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strong coulomb electron - electron ( e - e ) interactions can drive transitions from metallic to exotic insulating states , the most well known of which are the mott - hubbard semiconductor ( mhs ) and the wigner crystal ( wc ) . the mhs is a characteristic of systems with carrier concentration per site @xmath5 and is driven by strong onsite e - e repulsion , the hubbard @xmath6 interaction . depending upon the lattice structure the critical @xmath6 at which the metal - insulator ( mi ) transition occurs , @xmath7 , can be 0@xmath8 or finite @xcite . in contrast , the wc occurs in systems with @xmath9 , and is characterized by charge - ordering ( co ) , _ i.e. _ , a periodic arrangement of _ single _ charge carriers on the lattice . the wc is driven by strong onsite as well as inter - site coulomb interactions @xcite . although in principle the wc is likely at any arbitrary @xmath10 , it has been been studied most intensively for @xmath11 bipartite lattices , where the nearest neighbor ( n.n ) coulomb repulsion can drive the mi transition @xcite . the combined effects of e - e and electron - phonon ( e - p ) interactions are also of interest , usually in one dimension ( 1d ) , where the mhs can further exhibit the spin - peierls ( sp ) transition . importantly for our purpose here the above semiconducting states have been intensively studied over the past several decades , and are largely understood , although arguments regarding the magnitude of @xmath7 for formation of the mhs state in specific lattices or the detailed mechanism of the mi transition may continue to persist . in the present work , we discuss a new correlated - electron semiconductor , the paired - electron crystal ( pec ) , that occurs in @xmath11 systems in the presence of moderate to strong geometric lattice frustration @xcite . we believe that our work has direct application to 2:1 cationic or 1:2 anionic charge - transfer solids ( cts ) that exhibit correlated insulator sc transitions , and further applies to other inorganic strongly - correlated @xmath4-filled materials . the combined effects of e - e interactions and geometric lattice frustration are of strong current interest @xcite . the bulk of the work here is for @xmath5 , where the hubbard model in the limit @xmath12 reduces to the heisenberg spin hamiltonian . interest in the consequences of lattice frustration stems from the seminal proposal by anderson that the ground state of the heisenberg antiferromagnet ( haf ) model on a triangular lattice is a quantum spin liquid ( qsl ) with no spin ordering even at zero temperature@xcite . the type of wavefunction usually assumed to describe a qsl is often referred to as a resonating valence bond ( rvb ) state . whether or not rvb states appear in the square lattice for @xmath10 slightly different from 1 , and the relationship of such states to superconductivity ( sc ) in doped strongly correlated semiconductors remains contentious . an extension of the rvb theory of dopant - induced sc in @xmath13 is the proposal that frustration - induced sc occurs in the anisotropic triangular lattice within the simple hubbard model even for @xmath10 exactly 1 , where a narrow superconducting phase is straddled on both sides by broader paramagnetic metallic ( pm ) and antiferromagnetic ( afm ) insulator phases @xcite . it has been claimed that this transition explains the sc in the cts @xcite . recent numerical work by us and others , however , have determined that sc is absent within the @xmath5 triangular lattice hubbard model @xcite and the earlier results are artifacts of mean - field approximations . while the ground state of the haf on the isotropic triangular lattice is now known to be the ordered 120@xmath14 afm rather than the originally proposed qsl state @xcite , other frustrated lattices , most notably the kagom lattice @xcite , have been investigated in the search for qsl states . proposed ground states here include various types of qsl states @xcite as well as valence - bond solid ( vbs ) states @xcite . the literature on vbs states has a long history going back to the well known ghosh - majumdar model @xcite . the common theme in works on vbs is the frustration - driven transition from the afm state to a total spin @xmath15 singlet state . we have found a similar frustration - driven transition from the afm to a @xmath15 state in strongly correlated systems with @xmath11 , where reduction to a spin hamiltonian is not possible . in contrast to the voluminous literature on correlated and frustrated systems at @xmath5 , the literature on frustrated @xmath13 is relatively sparse and new . the discovery of sc@xcite in hydrated na@xmath16coo@xmath1 has spurred interest in correlated systems away from @xmath5 @xcite , although to the best of our knowledge only isotropic triangular lattices have been studied . we will specifically focus on @xmath11 within the present work  on the triangular lattice with varying anisotropy . in the square lattice limit at this density , we show that spontaneous in - phase dimerization occurs in the presence of electron - phonon interactions modulating n.n hopping integrals , leading to an _ effective _ @xmath5 system with one electron per dimer and afm order . this result is the origin of the so - called the dimer mott - hubbard model @xcite that is commonly used to describe the 2:1 cationic or 1:2 anionic organic cts . very recently we have proposed that under the influence of lattice frustration this dimer mott - hubbard afm state gives way to a spin - paired state that we termed the pec @xcite . the pec is different from any of the above more well known correlated semiconducting states in that it is a wc of heitler - london spin - singlets  simultaneously charged - ordered and spin @xmath15 . alternately , the pec is the @xmath11 equivalent of the @xmath5 vbs . a conceptually similar state was postulated for the electron gas many years back by moulopoulos and ashcroft @xcite . there is a fundamental similarity between this earlier work and ours , in that in both cases the pairing is driven by the exchange interaction . our previous work @xcite only considered a limited set of parameters , and the full phase diagram was not discussed . here we give a more complete phase diagram , including the competition with the wc state that was ignored before . as with the vbs state at @xmath5 , the pec is a consequence of frustration - induced quantum effects . its extraordinary stability at @xmath11 is a commensurability effect ( recall that mhs and wc formation also require commensurability ) .    in section [ model ] we introduce the model we consider . we include e - p interactions to stabilize the lattice dimerization that gives the effective @xmath5 dimer lattice . we present the physical mechanism behind the pec formation by briefly discussing simple molecular clusters , for which we show that spin - singlet formation in @xmath11 necessarily requires charge disproportionation . the charge disproportionation in the infinite one - dimensional ( 1d ) chain and the so - called zigzag ladder leads to periodic co , viz . , the simplest pecs . following the discussions of these simple cases , we introduce the two - dimensional ( 2d ) lattice that will be the focus of this work , and discuss the different possible phases . in section [ results ] we present out numerical results for the 2d system , covering a wide region of parameter space . in section [ discussion ] we discuss the relevance of our results for several classes of @xmath11 materials and the outlook for understanding unconventional sc . we particularly emphasize the cases of the organic charge - transfer solids cts @xmath0-(bedt - ttf)@xmath1cu@xmath1(cn)@xmath2 and etme@xmath2sb[pd(dmit)@xmath1]@xmath1 which have been described as qsls within the effective @xmath5 scenario @xcite . we believe frustration - induced charge disproportionation is an alternate possibility . the hamiltonian we consider contains electron hopping , semi - classical inter - site and onsite e - p couplings , and onsite and n.n . coulomb interactions : @xmath17 in eq . [ ham ] , @xmath18 indexes the different bond directions in the lattice ; for example @xmath19 in 1d and @xmath20 in the 2d square lattice . our actual calculations ( see below ) are for the anisotropic triangular lattice , @xmath21 . @xmath22 is the electron hopping between sites @xmath23 and @xmath24 with electron creation ( annihilation ) operators @xmath25 ( @xmath26 ) . @xmath27 is the inter - site e - p coupling constant , @xmath28 is the corresponding spring constant , and @xmath29 is the distortion of the bond between sites @xmath23 and @xmath24 . @xmath30 is the intra - site phonon coordinate and @xmath31 is the intra - site e - p coupling with corresponding spring constant @xmath32 . both @xmath29 and @xmath30 are determined self - consistently @xcite . @xmath27 are in general taken close to the minimum value needed for the transition to occur , our goal being the replication of the same instability from finite cluster calculations that would occur in the infinite system for @xmath33 coupling . @xmath6 and @xmath34 are on - site and n.n . coulomb interactions , respectively . the physically relevant range of @xmath34 is @xmath35 based on comparison between @xmath5 and @xmath11 cts @xcite . we first present a simple qualitative discussion of coupled spin - singlet and co formation at @xmath11 . the ideas are quite general and we argue that the mechanism is _ independent _ of dimensionality . the key requirement is that the density must be exactly @xmath11 , as the effect requires commensurability . consider a single dimer of two sites with one electron . the electron populations per site are 0.5 each , but the quantum mechanical wavefunction for the system is the superposition @xmath36 , where @xmath37 and @xmath38 are site charge densities . if one now brings two of these dimers together , as in insert ( a ) of fig . [ cofig ] , the composite wavefunction of the two - dimer system can be written as @xmath39 . if the two electrons are in a spin - singlet state then within the simple hubbard hamiltonian the configuration @xmath40 , in which singlet stabilization can occur from a single n.n . hop that creates a virtual double occupancy , must dominate over the configurations @xmath41 and @xmath42 , in which singlet stabilization requires two and three hops , respectively . thus as the singlet bond between the dimers gets stronger we expect a charge difference @xmath43 between sites belonging to the same dimer ( nominally between sites 1 and 2 , or between sites 3 and 4 in the linear chain of fig . [ cofig ] ) . while some charge disproportionation must occur in finite linear chains from end effects alone , we note that our proposed picture demands that similar charge disproportionation occurs between members of the same dimer even in the case of the periodic molecule shown in the insert ( b ) of fig . [ cofig ] . in this case the charges on the sites connected by the diagonal bond must be larger than 0.5 , while the charges on the two other sites must be smaller . importantly , the modulation of charge density , bond orders , as well as spin - singlet pairing all occur cooperatively , and any of these observables may be used as an order parameter in the case of a real transition . in fig . [ cofig ] we have plotted @xmath43 versus the hopping integral @xmath44 corresponding to the dotted bonds in the molecules shown in the insert , for the ground spin - singlet state . in both cases , as @xmath44 increased from zero @xmath43 becomes nonzero and increases with @xmath44 . importantly , in the spin - triplet @xmath45 state the sign of @xmath43 is reversed in the cyclic molecule , indicating repulsive interaction among the electrons . coupled co and singlet formation occur also in the sp state in 1d @xmath11 systems @xcite . a key difference from the sp transition in @xmath5 is that for @xmath11 , a mi transition first occurs at a intermediate temperature , followed by the sp transition at low temperature . for @xmath11 , the mi leads to either a bond - dimerized or to a charge - ordered wc state with equal bond lengths @xcite . provided the n.n . coulomb interaction is not too strong , a sp transition occurs from either insulating state @xcite , resulting in a ground state with period-4 co @xmath460110@xmath46 . this state is the simplest realization of the pec and may visualized as a second dimerization of dimer units of molecules ; the singlet bond giving the spin gap ( sg ) forms _ between _ adjacent dimer units . the intermediate temperature bond - dimerized states , wc state , and pec state with sg are all found experimentally in quasi-1d cts @xcite . zigzag ladder systems , coupled two - stack systems in which each site on one stack is coupled to two sites on the other stack , are a second realization of the pec state . cts zigzag ladder materials that are @xmath11 have been found with spin - gap transition temperatures much larger than in 1d @xmath11 sp materials @xcite . the insulating ground state in this case may be understood again as a pec state occurring in a zigzag ladder lattice , with singlet bonds oriented between the two chains @xcite . unlike the 1d case , bond orders are now modulated in several lattice directions , leading to a larger sg than for the 1d pec case @xcite . interestingly , in both the linear chain and the zigzag ladder , the @xmath11 pec is obtained by simply removing alternate spin - singlet bonds from the corresponding @xmath5 vbs , and replacing them with pairs of vacancies .      in this paper we will focus on ground state solutions of eq . [ ham ] in 2d in the presence of variable lattice frustration . the lattice we choose is a 2d square lattice with a single frustrating bond , as shown in fig . [ lattices](a ) . thus @xmath21 within eq . [ ham ] for this lattice . in most of the results we will present , @xmath47 , although we will also consider @xmath48 in some cases . energies will be given in units of @xmath49 . we will take the frustrating bond @xmath50 in the range @xmath51 , covering the wide region between the unfrustrated square lattice ( @xmath52 ) and the nearly isotropic triangular lattice ( @xmath53 ) . for the inter - site e - p coupling , unless denoted otherwise we choose @xmath54 and @xmath55 , with similarly identical spring constants @xmath56 . for all calculations we assume periodic boundary conditions . our calculations are largely for @xmath57 , but variable @xmath58 . in reference we presented limited numerical results for a select set of coulomb interaction parameters ( @xmath59 , @xmath60 , @xmath61 ) demonstrating transition from nel antiferromagnetism to the pec state in this lattice when @xmath44 exceeds a critical value @xmath62 . for completeness and for giving an introduction to the various competing states in 2d we briefly review these results here . for small @xmath44 , the self - consistent solution of eq . [ ham ] gives spontaneous dimerization along the @xmath63 axis as shown in fig . [ lattices](b ) . the dimerized lattice state is effectively @xmath64-filled with one carrier per dimer and has nel afm order between dimers for finite @xmath6 . the nel order is very clearly observable from the spin - spin correlations calculated for @xmath65 clusters @xcite . importantly , the charge density @xmath66 of all sites is exactly 0.5 . as @xmath44 increases , frustration reduces the strength of the antiferromagnetic correlations . provided the n.n . coulomb interaction @xmath34 is not too strong ( see below ) , at a critical @xmath67 antiferromagnetic correlations disappear and charge disproportionation develops , with the charge densities within each dimer becoming inequivalent . this is shown in fig . [ lattices](c ) . in this state , the charge densities follow the pattern @xmath461100@xmath46 along the @xmath63 and @xmath68 directions , and the pattern @xmath461010@xmath46 along the @xmath69 direction . the bond distortion is period-4 along @xmath63 and period-2 along @xmath69 . the strongest bond orders @xmath70 occur between adjacent charge - rich ` 11 ' sites in the @xmath68 direction . importantly , the spin - spin correlations change dramatically for @xmath71 . they are strongly negative between the bonded ` 11 ' sites along the @xmath68 direction ( see fig .  [ lattices](c ) ) , and are nearly zero between either member of the pair and all other sites , indicating the formation of spin - singlet bonds @xcite . any of the observables @xmath43 , bond order between the charge - rich sites , or z - z spin correlations , @xmath72 , may be used as order parameters for the pec state @xcite .    in section [ results ] we present further details of the pec phase and the full parameter dependence of eq . [ ham ] , with the goal of demonstrating that _ ( i ) the transition to the pec that we are interested in is driven by quantum effects due to frustration only ; and ( ii ) the pec occurs over a broad region of parameter space . _ given the number of parameters in eq . [ ham ] , it should be relatively easy to generate co driven by specific ( presumably artificial ) choices of @xmath34 . such classical results would be uninteresting . we therefore consider several distinct choices of coulomb interactions : ( i ) @xmath73 and all @xmath74 , ( ii ) @xmath73 , @xmath75 , and @xmath61 , ( iii ) @xmath73 and @xmath76 . pec formation occurs in all of these parameter regions . we also show that for sufficiently strong n.n . coulomb interactions corresponding to parameter region ( ii ) , the wc phase with checkerboard co ( fig . [ lattices](d)-(e ) ) is the ground state of eq . we also will consider other modifications of the basic lattice , viz . , sign of @xmath44 opposite site to @xmath49 , and @xmath77 . finally , we will argue that our results are not consequences of finite size effects and are to be expected in the thermodynamic limit . figs .  [ bigplot](a)-(d ) show the charge disproportionation @xmath79 , bond order @xmath80 , n.n . z - z spin - spin correlation @xmath72 , and spin gap @xmath81 as a function of @xmath44 . @xmath81 is defined as the excitation energy from the ground @xmath15 state to the lowest @xmath45 state . @xmath6 here is finite but all @xmath82 terms are zero . sites @xmath23 and @xmath24 in figs .  [ bigplot](b)-(c ) correspond to two ` 1 ' sites in the pec state connected by a @xmath44 bond ( filled circles connected by heavy lines in fig . [ lattices](c ) ) . as with the results in reference which included the @xmath83 and @xmath84 interactions , for small @xmath44 the charge density is 0.5 on all sites , and antiferromagnetic order can be seen in the spin - spin correlations ( not shown here , see fig . 3(a ) in reference ) . nonzero @xmath62 in fig . [ bigplot ] is a consequence of the nature of the diagonal bonds in fig . [ lattices](b ) ; the diagonal bonds inside each plaquette with two strong dimer bonds actually strengthen the afm , and only the inter - plaquette diagonal bonds have a frustrating effect . at small @xmath44 these two effects appear to cancel , and there is only a weak effect on the afm .    for @xmath71 , @xmath43 becomes nonzero , with the charge pattern as shown in fig . [ lattices](c ) . similarly , the bond order between paired ` 11 ' sites increases abruptly , and the z - z spin - spin correlation between these sites becomes strongly negative , and nearly zero with all other lattice sites ( see reference ) , indicating formation of a singlet bond . although @xmath81 is nonzero in all cases in a finite cluster , we nevertheless see a large jump in @xmath81 at @xmath62 , also indicating spin - singlet formation . the increase in bond order strength and strength of spin - spin correlation clearly follow the same pattern as @xmath79 . these results show that the n.n . coulomb interaction is not essential for formation of the pec state . unlike the wc phase where co is driven by the n.n . coulomb interaction , the pec state is a consequence of geometric lattice frustration . increasing @xmath6 moves @xmath62 to larger @xmath44 : @xmath6 tends to strengthen the afm phase and therefore increasing @xmath6 is expected to make the afm order persist for stronger lattice frustration . the transition also becomes more discontinuous to changes in @xmath44 as @xmath6 increases , suggesting it may be continuous for small @xmath6 and first order for large @xmath6 .    in both the 1d and in the zigzag ladder lattice , the pec state occurs unconditionally even in the noninteracting limit ( @xmath85 ) for any finite e - p coupling . in these two cases , the unconditional occurrence of pecs is a consequence of simple nesting . in contrast , in the isotropic @xmath11 2d band considered here , the lack of nesting forbids an unconditional peierls transition . in agreement with this , we found that for @xmath86 , the pec phase did not occur . instead , the self - consistent calculations converged to disordered states with no clear charge pattern . this is an indication that in the thermodynamic limit , the preferred ground state is one of uniform charge . this result is reminiscent of that in quantum spin systems : the simplest vbs transition , the sp transition , can be predicted from nesting behavior in the 1d xy model following jordan - wigner transformation @xcite . this is , however , not true in 2d frustrated spin systems . we next consider the effect of n.n . coulomb interactions @xmath75 , but @xmath61 . figs . [ bigplot](e)-(h ) show the same order parameters as in figs . [ bigplot](a)-(d ) for @xmath88 . comparing the data with and without @xmath82 , the effect of moderate @xmath82 is to _ strengthen _ the pec state the magnitude of all order parameters increase when @xmath89 for a fixed value of @xmath6 . for fixed @xmath6 , the afm pec boundary @xmath62 also moves to smaller @xmath44 with increasing @xmath82 . [ uv - phase ] shows the phase diagram in the @xmath44-@xmath6 plane for both the @xmath90 and @xmath89 cases . here , for each value of @xmath6 , @xmath44 was varied until the afm pec transition occurred , and the first @xmath44 where @xmath43 became nonzero was taken as the afm  pec boundary . one expects that when @xmath82 is above a critical value @xmath91 , the @xmath461100@xmath46 pec co will give way to the checkerboard wc state . in 1d , this transition occurs exactly at @xmath92 in the limit @xmath12 , and at a larger @xmath91 for finite @xmath6 @xcite . previous exact diagonalization for a 2d cluster@xcite ( without however e - p interactions or @xmath44 as considered here ) found @xmath93 for @xmath94 , and showed that @xmath91 increases when @xmath95 . [ wcplot ] shows the evolution of @xmath96n and diagonal bond orders with @xmath82 for @xmath97 . while difficult to see in fig . [ wcplot ] due to the choice of axis scales and parameters , in the pec region @xmath43 increases with increasing @xmath82 . for the parameters of fig . [ wcplot ] ( @xmath6=6 , @xmath98 , @xmath99 , @xmath100 , @xmath101 ) the charge order pattern changes from the pec ( fig .  [ lattices](c ) ) to the checkerboard wc ( fig .  [ lattices](d)-(e ) ) at @xmath102 . in addition , @xmath79 increases sharply when entering the wc phase . at larger @xmath103 a slight cusp occurs in the @xmath43 versus @xmath82 plot . at the same time , the pattern of bond orders changes : for @xmath104 in the wc phase the bond orders alternate strong - weak along the @xmath68 direction ( as shown in fig . [ lattices](e ) ) , while for @xmath105 the bond orders along @xmath68 are uniform ( fig . [ lattices](d ) ) . within the wc phase region there are therefore two sub - phases : a phase with equal length bonds in the diagonal @xmath68 ( @xmath44 ) directions , and a phase in which these bonds become dimerized . the added bond dimerization in the diagonal direction will result in a spin gap , and we denote this phase as the wigner crystal  spin gap ( wc - sg ) phase . a similar spin - gapped wc phase can be found in a small region of parameter space in the 1d model @xcite . [ vt - phase](a ) shows the resulting phase diagram in the @xmath44-@xmath82 plane for @xmath6=6 . the @xmath91 we find in the @xmath52 limit is slightly larger ( @xmath97 , @xmath106 ) than the results of reference ( @xmath94 , @xmath107 ) ; however , both the smaller @xmath6 here as well as the e - p coupling in eq . [ ham ] would be expected to increase @xmath91 . unlike the afm , pec , and wc phases , the wc - sg phase is limited to a relatively narrow range of parameters . larger lattice calculations are needed to confirm whether the wc - sg phases persists in the thermodynamic limit . the @xmath109 interaction destabilizes the checkerboard - pattern wc , leading to a metallic phase in the absence of e - p interactions @xcite . here we will consider parameters @xmath76 . for eq.[ham ] without e - p interactions , @xmath94 , and @xmath110 , exact diagonalization found@xcite that in this case a metallic phase exists for @xmath82 up to at least @xmath1115 . charge fluctuations within the metallic phase adjacent to the wc were speculated to cause a co - to - sc transition @xcite . in our calculations , summarized in the phase diagram in fig . [ vt - phase](b ) for @xmath97 , we also found that the wc phase does not occur , but rather than being metallic the system is insulating  either afm at small frustration or pec at large frustration . in this case pec transition can occur over a wide range of lattice frustration , @xmath112 . within the pec phase the co pattern remains the same for all @xmath82 , although @xmath43 increases with @xmath82 as in the @xmath61 case considered in the previous section . although our calculations are for one value of @xmath109 only , fig . [ vt - phase](b ) suggests that the pec region is broadened relative to that in fig . [ vt - phase](a ) for any @xmath113 . as shown in the previous section , variation of coulomb interactions can cause a substantial variation in the extent of frustration needed to form the pec state . next we show the effect of varying the one - electron parameters in eq . [ ham ] , @xmath114 and @xmath27 .    due to the lack of particle - hole symmetry in the anisotropic triangular lattice , differences might be expected when @xmath44 is taken as negative . however , as fig . [ uv - phase ] shows , we found only a small variation in the afm pec phase diagram when the sign of @xmath44 is changed . this is consistent with the expected mechanism for spin frustration in an effectively @xmath64-filled band : the frustrating exchange interaction is proportional to @xmath115 , so reversing the sign of @xmath44 should only change the effective frustration at higher order .    fig . [ uv - phase ] also shows the effect of changing @xmath116 . as expected for a cooperative transition , stronger e - p coupling increases the size of the pec region . the effect of anisotropy of the inter - site e - p interaction is shown in fig . [ txty - alpha - phase](a ) , where @xmath117 is fixed at 1.1 and @xmath118 is varied . increasing either @xmath117 or @xmath118 separately strengthens the pec , as shown in fig . [ txty - alpha - phase](a ) , where the @xmath119 ratio is varied . pec phase diagram is relatively insensitive to the ratio of @xmath120 . in fig . [ txty - alpha - phase](b ) , @xmath121 is fixed at 1 and the value of @xmath122 is varied  the resulting @xmath62 does not depend sensitively on the choice of @xmath120 .    summarizing sections [ results](a)-(d ) , we see the pec state in our calculations for a wide range of parameters . in every case , we first arrived at parameters that placed the system in the afm or wc phases in the @xmath52 limit , and then varied only @xmath44 . thus the transition to pec is a consequence of frustration alone . with variation of e - p coupling and coulomb interactions , the amount of frustration needed to drive the afm pec transition can vary over a sizable range of frustration . we will discuss the implications of this further in relationship to the cts materials further in section [ discussion ] . we have performed several checks on our calculations that indicate that the pec state found in our numerical calculations is an intrinsic property of eq . [ ham ] and _ not _ induced by finite size effects .    _ ( i ) noninteracting bandstructure : _ one common finite - size effect in numerical calculations are changes in the fermi - level degeneracy or level crossings of the noninteracting system . in the lattice of fig . [ lattices](a ) the fermi level degeneracy does not change throughout the range @xmath123 , remaining 2-fold degenerate throughout this range . this degeneracy is broken by @xmath63-axis dimerization , shown in fig . [ lattices](b ) , giving a nondegenerate fermi level for @xmath123 . at @xmath53 the degeneracy at the fermi level increases to 6-fold ; in the presence of interactions the ground state can become triplet ( @xmath124=1 ) for @xmath125 . hence we stop at @xmath126 where the ground state is @xmath15 .    _ ( ii ) interactions : _ as mentioned above , no transition occurs for @xmath85 . this further indicates that the transition is not a feature of the single - particle bandstructure .    _ ( iii ) commensurability : _ we have verified that the pec state does not occur for electron densities different from @xmath11 ; for example , no transition to pec or any `` nonmetallic '' state occurs for 6 or 10 electrons on the @xmath127 lattice . in the following we discuss how our theory applies to real materials , and may even give insight to the mechanism of correlated - electron superconductivity in @xmath11 materials . the superconducting organic cts share many characteristics of other strongly - correlated superconductors , in particular the high - t@xmath128 cuprates , including reduced dimensionality and the presence of afm near sc . at the same time , sc in the cts occurs under pressure at a constant carrier density of @xmath11 rather than under the influence of doping . a variety of exotic insulating states _ in addition to afm _ , including co @xcite and spin - gapped states @xcite , as well as possible qsl states @xcite , are proximate to the superconducting state in the cts . our work shows that only the afm phase is described by the dimer mott - hubbard model . since in all cases the experimental systems are structurally related , with identical or near - identical molecular components , we believe that the same mechanism of sc should apply to them . in reference we had pointed out how the pec concept can perhaps lead to such a unifying theory . here we expand on this theme . we briefly review here experimental evidence for pec formation in several 2d cts families .    _ ( i ) @xmath129-(et)@xmath1x : _ co corresponding to the pec and spin gap are found in the @xmath129-(et)@xmath1mm@xmath130(scn)@xmath131 family @xcite . in @xmath132=rbzn , the co occurs below the mi transition at @xmath133190k , while the sg appears below 20k @xcite . the charge order pattern in the co phase below the mi transition has been experimentally determined for @xmath132=rbzn and follows a horizontal stripe pattern ( see fig .  6 in reference ) . the horizontal co is definitely not the wc . rather , the co pattern is precisely as expected in the pec , with @xmath461100@xmath46 co along the two directions of largest hopping ( the @xmath134-directions in the @xmath129-(et)@xmath1x lattice ) , and @xmath461010@xmath46 order along the direction of weakest hopping ( @xmath135-direction in @xmath129-(et)@xmath1x ) . experiments have revealed that with decreasing temperature , the @xmath135-axis lattice parameter decreases @xcite . the decrease in the lattice parameter implies increased carrier hopping in this direction and therefore increased frustration within our theory , giving the transition to the singlet pec and observed spin gap .    _ ( ii ) @xmath116-(et)@xmath1x : _ the crystal structure of @xmath116-(et)@xmath1x is quite similar to that of @xmath129-(et)@xmath1x . the existence of a sg opening below 136k has been known for some time in @xmath116-(et)@xmath1i@xmath2 @xcite . below the 136k transition co is found which has been confirmed to be of the same pattern as in @xmath129-(et)@xmath1x ; see for example fig .  2 in reference .    _ ( iii ) @xmath31-(meso - dmet)@xmath1pf@xmath136 : _ this 2d cts exhibits a pressure - induced transition from co to sc @xcite . while the charge order pattern in this cts is referred to as `` checkerboard '' by the authors , the checkerboard pattern refers to meso - dmet _ dimers _ as units . in terms of meso - dmet _ monomer _ units , the co pattern is the same as the pec in fig . [ lattices](c ) , with @xmath461100@xmath46 in two directions and @xmath461010@xmath46 along the third direction ( see fig .  2 in reference . )    _ ( iv ) @xmath137-x[pd(dmit)@xmath1]@xmath1 : _ in this family the materials dmit molecules are arranged in dimers . the frustration varies with the cation @xmath138 , with the least frustrated in the series showing afm order @xcite . among the materials with larger frustration , @xmath139 has a sg transition at 25k to what has been described as a vbs state @xcite . the experimentally determined bond and charge distortion patterns ( see fig . 3(b ) in reference ) are exactly as expected for the pec , with period 4 charge and bond distortions along what is the @xmath63 axis in fig . [ lattices ] . the intradimer charge disproportionation in particular argues against the sg state from being a simple vbs , which would require equal charge densities on all the molecules . there has been much recent interest in these cts @xcite specifically because they present possible realizations of the long awaited qsl @xcite . in both cases the materials have nearly isotropic triangular lattices of dimer unit cells ( corresponding to fig . [ lattices](b ) with @xmath53 ) within an effective @xmath5 model . in @xmath0-(et)@xmath1cu@xmath1(cn)@xmath2 ( hereafter @xmath0-cn ) the estimate for the heisenberg exchange integral between n.n . dimers is @xmath140220250 k @xcite . @xmath141h nmr experiments find absence of long range magnetic order down to 32 mk @xcite . very similar behavior is also seen in etme@xmath2sb[pd(dmit)@xmath1]@xmath1 ( hereafter dmit - sb ) @xcite . the insulating ground state in @xmath0-cn is close to sc , transition to superconductivity occurring under moderate pressure @xcite . sc is found in the dmit family as well @xcite . recent experiments in both @xmath0-cn and dmit - sb have found peculiarities that appear to be unexpected within qsl theories . below we list experiments that seem to indicate that apparent qsl behavior at low temperatures is giving way to a `` hidden order '' , and perhaps even charge disproportionation , which by itself would be against spin - only models . \(i ) a second order phase transition is seen at 6 k in @xmath0-cn in measurements of heat - capacity c@xmath142 @xcite , @xmath143c nmr relaxation rate 1/t@xmath144 @xcite , and lattice expansion coefficients @xcite . the last experiment finds strong lattice effects at the transition , indicating possible role of charge degrees of freedom @xcite . a symmetry - breaking and/or topological ordering transition at t @xmath145 1 k has also been observed in dmit - sb @xcite . \(ii ) the specific heat c@xmath142 in @xmath0-cn is linear in t for t between 0.75 and 2.5 k , indicating a gapless energy spectrum @xcite . the sommerfeld coefficient @xmath146 is nonzero and large even at @xmath14775 mk @xcite . equally perplexingly , c@xmath142 is independent of magnetic field up to 8 t , indicating absence of zeeman coupling of spins to the field @xcite . in contrast , thermal conductivity measurements down to 80 mk indicate a spin gap @xcite . \(iii ) the temperature dependence of the thermal conductivity of dmit - sb suggests gapless mobile excitations , but magnetic field dependence of the thermal conductivity again indicates a gap @xcite . taken together , ( ii ) and ( iii ) suggest gapless spin - singlet excitations but gapped spin - triplet excitations in @xmath0-cn and perhaps also dmit - sb . it is conceivable that the gap in dmit - sb is nodal @xcite . \(iv ) measurements of dielectric response in @xmath0-cn have shown increasing and frequency - dependent dielectric constant below 60 k , and possible antiferroelectric ordering of dipoles at t@xmath1486 k @xcite . the latter requires unequal site charges on the molecules within the dimer unit cells @xcite . it may be relevant in this context that @xmath143c - nmr experiments on both @xmath0-cn @xcite and dmit - sb @xcite find unusual line broadenings at low t that can not be ascribed to disorder @xcite , which might also indicate charge disproportionation . similar line broadening at low t in etme@xmath2p[(dmit)@xmath1]@xmath1 occurs at the transition to the pec @xcite , which in turn gives way to superconductivity under pressure @xcite . although it is as yet not entirely clear whether or not existing spin - liquid models @xcite with appropriate modifications can explain the above anomalies , it appears that for both @xmath0-cn and dmit - sb , spin degrees of freedom alone can not describe the low temperature ( below 6k in @xmath0-cn and below 1k in dmit - sb ) properties . rather , any description of the ground state must involve charge as well as spin degrees of freedom . assuming that the above experiments and their interpretations are correct , the proper theoretical model should have built - in significant electron - lattice coupling and should lead to charge disproportionation and excitation energy spectrum with gapless singlet excitations and gapped spin excitations . we believe that the highly frustrated @xmath11 model satisfies all of the above criteria , in addition to providing the starting point for a theory of superconductivity in the cts ( see below ) . t - linear specific heat but gapped magnetic susceptibility were noted in nonmetallic vanadium bronzes@xcite as far back as 1978 , and at the time was considered to be a distinctive proof for bipolarons . the nonzero spin - singlet degeneracy within this model comes from tunneling motion of the bipolarons , causing them to `` flip flop '' between equivalent configurations @xcite . we propose that a similar mechanism is at play in the present case at large @xmath44 . particularly in @xmath0-cn the lattice structure and the orientations of molecules are such that the spin - singlet bonds , which occur between monomers belonging to two neighboring dimers can flip flop between the monomers ( see fig .  4 in reference . ) within our proposed picture for @xmath0-cn , the transition at 6k is to the spin - singlet charge - disproportionated state , with short range fluctuating order . magnetic excitations even in this state , however , require breaking the spin - singlet bonds . note that the observed difference between @xmath0-cn and dmit - sb is expected within the @xmath11 model , since the simple description as a triangular lattice of dimers is no longer enough and the detailed couplings between the monomers in the materials are indeed different because of their different crystal structures .      as mentioned above , numerical results for @xmath149 are not useful due to the highly degenerate ground state becoming spin - triplet in our finite clusters . we have suggested elsewhere that the occupied spin - singlet bonded ` 11 ' sites of the pec can be thought of as _ effective _ single sites doubly occupied by charge carriers , and similarly the pairs of ` 00 ' vacancies can be thought of as single vacant sites @xcite . such a mapping would transform the pec to an effective checkerboard co with alternate sites ( in the square lattice representation ) occupied by double occupancies and vacancies . the effective hamiltonian that describes the checkerboard co in this case is a @xmath5 extended hubbard model with weak _ attractive _ @xmath6 whose origin is the exchange interaction that stabilizes the singlet bond within the original @xmath11 hamiltonian . the n.n . interaction within the effective hamiltonian remains repulsive to simulate the checkerboard ordering of the effective doubly occupied sites . while such a mapping is not rigorous , similar mapping of n.n . spin - bonded sites to double occupancies has been routinely used in the literature on bipolaron models @xcite . the difference between our work and traditional bipolaron models is that the spin - singlet bonding within our work is driven primarily by afm correlations , while within the bipolaron models it is a consequence of effective attraction due to the overscreening of the e - e repulsion by e - p interactions @xcite . we have investigated the consequences of stronger frustration@xcite in the @xmath11 anisotropic triangular lattice within the effective @xmath5 extended hubbard model with attractive @xmath6 @xcite . we found a co - to - sc transition within the effective model @xcite , suggesting a pec - to - sc transition within the @xmath11 model with repulsive interactions . although this result is not a proof of transition to sc within the repulsive @xmath11 model , it nevertheless is instructive and provides the direction for future research . demonstration of sc within the actual model will require further work . what is interesting though is that the proposed scenario can give a unified approach to sc in all organic cts , irrespective of whether the insulating state proximate to sc is afm @xcite , co @xcite or vbs @xcite . recall that within existing mean field theories the afm - to - sc transition is driven by spin fluctuations @xcite , while the co - to - sc transition is driven by charge fluctuations @xcite . even if we ignore that recent precise numerical calculations @xcite have demonstrated the absence of sc within the proposed spin - fluctuation models in this context ( thereby raising doubts also about mean - field theory of charge - fluctuation mediated superconductivity ) , different mechanisms of sc for structurally related materials with identical or near - identical molecular components appear to be unrealistic . within our proposed scenario , the charge - ordered or vbs systems that exhibit sc are pecs in the semiconducting state , while the afm systems under pressure transform to pecs first and then to sc ( although due to actual crystal structures the `` width '' of the pec region could be narrow to vanishing within the latter ) . the sc phase within this scenario is a paired - electron liquid . there is considerable overlap between these ideas and the one proposed by moulopoulos and ashcroft for the continuous electron gas @xcite . the thrust of our work has been to understand within a unified theoretical approach the variety of exotic insulating states that are proximate to the superconducting state in 2:1 cationic or 1:2 anionic 2d cts . we have shown here that the peculiarities of these materials originate from the unique behavior of @xmath11 in the presence of both strong e - e interactions and lattice frustration . it is conceivable that the complex behavior of apparently unrelated inorganic families can be understood within the same broad theoretical approach . we list two such classes of materials below to point out the hitherto unnoticed similarities between them and @xmath11 cts . layered cobaltates na@xmath16coo@xmath1 have attracted wide attention because of their 2d structure , tunable carrier concentration , and the occurrence of sc @xcite . the co ions form a 2d triangular lattice , are in their low - spin state , and their valence ranges from co@xmath150 at @xmath151 to co@xmath152 at @xmath153 . charge carriers are @xmath154 holes on the co@xmath152 sites and the hole density @xmath155 . trigonal distortion splits the occupied @xmath156 orbitals into @xmath157 and @xmath158 orbitals . lda calculations have suggested that although for large @xmath63 the @xmath157 orbitals occur below the fermi level and an @xmath158-only description is valid , this description breaks down at small @xmath63 where @xmath157 orbitals can be nearly degenerate @xcite . in contrast , correlated - electron calculations find that the @xmath158 @xmath157 energy separation is positive and relatively large for all @xmath63 @xcite , suggesting that low energy excitations can likely be described within @xmath158-only single - band models . the temperature dependent magnetic susceptibility @xmath159 shows a peculiar @xmath10-dependence within the family , with small @xmath10 ( large @xmath63 ) exhibiting strongly correlated behavior and large @xmath10 ( small @xmath63 ) exhibiting weakly correlated behavior @xcite . we have recently shown that @xmath10-dependent @xmath159 , exactly as seen in the cobaltates , is expected within the single - band extended hubbard model on a triangular lattice @xcite . equally interestingly , @xmath159 behavior in na@xmath16coo@xmath1 is very similar to that in the family of cts as a whole , where also @xmath159 shows a systematic @xmath10-dependence that is understood within the single - band extended hubbard model @xcite . it is tempting to compare now the superconducting states in hydrated cobaltates and cts with this apparent similarity in mind . although superconductivity in na@xmath16coo@xmath1 @xmath160 @xmath69h@xmath1o occurs at @xmath161 @xcite , it is now established that the co - ion valency here is determined not only by the na content , but also by h@xmath2o@xmath8 ions . there have been several reports that sc here occurs over a very narrow range of hole density @xmath10 , and that maximum @xmath162 occurs at or very close to co - ion valency 3.5 + , corresponding to @xmath11 @xcite . if this is confirmed from future experimental work , it would appear that like the cts , cobaltates are yet another example of a frustrated 2d @xmath11 superconductor , suggesting that the mechanism of sc in the two families is related . the b sublattice in spinel compounds ab@xmath1x@xmath131 form a frustrated three - dimensional ( 3d ) pyrochlore lattice and usually consist of transition metal cations that possess partially filled @xmath156 @xmath163-orbitals . for integer occupancies of @xmath163-electrons per @xmath164 cation the geometrical degeneracy of the underlying lattice is often lifted by orbital ordering ( oo ) , leading to formation of spin - singlet dimers . only four of the many spinel compounds are superconducting , of which three have effective carrier density @xmath11 : liti@xmath1o@xmath131 , curh@xmath1s@xmath131 , and curh@xmath1se@xmath131 . in liti@xmath1o@xmath131 there is one @xmath163-electron per two ti@xmath165 ions ; in curh@xmath1s@xmath131 and curh@xmath1se@xmath131 the rh@xmath165 ions have average @xmath163-hole occupancy of @xmath64 . jahn - teller instability or oo can lead to occupancy of the same @xmath156 orbitals , making the filled bands exactly @xmath4-filled , as in the superconducting organic cts ( the apparently different spinel superconductor cuv@xmath1s@xmath131 , also has noninteger number of @xmath163-electrons per v@xmath165 ion . )    yet another similarity between the organics and the @xmath11 spinels is the proximity of the superconducting state to exotic semiconducting states . thus cuir@xmath1s@xmath131 and lirh@xmath1o@xmath131 , both isoelectronic with curh@xmath1s@xmath131 , undergo mi transitions that are accompanied by co. in cuir@xmath1s@xmath131 the ir ions are charge - ordered as ir@xmath150-ir@xmath150-ir@xmath152-ir@xmath152 along specific directions @xcite . this would correspond exactly to the @xmath460011@xmath46 co in our notation here . as in the 2d pec , these inorganic 3d systems are spin - gapped due to the formation of ir@xmath152-ir@xmath152 singlet bonds . it is interesting to recall that similar singlet bonds between ti@xmath150 ions had been proposed @xcite many years back within bipolaron theories of ti@xmath131o@xmath166 and liti@xmath1o@xmath131 . as explicitly shown in our work here , the singlet bond formation is a natural consequence of frustration and @xmath4-filling . work is currently in progress to extend the pec concept to the checkerboard lattice , thought to be 2d equivalents of the pyrochlore lattice , multiple orbitals per site @xcite . in summary , there is an extraordinarily strong tendency to form spin - singlets in systems with charge carrier concentration precisely @xmath64 . naturally , in @xmath11 this spin - singlet state is accompanied by co. the stability of the pec derives from the commensurability of the pec at @xmath11 . in the anisotropic triangular lattice , the pec consists of charge arrangements @xmath461100@xmath46 in two directions and @xmath461010@xmath46 in the third direction . thus although @xmath11 in principle is incommensurate on the triangular lattice , `` separately commensurate '' periodic charge arrangements are nevertheless possible . we thank m. abdel - jawad , h. fukuyama , k. kanoda , and r. kato for helpful discussions . this work was supported by the us department of energy grant de - fg02 - 06er46315 . rtc thanks the university of arizona for support while on sabbatical .
we present a study of the effects of simultaneous charge- and spin - frustration on the two - dimensional strongly correlated quarter - filled band on an anisotropic triangular lattice . 
 the broken - symmetry states that dominate in the weakly frustrated region near the rectangular lattice limit are the well known antiferromagnetic state with in - phase lattice dimerization along one direction , and the wigner crystal state with the checkerboard charge order . for moderate to strong frustration , however , the dominant phase is a novel spin - singlet paired - electron crystal ( pec ) , consisting of pairs of charge - rich sites separated by pairs of charge - poor sites . 
 the pec , with coexisting charge - order and spin - gap in two dimension , is the quarter - filled band equivalent of the valence bond solid ( vbs ) that can appear in the frustrated half - filled band within antiferromagnetic spin hamiltonians . 
 we discuss the phase diagram as a function of on - site and intersite coulomb interactions as well as electron - phonon coupling strength . 
 we speculate that the spin - bonded pairs of the pec can become mobile for even stronger frustration , giving rise to a paired - electron liquid . 
 we discuss the implications of the pec concept for understanding several classes of quarter - filled band materials that display unconventional superconductivity , focusing in particular on organic charge transfer solids . 
 our work points out the need to go beyond quantum spin liquid ( qsl ) concepts for highly frustrated organic charge - transfer solids such as @xmath0-(bedt - ttf)@xmath1cu@xmath1(cn)@xmath2 and etme@xmath2sb[pd(dmit)@xmath1]@xmath1 , which we believe show frustration - induced charge disproportionation at low temperatures . 
 we discuss possible application to layered cobaltates and @xmath4-filled band spinels .