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the theoretical prediction and subsequent experimental verification of the conducting surface state of a strong topological insulator ( ti ) continues to generate a lot of excitement in physics ; for reviews , see refs .  . in a ti , strong spin - orbit couplings and band inversion conspire to produce a time - reversal invariant topological state different from a conventional band insulator . using bi@xmath0se@xmath1 as a weakly correlated reference ti material with rather large bulk gap @xmath2  mev , surface probe experiments have provided strong evidence for the topologically protected gapless surface state.@xcite the measured spin texture of the surface state is consistent with predictions obtained for two - dimensional ( 2d ) massless dirac fermions . under this `` relativistic '' description , the spin direction is locked to the surface , and the surface state is stable against the effects of weak disorder and weak interactions ( topological protection).@xcite it is thus useful to first study the simplest case of a noninteracting disorder - free model , which is the case investigated below .    because of the residual bulk conductivity of the presently available ( nominally insulating ) ti samples , it has been difficult to experimentally extract the surface contribution to the electrical conductivity . one attempt to improve the situation is to consider mesoscopic samples , where the surface - to - volume ratio is more advantageous . in particular , thin - film geometries@xcite and quasi-1d nanowires ( `` ribbons'')@xcite have been studied experimentally . signatures for aharonov - bohm interference effects associated with the topological surface state in bi@xmath0se@xmath1 nanowires were reported,@xcite cf . also related experiments for sb@xmath0te@xmath1 nanowires.@xcite for infinitely long and circulary symmetric topological nanowires , band structure calculations predict a multi - channel waveguide where all surface modes are gapped because of spin - surface locking.@xcite with @xmath3 denoting the conserved momentum along the wire axis ( taken along the @xmath4 direction ) , and @xmath5 the half - integer total angular momentum , the dispersion relation of these modes is @xmath6 where @xmath7 throughout and @xmath8 for conduction and valence band , respectively . the fermi velocities @xmath9 and @xmath10 differ because the bulk dispersion relation is anisotropic , see below , and the nanowire radius is @xmath11 . note that there is a minimal gap @xmath12 for the surface modes since @xmath5 is half - integer . for reasonable values of @xmath11 , we have @xmath13 . experiments probing quantum dot physics in finite - length ti nanowires are expected to yield new insights into the exciting physics of tis , in close analogy to semiconductor nanowires and carbon nanotubes where such experiments have been highly successful.@xcite to prepare the ground , we here consider the band structure for ti quantum dots with a finite - length nanowire geometry . for a spherical ti dot , the band structure was worked out before.@xcite however , we draw attention to several features that only arise when the surface contains sharp edges , i.e. , non - differentiable parts , as is the case for the cylinder . we employ three different and independent approaches to understand ti quantum dot energy levels and their spin texture : ( 1 ) for a cylindrical ti nanowire of length @xmath14 and radius @xmath11 closed by flat caps , we have performed detailed numerical calculations for the energy spectrum and the spin texture of the eigenstates based on the effective low - energy theory of zhang _ et al._;@xcite material parameters were chosen for bi@xmath0se@xmath1 as quoted in ref .  . ( 2 ) an analytical approach starting from a surface dirac fermion description has been developed for the same geometry . most of our numerical results can thereby be quantitatively reproduced within an analytical theory . ( 3 ) we have also studied a microscopic tight - binding model for a strong ti in the finite - size nanowire geometry and find qualitatively similar results . the main conclusions reached from these three approaches are as follows : first , longitudinal momentum quantization implies a discrete sequence of energy levels which can ( roughly ) be approximated by letting @xmath15 ( @xmath16 ) in eq .  . remarkably , in addition we find unconventional `` zero - momentum '' states ( where , formally , @xmath17 ) . in such a state , the charge and spin densities are almost homogeneous along the @xmath4-direction . furthermore , there are subgap states energetically located within the surface mode gap ( @xmath18 ) . these states are localized near both caps and show interesting spin texture . second , we observe significant out - of - surface components for the spin density associated with all energy eigenstates . note that an out - of - plane spin texture is only expected when trigonal warping effects are important,@xcite see also very recent experimental results reporting such features.@xcite however , to lowest order in momentum ( around the @xmath19 point ) , trigonal warping can be neglected , while we persistently find broken spin - surface locking also at the lowest energy scales . our observations are instead related to the presence of non - differentiable sections of the surface , where the wavefunctions in the `` trunk '' and `` cap '' regions of the cylinder have to be matched . such non - differentiable surface parts also appear in the samples studied in refs .  , and therefore surface probe experiments for these devices could directly test our prediction of broken spin - surface locking . for differentiable closed surfaces , we expect spin - surface locking to stay intact . in fact , the explicit solution of the problem in a spherical geometry exhibits spin - surface locking.@xcite we note in passing that for a flat ti surface , a time - dependent out - of - plane spin component can also be generated by elastic disorder . however , this component will precess around the momentum - dependent spin - orbit axis ( which lies in the plane ) and averages to zero on time scales corresponding to the inverse fermi energy.@xcite    the structure of the remainder of this paper is as follows . in sec . [ sec2 ] , we describe the results of our numerical calculations for cylindrical ti nanowire dots based on the effective low - energy theory . in sec . [ sec3 ] , an analytical approach based on the surface dirac fermion picture is described for the same geometry . the spectrum and the spin density profile for the resulting eigenstates will be derived , and the results are compared to the numerical findings in sec . [ sec2 ] . in sec . [ sec4 ] , we turn to the tight - binding calculation and compare those computations to the previous results . finally , we conclude in sec . appendix [ appa ] contains a derivation of the surface dirac fermion hamiltonian for an infinitely long nanowire using the approach of ref .  . in this section , we compute the band structure of a cylindrical nanowire of length @xmath14 and radius @xmath11 from the effective low - energy theory of zhang _ _ et al.__@xcite using parameters for bi@xmath0se@xmath1 . up to terms of order @xmath20 with @xmath21 , the low - energy bulk hamiltonian employing the four bands energetically closest to the @xmath19 point is @xcite @xmath22\tau_x,\ ] ] where @xmath23 and @xmath24 , with @xmath25 . this also defines the fermi velocities @xmath26 ( in @xmath4 direction ) and @xmath27 ( in the @xmath28 plane ) , see eq .  . the basis states encoding the spin - parity structure , where pauli matrices @xmath29 ( @xmath30 ) act in spin ( parity ) space and @xmath31 ( @xmath32 ) denotes the respective unit matrix , are explicitly given in ref .  . this work also describes the extension of eq .   to the case of eight bands or towards including trigonal warping . using the parameters in ref .  , listed for convenience also in table [ tab0 ] , the criterion for a ti phase@xcite is satisfied . one therefore must have an odd number of conducting surface modes when boundaries are present .       here we analytically derive eq . ( [ hd ] ) for an infinite nanowire using the gap - inversion model of ref .  . our starting point is eq .   to linear order in @xmath33 ( we put @xmath34 here ) , @xmath35 \tau_x .\ ] ] following ref . , we assume that the gap parameter changes sign at @xmath36 , i.e. , @xmath37 with @xmath38 . for @xmath39 ( @xmath40 ) , the material is then in the topologically nontrivial ( trivial ) phase . in the 1d representation , for given @xmath5 and @xmath3 , solutions to the dirac equation take the form @xmath41 , with the 1d radial dirac equation @xmath42 here @xmath43 corresponds to @xmath44 \tau_x .\ ] ] for @xmath43 , there are two solutions @xmath45 where the first ( second ) spinor refers to parity ( spin ) space , @xmath46 , @xmath47 , and @xmath48 for @xmath40 , we have to replace @xmath49 and @xmath50 in the spin part , where @xmath51 and @xmath52 are modified bessel functions . the general solution is then given by @xmath53 where the coefficients @xmath54 and @xmath55 are obtained by requiring continuity of the wave function at @xmath36 . this results in a linear system of equations for the coefficients , which has a nontrivial solution under the condition @xmath56 where all bessel functions have the argument @xmath57 and @xmath58 is evaluated for @xmath59 . assuming @xmath60 and using the asymptotic form of the bessel functions , we find @xmath61 and thus @xmath62 . the corresponding @xmath43 wavefunctions , @xmath63 , are given by eq . with @xmath64 , @xmath65 , and @xmath66 for small @xmath67 , the effective surface hamiltonian , eq . ( [ hd ] ) , is obtained by projecting eq .   onto the subspace spanned by the above @xmath43 states . we thereby find @xmath68 where the @xmath69 are pauli matrices in the zero - momentum subspace . in this way , all combinations of spin - parity matrices , @xmath70 , can be represented in the truncated basis . for @xmath60 , we obtain the results in table  [ tab1 ] . of course , this representation is not single - valued , i.e. , there is no one - to - one correspondence between the @xmath71 and the @xmath72 matrices . in particular , the spin operators @xmath73 and the parity operator @xmath74 are entangled , since @xmath75 and @xmath76 . taking @xmath77 and @xmath78 , cf .  the last column of table [ tab1 ] , one gets the surface hamiltonian quoted in the main text , see eq .  . because this representation is multi - valued , the replacement of @xmath79 by products of spin and parity matrices causes a double - counting of states if used naively . this double - counting problem is , however , not severe and can be circumvented , as we discuss in sec . [ sec3c ] in the main text . we note that a theory for quantum dots based on the 2d semiconductors hgte and hgs ( which may have topological phases ) was recently put forward by n. malkova and g.w . bryant , phys . b * 82 * , 155314 ( 2010 ) .
we consider the energy spectrum and the spin - parity structure of the eigenstates for a quantum dot made of a strong topological insulator . 
 using the effective low - energy theory in a finite - length cylinder geometry , numerical calculations show that even at the lowest energy scales , the spin direction in a topologically protected surface mode is not locked to the surface . 
 we find `` zero - momentum '' modes , and subgap states localized near the `` caps '' of the dot . 
 both the energy spectrum and the spin texture of the eigenstates are basically reproduced from an analytical surface dirac fermion description . 
 our results are compared to microscopic calculations using a tight - binding model for a strong topological insulator in a finite - length nanowire geometry .