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the simplest model of strongly interacting bosons is the bose - hubbard model ( bhm ) , which has been used to describe superfluid helium@xcite , cooper pairs in thin granular superconducting films@xcite and josephson junction arrays@xcite . much theoretical work has concentrated on the phase diagram of the bhm in zero magnetic field@xcite because of the technical problems associated with introducing an external magnetic field ( such as the sign problem in quantum monte carlo simulations ) . experimentalists on the other hand have concentrated on studying systems in an external magnetic field because the phase transition can be tuned by adjusting the magnetic field rather than changing the samples measured@xcite . we study the bhm on two - dimensional lattices in a perpendicular magnetic field by extending the strong - coupling perturbation theory for the field - free case@xcite . this theoretical technique can incorporate magnetic - field dependence in a straightforward manner and is useful in studying field - tuned transitions . we concentrate on pure systems in this contribution and do not include any effects due to disorder . we present zero - temperature phase diagrams , study excitation - gap energies , calculate the dynamical critical exponent @xmath0 for small fields , and compare our theoretical results with experimental ones . the bhm contains the key physics of a many - particle bosonic system with competing potential and kinetic energy effects . the typical zero - temperature phase diagram for the non - magnetic case shows incompressible mott - insulating ( mi ) phases surrounded by compressible superfluid phases ( sf ) @xcite . the insulator to superfluid transitions at the tip of the lobes ( where the density remains constant ) are driven by quantum phase fluctuations , while those at the sides of the lobes ( where the density varies ) are driven by density fluctuations , i.e. particle or hole excitations . introducing a magnetic field is expected to increase the region of the mi phase because the localizing effect on the itinerant bosons reduces the stability of the sf phase . we consider bosons with total spin @xmath1 . the only effect of a perpendicular magnetic field @xmath2 is then on the orbital motion of the bosons , which effects changes in the phase of the hopping - matrix @xmath3 between lattice sites @xmath4 and @xmath5 . by choosing a landau - gauge for the vector - potential @xmath6 , the bose - hubbard hamiltonian in an external magnetic field becomes @xmath7 where the hopping matrix is nonzero only between nearest neighbors and is given by @xmath8 this hopping matrix is hermitian because @xmath9 is real and @xmath10 . here the boson creation operator for lattice site @xmath4 is @xmath11 , @xmath12 is the corresponding number operator , @xmath13 is the on - site repulsion of the bosons , and @xmath14 is the chemical potential . we choose @xmath13 to be our energy scale and measure all energies in units of @xmath13 . the magnetic flux quantum is given by @xmath15 , and the magnetic flux per plaquette @xmath16 is a measure of the strength of the magnetic field @xmath2 ( where @xmath17 is the lattice spacing ) . the form of the zero - temperature phase diagram can be understood by starting from the atomic limit@xcite , where the hopping @xmath9 is zero and every site is occupied by a fixed number of bosons @xmath18 . the energy to add one boson onto a site with @xmath18 bosons is @xmath19 , so that there is a finite energy gap when @xmath20 and the system is an incompressible mott insulator . when @xmath21 , then all states with a density between @xmath18 and @xmath22 bosons per site are degenerate in energy , and the system becomes a compressible fluid . as the strength of the hopping matrix elements increases , the range of the chemical potential about which the system is incompressible decreases . the mott insulator phase disappears at a critical value of the hopping matrix elements ( which depends on the strength of the magnetic field and the density of the insulating phase ) and the system becomes a superfluid ( see fig.[phasesqtri ] ) . we provide a systematic study of the bhm in a magnetic field by examining the system in a perturbative expansion about the atomic limit with the boson kinetic energy acting as the perturbation . the paper is organized as follows : section ii describes the formalism used in the strong - coupling perturbation theory in the presence of a magnetic field . section iii presents the results for the phase diagrams and the excitation energies in a magnetic field , and section iv contains the conclusions . our procedure is to calculate the ground - state energy of the mi phase with @xmath18 bosons per site @xmath23 , and of the excited states in the charge sector with one extra boson @xmath24 and one extra hole @xmath25 , in a rayleigh - schrdinger perturbative expansion in the hopping matrix element @xmath9 ( the kinetic energy is chosen as the perturbative part of the hamiltonian ) . when the energy of the mi and the state with one extra boson are equal , the system undergoes a phase transition from the incompressible mi phase to the compressible sf phase with density larger than @xmath18 . the similar occurs when the state with one extra hole is degenerate with the mi phase ( except now the density of the sf phase is less than @xmath18 ) . the detailed formalism of the strong - coupling expansion for the ground - state energies has already been presented@xcite . the only modifications of the previous calculations needed here are to take into account the fact that the hopping matrix now has a complex phase and the changes required for the nonbipartite hopping matrix of the triangular lattice . the important parameter that enters the results is the minimal eigenvalue @xmath26 of the kinetic energy matrix @xmath27 which includes a factor @xmath28 that depends on the magnetic field . this parameter determines how the degeneracy is lifted in the first - order secular equation for the energy of the excited states in the charge sector . formally this solution of the minimal eigenvalue is identical to finding the band minimum in the hofstadter problem@xcite . the mott phase diagram is determined by the two mott phase boundaries one for the particle excitations [ where @xmath29 and one for hole excitations [ where @xmath30 . for each value of @xmath9 there is a critical value of the chemical potential where the system changes phase from an incompressible to compressible fluid . the upper and lower curves for the mott phase lobe are then described by this critical value of the chemical potential @xmath31 . the results of our expansion through third - order are @xmath32\epsilon t^3\nonumber\\               & + & 12\delta_{lat}n_0(\frac{31}{4}n_0 ^ 2+\frac{21}{2}n_0 + 3 )                     \cos(2\pi\phi)t^3+o(t^4)\nonumber\\     \mu_{h}(t)&=&n_0 - 1-\epsilon n_0 t+n_0(n_0 + 1)\epsilon^2 t^2-                      \frac{n_0 + 1}{2 }                      ( 5n_0 + 1)zt^2\nonumber\\               & + & n_0(n_0 + 1)[(2n_0 + 1)(-2-\epsilon^2)+z(\frac{25}{4}n_0                    + \frac{11}{4})]\epsilon t^3\nonumber\\               & -&12\delta_{lat } ( n_0 + 1)(\frac{31}{4}n_0 ^ 2 + 5n_0+\frac{1}{4 } )                     \cos(2\pi\phi)t^3+o(t^4 ) , \end{aligned}\ ] ] where @xmath33 or @xmath34 for the square or the triangular lattice , respectively , and @xmath35 is the corresponding number of nearest neighbors ( 4 or 6 , respectively ) . these results have been verified by both analytical small - cluster calculations and by numerical cluster expansions . the magnetic field appears in two places : ( 1 ) the magnetic field couples to the orbital current of the bosons as the particle or hole travels around a lattice plaquette and encloses the flux @xmath36 . in our third - order calculation , the orbital coupling only enters for the triangular lattice ( it enters at fourth order for the square lattice as four hops are required to enclose a plaquette ) . ( 2 ) the other effect of the magnetic field is to change the minimal energy of the extra particle or hole moving in the mott phase background . although the location of this mininum in the brillouin zone is gauge - dependent , its value is not ( thus @xmath37 ) . we consider a rational flux @xmath38 . to find the minimal eigenvalues @xmath26 of the hopping matrix @xmath39 , we follow bellisard @xcite and hasegawa@xcite . in the landau gauge , the system maintains its translational invariance in the @xmath40-direction while it requires @xmath41 steps for translational invariance in the @xmath42-direction . for the square lattice , the period @xmath41 equals @xmath43 , for the triangular lattice , @xmath44 for even @xmath43 and @xmath45 for odd @xmath43 . a fourier - transformation now changes @xmath39 to the following @xmath46-matrix @xmath47 : @xmath48 with @xmath49 the eigenvalues within the corresponding brillouin - zone @xmath50 determine the @xmath41 energy bands of a boson moving in the magnetic field . the minimal eigenvalue of the band - structure @xmath26 is shown in fig . [ butterfly ] as a function of the magnetic flux per plaquette @xmath36 . this is the parameter that enters eq . ( [ mueq ] ) to determine the mott phase boundary . notice how on the square lattice the largest dip is at @xmath51 followed by 1/3 , 1/4 , and 2/5 , ... while on the triangular lattice the sequence corresponds to 1/2 , 1/4 , 1/3 , and 3/8 , ... the relation @xmath52 holds for all lattices , since the flux of @xmath53 is equivalent to a flux of 0 . hence , the maximal magnetic field that can be applied corresponds to @xmath51 . this maximal field configuration is realized by all real hopping matrix elements for each lattice : on the square lattice one takes three positive and one negative matrix element on each plaquette , while on the triangular lattice one takes all matrix elements to be negative . in particular , the `` fully frustrated '' case of @xmath51 on a triangular lattice is the only nonzero magnetic - field case that can be easily treated by a high - order expansion@xcite because it maintains the full periodicity of the triangular lattice in zero magnetic field .    rather than plotting the phase diagrams for just a third - order expansion , we choose to extrapolate our results using knowledge about the overall structure of the phase diagram . there are numerous ways in which one can envision extrapolating the results of our third - order expansion to higher order . it has been demonstrated by elstner and monien that a pade analysis of the strong coupling perturbation series yields rapid convergence in zero field @xcite . we apply this method to the magnetic - field case for small magnetic fields by using a pade approximant@xcite for the logarithmic derivative of the difference in particle and hole mott phase boundaries @xmath54 . we assume the same behavior as found in the zero - field case , where the tip of the mott lobe has a power law `` critical point '' with an exponent @xmath0 , @xmath55 . then the pade analysis fits @xmath56 to estimate the critical point and the dynamical critical exponent ( the pole determines @xmath57 and the residue determines @xmath0 ) . the rest of the pade approximant determines @xmath58 , which then allows @xmath59 to be constructed . a similar pade analysis for the midline of the mott lobe @xmath60 $ ] ( which is a regular function of @xmath9 ) then allows the entire phase diagram to be constructed . atomic systems with a lattice spacing @xmath17 of around @xmath61 require a field of @xmath62 tesla for a half flux quantum per plaquette . this is too large a field to be accessed experimentally , hence atomic systems always lie in the low - field region , where the perturbation theory is most accurate . however for macroscopic lattice systems , with @xmath63 ( as in a josephson - junction array ) the whole magnetic field range is attainable by experiments with fields as low as @xmath64 gauss . we show below how our theoretical results compare to the superfluid - insulator transitions on two dimensional josephson - junction arrays @xcite . ( 60,50)(0,0 ) ( 21,0 ) ( 86,0 )    fig . [ phasesqtri ] presents the phase diagram of the first three lobes for the square lattice [ phasesqtri](a ) and the triangular lattice [ phasesqtri](b ) in the low magnetic - field region . the incompressible , mott - insulating phase grows in size when the magnetic field increases from zero . this shows , as expected , the localizing effect of the magnetic field on the itinerant bosons . it appears that the shape of the lobes changes from a simple power - law dependence at the tip to something else . there are three possibilities for the shape of the phase lobe at the tip : ( i ) it remains power - law - like with an exponent that may depend on the magnetic field ; ( ii ) it has a more exotic `` cusp - like '' dependence as seen in the kosterlitz - thouless transition ; or ( iii ) it has a discontinuous change in slope at the tip , and thereby has a `` first - order - like '' shape corresponding to the crossing of the two curves representing the upper and lower phase lobes , with no `` critical '' behavior at the tip ( this last result corresponds to @xmath65 ) . there is theoretical evidence in favor of this last conclusion , which is a similar behavior to what happens to the mi phase in the presence of disorder , but here the explanation is different@xcite . if @xmath66 is expressed in lowest terms , then the order parameter requires @xmath43 components to describe it . as @xmath43 increases , it is more likely that the transition is first - order rather than second - order , and hence one would expect that the tip has a slope discontinuity immediately upon the introduction of the magnetic field , since small fields correspond to large values of @xmath43 . the pade analysis given above assumes that ( i ) holds , but as @xmath0 approaches 1 for larger fields , the system crosses over to the behavior of ( iii ) unless something more exotic like ( ii ) intervenes . we can not distinguish between scenarios ( i ) or ( iii ) with the low - order expansions presented here , but our results suggest scenario ( iii ) is correct , because the exponent @xmath0 rapidly approaches 1 as the field is increased . clearly further theoretical analysis is needed to decide this issue .    for larger magnetic fields , we find that the two boundaries of the mott phase lobe no longer cross at a critical value of @xmath9 , but rather they `` repel '' each other . this indicates that the tip of the mi phase lobe has moved out to such a large value of @xmath9 that the perturbation theory needs to be carried out to higher order to achieve proper convergence . such an exercise has been already carried out through a linked - cluster expansion for the fully frustrated case ( @xmath51 ) on a triangular lattice@xcite . the calculation from 3rd to 11th order converges toward a `` first - order '' transition @xmath67 at @xmath68 which yields a much larger insulating phase regime than with no magnetic field where @xmath69 . a comparable experimental result is found@xcite where the insulator to superfluid transitions on josephson - junction arrays were studied by changing the ratio of the josephson coupling energy @xmath70 and the charging energy @xmath71 . since @xmath70 is the energy scale for the superconducting coupling between the islands and @xmath71 is the scale for the interaction between the charge carriers , they can be related to the hopping energy @xmath9 and the on - site repulsion @xmath13 , respectively . the results for small temperatures can be extrapolated to @xmath72 and show a transition at larger @xmath73 for the @xmath51 case than for the @xmath74 case , which qualitatively matches with our calculations ( since @xmath75 increases with magnetic field ) . the experimental data can be analyzed in one of two dual pictures with the bose hubbard model . in the first view , the bosons are the cooper pairs on each island , and the main effect of the magnetic field is to modify the hopping matrix . vortices appear in this case as supercurrent loops in the system . while the density of vortices increases as the magnetic field is modified , the density of cooper pairs remains essentially constant over the low field ranges ( on the order of gauss ) explored in the experiments . the second picture employs the duality transformation that views the vortices themselves as the bosonic particles@xcite . the mobility of the vortices is determined by @xmath71 and their interaction by @xmath70 , so the roles of those parameters are reversed in this case . the magnetic field then takes on the role of the chemical potential . we will not employ this second picture here , which has been used to evaluate the mott insulating phase of the vortices in the quasi - one - dimensional josephson junction arrays@xcite .    in the following we compare small - field results . as shown in fig . [ phasesqtri ] the mi phase area and the location of the tip of the lobe increase with increasing magnetic field up to about @xmath76 ( for the square lattice ) . the measurements of @xcite show @xmath77 and @xmath78 for @xmath74 and @xmath79 respectively , which is similar to our results with @xmath57 increasing by about 45% from 0.057 to 0.083 as @xmath80 increases from 0 to 0.1 . in addition , the tip location for larger field strengths saturates in the experiments , which is a result that we can not confirm , because our analysis through third - order in @xmath9 fails when @xmath57 becomes too large . ( 60,50)(0,0 ) ( 21,0 ) ( 86,0 )    [ deltamutri ]    in fig . [ deltamutri](a ) we show the evolution of the excitation - gap energies @xmath59 for fixed hopping @xmath9 as a function of magnetic field . initially the gap energy increases which indicates the increasing stability of the mi phase in a weak magnetic field . for larger magnetic fields , we see dips in the gap energy around rational magnetic fluxes such as @xmath81 or @xmath82 ( which can be reliably calculated for small values of @xmath9 ) . a similar commensurable structure is found in experiments that measure the zero - bias resistance @xmath83 of a josephson - junction array in a magnetic field@xcite . a small @xmath84 indicates proximity to the sf phase . at rational @xmath80 the vortices form a lattice which is commensurable with the josephson - junction array and favors a pinning of the vortices , leading to a decrease of the zero - bias resistance . qualitatively , we see the same behavior in the excitation gap energies and the zero - bias resistance . notice how the dips track closely with the dips seen in the minimal eigenvalue of the hopping matrix as shown in fig . [ butterfly ] . this behavior explains the dips seen in the experimental data on the triangular lattice that are most prevalent at 1/2 , followed by 1/4 , 1/3 , 3/8 , and so on . the experimental data@xcite also shows that the system has four different regions of superconducting stability for one value of @xmath9 , centered at 0 , 1/2 , 1/4 and 1/3 . while our results in fig . [ deltamutri](a ) only show superconductivity around @xmath74 [ where @xmath85 , we can see that if @xmath9 was increased , and the calculations carried out to higher order , it is likely that we would see additional superconducting regions appearing ( we believe first around 1/2 , followed by 1/4 and 1/3 ) . finally we study the magnetic - field dependence of the dynamical critical exponent @xmath0 ( eq .  [ znuformel ] ) for small fields in fig . [ exponent](b ) . the bhm in zero magnetic field can be mapped onto a three - dimensional xy model which has @xmath86 independent of the lattice structure@xcite . by using the pade analysis on our third - order expansion , we obtain @xmath87 for the square lattice and @xmath88 for the triangular lattice at @xmath74 . a pade analysis of a tenth - order expansion@xcite yields @xmath89 for both lattice types , which shows the convergence of higher - order calculations in our method . as seen in fig . [ deltamutri ] , the dynamical exponent appears to increase as the magnetic field increases . it is , however , difficult to conclude whether @xmath0 remains equal to it s zero - field value , increases with magnetic field , or immediately jumps to one upon the onset of a magnetic field , solely on the basis of this third - order analysis . but , the fact that the most likely point to have @xmath90 is @xmath51 ( because it requires only two components for the order parameter ) , and a higher - order expansion predicts @xmath65 there , leads us to conjecture that @xmath65 for all nonzero magnetic fields . this latter result is supported by monte carlo data on the antiferromagnetic xy model for stacked triangular planes , which have a weakly first - order transition rather than a critical point@xcite . in conclusion , we applied a strong - coupling @xmath91-expansion ( up to third order ) to study the insulator - superfluid phase transitions of the two - dimensional bose - hubbard model under the influence of a magnetic field . although our analysis is limited , we are able to produce reasonable results that both agree with physical intuition and with experiments . we found that the mott insulating phase enlarges with an increasing magnetic field . this is explained by the localizing effect of the magnetic field on the moving bosons . qualitative agreement in the increase of the critical hopping energy @xmath57 was found with experimental results on josephson junction arrays@xcite . for small magnetic fields , we find a power - law behavior of the gap energy close to the critical point ( @xmath92 ) with a dynamical critical exponent that increases with @xmath80 . for larger magnetic fields , we find a repulsion of the two phase boundaries , which indicates a change from the power - law like behavior to either kosterlitz - thouless behavior or to a `` first - order '' transition . our results are also consistent with the `` critical point '' immediately changing to a `` first - order '' discontinuous change in the slope of the phase boundaries as the magnetic field is turned on . we found that the gap energies for small fixed hopping and variable magnetic field illustrate commensurability effects for rational fluxes which is also seen in josephson - junction array experiments . more work is needed to understand the change in character of the insulator to superfluid phase transition as a magnetic field is introduced : does the system have a field - dependent power - law dependence which crosses over to a `` first - order '' ( or more exotic cusp - like shape ) as the field increases , or does it immediately become `` first - order '' in a field . higher - order calculations are needed to decide this issue . it would also be interesting to extend the scaling analysis of the bhm to include its behavior in an external magnetic field . we would like to thank m. ma for useful and interesting discussions . j.k.f . acknowledges support from an onr yip grant n000149610828 and from the petroleum research fund administered by the american chemical society ( acs - prf 29623-gb6 ) . h.  m. jaeger , d.  b. haviland , b.  g. orr and a.  m. goldman , phys . b * 40 * , 182 ( 1989 ) ; y.  liu and k.  a. mcgreer , b.  nease , d.  b.  haviland , g.  martinez , j.  w.  halley , and a.  m.  goldman , phys . rev . lett . * 67 * , 2068 ( 1991 ) . h.  s.  j. van der zant , f.  c. fritschy , w.  j. elion , l.  j. geerligs , and j.  e. mooij , phys . lett . * 69 * , 2971 ( 1992 ) ; h.  s.  j. van der zant , w.  j. elion , l.  j. geerligs and j.  e. mooij , phys . b * 54 * , 10081 ( 1996 ) . b. pannetier , j. chaussy , r. rammal and j.  c. villegier , phys . lett . * 53 * , 1845 ( 1984 ) ; a.  f.  hebard and m.  a.  paalanen , ibid . * 65 * , 927 ( 1990 ) ; a. yazdani and a. kapitulnik , ibid . * 74 * , 3037 ( 1995 ) . a.  van oudenaarden and j.  e. mooij , phys . lett . * 76 * , 4947 ( 1996 ) ; a.  van oudenaarden , s.  j.  k.  vrdy , and j.  e.  mooij , ibid . * 77 * , 4257 ( 1996 ) ; a.  van oudenaarden , b.  van leeuwen , m.  p. m.  robbens , and j.  e.  mooij , phys . b * 57 * , 11684 ( 1998 ) .
the bose - hubbard model in an external magnetic field is investigated with strong - coupling perturbation theory . 
 the lowest - order secular equation leads to the problem of a charged particle moving on a lattice in the presence of a magnetic field , which was first treated by hofstadter . 
 we present phase diagrams for the two - dimensional square and triangular lattices , showing a change in shape of the phase lobes away from the well - known power - law behavior in zero magnetic field . 
 some qualitative agreement with experimental work on josephson - junction arrays is found for the insulating phase behavior at small fields .