for the 2d screening @xcite . this potential has been known for more than three decades @xcite but some of its unusual properties were only discovered recently , e.g. the existence of a remarkably simple relation between @xmath113 and the number of bound states . namely , with decreasing screening , new bound states appear at the critical integer values of the screening length given by the simple formula@xcite @xmath116 where @xmath5 is the angular momentum and @xmath117 indicates the number of nonzero nodes of the radial wave function . equation ( [ eq22 ] ) shows that several bound states corresponding to the given value of @xmath118 appear simultaneously . this degeneracy is different from the degeneracy for the bound states of the unscreened 2d exciton ( or hydrogen atom ) , for which the states with the same value of @xmath119 are degenerate @xcite . the hidden symmetry that underlines this new degeneracy has not been fully understood yet , and eq . ( [ eq22 ] ) still lacks a rigorous analytical derivation . we also consider the low - density ( nondegenerate ) limit , when there is no pauli blocking and the self - energy correction @xcite to the beth - uhlenbeck formula can be neglected and eqs . ( [ eq2],[eq3 ] ) and eq . ( [ eq6 ] ) can be used for @xmath120 and @xmath121 respectively . the shortcomings of this model for the quantitative description of a real system of photoexcited electrons and holes in semiconductor quantum wells are self - evident ; however , it does provide a tractable model containing all the salient features of the system .    to find the second virial coefficient given by eq . ( [ eq20 ] ) one must calculate the binding energies and scattering phase shifts entering the partition functions @xmath120 and @xmath121 . we use for this purpose the 2d modification of the variable - phase method @xcite known from scattering theory . in this method the scattering phase shift and the function defining bound - state energies can be obtained as a large distance limit of the phase function , which satisfies the first - order , nonlinear riccati equation originating from the radial schrdinger equation . the variable - phase method application to scattering and bound states in the screened coulomb potential ( [ eq21 ] ) is described in detail in ref . the method is especially effective for calculation of shallow - state binding energies and low - energy scattering phase shifts . figure [ fig2 ] shows the results from the calculation of the electron - hole part of the partition function , @xmath122 , which contains both the bound state sum and the scattering phase shift integral . in this figure @xmath120 is plotted as a function of the inverse screening wave number @xmath123 . 16truecm    for the electron - hole plasma @xmath113 is a function of carrier density @xcite , and in the purely 2d case @xmath124 for low densities and is independent of density , @xmath125 for @xmath126 , in the degenerate limit . here we treat @xmath113 as a parameter characterizing the strength of the screened interaction potential eq . ( [ eq22 ] ) . to emphasize the role of scattering we show on the same plot the bound - state sum , @xmath127 , which exhibits jumps whenever new bound states appear [ i.e. when @xmath113 satisfies eq . ( [ eq22 ] ) ] . these jumps become higher with increasing screening length @xmath128 since several bound states appear simultaneously . as can be seen the additional scattering phase shift contributions completely remove these jumps . the partition function is plotted for two values of the ratio of @xmath129 to the excitonic rydberg , @xmath130 and @xmath131 , which roughly correspond to znse ( or gan ) and gaas at room temperature . one can see that for high temperature ( or low binding energy ) the bound - state contributions to @xmath120 are suppressed by the scattering phase shift integral more strongly than in the @xmath132 case .    in fig . [ fig3 ] the second virial coefficient @xmath133 ( scaled by @xmath134 ) is plotted versus the screening wave number @xmath113 for two different values of @xmath135 . 16truecm    equation ( [ eq20 ] ) is used for the calculation of @xmath136 , and the repulsion ( @xmath137 ) term partially compensates the @xmath120 term . this compensation is especially significant in the high - temperature case @xmath138 , in which the 2d electron - hole plasma behaves much like an ideal gas over a wide range of screening wave vectors . for @xmath132 the electron - hole attraction term dominates and the plasma is strongly correlated for all values of @xmath139 . in this case a small statistical repulsion ( @xmath140 , horizontal line in fig .  [ fig3 ] ) , which is due to the fermionic nature of electrons and holes , can be completely neglected . thus , at room temperature the electron - hole plasma in gaas - based quantum wells can be treated as an ideal gas , whereas in wide - gap semiconductors ( e.g. , znse or gan ) due to the high value of ry@xmath115 the 2d electron - hole plasma is strongly correlated and excitonic effects are important for its thermodynamic properties . in this paper we show that a proper account of scattering eliminates discontinuities in thermodynamic properties of the nonideal 2d gas whenever extra bound states appear with small increase of the strength of an attractive potential . this treatment provides a heuristic proof of levinson s theorem in two dimensions . we trace the way in which the bound - state contribution to the partition function of the 2d gas , with a weak short - range attraction between its particles , vanishes when the binding energy decreases . a weak @xmath90 binding energy dependence of the second virial coefficient of such a gas is found for @xmath141 . a consistent treatment of bound and scattering states in a screened coulomb potential allows us to calculate the quantum - mechanical second virial coefficient of the dilute 2d electron - hole plasma and to establish the difference between the nearly ideal electron - hole gas in gaas and the strongly correlated exciton / free - carrier plasma in wide - gap semiconductors . the 2d electron - hole plasma was considered in the low - density nondegenerate limit only . transition to the strongly degenerate fermi limit and related questions of pauli blocking and self - energy corrections to the beth - uhlenbeck formula in 2d remain the subject of further research .      also at a. f. ioffe physico - technical institute , st.petersburg , russia . , edited by r. e. prange and s. m. girvin ( springer , new york , 1990 ) . t. ando , a. b. fowler , and f. stern , rev . mod . phys . * 54 * , 437 ( 1982 ) . w. w. chow , s. w. koch , and m. sargent iii , _ semiconductor - laser physics _ ( springer , berlin , 1994 ) . a. i. larkin , zh . 38 , 1896 ( 1960 ) [ sov . jetp * 11 * , 1363 ( 1960 ) ] . f. j. rogers , h. c. graboske jr , and h. e. dewitt , phys . 34a * , 127 ( 1971 ) ; f. j. rogers , phys . rev . a * 4 * , 1145 ( 1971 ) . m. e. portnoi and i. galbraith , solid state commun . , * 103 * , 325 ( 1997 ) ; cond - mat/9705136 . lin , phys . rev . a * 56 * , 1938 ( 1997 ) . f. stern and w. e. howard , phys . rev . * 163 * , 816 ( 1967 ) . e. beth and g. e. uhlenbeck , physica ( amsterdam ) * 4 * , 915 ( 1937 ) ; see also , e.g. , l. d. landau and e. m. lifshitz , _ statistical physics _ ( pergamon , new york , 1980 ) , pt . n. levinson , k. dan . selsk . mat . . medd . * 25 * , 3 ( 1949 ) . r. g. newton , _ scattering theory of waves and particles _ ( mcgraw - hill , new york , 1966 ) . ma , phys . 76 * , 3654 ( 1996 ) ; n. poliatzky , _ ibid . _ * 76 * , 3655 ( 1996 ) ; f. vidal and j. letourneaux , phys . c * 45 * , 418 ( 1992 ) ; l. rosenberg and l. spruch , phys . a * 54 * , 4978 ( 1996 ) ; * 54 * , 4985 ( 1996 ) ; k. a. kiers and w. van dijk , j. math . phys . * 37 * , 6033 ( 1996 ) ; m. s. debianchi , _ ibid . _ * 35 * , 2719 ( 1994 ) ; v. milanovi _ et al . a * 170 * , 127 ( 1992 ) ; v. milanovi _ et al . _ , a * 25 * , l1305 ( 1992 ) ; m. diventra _ et al . _ , b * 55 * , 1353 ( 1997 ) ; p. a. martin and m. s. debianchi , europhys . lett . * 34 * , 639 ( 1996 ) , and references therein . r. l. siddon and m. schick , phys . rev . a * 9 * , 907 ( 1973 ) ; j. g. dash , _ films on solid surfaces _ ( academic , new york , 1975 ) . p. giacconi , f. matoni , and r. soldati , phys . b * 53 * , 10065 ( 1996 ) , and references therein . p. arovas , r. schrieffer , f. wilczek , and a. zee , nucl . b * 251 * , 117 ( 1985 ) . for a review on anyons see , e.g. , _ fractional statistics and anyon superconductivity _ , ed . by f. wilczek ( world scientific , singapore , 1990 ) . j. e. kilpatrick , j. chem . * 21 * , 274 ( 1953 ) ; j. e. kilpatrick , w. e. keller , e. f. hammel , and n. metropolis , phys . 94 * , 1103 ( 1954 ) . l. d. landau and e. m. lifshitz , _ quantum mechanics ( non - relativistic theory ) _ ( butterworth - heinemann , oxford , 1997 ) . m. e. portnoi , pisma zh . 14 , 1252 ( 1988 ) [ sov * 14 * , 547 ( 1988 ) ] . it is interesting to note that the expression for the scattering cross - section for resonant @xmath25-wave scattering in two dimensions , @xmath142 $ ] , despite being quite different from its 3d equivalent @xcite , nevertheless also has a pole at @xmath143 . g. h. hardy , _ ramanujan _ ( cambridge university press , cambridge , 1940 ) , p. 196 . i. s. gradshtein and i. m. ryzhik , _ tables of integrals , series , and products _ ( academic press , new york , 1994 ) . m. e. portnoi and i. galbraith , unpublished . d. boll , ann . 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we consider scattering state contributions to the partition function of a two - dimensional ( 2d ) plasma in addition to the bound - state sum . 
 a partition function continuity requirement is used to provide a statistical mechanical heuristic proof of levinson s theorem in two dimensions . 
 we show that a proper account of scattering eliminates singularities in thermodynamic properties of the nonideal 2d gas caused by the emergence of additional bound states as the strength of an attractive potential is increased . 
 the bound - state contribution to the partition function of the 2d gas , with a weak short - range attraction between its particles , is found to vanish logarithmically as the binding energy decreases . 
 a consistent treatment of bound and scattering states in a screened coulomb potential allowed us to calculate the quantum - mechanical second virial coefficient of the dilute 2d electron - hole plasma and to establish the difference between the nearly ideal electron - hole gas in gaas and the strongly correlated exciton / free - carrier plasma in wide - gap semiconductors such as znse or gan .