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in our previous paper devoted to the mean - field analysis of the phase structure of the higgs - yukawa systems @xcite we have concluded that intersection of the critical line separating the ferromagnetic ( fm ) and paramagnetic ( pm ) phases with the line separating the antiferromagnetic ( am ) and pm phases always leads to appearing the ferrimagnetic ( fi ) phase , and that the slopes of the critical lines in the point of their intersection ( denote it @xmath0 has been employed in @xcite for another point . ] ) are continuous .
these statements have been criticized recently in @xcite where it was argued that the slopes are in general discontinuous and that the fi phase may not appear at all in the point @xmath0 .    in this paper
we demonstrate that in reference to the systems considered in @xcite this criticism is justified only in part .
namely , considering within the mean - field approximation the weak coupling regime for the u(1 ) systems we show that in all the cases when the intersection of the fm - pm and am - pm critical lines occurs , the fi phase does appear , and that the slope of the am - pm  fi - fm line is always continuous .
the reason for the latter fact is that the critical exponents of the magnetization ( v.e.v . of the higgs field ) in this approximation is grater than @xmath1 .
the slope of the fm - pm  fi - am line generally changes in the point @xmath0
. however the magnitude of this change depends strongly on the number of the fermion flavours .
we show that for given system ( form of the lattice fermion action ) there exists such number @xmath2 ( generally noninteger ) , that for the number of the fermion flavours @xmath3 the slope of this line is continuous , too .
the qualitative picture for the phase diagrams is shown in fig .
1 . by a product
, we find that in the fi phase in the vicinity of the point @xmath0 the magnetization and the staggered magnetization align orthogonally to each other in the group space .
we want to emphasize that all the results of @xcite concerning the phase structure of the systems in the regions bordering on the pm phase are not affected .
the slopes of the critical lines in those regions are determined by the terms in the free energy quadratic in the mean fields .
such terms has been calculated in the ladder approximation developed in @xcite . for examination of
the region bordering on the fi phase one needs to know the free energy up to the fourth order in the mean fields @xcite . in this case
the summation of the ladder diagrams becomes too complicated and the approximation loses its advantages . at the same time , unjustified neglecting the ladder diagrams may lead to obviously incorrect results : for example , in the case of the su(2 ) system with the naive fermions the fi phase gets lost .
therefore we consider here the u(1 ) systems for which the ladder diagrams do not contribute to the free energy at least up to the terms quadratic in the mean fields .
we limit ourselves to the consideration of the weak coupling regime ; the examination of the strong coupling regime can be done easily in analogous way .
our basic point is that the free energy for the u(1 ) systems in four dimensions up to terms quartic in the mean fields has the form ( all the notations as in @xcite ) @xmath4 where coefficients @xmath5 to @xmath6 are functions of @xmath7 and @xmath8 , and @xmath9 , @xmath10
.    the coefficients @xmath5 to @xmath11 and @xmath12 are calculated directly from the eqs .
( 8)(15 ) of @xcite and read as @xmath13 + y^4 n_f
\int_p k^{2}(p ) k(p ) \cdot k(p+\pi ) , \nonumber \\
c^{s}(\kappa , y)&=&- \frac{3}{4 } + y^2 n_f \int_p
\left[k^{2}(p ) + k(p ) \cdot k(p+\pi)\right ] + \frac{1}{2 } y^4 n_f \int_p k^{2}(p ) k^{2}(p+\pi ) , \nonumber \\ c^{v}(\kappa , y)&=&- \frac{1}{2 } y^4 n_f \int_p k^{2}(p ) k^{2}(p+\pi ) , \nonumber \\
e(\kappa , y)&=&- \frac{9}{8 } - 12 \kappa + 3 y^2 n_f \int_p k(p ) \cdot k(p+\pi ) \nonumber \\ & & + \frac{3}{2 } y^4 n_f \int_p \left[2 \left(k(p ) \cdot k(p+\pi)\right)^2 - k^{2}(p ) k^{2}(p+\pi)\right],\end{aligned}\ ] ] where @xmath14 , so that @xmath15 , @xmath16 @xcite ( we omit the superscripts @xmath17 meaning the weak coupling regime ) .
the coefficients @xmath6 and @xmath18 in ( 1 ) can not be calculated by the term - wise integration of the standard weak coupling expansion of the logarithm of the fermion determinant : the integrals of the @xmath19 terms beginning from @xmath20 have infrared divergences , since @xmath21 at @xmath22 .
this is the reason for appearing the logarithmic term in ( 1 ) .
it it this term that differs our results from those of ref .
@xcite .    to estimate the coefficient @xmath6 , note that the contribution of the fermion determinant to the free energy at @xmath23 is given by the expression @xmath24.\ ] ]
the integral is finite and certainly depends on the form of the function @xmath25 .
the point , however , is that the term @xmath26 comes from the infrared region @xmath27 which is not sensitive to the detailed form of the function @xmath25 .
therefore , to find the coefficient @xmath6 we can consider the integral with the same infrared properties @xmath28 where @xmath29 is an ultraviolet cutoff . from
( 4 ) , taking into account the relation @xmath30 , we find @xmath31    since we are interested in the domain of the phase diagram where both @xmath32 and @xmath33 tend to zero , and therefore @xmath34 , one can neglect the term @xmath35 in ( 1 ) .
the fm - pm and am - pm critical lines are determined by the equations @xmath36 and @xmath37 , respectively , so that the coordinates of the point @xmath0 are @xmath38 } , \quad y^{2}_{b } = \frac{1}{n_f [ g(0 ) + g(\pi)]}.\ ] ]    in the vicinity of the point @xmath0 we have ( in the notations of @xcite ) @xmath39 with @xmath40 and @xmath41 or @xmath42 .    from ( 2 ) and ( 5 )
it follows that @xmath43 and @xmath44 .
hence , in the fi phase ( if it exists ) the mean fields @xmath32 and @xmath33 align in such a way that @xmath45 and @xmath46 , i.e. orthogonally to each other in the group space .
therefore , in the vicinity of the point @xmath0 the expression ( 1 ) for the free energy is reduced to @xmath47   \ , h^2 + \frac{1}{2 } [ b_{\kappa } ( \kappa - \kappa_b ) + b_y ( y - y_b ) ]   \ , h^{2}_{st } \nonumber \\ & & \mbox{}+ \frac{1}{4 } c_b \ , h^2 h^{2}_{st } + \frac{1}{24 } e_b \ , h^{4}_{st } + \frac{1}{24 } f_b \ , h^4 \ln h^2 + o(\delta^2),\end{aligned}\ ] ] where @xmath48 . from ( 9 ) necessary conditions for the existence of the fi phase
follow : @xmath49 ( provided @xmath50 , @xmath51 ) . hence , quite near the point b , where @xmath52 , the fi phase can exist in fact at any value of the coefficient @xmath53 . from ( 5 ) it follows that the third condition in ( 10 ) is satisfied always , and we shall demonstrate in the next section that the second condition for the systems considered in @xcite is satisfied , too .
now , following the consideration of ref .
@xcite setting up the relation between the slopes of the critical lines and the critical exponents of the magnetizations , one can easily find the slopes at the point @xmath0 .
consider first the @xmath54 critical line .
near the point @xmath0 it is determined by the equation @xmath55 from ( 9 ) it follows that nonzero value of the staggered magnetization is determined by the equation @xmath56.\ ] ] then , from ( 11 ) and ( 12 ) , in full correspondence with the results of ref .
@xcite , we have @xmath57    the things are different in the case of the @xmath23 critical line .
indeed , though this line is determined by the equation similar to ( 11 ) : @xmath58 the nonzero value of the magnetization is determined by the equation ( @xmath59 ) @xmath60.\ ] ] from ( 16 ) it follows that @xmath61 , i.e. the critical exponents of the magnetization is grater than @xmath1 .
therefore we have in this case @xmath62 i.e. the slope of the @xmath23 critical line does not change in the point @xmath0 .
it is seen now that the fi phase appears always , provided the conditions ( 10 ) are fulfilled .
the change of the slope of the @xmath54 critical line in the point @xmath0 is determined mainly by the value of @xmath53 : if @xmath63 the slope does not change ; if @xmath64 the domain with the fi phase gets narrow and in the limit @xmath65 it shrinks ; on the contrary , if @xmath66 the domain with the fi phase becomes wider and in the limit @xmath67 it occupies near the point @xmath0 all the region between the fi - fm and am - pm lines ( fig .
1 ) .    we shall demonstrate now that all the three variants : @xmath63 , @xmath68 , and @xmath66 are realised for the systems considered in @xcite .
the fm - pm and am - pm lines intersect in three of those systems : in the systems with the slac , weyl , and the mirror fermion actions . in these cases we have : @xmath69 , @xmath70 , @xmath71 for the slac action ; @xmath72 , @xmath73 , @xmath74 for the weyl action ; @xmath75 , @xmath76 , @xmath77 for the mirror fermion action ( the values of @xmath78 and @xmath79 are given in @xcite ) .
although @xmath80 depends on the number of flavours @xmath81 , in all these cases @xmath82 for any @xmath81 and therefore the fi phase appears in all of these systems .
the sign and the value of @xmath53 however depends strongly on @xmath81 : this is due to the fact that the @xmath83 and @xmath84 terms in @xmath53 are independent of @xmath81 , while the @xmath85 terms are @xmath86 . moreover , for given system there exists such number @xmath2 , that @xmath63 if @xmath3 , @xmath87 if @xmath88 , and @xmath66 if @xmath89 . for our systems
we find : @xmath90 the corresponding phase diagrams are shown qualitatively in fig
thus , we have demonstrated that in the u(1 ) higgs - yukawa systems with the slac , weyl , or mirror fermion actions the fi phase does appear in the weak coupling regime in the point @xmath0 . in the fi phase ,
at least quite near this point , the magnetization @xmath32 and the staggered magnetization @xmath33 align orthogonally to each other in the group space .
due to the fact that the free energy involves the logarithmic term , that , in turn , leads to the critical exponents of the magnetization @xmath32 grater than @xmath1 , the @xmath23 critical line ( the am - pm  fi - fm line ) is continuous together with its first derivative in the point @xmath0 .
the derivative of the @xmath54 critical line ( fm - pm  fi - am line ) is generally discontinuous in the point @xmath0 .
however the discontinuity depends strongly on the number of the fermion flavours @xmath81 , so that in each system there exist such number @xmath2 , that for @xmath3 the line is smooth , too .    to conclude ,
note that for the systems in which one has @xmath91 ( as it is , for example , for the naive fermions ) , the logarithmic terms in the free energy appear also for the staggered magnetization .
this means that the critical exponents for both @xmath32 and @xmath33 are grater than @xmath1 .
therefore , in the cases when the fm - pm and am - pm lines intersect in such systems , the slopes of both critical lines are continuous in the point @xmath0 .
this may occur , for example , in the case of the su(2 ) system with the naive fermions .
we admit , however , that to make definite conclusion on the existence of the fi phase in the su(2 ) systems a special investigation is necessary .
i am grateful to s.  tominaga for the numerical calculations of the integrals @xmath92 to @xmath93 and to j.  l.  alonso for a correspondence .
the work was partly supported by the russian basic research fund under grant 95 - 02 - 03868a .
99 s.  tominaga and s.  v.  zenkin , phys . rev . * d 50 * , 3387 ( 1994 ) j.  l.  alonso , ph .
boucaud , and a.  j.  van  der  sijs , report dftuz/95/07 , lpte  orsay-95/13 , hep - lat/9503006 s.  v.  zenkin , mod .
. lett . * a 9 * , 983 , ( 1994 )
we calculate within a mean - field approximation the slopes of the critical lines near the point of appearing the ferrimagnetic phase for the u(1 ) systems in the weak coupling regime . 
 it is demonstrated that the slope of one of the critical line is continuous , while change of the slope of the other depends strongly on the number of the fermion flavours . 
 we also find that in the ferrimagnetic phase near such a point the magnetization and the staggered magnetization align orthogonally to each other . 
 + pasc numbers : 11.15.ha , 05.70.fh , 14.80.bn    -0.6 in -1.7 in -10 mm -10 mm 6.5 in    * on behaviour of critical lines near ferrimagnetic *    * phase in higgs - yukawa systems *    1.0truecm    sergei v. zenkin    0.2 cm    institute for nuclear research of the russian academy of sciences    60th october anniversary prospect 7a , 117312 moscow , russia