You are an expert at summarizing long articles. Proceed to summarize the following text: 
recently , there has been a growing interest in the study of interfaces with non - trivial geometry . such interfaces arise in a variety of situations including domain walls in random magnets,@xcite fluid invasion in porous media@xcite , spreading on heterogeneous surfaces@xcite , membranes and vesicles in biology,@xcite and epitaxial growth in materials science.@xcite    in connection with such problems it is natural to ask whether quantum effects play a significant role . for static properties it is well established that in nonrandom systems , such as a spin @xmath0 antiferromagnet , with only a nearest neighbor exchange interaction , @xmath10 , there are various regimes . near the critical temperature at @xmath11 , thermal fluctuations are dominant . in the ordered phase , as long as @xmath12 , quantum effects due to the finiteness of @xmath0 are unimportant . for @xmath13 one is in a quantum regime , where the quantum statistics of spin waves and their interactions leads to dependences on @xmath0 and @xmath14 not present in the classical ( @xmath15 ) limit . heuristic arguments indicate that quantum effects could influence the nature of domain walls in spin systems . in the picture in which the trajectory of the domain wall is likened to the trajectory of a particle in space as a function of time , quantum effects cause a smearing out of the trajectory . for such a system analytic and numerical work is obviously very difficult . accordingly , we have been led to carry out a program of analytic work for quantum domain walls in one dimension . for this purpose we consider domain walls at zero temperature @xmath14 in the spin @xmath0 ising model ( with nearest - neighbor exchange coupling @xmath10 ) in a transverse field , @xmath5 , whose hamiltonian is @xmath16 where @xmath17 is a quantum spin @xmath0 operator at site @xmath18 .    to study domain walls in this model we introduce boundary conditions in which the spin at one end of the chain is fixed to be `` up '' and that at the other end is fixed to be `` down . '' this model has some interest in its own right . for @xmath19 its properties can , in principle , be related to those of the associated free fermion model obtained via the jordan - wigner@xcite transformation . however , with domain - wall boundary conditions , this relation is not easy to implement . accordingly , we approach the analysis of the properties of domain walls in this model via perturbation theory for @xmath20 and more generally via numerical solution for the wave functions and energies of the ground state and the low - lying excited states . needless to say , some of the properties of this model in one dimension can not be extrapolated to higher dimensional systems . however , in most cases , one can safely say which features can be so extrapolated and which can not . we may summarize our results for the one dimensional model as follows . the magnetization profile has a width of order the length of the chain . the low - lying excited states comprise a manifold of `` particle '' states , which results when the center of the wall propagates from site to site . these results are easily understood within perturbation theory in @xmath21 . in the classical limit the width of the wall is of order the correlation length , i. e. it is of order a lattice constant , as long as one is far from the critical regime at @xmath22 , above which long range order disappears . in the classical limit and for small @xmath21 , we evaluate a barrier energy , analogous to the peierls - nabarro energy,@xcite which prevents the free motion of the domain wall . when the correlation length becomes very large this barrier energy becomes small and since it is harder to calculate in this limit , we did not attempt such calculations . it would be interesting to calculate this energy for a quantum system , but in the present case , since the width of the wall is of order @xmath7 , we can say that this barrier energy vanishes in the thermodynamic limit . also , in the large @xmath0 limit , we find that the quantum hopping of the domain wall from one site to the next is actually analogous to a tunneling process , so that the hopping matrix element is not of order @xmath5 , as it is for the low spin case , but is now of order @xmath23 , where @xmath24 . it would be of some interest to recover this result within a field theory of one space and one time dimension to describe such a quantum effect . briefly , this paper is organized as follows . in sec . ii we define the class of models we will analyze in one dimension . in sec . iii we give numerical results for the magnetization profile for @xmath19 , @xmath25 and @xmath26 and compare these to analytic results we obtain using perturbation theory in the small @xmath21 limit . in sec . iv we give analytic results for large @xmath0 . the classical results for @xmath27 are given both in the continuum limit ( i.e. , when the wall is very broad ) and in the discrete limit ( when the wall is very narrow ) . here we also analyze the quantum system for large @xmath0 . in sec . v we give various results concerning the nature of the energy spectrum in the presence of a domain wall . we give numerical and analytic results for the nature of the particle spectrum caused by the matrix element which allows the wall to hop from one site to the next . finally , in sec . vi we summarize the conclusions to be drawn from our work . the first one - dimensional model we consider is the spin @xmath0 ising model in a transverse field , with the hamiltonian @xmath28 this formulation of the model has the advantages that a ) the domain wall energy ( for @xmath29 ) is independent of the value of @xmath0 and is equal to @xmath10 , and b ) the mean - field transition temperature @xmath30 ( at which significant correlations begin to develop between neighboring spins ) is of order @xmath10 independent of @xmath0 . to discuss domain walls , the spins @xmath31 and @xmath32 will be fixed by boundary conditions , as discussed below . thus @xmath7 is the number of `` active '' spins in the chain . we will often use the notation @xmath33 . we will also consider the yz " model in a transverse field , for which the hamiltonian is @xmath34 the most important difference between these two models is that the interaction term ( proportional to @xmath35 ) in @xmath36 allows the spins to tip away from the @xmath4-axis . for @xmath29 and with free - end boundary conditions , the  y - z " model with @xmath37 has the continuous @xmath38 symmetry instead of the discrete @xmath39 symmetry of the ising model which results when @xmath40 . in fact , we recall the ground state phase diagram@xcite of this more general model , shown in fig . 1 for spin @xmath41 , where one sees ( at zero temperature ) the disordered phase ( d ) , the ordered ferromagnetic ising phase ( f ) , and the ordered oscillatory phase ( o ) . in both ordered phases the spontaneous magnetization @xmath42 is non - vanishing ( where @xmath43 denotes the thermodynamic average at temperature @xmath14 ) . however , the ( connected ) two - point correlation function , @xmath44 , behaves as follows@xcite for @xmath45 : @xmath46 where the correlation length @xmath47 is a known function of @xmath48 and @xmath35 in each of the three phases , @xmath49 is a constant and @xmath50 . when going from phase f to phase o , there is a new diverging length @xmath51 , which is not simply related to a gap in the spectrum of @xmath36 ( as is @xmath47 ) , but indicates the wavelength at which the correlation function at large distances oscillates under an exponentially decaying envelope .    to discuss the width and energy of a domain wall , we shall work with `` up - down '' boundary conditions , in which we require @xmath52 we can then study the profile of the wall by evaluating    @xmath53    where @xmath54 denotes the ground state . as we discuss in more detail below , in order to obtain a spin profile for a _ quantum _ system , antiperiodic boundary conditions which introduce a boundary coupling @xmath55 _ can not _ be used in a naive way . however , for a classical system such antiperiodic boundary conditions will prove convenient . the energy of the domain wall , @xmath56 , is defined to be @xmath57 , where @xmath58 is the energy with `` up - down '' or antiperiodic boundary conditions and @xmath59 that with `` up - up '' or periodic boundary conditions . we begin by presenting numerical results for the profile function @xmath62 for the ferromagnetic phase f. for general spin we are constrained to rather small systems , since the number of quantum states is @xmath63 for @xmath7 sites ( excluding the fixed boundary spins ) . so why not use translational symmetry to reduce the number of states ? indeed , that would be possible for the special case of antiperiodic boundary conditions ( with @xmath64 ) mentioned in the previous section . these boundary conditions are peculiar since @xmath65 commutes with the modified translation operator @xmath66 where @xmath67 is the usual translation operator and @xmath68 changes the sign of @xmath69 . in addition , @xmath65 can be decomposed ( for any @xmath70 ) into a block diagonal form . the corresponding states are said to be in the even or odd sector , respectively , depending on their parity under spin reversal . furthermore , @xmath71 in that case and one should consider instead the quantity @xmath72 where @xmath54 and @xmath73 are the ground states in the even and odd sectors ( with respect to the ising symmetry ) of the model , respectively . it is well - known that at least for _ periodic _ boundary conditions this defines@xcite a sensible finite - lattice approximation to the order parameter ( see ref . and references therein ) . however , for @xmath64 , even for @xmath7 finite , there are two _ degenerate _ states @xmath73 , @xmath74 in the odd sector , one of which gives rise to the desired wall profile , while the profile of the other one is essentially flat . on the other hand , for @xmath75 , one does get a wall profile , but its shape depends on the value of @xmath76 . we avoid these problems by choosing the `` up - down '' boundary conditions defined above in eq . ( [ bc ] ) . furthermore , since we are interested in situations far from criticality , the convergence of the finite - size data with @xmath77 is usually quite rapid so that the moderate sizes achieved are sufficient for our purposes .    the determination of the lowest eigenstates of the hamiltonians @xmath78 and @xmath36 using the lanczos algorithm is fairly standard@xcite . from the ground state @xmath54 we calculate the local magnetization @xmath79 @xmath80    our results pertaining to the ferromagnetic phase f are as follows . in fig . 2 , we display @xmath79 for @xmath81 and @xmath82 and @xmath41 for various values of @xmath7 . with the exception of the smallest @xmath7-values used , we first observe that the data collapse onto a single curve , which implies that the @xmath7 considered are already sufficiently large as compared to the correlation length that they represent the @xmath83 limit . second , we note that the profile is quite wide and a continuous function of @xmath84 . we find a similar data collapse for all the situations we are going to consider in the sequel . next , we show in fig . 3 , that the overall shape of the profile is not a peculiarity of having spin @xmath41 . rather , we see that the profiles obtained with the parameters @xmath82 and @xmath81 ( fig . 3a ) and @xmath85 ( fig . 3b ) both spins @xmath25 and @xmath86 are very close to the one for @xmath41 . nevertheless , the finer details of the respective shapes become different as @xmath5 increases . finally , in fig . 4 , we show the effect of varying @xmath48 in the @xmath41 case . note that the profiles for a transverse field as small as @xmath87 and as large as @xmath81 are superimposed onto each other . the fact that the width @xmath88 of the domain wall is proportional to the size of the system is a peculiarity of a one dimensional quantum model . in higher dimensions we would expect @xmath88 to remain finite as @xmath83 except possibly at a critical point where the correlation length diverges . we shall turn to a quantitative explanation of these findings below . profiles for the oscillatory phase are discussed in subsection c.      in order to gain some understanding of the results of the last section , we now present some simple approximate calculations . as it turns out , most of the physics of the problem for @xmath48 small is conveniently described in terms of degenerate perturbation theory involving the manifold @xmath89 of the @xmath90 ground states . this procedure avoids the additional technical complications of the free fermion method for `` up - down '' boundary conditions . the boundary conditions are that spin @xmath31 is fixed to have @xmath91 and spin @xmath32 has @xmath92 . for @xmath19 the manifold @xmath93 contains the states @xmath94 , for @xmath95 , where @xmath96 denotes the quantum state ( shown in fig . 5 ) in which spins @xmath97 with @xmath98 have their @xmath4-component of spin equal to @xmath99 and those with @xmath100 have their @xmath4-component of spin equal to @xmath101 . the only nonzero matrix elements of the hamiltonian within the ground manifold are @xmath102 considering only the manifold @xmath93 , one finds the corresponding eigenvectors @xmath103 , and eigenenergies @xmath59 ( for @xmath104 ) to be @xmath105 |n \rangle \ , \ \ \ \ e_p = e_0 + h \cos [ p \pi / ( l+2 ) ] \ , \ ] ] where @xmath106 is a normalization constant . perturbative corrections to the energy will occur at order @xmath107 and to the wavefunction at order @xmath21 . we may calculate the profile function within the small @xmath48 approximation : @xmath108 - \sum_{k=1}^n \sin^2 [ kp \pi / ( l+2 ) ] \over \sum_{k=1}^{l+1 } \sin^2 [ kp \pi / ( l+2 ) ] } \nonumber \\ & = & 1 - 2 { \sum_{k=1}^n \sin^2 [ kp \pi / ( l+2 ) ] \over \sum_{k=1}^{l+1 } \sin^2 [ kp \pi / ( l+2 ) ] } \nonumber \\ & = & 1 - 2 { ( 2n+1 ) \sin { p \pi \over l+2 } - \sin { ( 2n+1)p \pi \over l+2 } \over ( 2l+3 ) \sin { p \pi \over l+2 } - \sin { ( 2l+3)p \pi \over l+2 } } \ .\end{aligned}\ ] ] we set @xmath109 and work in the limit of infinite @xmath7 . thereby we find @xmath110 in fig . 3 , @xmath111 is shown together with the numerically determined ground - state profiles @xmath79 for @xmath41 , @xmath25 and @xmath86 and we find a nice qualitative agreement ( even for moderately large values of @xmath48 ) . the above calculation can be generalized to larger @xmath0 . consider a state in which the wall is between lattice sites . in this state let all spins to the left of the wall have @xmath91 and those to the right have @xmath92 . note that it is possible to change the value of @xmath6 for _ either one _ ( but not simultaneously both ) of the spins adjacent to the wall without changing the unperturbed ( @xmath90 ) energy . for spin @xmath25 since the nonzero matrix elements of the perturbation due to the transverse field are all the same , the problem is totally equivalent to a spin 1/2 chain of twice the length . this observation explains the fact that our results for @xmath19 and @xmath25 are indistinguishable . when @xmath112 , one has to account for the fact that to move the wall through one lattice constant involves matrix elements which depend on the initial and final values of @xmath6 . this case will be considered later . the discussion has been so far restricted to the ferromagnetic phase f. the oscillatory phase is distinguished from it by showing a non - monotonic decrease of the correlation function [ see eq . ( [ corrs ] ) ] . how does this behavior manifest itself on the level of the magnetization profile @xmath79 ?    to answer this question , we display in fig . 6a , for @xmath41 , @xmath79 for @xmath113 and @xmath114 and for @xmath7 _ even_. indeed , this profile is distinct form the ones observed previously in the phase f. first , finite - size effects are much larger than in the situations discussed before . second , @xmath79 displays , at least for @xmath7 finite , a step - like behavior and it looks as if the system was built out of hard objects each occupying two lattice sites . when @xmath48 is increased , these composites soften until they completely melt at the transition towards the f phase . this picture is modified in interesting ways for @xmath7 _ odd_. in fact , the calculation of @xmath79 requires a little more care in this case . for @xmath7 even , the system has a single well - defined ground state separated by a gap at least of order @xmath115 ( see sec . v ) from the excited states . that is not so for @xmath7 odd . rather , for @xmath90 but @xmath116 , one finds that the ground state is twofold degenerate . that is a new degeneracy which has nothing to do with the ferromagnetic ordering of the system . even when @xmath117 , the two lowest states in @xmath89 remain much closer to each other than with the other states , the latter one having gaps of order @xmath118 . this new ( near ) degeneracy implies that one must reconsider the calculation of the magnetization profile @xmath79 . rather than blindly using eq . ( [ magpro ] ) , we take the _ two _ ground states @xmath54 and @xmath119 to be nearly degenerate . then we construct the matrix @xmath120 and find its two eigenvalues @xmath121 . each of those represents a magnetization profile and one of them is shown in fig . the other profile is obtained by reflection around @xmath122 . the asymmetry in fig . 6b , observed for @xmath7 odd and finite , is easily understood in terms of the composite objects introduced above . note that within the manifold @xmath93 , the @xmath123 term in the hamiltonian moves the wall through two lattice spaces . since we are discussing @xmath90 , this is the only kinetic energy of the domain wall . thus , if we write @xmath124 , where @xmath125 ( @xmath126 ) is the submanifold of state in which there are an even ( odd ) number of up spins , then there are no matrix elements between these two submanifolds . furthermore , these two submanifolds are related to one another by spatial inversion ( accompanied by @xmath127 ) . thus , for @xmath7 odd , the problem decomposes into two identical eigenvalue problems ( hence the twofold degeneracy of the energies ) . the matrix to be diagonalized is exactly the one considered in sec . iii.b , but now for @xmath128 sites and with @xmath129 . the two sets of eigenstates are thus @xmath130 ) diagonal . writing @xmath131 and @xmath132 , respectively , we find for the profile for the @xmath133 set of eigenstates ( indicated by a subscript `` @xmath133 '' ) @xmath134 in particular , this reproduces the steps in @xmath135 observed numerically . now , taking the continuum limit , it is easy to see that the width of the terraces decreases as @xmath77 and that @xmath136 . we therefore suggest that the phenomenon observed in fig . 7 is a novel type of finite - size effect . as @xmath77 , the step - like and asymmetric profiles found for @xmath7 finite will converge towards a smooth and symmetric limit function .    at least for @xmath137 , this limit function appears to show a simple relationship with the profiles previously found in the f phase . indeed , our data suggest the following . the profiles for @xmath7 large as found for @xmath138 are very close to those obtained for @xmath139 . for small @xmath35 our analytic work shows this to be true with @xmath140 . for larger @xmath35 we have the phenomenological result , @xmath141 evidence for this is provided in fig . 7 , where some profiles found for @xmath90 and @xmath116 ( data points ) are collapsed with profiles calculated with @xmath142 . in this section we obtain several analytic results in the large @xmath0 limit . we first analyze the classical ( @xmath15 ) system . then we treat the quantum system with @xmath0 large , but not infinite .    to treat the classical ( @xmath27 ) system , we write @xmath143 where @xmath144 is a continuous classical variable . then the energy of the classical system is @xmath145 for the classical calculation without a wall , we will adopt periodic boundary conditions , so that @xmath146 and the last term in eq . ( [ eclass ] ) runs from @xmath147 to @xmath148 . in that case @xmath149 , where , for convenience , we set @xmath150 where @xmath151 is the critical value of @xmath48 above which long - range ising order disappears . for the periodic chain the energy per spin , @xmath152 , is given by @xmath153    in the classical calculations we will use two types of boundary conditions to generate a wall . the first type of boundary condition is used for the continuum calculation . here the number of sites is infinite and hence we set @xmath154 . this continuum calculation is valid when the angle @xmath144 changes slowly with position , as it does for @xmath155 . the second type of boundary condition is that used in the discrete calculation . here it is convenient to use antiperiodic boundary conditions in which spins at opposite ends of the chain are coupled antiferromagnetically , with @xmath64 and we require @xmath156 .    in any case , the ground state energy is found by minimizing @xmath58 with respect to the @xmath144 s . apart from end effects this minimization yields the following conditions ( for @xmath157 ) which characterize the exact ground state , @xmath158      for small @xmath159 the angle @xmath144 will change abruptly at the domain wall and we will treat this case in the next subsection . as @xmath159 increases ( up to the critical value @xmath160 ) , @xmath161 will become a smoother function of @xmath18 . in that case , a continuum approximation for @xmath161 makes sense . to treat the continuum limit for @xmath155 , we set @xmath162 , where @xmath163 , whence eq . ( [ exact ] ) becomes @xmath164 \ .\ ] ] the continuum limit of this equation is @xmath165 \ .\ ] ] the solution to this differential equation is reduced to quadratures : @xmath166    we now discuss the significance of this result . first we analyze the behavior for large @xmath2 . at large @xmath167 , @xmath168 approaches @xmath169 , where @xmath170 . thus for large @xmath167 we set @xmath171 $ ] . keeping only the dominant terms in eq . ( [ x ] ) , we obtain @xmath172 this gives @xmath173 where @xmath174 and @xmath47 is the correlation length , given by @xmath175 thus , for @xmath176 , @xmath177 with @xmath178 , @xmath179 , and @xmath180 . next let us see the behavior of the solution near @xmath181 . to do that , differentiate the solution above with respect to @xmath2 at @xmath181 : @xmath182 where we used @xmath183 . this gives @xmath184_{x=0 } = { \sqrt 2 \over a } ( 1-\tilde h ) \ .\ ] ] to estimate the width , @xmath185 , of the wall , we note that over a distance @xmath185 , the angle @xmath144 varies from approximately @xmath186 to approximately @xmath187 . this reasoning indicates that @xmath188_{x=0 } = 2 \phi_{\rm eq } \ , \ ] ] for @xmath189 , we have @xmath190 , so that @xmath191 thus for @xmath192 , @xmath185 is of order @xmath47 , as one might expect .      in this and the next subsection we look at the discrete equations , which are the appropriate ones for small @xmath159 . since we are dealing with a classical system , it is convenient to use antiperiodic boundary conditions in which spins at opposite ends of the chain are coupled antiferromagnetically , with @xmath64 . first we consider the case when the center of the wall is at a lattice site . to treat this case , we consider a chain with an odd number of spins . number the sites @xmath193 and fix @xmath194 . then @xmath195 and the energy @xmath196 for this chain of @xmath197 sites is @xmath198 the @xmath161 are determined by eq . ( [ exact ] ) for @xmath199 , by @xmath200 and by @xmath201 to obtain the solutions for the @xmath144 s at small @xmath48 we write @xmath202 for @xmath203 . then for @xmath204 @xmath205 \ , \ \ \ \tilde h \cot \epsilon_n = { 1 \over 2 } [ \cos \epsilon_{n-1 } + \cos \epsilon_n ] \ .\ ] ] to obtain the energy at order @xmath159 we only need to solve the equations correct to order @xmath159 , in which case @xmath206 then , to find the energy to order @xmath159 , we may set @xmath207 , and @xmath208 , so that @xmath209 the wall energy @xmath210 is given by the large @xmath211 limit of @xmath212 where @xmath152 is the energy per site of the uniform chain . thus @xmath213      to treat the case when the center of the wall is midway between two lattice sites , we consider a chain of @xmath214 sites , numbered @xmath215 . we require that @xmath195 and we use the same antiperiodic boundary conditions as in the previous case . then the energy is @xmath216 as before , the wall energy , @xmath217 is obtained via @xmath218 the @xmath161 are determined by eq . ( [ exact ] ) for @xmath199 , by eq . ( [ phin ] ) , and by @xmath219 to analyze these equations , set @xmath220 . then , to first order in @xmath221 we have ( for @xmath222 ) @xmath223\ ] ] or @xmath224 we expect an exponentially decaying solution for @xmath221 for @xmath225 . so set @xmath226 so that @xmath227 from which we get @xmath228 for small @xmath159 , we get @xmath229 . for @xmath176 , we get @xmath230 , in agreement with the continuum result . these results hold irrespective of the position of the center of the wall . we now analyze the solution near the wall for small @xmath159 . to do that we first look at eq . ( [ phiref ] ) for @xmath231 when we set @xmath232 : @xmath233 \ .\ ] ] but @xmath234 , so @xmath235 = { 1 \over 4 } \epsilon_1 ^ 2 \ .\ ] ] so @xmath236 from here on one should interpret @xmath237 to mean @xmath238 . next , look at the equation for @xmath239 : @xmath240 \approx   1 \ .\ ] ] in this way we find that @xmath241 , for @xmath242 . a more systematic approach shows that we can write the solution for the @xmath35 s as @xmath243 \ , \ \ \ \ \ \epsilon_n = \epsilon_{\rm eq } + a_n(4\tilde h)^{2n-(7/3)}f_n[(4\tilde h)^{2/3 } ] \ , \ \ \ \ \   n > 1 \ , \ ] ] where @xmath244 , @xmath245 , and the functions @xmath246 are analytic functions such that @xmath247 .    to calculate @xmath248 , note that , up to order @xmath249 in the energy , we may write @xmath250 ^ 2 + { 1 \over 24 } ( 4\tilde h)^{4/3 } [ f_1(\tilde h^{2/3})]^4 \dots \nonumber \\ \sin \phi_j & = & \cos \epsilon_j = 1 \ , \ \ \ \ \ \ j>1 \ ; \nonumber \\ \phi_1 & = & \sin \epsilon_1 = ( 4\tilde h)^{1/3 } f_1 ( \tilde h^{2/3 } ) + { \rm o}(\tilde h ) \nonumber \\ \phi_j & = & \sin \epsilon_j = { \rm o}(\tilde h ) \ , \ \ \ \ \ \ j>1 \ .\end{aligned}\ ] ] putting these evaluations into eq . ( [ eprpr ] ) , we obtain @xmath251 ^ 2 + { 1 \over 24 } ( 4\tilde h)^{4/3 } [ f_1(\tilde h^{2/3})]^4 + ( n-2 ) \nonumber \\ & & \ \ - \frac{1}{2}\biggl [ 1 - ( 4\tilde h)^{2/3 } [ f_1(\tilde h^{2/3})]^2 + { 1 \over 3 } ( 4 \tilde h)^{4/3 } f_1(\tilde h^{2/3})]^4 \biggr ] + \frac{1}{2 } + 2\tilde h [ ( 4\tilde h)^{1/3 } f_1(\tilde h^{2/3 } ) ] \nonumber \\ & = & n-1 + ( 3/8 ) ( 4\tilde h)^{4/3 } + { \rm o}(\tilde h^2 ) \ .\end{aligned}\ ] ] where we used the definition that @xmath252 . then the wall energy is @xmath253 note that this result is _ not _ identical to the case when the center of the wall is at a lattice site . the physics of this result will be discussed in more detail in the next section . we now consider the quantum chain for large @xmath0 . for a chain of @xmath7 spins of magnitude @xmath0 , there are @xmath254 states which have the same @xmath90 ground state energy and which therefore must be treated within degenerate perturbation theory . numerical studies show that for @xmath255 , this manifold of states splits into bands which are slightly different depending on whether @xmath256 is even or odd . for @xmath0 half integer there are @xmath256 bands , each having @xmath7 states except for the middle band which has @xmath257 states . approximately , the centers of these bands are at energies @xmath258 where @xmath259 assumes the values @xmath260 , @xmath261 , @xmath262 , @xmath263 @xmath264 . for integer @xmath0 one has @xmath265 bands , each having @xmath7 states except for the middle band which has @xmath266 states . in this case the centers of the bands are given by eq . ( [ center ] ) , but @xmath259 assumes the values @xmath260 , @xmath267 , @xmath268 , @xmath263 , @xmath264 . as an example , in fig . 8 we show the density of states for @xmath269 within the manifold @xmath93 . as fig . 8 illustrates , the bands become wider and the gaps between bands become smaller as one approaches zero energy ( about which the levels occur symmetrically ) . a qualitative explanation of the occurrence of such band gaps is as follows . if the spin 1/2 case is likened to a hopping model , then the spin @xmath0 case is analogous to a hopping model in which there is a periodically variable hopping matrix element . this problem is therefore analogous to that of an electron in a weak periodic potential , where one knows@xcite that even for arbitrarily weak potentials one has band gaps . we will now demonstrate the existence of the band gaps and obtain quantitative information on the band widths for @xmath270 as follows . we will give a construction for the outermost ( and narrowest ) sub - band , for which @xmath271 in eq . ( [ center ] ) . consider the submanifold of states , @xmath272 , which contains the states , shown in fig . 5 , @xmath273 ( for @xmath274 . these states are defined so that all spins ( if any ) having @xmath275 have @xmath276 , all spins ( if any ) having @xmath277 have @xmath278 , and @xmath279 . one may verify that the state @xmath96 is in the subspace @xmath93 of states of minimum energy eigenstates , @xmath280 , of the @xmath90 hamiltonian , i.e. , that @xmath281 . the states @xmath96 are not orthogonal to one another , but do have an overlap which is small for large @xmath0 and is given by @xmath282 for @xmath283 . rose@xcite shows that @xmath284 where @xmath285 is a rotation matrix . therefore , we use states which are orthonormalized to leading order in @xmath286 , @xmath287 where we interpret @xmath288 and @xmath289 to be zero . we now work only to first order in @xmath286 . then one notes that unless @xmath290 , @xmath291 has zero overlap with the subspace @xmath93 . to see this , note that @xmath292 . therefore we write , correct to leading order in @xmath286 , @xmath293 where @xmath294 . also @xmath295 thus we expect this subspace of states with @xmath296 to form a sub - band centered at energy @xmath297 with a width determined by the hopping matrix element@xcite @xmath298 . thus the width of this band should be @xmath299 . numerically , for @xmath300 we found a band width of @xmath301 in excellent agreement with our calculation . we thus have an analytic description of the outermost two sub - bands corresponding to @xmath302 . we did not consider an analysis of the remaining inner sub - bands . note that the results shown in fig . 8 are not for asymptotically large @xmath0 . we did check that when @xmath0 is large enough , the outermost subband does become symmetric , as one would expect for a one - dimensional density of states corresponding to eqs . ( [ hoff ] ) and ( [ hdiag ] ) .    in a sense , one sees from this calculation that for the wall to move one lattice spacing , it must tunnel through the phase space of spin states indicated by the result of eq . ( [ tunnel ] ) . in this connection a nice analogy has been suggested by stinchcombe.@xcite consider a rotor in a strong @xmath303 potential . on knows from exact solutions of the associated eigenvalue problem that one has a limit in which one has harmonic oscillator levels in the bottom of the potential well , all of which are doublets with a small tunnel splitting.@xcite it is tempting to think that one can provide a field theory in one space and time dimension in which the single potential minimum of the rotor becomes a periodic potential in a lattice . then the bands would reflect the tunnel splitting . so far , we have considered the form of the profiles @xmath79 and have seen that at least qualitatively , their form can be understood from simple degenerate perturbation theory . the manifold @xmath89 considered consists of those states which give the lowest energies in the @xmath137 limit . for the case of @xmath41 , for @xmath90 these are the states shown in fig . 5 , namely , @xmath305 and for @xmath7 sites , there are @xmath257 of them . for @xmath25 one starts at @xmath90 from the states @xmath306 and one gets a subspace of @xmath307 states . all these states have in common that they contain a single wall . states which do not belong to this `` one - particle subspace '' @xmath89 have a large gap with the ground state . it is now remarkable that this clear separation of the subspace @xmath89 from all other states remains intact even for _ finite _ values of @xmath48 . the subspace @xmath89 has a simple analogy for ( anti)periodic boundary conditions . in these cases , @xmath65 commutes with the ( modified ) translation operator @xmath67 ( or @xmath308 ) and is thus decomposed into block diagonal form , the blocks being labeled by the eigenvalues of @xmath67 ( or @xmath308 ) . @xmath89 corresponds to the set of lowest eigenstates of the blocks of @xmath65 .    while the gap of the lowest excited state _ outside _ @xmath89 with the ground state is finite ( e.g. @xmath309 for @xmath41 and when @xmath82 ) , the gaps inside @xmath89 scale with @xmath304 . if @xmath310 denotes the @xmath311 gap , we expect the scaling @xmath312 with @xmath313 , as obtained in eq . ( [ pertth ] ) . finite - size estimates for @xmath314 are then obtained from , with @xmath315 @xmath316 the extrapolation towards the limit @xmath317 was carried out with the bst extrapolation algorithm@xcite . we illustrate the convergence of the finite - size data with a few examples for @xmath25 in table  [ tab1 ] . the convergence towards @xmath313 is even clearer for @xmath41 . we now consider the finite - size amplitudes @xmath318 the motivation for this comes from systems being precisely at a critical point . in that case , it is known that for one - dimensional quantum chains finite - size amplitudes defined as @xmath319 are related@xcite to the anomalous dimensions of the scaling operators of the model and this leads to simple patterns of the spectrum of the amplitudes ( see e.g. ref . for more information ) . although we are not working here at a critical point , we observe a simple structure of the excited states within @xmath89 for @xmath77 out of the critical region . namely @xmath320 evidence for this is presented in table  [ tab2 ] for @xmath41 and table  [ tab3 ] for @xmath25 which give the bst extrapolated@xcite estimates for the @xmath321 . it is remarkable that the _ same _ expression should hold for the gaps ratios of both the @xmath41 and the @xmath25 models , even though @xmath48 is not very small and/or @xmath116 . for @xmath48 small and @xmath82 , this result can be recovered from the energies ( [ pertth ] ) in lowest order of perturbation theory .      turning to the oscillatory phase , we note that the spectrum of the states within the subspace @xmath89 is different from that in the f phase . here there are important differences between @xmath7 even and @xmath7 odd .    for simplicity , we restrict attention to spin @xmath41 and first concentrate on the line @xmath90 . the structure of the levels we are going to find is shown in fig . 9 . for @xmath7 even , the spectrum of the low - lying states corresponding to @xmath89 is grouped into doublets . while the doublet splitting is of order @xmath115 , the gaps between doublets are of order @xmath118 . for @xmath7 odd , on the other hand , the energy levels are doubly degenerate , with gaps between them of order @xmath118 .    to see this , we look for a scaling behavior of the energy gaps @xmath323 and define finite - size estimates for the exponent @xmath314 using ( [ thetafs ] ) . in table [ tab5 ] , we give our results for @xmath324 . for @xmath7 even , estimates of @xmath314 from the lowest six gaps ( @xmath325 ) are given . ( for some of the higher gaps , the finite - size data for @xmath7 small are not yet in the asymptotic regime and are thus discarded . ) we observe that the lowest gap scales with an exponent @xmath326 , while all the higher gaps scale with @xmath313 . for @xmath7 odd , on the other hand , each energy level is doubly degenerate even for finite @xmath7 , i.e. @xmath327 , @xmath328 . estimates of @xmath314 for @xmath329 are given in table  [ tab5 ] and we find @xmath313 throughout . similar results are also found for @xmath330 and @xmath331 . next , we study the finite - size amplitudes @xmath332 , with @xmath315 . estimates for the lowest six amplitudes extrapolated to @xmath77 are shown in table  [ tab6 ] , obtained for @xmath7 even or odd , respectively . we find that for @xmath7 even , the estimates for pairs of amplitudes are quite close to each other and are consistent with being equal . furthermore , their numerical values are near to the ones found for @xmath7 odd . taken together , the present data suggest the simple picture that in the @xmath77 limit , the amplitudes @xmath321 should become doubly degenerate and independent of the evenness or oddness of @xmath7 . finally , when looking for the amplitude ratios @xmath333 , we find the following pattern @xmath334 which is consistent with the very same structure ( [ amprat ] ) of the low - lying amplitudes found on the @xmath82 line this finding is certainly consistent with the relationship between the order parameter profiles observed earlier ( see fig . 7 ) between systems on the @xmath90 and @xmath82 lines . these findings can be reproduced from the perturbative arguments of section iii.b when @xmath35 is small . for @xmath7 odd , the eigenvalue problem then reduces to one treated there , but with @xmath335 sites and @xmath336 and each state being twofold degenerate . for @xmath7 even , we get two distinct problems , one with @xmath337 sites and one with @xmath338 sites . using the expression ( [ pertth ] ) for the energies , we find pairs of levels with a splitting of order @xmath115 between them . qualitatively , the same structure persists throughout the o phase . we illustrate this in fig .  10 , where we show for @xmath339 and @xmath340 sites the dependence of the first few eigenvalues on @xmath48 . because @xmath7 is not very large , the statements we shall make about the spectrum apply most strongly to the lowest levels and less well to the higher levels . we see that inside the o phase , the lowest energies occur in pairs which oscillate around another . in the ground state , there are for @xmath7 sites @xmath335 level crossings for @xmath7 odd and @xmath337 level crossings for @xmath7 even , respectively . these level crossing do not continue into the f phase . finally , we point out that at the f / d transition , the energies combine into a @xmath341 representation of the @xmath342 virasoro algebra , as predicted from conformal invariance@xcite . in agreement with that prediction , we observe in fig . 10 close to the critical point @xmath343 an ( approximately ) equidistant level spacing with the degeneracies @xmath344 . ( actually , because of finite size effects , this structure is realized at @xmath345 rather than at @xmath346 . ) this kind of level crossings in the ground state of finite - lattice quantum system has been first investigated for _ periodic _ boundary conditions@xcite . there is was noted that the level crossings always occurred between the @xmath39 even and odd sectors of the model . here we find that the level crossing persist even if the @xmath39 symmetry is broken by our chosen `` up - down '' boundary conditions ( [ bc ] ) . applying finite - size scaling@xcite to the location @xmath347 of the level crossings , we expect ( but did not test ) a scaling @xmath348 . for periodic boundary conditions it is known@xcite that @xmath178 . ( @xmath349 gives the o / f transition line ) . it is obvious that the fact that the width of the wall for the quantum model is of order the system size is a result of a quantum superposition of states each of which have a narrow wall . it is tempting , therefore , to introduce a measure of the  intrinsic " width of the wall , which is the minimum width obtainable within the subspace of states under consideration . we emphasize that this concept depends on the subspace of states begin considered . as we vary @xmath35 , the strength of the @xmath3-@xmath3 coupling , it seems plausible that the character of the low lying  particle " states may vary . thus , it would be of interest to introduce a measure of the intrinsic width applicable to this case . for this purpose consider the quantity @xmath350 defined as @xmath351 where the maximum is to be taken over all possible quantum states @xmath96 in the subspace @xmath93 . ( as long as the site @xmath97 is not near an end , @xmath350 will depend only weakly on @xmath97 . ) if the states consist of sharp walls , then @xmath350 will be unity . so we use this quantity to define the intrinsic width , @xmath352 , via @xmath353 . this type of definition is somewhat similar to the inverse participation ratio introduced to characterize localized and extended states of an electron in a random potential.@xcite here we take the subspace @xmath93 to be the  particle " subspace of states . this definition is appropriate as long as the subspace of particle states can be unambiguously separated from the continuum . this requirement suggests that @xmath354 . but , we can also use this idea when @xmath35 is nonzero . we should mention that the maximization required in eq . ( [ max ] ) is easy to carry out . one simply forms the matrix , @xmath355 , ( in any representation ) where @xmath356 and @xmath350 is the largest eigenvalue of @xmath355 . when @xmath350 is less than unity , one also obtains information on the shape of the wall via the eigenvalue spectrum of @xmath355 . for instance , if the wall has width 3 , and if the three largest eigenvalues of @xmath355 are degenerate , then one would conclude that the intrinsic wall profile is linear . one context in which this concept provides some information is when we consider the large @xmath0 limit of the transverse ising model . in that case we have already seen that the low lying particle spectrum splits into bands corresponding roughly to values of @xmath357 . when we apply eq . ( [ max ] ) taking the subspace @xmath93 to be the lowest band of @xmath7 states with @xmath296 , we expect to find @xmath358 . we in fact verified this expectation in our numerical calculations . they showed that one obtains two nearly degenerate maximal eigenvalues of magnitude essentially equal to @xmath359 . there are two eigenvalues because there are two wavefunctions which can maximize @xmath350 since the minimum intrinsic wall width is two lattice spacings . we see then , that the prescription of eq . ( [ max ] ) coincides with the perturbative calculations for small @xmath48 . first we summarize the result for the classical ( @xmath15 ) system . note that @xmath360 , for small @xmath5 . so to move the wall through one or more lattice constants requires an energy @xmath361 \ .\ ] ] we have numerically studied the energies of the classical domain wall when its center is either at a lattice site or midway between lattice sites . we find that the barrier energy @xmath362 does not change sign as @xmath5 is increased as might be suggested by eq . ( [ barrier ] ) , but rather @xmath362 decreases monotonically to zero as @xmath5 is increased towards its critical value of unity . we carried out similar calculations for quantum systems with @xmath41 and @xmath25 . we found that the difference in energy when the center of the wall moved from a lattice site to a point midway between lattice sites was too small to be accessible to double precision arithmetic . since , for a chain of length @xmath363 , we can not imagine a parameter on such a scale , we assert that the quantum barrier energy is zero . practically , we performed this test as follows . for sufficiently large lattice sizes @xmath7 , we expect for the ground state energies @xmath364 ( i.e. wall between two sites ) and @xmath365 ( i.e. wall at a site ) @xmath366 where @xmath367 is the ground state energy per spin and independent of the boundary conditions . for @xmath41 , @xmath368 where @xmath58 is a complete elliptic integral@xcite . the desired wall energy difference should be proportional to @xmath369 . this in turn can be estimated from the quantities @xmath370 and we note that @xmath371 and @xmath372 should have opposite signs . we give in table  [ tab4 ] estimates for both @xmath371 and @xmath372 for both @xmath41 and @xmath25 for @xmath81 . we chose this value of @xmath48 to be small enough that is was for from criticality but large enough so that @xmath373 should be easily observable if nonzero . taking the entries at face value , the wall energies must be less than @xmath374 . moreover , since @xmath375 and @xmath376 should have opposite signs , the values given in table  [ tab4 ] appear to be entirely due to the leading finite - size corrections and we find that @xmath377 decrease rapidly with increasing @xmath211 . so we get an upper bound @xmath378 for both @xmath41 and @xmath25 . this result is not surprising . in the limit when the width of the domain wall diverges ( as it does for @xmath83 ) , the barrier energy should vanish . whether it vanishes algebraically or exponentially would require further numerical or analytic analysis . here we discuss the various regimes which exist at at large @xmath0 . it is clear that bands of energy levels exist on various scales . it is useful to discuss the consequences of these energy scales in terms of scaling functions . for notational simplicity , in this subsection @xmath379 will denote a scaling function of @xmath2 which , in general , will be different in different appearances . we first discuss the thermodynamic properties of the wall . for this purpose we will consider the entropy associated with the wall , @xmath380 , defined as the entropy with `` up - down '' boundary conditions minus that with `` up - up '' boundary conditions . we emphasize that the scaling of the bulk thermodynamics , i.e. , the extensive part of the thermodynamic potentials is independent of the boundary conditions and only depends on the bulk thermodynamic variables.@xcite at temperatures small compared to @xmath8 the entropy with `` up - up '' boundary conditions will be zero , because the ground state is nondegenerate . the entropy with `` up - down '' boundary conditions will be that due to the band of states with just one domain wall . first there is an extreme low  temperature limit in which the entropy is vanishingly small : @xmath381 where we always set @xmath382 , the boltzmann constant equal to unity . in this limit only the lowest single state of the tunneling band found in sec . ivb comes into play . at higher temperatures the tunneling band is activated , so that @xmath383 here @xmath384 is the entropy of a one dimensional tight binding band of @xmath7 sites . by noting the form of the density of states one can see that for small @xmath2 , @xmath385 and as @xmath386 , @xmath387 . the upper limit of this regime is defined so that essentially only the lowest subband of sec . ivb is activated . next , there is a regime when the temperature is high enough that all states of these subbands are equally populated , but only one domain wall exists . in this discussion it is assumed that @xmath388 , so that the band of particle states is well separated from the states with more than one wall . in this regime one has @xmath389 in this case , @xmath387 for @xmath386 and @xmath390 for @xmath391 . finally , we have the classical regime when @xmath392 . in that regime any effects due to the fact that @xmath393 and @xmath394 do not commute with one another becomes unimportant . in that regime then , the partition function for the quantum model of eq . ( [ hamzz ] ) is the same [ up to corrections of relative order @xmath395 as that of the classical model in which the operators are interpreted as classical variables according to eq . ( [ class ] ) . the effect of a random potential is somewhat different . in the present context a random potential would be created by a field which independently for each site assumes random values from a distribution of width @xmath396 . it is known@xcite that in the presence of a random potential all states are localized . that means that in the presence of a random field the eigenfunctions are localized and therefore the width of the wall in the ground state remains finite in the limit @xmath83 . nevertheless , the width of the wall , @xmath185 , can be expressed in terms of scaling functions of the same variables as describe the thermodynamics except that here @xmath396 replaces @xmath14 . thus @xmath397 and @xmath398 since we know that @xmath185 can not be less than the intrinsic width , we see that @xmath399 . similarly , we have @xmath400 in this case @xmath401 and @xmath402 , the intrinsic wall width in this regime . for any larger @xmath396 , @xmath403 remains equal to unity . we may summarize our conclusions for the ising model in a transverse field as follows .    1 . although an exact analysis of domain walls in the transverse ising model can in principle be accomplished using the jordan - wigner transformation , it is much simpler to consider the nature of degenerate perturbation theory in the presence of `` up - down '' boundary conditions . within degenerate perturbation theory for @xmath19 and @xmath25 the hamiltonian governing the position of the center of the domain wall is isomorphic with a hopping model with a hopping matrix element @xmath404 . we verified this in detail with analysis of the scaling of the energy levels with chain length , @xmath7 . 2 .   for spins @xmath405 , @xmath60 and @xmath406 the ground state profiles when spins at opposite ends of a chain of length @xmath7 are fixed to be antiparallel show relatively little dependence on the transverse field @xmath48 ( as long as @xmath48 is less than the critical value , @xmath407 , above which ordering in @xmath6 is destroyed ) . for small @xmath48 the profiles at large but finite @xmath0 are not very different from those at small @xmath0 . in all cases , the profiles are quite well obtained from the hopping model . 3 .   the profiles in the oscillatory phase show much larger finite size corrections than those in the ferromagnetic phase . in the limit @xmath83 the profiles in the oscillatory phase may still show some oscillatory behavior , unlike in the ferromagnetic phase . also , our results for the profiles show an interesting finite - size effect in which spatial parity appears to be broken . 4 .   in the spectrum of the oscillatory phase at @xmath90 one finds an unusual pattern for the gaps between the ground state and the first few excited states . while for @xmath7 even , the energies form doublets , each with a splitting which scales with @xmath7 as @xmath408 , and with gaps between the doublets which scale as @xmath304 . for @xmath7 odd , the energies are doubly degenerate and the gaps scale with @xmath7 as @xmath304 . this behavior was explained in terms of dimer excitations which occur in the limit @xmath409 when @xmath410 . 5 .   for large @xmath0 the quantum transverse ising model the hamiltonian describing the position of the center of the domain wall gives rise to band splittings analogous to those found for an electron in a periodic potential@xcite . the lowest sub - band is described by a hopping model in which the center of the domain wall tunnels from one site to a neighboring sites . the associated tunneling matrix element is of order @xmath411 , where @xmath412 . it would be interesting to recover this result within a field theory for one space and one time dimension . 6 .   for the classical ( @xmath27 ) model the domain wall is extremely narrow for small @xmath48 . we have determined the domain wall energy for the case when i ) the center of the domain wall is at a lattice site and ii ) the center of the domain wall is midway between two lattice sites . the difference between these energies is the pinning energy which must be exceeded to move the domain wall through the lattice . for the quantum model we find this pinning energy vanishes as @xmath83 due to the fact that the width of the domain wall diverges in this limit . 7 .   in attempting to relate quantum problems to classical problems it is necessary to define an intrinsic width , @xmath352 , of a quantum domain wall . we have defined @xmath352 as the minimum possible width attainable using wave functions from a manifold @xmath93 . at temperatures large compared to the band but small compared to the activation energy for creation of an additional wall , the manifold @xmath93 is the hopping band . in that case , the intrinsic width of the quantum system is essentially equal to the width of the analogous classical domain wall . * acknowledgements * the authors appreciate helpful advice from d.b . abraham , r. b. stinchcombe , t. c. lubensky , i. affleck and j.l . abh thanks the department of theoretical physics of oxford university for its kind hospitality . abh is supported by an epsrc visiting fellowship and mh is supported by a grant from the ec programme  human capital and mobility " . mc acknowledges support by the polish agency kbn ( grant number 2 p302 127 07 ) . f. family and t. vicsek , editors , `` dynamics of fractal surfaces '' , world scientific , singapore ( 1991 ) ; g. forgacs , r. lipowsky and th . m. nieuwenhuizen , in `` phase transitions and critical phenomena '' , edited by c. domb and j. l. lebowitz , vol . 14 , academic press , ny ( 1991 ) . this argument is correct to leading order in @xmath286 . it neglects overlap between @xmath413 and other states of @xmath93 , notably states similar to @xmath96 , but in which @xmath414 has the value @xmath415 . however the band of the @xmath416 states is separated from the band of @xmath296 states by an energy approximately equal to @xmath1 . but since the relevant matrix element is of order @xmath417 and is off - diagonal , it will enter at order @xmath418 \sim s\tau^2 $ ] . thus this effect is negligible compared to the terms we do treat , at least for large @xmath0 .                m. n. barber in _ phase transitions and critical phenomena _ , vol . c. domb and j. lebowitz , ( academic press new york 1983 ) , p. 145 ; v. privman ( ed ) , _ finite size scaling and numerical simulation of statistical systems _ , ( world scientific , singapore 1990 ) . .finite - size estimates of the exponent @xmath314 for the spin1 model for two values of @xmath48 . the line labeled @xmath419 gives the @xmath77 estimates and the numbers in brackets give the estimated uncertainty in the last given digit . [ tab1 ] [ cols="^,^,^,^,^,^,^",options="header " , ]            comparison of the magnetization profiles @xmath79 for the spin-@xmath405 ( boxes ) , the spin-1 ( full circles ) and the spin-@xmath406 ( open circles ) models . the full curve gives the profile of eq . ( [ simpro ] ) which is correct to first order in @xmath48 . the data are for @xmath82 and a ) @xmath81 and b ) @xmath85 .            density of states ( per site ) @xmath424 as a function of energy @xmath58 ( relative to the center of the manifold ) within the manifold @xmath93 for @xmath269 . the middle subband contains @xmath266 states . the other subbands contain @xmath7 states . the asymmetry of the subbands rapidly decreases as @xmath0 increases . inside each subband , near its edge at @xmath425 , @xmath426 for @xmath83 . spectrum of @xmath36 for spin @xmath41 and @xmath339 as a function of @xmath48 for @xmath340 sites . the two vertical lines indicate the location of the transitions between the o / f phases and the f / d phases , respectively , for @xmath77 .
we discuss several problems concerning domain walls in the spin @xmath0 ising model at zero temperature in a magnetic field , @xmath1 , applied in the @xmath2 direction . 
 some results are also given for the planar ( @xmath3-@xmath4 ) model in a transverse field . 
 we treat the quantum problem in one dimension by perturbation theory at small @xmath5 and numerically over a large range of @xmath5 . 
 we obtain the spin density profile by fixing the spins at opposite ends of the chain to have opposite signs of @xmath6 . 
 one dimension is special in that there the quantum width of the wall is proportional to the size @xmath7 of the system . 
 we also study the quantitative features of the ` particle ' band which extends up to energies of order @xmath5 above the ground state . except for the planar limit 
 , this particle band is well separated from excitations having energy @xmath8 involving creation of more walls . at large @xmath0 
 this particle band develops energy gaps and the lowest sub - band has tunnel splittings of order @xmath9 . 
 this scale of energy gives rise to anomalous scaling with respect to a ) finite size , b ) temperature , or c ) random potentials . 
 the intrinsic width of the domain wall and the pinning energy are also defined and calculated in certain limiting cases . 
 the general conclusion is that quantum effects prevent the wall from being sharp and in higher dimension would prevent sudden excursions in the configuration of the wall .