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ultra cold electrons in two dimensions in a strong perpendicular magnetic field , the quantum hall ( qh ) system @xcite , exhibits a fascinating phase diagram including phases with fractionalized excitations and topological order @xcite . ever since its discovery three decades ago , the qh system has inspired a huge amount of experimental and theoretical effort , not least due to its richness in phenomenology and mathematical structure . new developments include the observation of the fractional quantum hall effect in graphene @xcite and ideas of applications in the context of topological quantum computing @xcite . moreover , it has been realized that the theoretical description of a system of rapidly rotating bosons is formally very similar to that of an electron gas in a magnetic field @xcite . a key property of fractional quantum hall ( fqh ) states is their topological order @xcite . one consequence thereof is that their physical properties are insensitive to smooth deformations of the manifold on which we choose to study them . this fact has been exploited in a series of recent studies of the interacting many - body problem in the limit geometry of a thin torus , referred to as the tao - thouless ( tt ) limit @xcite ( see also refs . @xcite for precursory studies and ref . @xcite for some recent related approaches ) .    in this proceeding we study the fqh system beyond the tt limit and argue that some characteristics that are not readily understood in the tt limit states are however manifest in the leading quantum fluctuations away from this limit . while several of the ideas presented here have been published earlier @xcite , we corroborate these results and also present a number of new results . we study the structure of excitation spectra of two - dimensional electrons in a magnetic field with coulomb interactions as a function of the torus circumference , @xmath8 , and contrast the cases of even and odd denominator filling fractions . for @xmath6 we interpret these results in terms of a deforming fermi sea @xcite . then , by extending the results of refs . @xcite for the @xmath9 fqh state , we further establish that fqh systems in a jain sequence@xcite , which has filling factor @xmath0 in the lowest landau level can be described by @xmath2 spin chains with @xmath3 sites unit cell . the remainder of this proceeding is organized as follows . in section [ model ] we review the lattice describing interacting particles in a landau level on the torus and present numerical spectra thereof as a function of the circumference , @xmath8 . in section [ beyond ] , we briefly motivate and restate known results about the energy spectra in the tt limit and then introduce spin models that provide an intuitive understanding of the numerical data away from this limit . in particular , these considerations shed light on why odd denominator fractions remain gapped while even denominator states tend to be gapless . we consider a model of @xmath10 interacting electrons on a torus pierced by @xmath11 flux quanta . in the landau gauge , a complete basis of @xmath11 degenerate single - particle states in the lowest landau level , labeled by @xmath12 , can be chosen as @xmath13 where @xmath14 are the circumferences of the torus , @xmath15 the corresponding coordinates , and @xmath16 the momentum along the @xmath8-cycle . we have set the magnetic length @xmath17 equal to unity .    in this basis , any translation - invariant two - dimensional two - body interaction hamiltonian assumes the following model on a one - dimensional discrete lattice , @xmath18 where the matrix - element @xmath19 specifies the amplitude for a process where particles with separation @xmath20 hop @xmath21 steps to a separation @xmath22 ( note that @xmath21 can be negative ) . the number of the lattice sites is fixed by the area , @xmath23 . at the filling @xmath24 the hamiltonian commutes with the center - of - mass magnetic translations @xcite @xmath25 and @xmath26 along the cycles , which implies , in particular , that the total momentum @xmath27 along the @xmath8-cycle is conserved modulo @xmath11 in this gauge . in this system ( [ 1d_model ] ) , two conservation numbers are given as @xmath28 where @xmath29 and @xmath30 .    we have calculated energy spectra of this system for @xmath31 using exact diagonalization in subspaces labeled by @xmath32 . using these symmetries , the hilbert space splits into @xmath33 sectors that are all of comparable size . this enable us to treat larger systems . the matrix elements of @xmath19 are calculated assuming coulomb interaction between the electrons . for the half - filled landau level @xmath6 , we have calculated the energies up to @xmath34 which is larger than the former analysis @xcite ( see fig .  [ spectra12 ] ) . increasing @xmath8 from the tt limit ( @xmath35 ) , the first level - crossing in the ground state energy takes place at @xmath36 which corresponds to a phase transition from a charge - density - wave ( cdw ) to a gapless phase . further level crossings occur among different gapless states , and finally @xmath8 arrives at the duality point @xmath37 ( @xmath38 for @xmath39 , @xmath40 for @xmath34 and @xmath41 for @xmath42 ) . the spectrum is invariant under @xmath43 ( although quantum numbers change ) and hence we only present data for @xmath44 . we have calculated the energies of the @xmath9 and @xmath45 cases , and confirmed that energy gaps do not close as @xmath8 changes , and level - crossings occur only in excited states . this is consistent with the fact that these states have energy gaps in the bulk systems as is evident in the thin - torus limit ( see below ) .    ) for ( a ) @xmath9 with @xmath46 and ( b ) @xmath45 with @xmath39 . the energy gaps never close as functions of the circumference @xmath8 of the torus.,title="fig:",width=302 ] ) for ( a ) @xmath9 with @xmath46 and ( b ) @xmath45 with @xmath39 . the energy gaps never close as functions of the circumference @xmath8 of the torus.,title="fig:",width=264 ]    ) for ( a ) @xmath9 with @xmath46 and ( b ) @xmath45 with @xmath39 . the energy gaps never close as functions of the circumference @xmath8 of the torus.,title="fig:",width=283 ] ) for ( a ) @xmath9 with @xmath46 and ( b ) @xmath45 with @xmath39 . the energy gaps never close as functions of the circumference @xmath8 of the torus.,title="fig:",width=283 ] from the shape of the single particle states ( [ psik ] ) it follows that the distance between adjacent states , the lattice constant @xmath47 , depends on @xmath8 while the extent of the wave functions is constant . for small @xmath8 the overlap between different single particle states decreases rapidly and the matrix elements @xmath19 simplify considerably . as @xmath5 one finds that @xmath48 thus the @xmath49 terms are exponentially suppressed for generic interactions in this limit . the remaining ( @xmath50 ) problem is exactly solvable : ground states at any @xmath24 are gapped periodic crystals ( with a unit cell of @xmath3 electrons on @xmath51 sites ) and the fractionally charged excitations appear as domain walls between degenerate ground states . the fractal structure ( a.k.a . devils staircase ) of the abelian haldane - halperin hierarchy @xcite is manifest for generic convex two - body interactions and provides an explicit estimate for the relative stability of odd denominator states in excellent agreement with experiment @xcite . the leading fluctuations are the @xmath52 terms describing correlated hopping of two particles one site each in opposite directions . in a series of earlier investigations , some of us have shown that these fluctuations ultimately lead to a melting of the gapped state for @xmath6 resulting in a gapless state @xcite , while for @xmath9 the excitation structure and gap is essentially unaltered , even for very strong fluctuations @xcite . below , we discuss how the numerical results at finite @xmath8 can be interpreted in terms of effective spin chains and extend this picture to include the most prominent ( @xmath52 ) jain series , @xmath0 and @xmath53 . the strategy is as follows . first , we truncate the interaction to only include the leading @xmath52 term , @xmath54 . this is especially sensible for a pseudo potential interaction where @xmath55 with @xmath56 being a polynomial in @xmath57 and @xmath21 . in this case our expansion in @xmath4 is well controlled , and we expect it to capture the physics also for more general interactions . second , we restrict the local hilbert space by including only local configurations that correspond to the tt ground state and local applications of @xmath54 . this naturally leads to a spin chain representation of the hamiltonian and enables us to deform it to connect to well studied spin chain models and obtain physical insights by drawing analogies between the physics of spin chains and fqh states . the electron system on a torus can be described in terms of @xmath7 variables in the following way @xcite : in the vicinity of the tt limit , the relevant physics is captured by a model containing the three dominant terms as @xmath58 ,   \label{truncated_ham}\ ] ] where @xmath59 . in the tt limit , the ground state is the charge - density - wave ( cdw ) state where electrons are located on every two sites @xmath60 , since the matrix elements of the electrostatic term @xmath61 is dominant . away from the tt limit , the competition between @xmath61 , @xmath62 and @xmath63 can be included as an interaction of the local spin states @xmath64 and @xmath65 which means that @xmath66 , @xmath67 , @xmath68 , @xmath69 , where @xmath70 is the index of the unit cell . the effective spin hamiltonian is then given by the @xmath7 xxz chain , @xmath71 where @xmath72 . it is well known that this model has a phase transition from a ferromagnetic ( cdw ) phase to a gapless ( tomonaga - luttinger liquid ) phase at @xmath73 . this describes the phase transition from the tt state to the gapless phase for the half - filled landau level @xmath6 . the obtained solution corresponds to the ( first ) elongated fermi seas in fig . [ spectra12 ] appearing for @xmath74 . next we consider spin - mapping for @xmath9 state . in the tt limit , the ground state of the system is the three - fold degenerate cdw state @xmath75 where the underlines denote unit cells . to capture the effect of the leading quantum fluctuations , we introduce @xmath2 spin variables as @xmath76 , where the operators are related as @xmath77 in this mapping , the three - fold degeneracy of the ground state is hidden . the degenerate states have different center - of - mass quantum numbers and there are no nonvanishing matrix elements between states in different momentum sectors . it follows that there are three distinct sectors which are described by the same effective hamiltonian , @xmath78[1-(s^z_{n+1})^2]+\mbox{h.c . } \label{s=1_chain_for_nu=1/3}\ ] ] this model is given by four spin interactions or the xy model with projectors to the state @xmath79 . note that this hamiltonian does not have the space inversion and spin reversal symmetries : the exchange process @xmath80 exists , but @xmath81 does not . the spin rotational symmetry is also broken . actually , numerical analysis shows that this hamiltonian well describes the fqh state around @xmath82 , by checking the overlap between the truncated hamiltonian and full hamiltonian with coulomb interaction and the trugmann - kivelson type potential @xcite which provides an exact parent hamiltonian for the laughlin state . there is also a term given by a function @xmath83 which stems from the electrostatic terms , but this term does not alter any essential features of the present model ( [ s=1_chain_for_nu=1/3 ] ) @xcite . in refs . @xcite , this model with extra parameters @xmath84 has been analyzed numerically to study adiabatic continuity from a conventional spin chain ( @xmath85 ) . for @xmath85 , it is known that there is a phase transition at @xmath86 between the haldane ( @xmath87 ) and large-@xmath88 ( @xmath89 ) phases @xcite . in refs . @xcite it was shown that the ground state of the model ( [ s=1_chain_for_nu=1/3 ] ) is smoothly connected both to the haldane phase and the large-@xmath88 phase without closing the energy gap under the change of parameters . this indicates that the spin model describing the @xmath9 fqh state has the nature of these two phases simultaneously , and there are indeed some similarities between the @xmath9 fqh effect and the @xmath2 haldane - gap state as noticed early on @xcite . the reason for the absence of a phase transition is due to the breaking of the discrete symmetries . according to ref . @xcite , haldane and large-@xmath88 phases are smoothly connected if the dihedral group , time reversal and parity symmetries are broken . since the present model ( [ s1extended ] ) with @xmath90 breaks these three symmetries , our result is consistent with this general argument . most properties of a finite chain of this model ( [ s1extended ] ) are similar to those of a usual large-@xmath88 phase where no edge spins appear , however this is not inconsistent with the absence of a phase transition @xcite . let us now consider extentions of the above @xmath2 mapping to other filling factors . as discussed by jain @xcite , @xmath91 fqh states may be described by the composite fermion picture , where @xmath92 quantum flux are attached to non - interacting electrons of the @xmath3-th landau level , and projection onto the lowest landau level . for definiteness , we will consider the positive @xmath52 jain sequence @xmath0 , @xmath93 . the results are easily transfered to the negative jain series by noting that our formulation is particle - hole symmetric , hence the @xmath0 is equivalent to the @xmath94 case . this is in contrast to the wave function based approaches and the composite fermion phenomenology , which lack this symmetry . before giving the general result , let us consider the @xmath45 state ( @xmath95 ) as the simplest example . in the tt limit , the ground state is given by the cdw state with the configuration @xmath96 . when @xmath54 and @xmath97 are applied to this cdw state several times , one finds that the number of electrons in each unit cell is always conserved , and the configuration @xmath98 never appears . therefore , configurations such as @xmath99 , @xmath100 , @xmath101 , @xmath102 are absent in the truncated hilbert space . this means that the @xmath45 states can be mapped to two @xmath2 variables by inserting @xmath103 appropriately ( between the two @xmath104 s ) in each unit cell : @xmath10510 \rangle \rightarrow |00    \rangle,\\   &    | 00110 \rangle \rightarrow | 00[10]10 \rangle \rightarrow    |\mathsf{-}0 \rangle,\\   &    | 01100 \rangle \rightarrow | 01[01]00 \rangle    \rightarrow |0\mathsf{+ } \rangle.\end{aligned}\ ] ] the relationship between fermion and spin-@xmath104 operators ( @xmath106 , @xmath107 ) for the @xmath45 state is summarized in table  [ stmapping ] . we now obtain an effective spin chain as @xmath108   + \mbox{h.c.}\end{aligned}\ ] ]    .relationship between fermion and spin-@xmath104 operators for the @xmath45 state . [ cols="^,^,^,^ " , ]     similarly , we can extend the @xmath2 spin mapping for general @xmath3 . for @xmath93 cases , the configuration in the tt limit is given by @xmath109 , and we obtain @xmath2 chains with @xmath3-sites unit cell ( see also fig .  [ s1_chain_p_unit ] ) , @xmath110+\mbox{h.c . } \label{s=1mapping}\ ] ] where @xmath111 means @xmath2 operators of @xmath112-th site in @xmath70-th unit cell . we have again ignored the contribution from the electrostatic terms , for simplicity . note that this model includes the @xmath9 model in eq . ( [ s=1_chain_for_nu=1/3 ] ) , but is not appropriate for the @xmath6 state . since this model also breaks the three discrete symmetries mentioned above , we expect that it has similar low - energy properties as @xmath113 quantum spin chains with haldane gaps . for @xmath114 cases , effective spin chains can be obtained by replacing @xmath3 in eq . ( [ s=1mapping ] ) by @xmath115 . the detailed derivation and further analysis of this model will be published elsewhere @xcite . ( 0,-2)(27,1 ) ( 0,0 ) ( 1.5,-1.5)@xmath116 ( 0,0)0.3(1.5,0)0.3(3,0)0.3 ( 0.3,0)(1.2,0 ) ( 1.8,0)(2.7,0 ) ( -0.75,0)(-0.3,0 ) ( 3.3,0)(3.75,0 ) ( 6,0 ) ( 1.75,-1.5)@xmath117 ( 0,0)(2.5,0)2 ( 0.5,0)(1.0,0.8 ) ( 0,0)0.3(1,0)0.3 ( 0.3,0)(0.7,0 ) ( -0.75,0)(-0.3,0 ) ( 1.3,0)(2.2,0 ) ( 3.8,0)(4.25,0 ) ( 13,0 ) ( 2.75,-1.5)@xmath118 ( 0,0)(3.5,0)2 ( 1.0,0)(1.5,0.8 ) ( 0,0)0.3(1,0)0.3(2,0)0.3 ( 0.3,0)(0.7,0 ) ( 1.3,0)(1.7,0 ) ( -0.75,0)(-0.3,0 ) ( 2.3,0)(3.2,0 ) ( 5.8,0)(6.25,0 ) ( 23,0 ) ( 1.5,-1.5)@xmath119 ( 0,0)(4,0)1 ( 1.5,0)(2.0,0.8 ) ( 0,0)0.3(1,0)0.3 ( 2,0)0.3(3,0)0.3 ( 0.3,0)(0.7,0 ) ( 1.3,0)(1.7,0 ) ( 2.3,0)(2.7,0 ) ( -0.75,0)(-0.3,0 ) ( 3.3,0)(3.75,0 ) we have studied the fractional quantum hall ( fqh ) effect based on the one - dimensional description with torus boundary conditions and illustrated how energy spectra of interacting electrons in a magnetic field changes as functions of the circumference of the torus for various filing factors using exact diagonalization . in an effort to understand these results microscopically , we discussed spin chain mappings of the fqh states . by extending earlier results for the @xmath6 and the @xmath9 cases , we mapped the jain sequences @xmath0 and @xmath120 to @xmath2 chains with @xmath3 spins in the unit cell that we conjecture to be gapped for all @xmath3 . this further strengthens the known analogies between the fqh physics and the quantum spin chains . we have also studied the gapless state at @xmath6 numerically . by going to slightly larger system sizes than before ( from @xmath121 to @xmath122 particles ) , we confirmed that the evolution from the thin torus to the bulk is , after an initial first order transition that is accurately describe by an @xmath7 spin chain model , smooth and described by a natural deformation of the quasi - particle fermi sea . we thank a.  karlhede and j.  suorsa for related collaborations , and k.  okamoto and m.  oshikawa for discussions . m. n. acknowledges support from global center of excellence program `` nanoscience and quantum physics '' of the tokyo institute of technology by mext . 9                    wilkin n k , gunn j m f and smith r a 1998 _ phys . lett . _ * 80 * 2265 ; cooper n r and wilkin n k 1999 _ phys rev . _ b * 60 * 16279(r ) ; regnault n and jolicoeur th 2003 _ phys . rev . lett . _ * 91 * 030402 ; for a recent review see viefers s 2008 _ j. phys . : condens . matter _ * 20 * 123202 . bergholtz e j , kailasvuori j , wikberg e , hansson t h and karlhede a 2006 _ phys . _ b * 74 * 081308(r ) ; seidel a and lee d h 2006 _ phys . lett . _ * 97 * 056804 ; read n 2006 _ phys . _ b * 73 * 245334 ; ardonne e , bergholtz e j , kailasvuori j and wikberg e 2008 _ j. stat . _ p04016 ; ardonne e 2009 _ phys . lett . _ * 102 * , 180401 ; seidel a 2010 _ phys . lett._*105 * , 026802 . wen x -g and wang z , 2008 _ phys . _ b * 78 * 155109 ; barkeshli m and wen x -g 2009 _ phys . _ b * 79 * 195132 ; bernevig b a , gurarie v and simon s h 2009 _ j. phys . a : math . theor . _ * 42 * 245206 ; bernevig b a and haldane f d m 2009 _ phys . lett . _ * 102 * 066802 ; hermanns m 2010 _ phys . lett . _ * 104 * , 056803 ; thomale r , estienne b , regnault n and bernevig b a arxiv:1010.4837 . pan w , strmer h l , tsui d c , pfeiffer l n , baldwin k w and west k w 2003 _ phys . lett . _ * 90 * 016801 ; pan w , xia j s , strmer h l , tsui d c , vicente c , adams e d , sullivan n s , pfeiffer l n , baldwin k w and west k w 2008 _ phys . _ b * 77 * 075307 . pollmann f , berg e , turner a and oshikawa m , arxiv:0909.4059 ( unpublished ) ; pollmann f , turner a , berg e and oshikawa m , 2010 _ phys . _ b * 81 * 064439 . wang z y , nakamura m and bergholtz e j 2011 _ in preparation .
we discuss the relationship between the fractional quantum hall effect in the vicinity of the thin - torus , a.k.a . 
 tao - thouless ( tt ) , limit and quantum spin chains . 
 we argue that the energetics of fractional quantum hall states in jain sequence at filling fraction @xmath0 ( and @xmath1 ) in the lowest landau level is captured by @xmath2 spin chains with @xmath3 spins in the unit cell . 
 these spin chains naturally arise at sub - leading order in @xmath4 which serves as an expansion parameter away from the tt limit ( @xmath5 ) . 
 we also corroborate earlier results on the smooth fermi surface deformation of the gapless state at @xmath6 , interpolating between a state described by a critical @xmath7 chain and the bulk .