You are an expert at summarizing long articles. Proceed to summarize the following text: 
a skyrmion , first introduced in high - energy physics in the description of baryons,@xcite is a topologically non - trivial configuration of a continuous field @xmath2 , which is localized , in the sense that its gradient @xmath3 is significant only in a finite spatial region @xmath4 . far away from @xmath4 , the field @xmath2 approaches a uniform configuration , which is conveniently described by a boundary condition at spatial infinity.@xcite the euclidean action @xmath5 $ ] usually has the form of non - linear sigma model ( nlsm ) , with the lower bounds of energy being guaranteed by the bogomolnyi  prasad  sommerfield inequality . @xcite such lower bound is proportional to @xmath6  the absolute value of an integer `` topological charge '' @xmath7 , which classifies the field @xmath2 as a continuous mapping . in the work of skyrme , @xcite @xmath2 maps the ( compactified ) 3 + 1 dimensional space - time to the manifold of lie group @xmath8 ( which is isomorphic to @xmath9 , the 3-sphere ) , and the third homotopy group @xmath10 provides the topological charge @xmath7 , which was identified as particle number in that work . after its introduction in high - energy physics , skyrmion physics regained interest in the context of condensed - matter physics as a topological object in a two - dimensional ferromagnetic state . they have been discussed in quantum hall systems,@xcite and in chiral magnets@xcite and magnetic thin films.@xcite while skyrmions have been experimentally viewed in scanning - tunnelling microscopy and spectroscopy ( stm / sts ) in the latter two systems,@xcite skyrmions in quantum hall systems have mainly been identified through their magnetic properties in nuclear - magnetic - resonance experiments,@xcite and a spectroscopic identification is yet outstanding . furthermore , skyrmions have been used to describe special configuration of the superconducting order parameter in superconductors.@xcite the magnetization or the superconducting order parameter , denoted as @xmath2 , usually maps the ( compactified ) two - dimensional plane to the bloch sphere @xmath11 . the second homotopy group @xmath12 provides the topological charge @xmath7 , which is the degree of the mapping between two 2-spheres . a more familiar case is the vortex configuration of the complex scalar field @xmath2 on two - dimensional plane , which is punctured by the vortex core . the phase of the complex scalar field lives in a ring @xmath13 . then the homotopy group @xmath14 provides us the topological charge @xmath7 , known as winding number . some more exotic cases appeal to the second homotopy group of@xcite @xmath15 that allows for establish of skyrmions in @xmath16-component systems , such as in refs . [ ] for @xmath17 and [ ] for @xmath18 , or , to the homotopy group @xmath19 in ref . [ ] . in the present study , we choose a @xmath0 field on two - dimensional plane to model the static configuration of qh skyrmions in graphene monolayer . the distinguished feature of quantum hall ( qh ) skyrmion is that it carries _ electric _ charge of @xmath20 , where @xmath7 is its _ topological _ charge and @xmath21 is the charge of an electron . moreover , such equality holds locally , i.e. the electric charge is smeared in the texture of the qh skyrmion , the excess charge density @xmath22 in a qh skyrmion is equal@xcite to @xmath23 , where @xmath24 characterizes the spatial variation of the texture in the qh skyrmion , and the integral @xmath25 reproduces its topological charge @xmath7 . as a consequence of the smeared charge density , a qh skyrmion has a lower energy in the @xmath1 landau level ( ll ) than the quasiparticle of the same charge , therefore reduces the charge transport gap . a lattice of qh skyrmions also has lower energy than the corresponding wigner crystal.@xcite we finally notice that , as a consequence of the intimate relation between the topological and the electric charge , a skyrmion is a fermionic excitation in qh systems , in contrast to most of the aforementioned magnetic systems .    in this work , we study the qh skyrmions in the @xmath1 ll in graphene monolayer . besides the robust quantum hall effects in graphene in a wide range of temperature@xcite and magnetic field,@xcite there are several advantages to choose graphene as the host system for qh skyrmions . first , the dirac valley of electron states can be described by a pseudospin degree of freedom , and in the presence of strong perpendicular magnetic field the coulomb interaction is approximately symmetric@xcite under global @xmath26 transformations in the spin@xmath27valley space . in contrast , the layer pseudospin in double - layer qh system does not enjoy such a symmetric coulomb interaction in the spin@xmath27valley space , because the layer separation always breaks the pseudospin symmetry of the coulomb interaction . in our approach , the @xmath26 spin - valley symmetry is explicitly broken by terms at lower energy scales ( compared to the coulomb energy ) , namely the zeeman coupling that couples directly to the spin , and the valley pseudospin anisotropy . as a second advantage of the graphene monolayer , one may point out that the valley degree of freedom coincides with the sublattice degree of freedom for the electron states in the @xmath1 ll . such a coincidence provides a convenient way to probe the @xmath28-component of the valley pseudospin directly  it can be read off from the sublattice occupation , thus allowing a direct imaging of the pseudospin texture via stm / sts experiments . moreover , the spin texture in graphene under qh conditions can also be imaged by spin - resolved stm / sts experiments . the combination of the lattice - resolved images for spin and pseudospin textures thus allows for an identification of the various skyrmion types in graphene . the fact that graphene is naturally a surface system renders these states spectroscopically more accessible than qh systems in semiconductor heterostructures , where the two - dimensional electron system appears at the interface between two types of semiconductors . for stm / sts experiments in semiconductor heterostructures , two - dimensional electron systems have been realized on an n - doped insb surface@xcite albeit with a mobility that does not reach that of the more common gaas hetero - structures or graphene . our work completes our previously obtained phase diagram for qh skyrmions in graphene@xcite in several respects . apart from a full description of the fm states one encounters at @xmath29 ( and @xmath30 by particle - hole symmetry ) , we provide here a detailed characterization of skyrmions with entanglement . in addition to a pure entanglement skyrmion , one obtains an exotic type of _ deflated _ pseudospin skyrmions with partial entanglement . we characterize all encountered skyrmion types with the help of three different bloch spheres that describe spin , valley pseudospin and entanglement , respectively . furthermore , all ferromagnetic and skyrmion states are analysed from their appearance in lattice - resolved density plots that may use as a guideline in an experimental stm / sts investigation . finally , we discuss the scaling relations of the skyrmion size and energy close to the transitions between different underlying fm states while a critical behavior is obtained in the case of a symmetry restoration at the transition , we find that the scaling is truncated in the other cases where a ( subordinate ) symmetry is not fully restored . these scaling relations are then discussed in the context of further modifications of the anisotropy parameters that we vary to obtain the skyrmion phase diagram , e.g. in the case of a tilted sample . the paper is organized as follows . in sec . [ sec : quantum - hall - ferromagnetic - state ] we propose an anisotropic energy for the ferromagnetic state in the @xmath1 ll at quarter filling , and discuss the @xmath0 representation of the qh ferromagnetic states . there are four types of ferromagnetic ground states for the anisotropic hamiltonian . each type is described by a @xmath0 spinor that contains six angular variables and visualized on the spin , pseudospin and entanglement bloch spheres . we also visualize them on the honeycomb lattice . in sec . [ sec : cp3-skyrmion - of - charge - q-1 ] we discuss the energy functional and ansatz for the @xmath0 skyrmion on top of the qh ferromagnetic states , and present the skyrmion type diagram , which is obtained by minimization of the energy functional with the ansatz and under various input parameters of the model . the detailed analysis of the minimization result is presented in sec . [ sec:4-types - skyrmion ] . we visualize the skyrmion @xmath0-field on three bloch spheres in a same manner as for the qh ferromagnetic states . these visualizations explicitly demonstrate that the skyrmions can be viewed as a wrapping of the @xmath31-plane on bloch spheres . besides , the @xmath0-skyrmion is also visualized on the honeycomb lattice , in order to show the difference between two types of skyrmions . the lattice - resolved profiles of the @xmath0-skyrmion serve as references for the stm / sts imaging of spin - pseudospin textures in graphene monolayer under qh conditions . the qualitative discussion on the @xmath0-skyrmions are followed by quantitative discussions in sec . [ sec : size_and_energy ] , where we present the dependence of size and energy of the @xmath0-skyrmions on the input parameters of the model . finally in sec . [ sec : modification - of - the - anisotropy - energy ] , we discuss three ways to modify the anisotropic energy of the @xmath0-skyrmions . we demonstrate how the type of a @xmath0-skyrmion is altered by these modifications . these modifications are relevant in certain experimental regimes . a multicomponent quantum hall system is characterized by the @xmath32-fold degeneracy of the landau levels which originates from internal degrees of freedom of 2d electrons in a strong magnetic field . for example , when the zeeman energy is much smaller than the coulomb interaction , one may first neglect the zeeman coupling of the electron spin to the applied magnetic field and consider spin as an internal degree of freedom which yields a two - fold degeneracy of the landau levels . at half - filling of such ll ( i.e. integer filling of landau sublevel ) , the major cause for the complete spin polarization in the ground state is the coulomb interaction , which is larger than the zeeman energy and prefers a totally asymmetric orbital part of the many - particle wave function , so that the electrons are separated as far as possible , and the spinor part is totally symmetric . the zeeman coupling at a smaller energy scale then conspires with the coulomb energy and orients the polarized spins along the applied magnetic field . in this way , the electrons in a single ll form a _ quantum hall ferromagnet _ ( qhfm ) .    in graphene monolayer , the internal degrees of freedom of the landau levels are doubled by the valley degeneracy , which is described by a _ pseudospin _ analogous to the electron spin . the coulomb interaction has approximate @xmath26 symmetry,@xcite which is broken by spin / pseudospin anisotropic interactions at smaller energy scale .    at quarter filling , i.e. when one of the four spin - valley sublevels is completely filled ( @xmath29 for electron filling ) or completely empty ( @xmath33 for hole filling ) , the ground state @xmath34 for the electron system is the @xmath26 quantum hall ferromagnetic ( qhfm ) state@xcite @xmath35 where @xmath36 creates an electron in the @xmath1 ll with spin @xmath37 ( @xmath38 ) and pseudospin @xmath39 ( @xmath40 ) . the quantum number @xmath41 is the landau orbit , which characterizes the ll degeneracy . it is related to the underlying translation invariance while this invariance is , strictly speaking , broken by the position - dependent vector potential , the physical magnetic field is constant in the entire plane . the kinetic energy of an electron in this field does therefore not depend on the center of its quantized cyclotron motion . the landau orbit @xmath41 is precisely the quantum number associated with the position of this center . in the case of one electron per landau orbit , which we consider here , the coefficients @xmath42 satisfy @xmath43 . to model electron states with spatially homogeneous spin - pseudospin magnetization at a length - scale larger than @xmath44 , we assume that @xmath45 do not carry landau orbit index @xmath41 . since @xmath46 and @xmath47 ( @xmath48 is a real constant ) correspond to the same quantum state , the coefficients in eq . ( [ eq : qhfm ] ) can be uniquely represented by a cp@xmath49-spinor @xmath50 , which is normalized @xmath51 and is equivalent to @xmath52 . ( see appendix . [ subsec : cp3-spinor - and - nlsm ] ) the meaning of the four components in the cp@xmath49-spinor is inherited from @xmath46 .    in this section , based on a general proposal of valley pseudospin anisotropy , we discuss different types of qhfm ground states at quarter filling ( @xmath29 for electron filling and @xmath33 for hole filling ) of the four - fold degenerate @xmath1 ll in graphene monolayer . we propose the anisotropic energy in sec . [ subsec : anisotropic - energy ] , and then discuss the parametrization of cp@xmath49-spinor in sec . [ subsec : parametrization ] . the four types of qhfm ground state are discussed in sec . [ subsec : qhfm - ground - states ] . we schematically visualize the qhfm states on the honeycomb lattice of graphene in sec . [ subsec : qhfm - visualization ] . the electrons are restricted to the @xmath1 ll in our problem and hence the kinetic energy is quenched and set to a constant . due to the @xmath26 symmetry of the interaction hamiltonian@xcite @xmath53 the coulomb energy @xmath54=\left\langle f\right|h_{\rm c}\left|f\right\rangle$ ] does not depend on the qhfm state @xmath46 . the @xmath26 symmetry is explicitly broken by the spin - pseudospin anisotropies @xcite @xmath55 where @xmath56 and @xmath57 are spin and pseudospin density operators . the first terms reflect a pseudospin anisotropy that can be generated , e.g. by short - range interactions of hubbard type@xcite ( @xmath58 and @xmath59 ) or an out - of - plane@xcite ( also contributing to @xmath59 ) and in - plane kekul - type @xcite lattice deformation due to electron - phonon couplings . the coefficients @xmath60 have the dimension of energy , and @xmath61 are proportional to the perpendicular component @xmath62 of the applied magnetic field , whereas @xmath63 is proportional to the total applied magnetic field @xmath64 . their numerical values are estimated as @xmath65k$ ] , @xmath66k$ ] , @xmath67k$ ] and @xmath68k$ ] respectively in ref . @xmath69 explicitly breaks the @xmath26 symmetry down to u@xmath70u@xmath71u@xmath72 . while the u@xmath73 and the u@xmath74 symmetries reflect rotations around the @xmath28-quantization axes of the spin and pseudospin , respectively , @xmath75 indicates that the two orientations @xmath28 and @xmath76 of the pseudospin are equivalent . they are not equivalent in the spin channel due to the zeeman coupling . the u@xmath77 symmetry is more subtle and related to spin - pseudospin entanglement,@xcite as we discuss in more detail below . the system energy is determined by minimization of the anisotropic energy @xmath78=\left\langle f\right|h_{\rm a}\left|f\right\rangle$ ] , which we propose to be the following form : @xmath79 = a_{0}\frac{\delta_{\rm z}}{2}\left\ { u_{\perp}\left(m_{\rm px}^{2}+m_{\rm py}^{2}\right)+u_{\rm z}m_{\rm pz}^{2}-m_{\rm sz } \right\}.\ ] ] where @xmath80 is a dimensionless area to ensure an extensive energy , @xmath81 , and the dimensionless parameters @xmath82 and @xmath83 characterize the pseudospin anisotropy . in the present work , @xmath82 and @xmath83 are treated as control parameters that generate the different phases even if they may be more difficult to vary in a typical situation . in the above equation , we denote the spin and pseudospin magnetizations as @xmath84 respectively . they can be expanded explicitly in the components of the @xmath85-spinor @xmath86 . for instance , the @xmath28-component of the spin magnetization @xmath87 is @xmath88 and the @xmath28-component of the pseudospin magnetization @xmath89 is @xmath90 where we use the labels @xmath91 , @xmath92 , @xmath93 and @xmath94 for the index @xmath95 of the components of @xmath86 to emphasize their meaning . it is essential for later discussions to use the schmidt decomposition to parametrize a general cp@xmath49-spinor @xmath96 with six real parameters , namely @xmath97 $ ] and @xmath98:@xcite @xmath99 where @xmath100 and @xmath101 , @xmath102 are the pauli matrices , @xmath103 is the unit vector for the direction of magnetization . the subscript @xmath104 ( @xmath105 ) stands for spin ( pseudospin ) . the pauli matrices for spin and pseudospin are designed as @xmath106 and @xmath107 respectively , in agreement with the earlier convention for the cp@xmath49-spinor @xmath86 . in the present section , where we discuss qhfm states , the real parameters @xmath108 , and @xmath109 are constant in space , i.e. independent of the landau orbit @xmath41 , while we consider an explicit position dependence in the following section on skyrmions . to simplify notations , we therefore omit here to write explicitly the argument @xmath110 in the spinors and parameters . under the parametrization eq . ( [ eq : parametrizationz ] ) , the spin and the pseudospin magnetization for the cp@xmath49-spinor @xmath96 are @xmath111 respectively . the meaning of @xmath112 and @xmath113 in the parametrization eq . ( [ eq : parametrizationz ] ) is evident from the above equations they are the polar and azimuthal angle of the spin and pseudospin magnetizations . as we discuss in detail later , the parameter @xmath114 enriches the types of fm ground state of the system . it can be understood in terms of `` entanglement''.@xcite it is possible to rewrite the state @xmath96 into a direct product of spinors for electron spin and and valley pseudospin _ only _ when @xmath115 or @xmath116 . in this case the spin and pseudospin are _ unentangled_. otherwise , @xmath96 is a superposition of two `` product states '' carrying opposite spin and pseudospin . consequently , as indicated in eq . ( [ eq : si_y ] ) and eq .  ( [ eq : ti_y ] ) , the spin and pseudospin magnetizations have magnitudes smaller than @xmath117 . remarkably , these magnitudes are equal to @xmath118 , because in a @xmath0-spinor the spin and pseudospin are on equal footing , analogous to the two entangled @xmath119-spins in the bell state.@xcite since we have specified the forms of @xmath120 and @xmath121 in eq . ( [ eq : psi_basis ] ) and eq . ( [ eq : chi_basis ] ) , the coefficients in eq . ( [ eq : parametrizationz ] ) are complex numbers in general and the parameter @xmath109 is just the relative phase of these complex coefficients . throughout this paper , we use the term `` entanglement '' as a synonym of the magnitude of the spin or pseudospin magnetizations : `` maximal entanglement '' means @xmath122 ; `` unentangled '' refers to @xmath115 or @xmath116 . four types of qhfm ground states on the @xmath123 plane . they are listed in tab . [ tab : fm - tab ] . ]    applying the parametrization in eq . ( [ eq : parametrizationz ] ) to the cp@xmath49-spinor @xmath86 representing the ferromagnetic state ( [ eq : qhfm ] ) , we minimize @xmath78 $ ] with respect to the six parameters of @xmath86 and obtain four types of qhfm ground states , which are displayed in fig . [ fig : fm - region ] and tab . [ tab : fm - tab ] . due to the zeeman coupling , a cp@xmath49-spinor @xmath86 with polarized spin along the @xmath28-axis is always energetically favourable , independent of the pseudospin and entanglement . it follows that the optimal value is @xmath124 in _ all regions _ throughout the @xmath123 plane , if we assume @xmath125 $ ] . moreover , when @xmath126 ( region 1 , 3 ) , it is energetically favourable to orient the pseudospin along the @xmath28 ( or @xmath76 ) axis and one thus obtains an easy - axis fm state with optimal value @xmath127 or @xmath116 . on the contrary , when @xmath128 ( region 2 , 4 ) , a pseudospin magnetization in the @xmath31-plane is favoured , and one obtains an easy - plane fm state with optimal value @xmath129 and arbitrary optimal value for @xmath130 . on the line @xmath131 , the @xmath8 symmetry of pseudospin is restored and the optimal values for @xmath132 and @xmath130 are arbitrary . @xmath126 , @xmath133 unentangled easy - plane pspin fm & ( 2 ) @xmath128 , @xmath134 entangled fm with & ( 3 ) @xmath135preferential easy - axis pspin & entangled fm with & ( 4 ) @xmath136preferential easy - plane pspin &    entanglement enriches the types of fm states because tuning the parameter @xmath114 allows the magnitudes of spin and pseudospin magnetization to decrease and thus to lower the anisotropic energy @xmath137 . when @xmath138 , the optimal value of @xmath114 is always 0 . this is easily understood : since at least one of the parameters @xmath139 or @xmath83 is negative , @xmath140 is minimized when the spin and pseudospin magnetizations are maximal , i.e. when @xmath141 . for @xmath142 , the zeeman energy is crucial for the presence of unentangled fm states , which are of the same type as in the previous case . were there no term proportional to @xmath143 in the expression of @xmath137 , all the other terms related to pseudospin would be non - negative , and the minimization would result in a spinor with @xmath144 ( maximal entanglement or @xmath122 ) for all possible @xmath145 in the 1-quadrant . the term proportional to @xmath143 prefers larger magnitudes of @xmath146 along the applied magnetic field . since @xmath147 always holds , there is a competition between the tendency towards small magnitude of pseudospin magnetization due to the pseudospin contribution to @xmath137 , and the tendency towards large magnitude of the spin magnetization due to the spin contribution . here in the case of @xmath142 , spin contributes more , so that a maximal value of @xmath148 is favoured , very much as for negative values of @xmath139 and @xmath83 , and spin and pseudospin remain unentangled . the above - mentioned energy competition yields different results when @xmath149 . in this case the pseudospin contribution to @xmath137 is positive , which can be lowered not only by proper choice of the direction ( @xmath113 ) of pseudospin magnetization , but also by shrinking the magnitude @xmath150 of pseudospin magnetization . although the later yield a less negative value of the spin contribution to @xmath137 due to the identity @xmath147 , the reduction of energy contribution from pseudospin overcomes the increase of the contribution from spin , so that the overall minimum of @xmath137 is reached when @xmath151      as a consequence of the identity between valley pseudospin and sublattice index in @xmath1 , the different qhfm bear a clear fingerprint in the spin - polarized electronic occupation of the two graphene sublattices . the four types of qhfm ground states are visualized in figs . [ fig : qhfm_uea ] , [ fig : qhfm_uep ] , [ fig : qhfm_eea ] and [ fig : qhfm_eep ] . the upper parts ( panels a , b and c ) in each figure are the bloch sphere representations for the cp@xmath49-spinor @xmath86 of the qhfm ground state . in particular , panel ( a ) shows the spin magnetization @xmath152 ( eq . [ eq : si_y ] ) of the cp@xmath49-spinor @xmath86 in the spin bloch sphere . for the cp@xmath49-spinor @xmath86 corresponding to the unentangled qhfm states , with an easy - plane ( fig .  [ fig : qhfm_uep ] ) and an easy - axis magnetization ( fig .  [ fig : qhfm_uea ] ) , @xmath146 is a unit vector and its arrowhead is located _ on _ the spin bloch sphere . in contrast , for the cp@xmath49-spinor @xmath86 corresponding to the entangled qhfm states , again with an easy - plane ( fig .  [ fig : qhfm_eep ] ) and an easy - axis magnetization ( fig .  [ fig : qhfm_eea ] ) , the vector @xmath146 has a magnitude @xmath153 smaller than @xmath117 , and explores the _ interior _ of the spin bloch sphere . the appearance of the pseudospin magnetization @xmath154 in the pseudospin bloch sphere in panel ( b ) can be understood similarly . in the case of an easy - axis pseudospin ferromagnetic state that corresponds to the electronic occupation of a single sublattice , the arrow points along the @xmath28-axis , while a balanced occupation of both sublattices goes along with pseudospin magnetization pointing to the equator of the bloch sphere . again , in the presence of spin - pseudospin entanglement , the arrow starts to exploit the inside of the sphere . panel ( c ) shows the _ entanglement vector _ @xmath155 for a cp@xmath49-spinor @xmath86 of the qhfm state , where @xmath114 and @xmath109 are obtained by the parametrization of @xmath86 with eq . [ eq : parametrizationz ] . the bounding sphere of the entanglement vector is called the _ entanglement bloch sphere _ by analogy . for the cp@xmath49-spinor @xmath86 corresponding to the unentangled qhfm states , both with easy - plane ( fig . [ fig : qhfm_uep ] ) and easy - axis magnetization ( fig . [ fig : qhfm_uea ] ) , the entanglement vector @xmath156 points to the north pole of the entanglement bloch sphere . meanwhile , for the cp@xmath49-spinor @xmath86 corresponding to the entangled qhfm states ( figs . [ fig : qhfm_eep ] and [ fig : qhfm_eea ] ) , the direction of @xmath156 is generic due to a non - vanishing value of @xmath114 .    [ cols="^,^,^ " , ]     visualization of the deflated cp@xmath49-pseudospin skyrmion with entangled easy - axis fm background ( point 12 in fig.[fig : skyrmion_type_diagram ] ) . conventions are the same as in fig . [ fig : pt4_spin_skyrmion ] . ] we comment on the evidence of entanglement in a cp@xmath49-field @xmath157 . although by examining the local patterns of the @xmath158 and @xmath159 profiles around a generic point @xmath160 in the @xmath31-plane , it is not possible to tell whether a cp@xmath49-field @xmath161 carries entanglement , it is possible to do so by examine those profiles at large scale over the entire texture . as we have discussed in detail , the factorizable skyrmions do not carry entanglement at any point in the texture , and the evidence is clear in the @xmath158 and @xmath159 profiles  for a cp@xmath49-spin skyrmion , the pattern of sublattice occupation is uniform over the @xmath158 profile because of the constant pseudospin component , whereas for a cp@xmath49-pseudospin skyrmion , the two profiles are identical because of the constant spin component with polarization along the @xmath28-direction in the space of spin magnetization . in contrast to the factorizable skyrmions , the entanglement skyrmions ( cp@xmath49-entanglement skyrmions , deflated cp@xmath49-pseudospin skyrmions ) embedded in the easy - axis pseudospin fm background usually have an `` anti - ferrimagnetic '' pattern in the @xmath159 profile , i.e. the spin magnetizations on two sublattices point in opposite directions . the cp@xmath49-entanglement skyrmion and the cp@xmath49-spin skyrmion which are embedded in the same easy - plane pseudospin fm background can not be distinguished by their @xmath158 and @xmath159 profiles . but we can still make a difference by comparing their @xmath162 profiles . in the previous section , we concentrated on a classification of the different skyrmion types one encounters in graphene qhfm in the @xmath1 ll . the present section is devoted to a more quantitative analysis of the skyrmions size and energy , namely in the vicinity of phase transitions between the different fm background states . size of optimal skyrmions at @xmath163 $ ] and @xmath164 . black dashed lines highlight the boundaries of four regions for the four types of fm background of a skyrmion . ] the size of a cp@xmath49 skyrmion is characterized by the optimal value of the @xmath165 parameter in the skyrmion ansatz . as discussed in sec . [ subsec : skyrmion - ansatz ] , it is determined by the competition between the coulomb energy @xmath166 and the excess anisotropic energy @xmath167 . to be more precise , according to eq . ( [ eq : lambda_dep ] ) the optimal value of the @xmath165 parameter in the skyrmion ansatz is obtained by solving @xmath168 , which gives @xmath169}{2e_{\rm a}'[\tilde{z}_{\rm sk}]}\right)^{\frac{1}{3 } } , \label{eq : lambda_0}\ ] ] where @xmath170 is the @xmath0-field for an optimal skyrmion , but @xmath165 has been set to @xmath117 . it has the same meaning as in eq . ( [ eq : lambda_dep ] ) in sec . [ subsec : radial - deformation ] . we define the skyrmion size @xmath171 ( in unit of @xmath44 ) by averaging @xmath172 over the topological charge density @xmath24 @xmath173 because @xmath24 can be viewed as the excess charge density induced by the texture in a cp@xmath49 skyrmion and is thus a good measure for the spatial extent of the texture . for a radially deformed skyrmion , we have @xmath174 with an @xmath172-dependent size parameter @xmath175 and topological charge density @xmath176^{2}}\left(1+\frac{2r^{2}}{\kappa\lambda_{0}^{2}}\right ) . \label{eq : rho - topo - deformed}\ ] ] the coefficient @xmath177 is a monotonically increasing function of the radial deformation parameter @xmath178 . its actual form depends on the radial deformation and is _ not _ related to the energy minimization . in the scale - invariant limit @xmath179 we have @xmath180 . away from this limit , @xmath181 does not change much : for instance @xmath182 . [ fig : skyrmion_size_all ] displays the size @xmath171 of optimal skyrmions obtained in sec . [ subsec : minimization - results ] ( @xmath183 $ ] , @xmath164 ) . one notices generally an increase in the skyrmion size for @xmath145 close to a transition between different ferromagnetic backgrounds . these transitions are indicated by dotted black lines in fig . [ fig : skyrmion_size_all ] . in order to investigate in more detail the increase in skyrmion size , we plot , in fig . [ fig : skyrmsize ] , the skyrmion size @xmath171 as a function of @xmath184 along line a and line b. here , @xmath185 denotes the value at the border between two regions of different fm background , which is @xmath186 for line a separating the unentangled easy - axis from the unentangled easy - plane fm and @xmath187 for line b separating the unentangled from the entangled easy - axis fm . one sees in fig . [ fig : skyrmsize ] that along line a , the skyrmion size diverges ( red curves ) when approaching the transition at @xmath188 . this is expected because , at the transition , the pseudospin texture costs the same amount of anisotropic energy as the state with uniform pseudospin magnetization , giving a vanishing excess anisotropic energy @xmath189 = e[\check{z}_{\rm sk } ] - e_{\rm a}[f]$ ] and thus a divergent @xmath165 . physically this means that the pseudospin @xmath8 symmetry is restored when @xmath131 , and the coulomb energy of the skyrmion tends to inflate the skyrmion to infinite size . in the vicinity of the point @xmath190 , @xmath189 $ ] is proportional to @xmath191 . consequently , @xmath189 \propto |u_{\rm z}-u_{\rm z0}|$ ] with @xmath192 , then from eq .  ( [ eq : lambda_0 ] ) we know @xmath193 . therefore along line a , the skyrmion size @xmath171 scales as @xmath194 . the exponent @xmath195 is extracted from the red lines of the log - log plot in the inset of fig . [ fig : skyrmsize ] , in agreement with the above scaling argument . the divergence of the skyrmion size at the line @xmath196 reflects the underlying transition between an easy - axis and an easy - plane pseudospin fm background , as long as both are unentangled . skyrmion size ( in unit of magnetic length @xmath44 ) as a function of @xmath83 , along the lines a and b in fig . [ fig : skyrmion_size_all ] . ]    the blue curves in fig . [ fig : skyrmsize ] show the evolution of the skyrmion size along line b cutting the border @xmath187 , where one also notices an enhanced albeit non - divergent size . in contrast to the transition at @xmath188 , there is no symmetry restoration that would lead to a divergent skyrmion size , i.e. the anisotropy energy associated with the pseudospin remains finite here . this yields a `` truncated power law '' , @xmath197 that is now cut off by a non - zero constant constant @xmath198 . as one may expect from our simplified scaling analysis , the exponent @xmath199 is again close to @xmath200 . even if there is no fully developed divergence in the skyrmion size , its increase unveils again a transition between different underlying fm background states . the effect of disorder on skyrmions becomes significant when the length scale of the disorder potential is comparable to the skyrmion size . in such conditions , the charge - carrying skyrmion is trapped to the disorder potential that is coupled to electric charge . moreover , if there is also pseudospin disorder in the hosting system , then a @xmath0-pseudospin skyrmion is frustrated by the two types of disorders . in the experiments where one can control the parameters @xmath82 and @xmath83 , the relation between the skyrmion size @xmath171 and the disorder length scale can be revealed in transport measurements . energy of optimal skyrmions at @xmath163 $ ] and @xmath164 . the dimensionless value @xmath201 is plotted in the figure , with the numerical value of @xmath202 . black dashed lines highlight the boundaries of four regions for the four types of fm background of a skyrmion . ]      as discussed in sec . [ subsec : skyrmion - ansatz ] and sec .  [ subsec : radial - deformation ] , all scale - invariant @xmath0-skyrmions carrying topological charge @xmath203 have size - independent energy @xmath204 , and size - dependent energy @xmath205 . the skyrmion size is determined by the competition between the coulomb energy @xmath166 and the anisotropic energy @xmath137 . the radius deformation of skyrmion is introduced to make @xmath137 finite , at a cost of slight increase of @xmath206 and @xmath166 . after minimization of the skyrmion energy at @xmath164 and @xmath183 $ ] , we find that the ratio @xmath207 shown in fig .  [ fig : skyrmion_energy_all ] is between @xmath208 and @xmath209 , whereas the ratio @xmath210 is always close to @xmath117 . thus the coulomb energy and anisotropic energy have significant contribution to the total energy of a @xmath0-skyrmion described by our model . despite of such contribution , a @xmath0-skyrmion still has lower energy compared to the quasiparticle with a single spin / pseudospin flip on top of the qhfm state at quarter filling of the @xmath1 ll . figure  [ fig : skyrmion_energy_all ] shows the value @xmath211/4\pi\rho_s-1 $ ] , i.e. the relative energy difference between the optimal skyrmion @xmath212 , at different @xmath82 and @xmath83 , and the scale - invariant skyrmion , with energy @xmath213 . we observe that the energy of an optimal skyrmion does not depend on @xmath82 and @xmath83 except when they are close to the lines @xmath214 or @xmath187 , as well as close to the line @xmath215 . as shown in the skyrmion phase diagram , fig . [ fig : skyrmion_type_diagram ] , one obtains pseudospin skyrmions in the vicinity of the latter line . their energy naturally depends on @xmath82 and @xmath216 in contrast to that of spin skyrmions that are encountered away from this line and that only depend on @xmath217 , which is constant here . the energy of the spin skyrmions thus remains constant , as seen in the light orange parts of fig . [ fig : skyrmion_energy_all ] that match the ( light red ) parts of the skyrmion phase diagram ( fig . [ fig : skyrmion_type_diagram ] ) where one obtains pure spin skyrmions . together with the plot of skyrmion size in fig . [ fig : skyrmion_size_all ] , we also observed that , for different anisotropy parameter @xmath82 and @xmath83 , an _ optimal _ skyrmion has a lower _ optimal _ energy when its _ optimal _ size is larger . the lowest value of @xmath211 $ ] is @xmath213 , which is achieved by the optimal skyrmion @xmath218 obtained at @xmath219 . at this condition , the size of the optimal skyrmion diverges and its coulomb energy vanishes , because the skyrmion energy has a pseudospin @xmath8 symmetry . the correlation between skyrmion energy and skyrmion size still holds when we vary the perpendicular magnetic field @xmath220 , which we choose here to be identical to the total magnetic field , i.e. we discard a tilt of field described in the following section . this is shown in fig . [ fig : energy_size_b ] , where we plot in a log - log scale the numerically obtained size and energy of four types of optimal skyrmions under @xmath221 . different types of optimal skyrmion have different value of energy and size , but the shape of the lines are similar , which root from same scaling relations . indeed eq .  ( [ eq : lambda_0 ] ) implies that the skyrmion size ( in unit of @xmath44 ) scales as @xmath222 , and one obtains from eq . ( [ eq : delta_z - b ] ) then gives @xmath223 in natural units that are independent of the magnetic field . one notices that the numerically obtained exponents in fig . [ fig : energy_size_b](a ) agree to great accuracy ( within 2% ) with that obtained from our simple scaling analysis even if the latter does not take into account the radial deformation in eq . ( [ eq : lambda_r ] ) . similarly , insert eq . ( [ eq : lambda_0 ] ) to eq . ( [ eq : lambda_dep ] ) , we obtain the scaling of the skyrmion energy @xmath224 or , in natural @xmath225-independent units @xmath226 which again agrees well ( within 2% ) with the numerically obtained scaling of the skyrmion energy plotted in fig . [ fig : energy_size_b](b ) . in the previous sections , we discussed the skyrmion type diagram , which provides a global view of the possible types of skyrmions at quarter filling of the @xmath1 ll of graphene monolayer . in this section we show how this diagram is changed by modifications of the anisotropic energy . we discuss three modifications to the anisotropic energy , each corresponding to one of the three contributions ( namely the in - plane pseudospin contribution , @xmath28-component pseudospin contribution and spin contribution ) . the common way to modify the anisotropic energy is by sample tilting , where the sample is tilted away from the upright position while increasing the total magnetic field @xmath64 , so that the strength of the perpendicular component @xmath220 is kept to a constant value . the electron spin is coupled to the total applied magnetic field , while the pseudospin contribution to the anisotropic energy is proportional to the component perpendicular to the sample . therefore , sample tilting guarantees that the pseudospin contribution to the anisotropic energy is unaltered , but the electron spin contribution is increasing . we further discuss terms that breaks the u@xmath227 symmetry between the @xmath228- and @xmath229-components of the pseudospin , whose contribution to the anisotropic energy is changed accordingly . at last we discuss the pseudospin analogue of the zeeman coupling , i.e. the term proportional to @xmath230 , which describes the sublattice asymmetry and that could eventually be induced by substrates that are roughly commensurate with the carbon spacing in graphene ( e.g. boron - nitride substrates ) .      the effect of sample tilting by an angle @xmath231 is the rescaling of the parameters @xmath139 , @xmath83 and @xmath217 in the anisotropic energy . if we maintain our definition of the effective parameters @xmath232 in terms of the perpendicular component @xmath220 of the magnetic field , @xmath233 , and use the zeeman coupling , sensitive to the total magnetic field @xmath64 , as the overall energy scale , one obtains from eq . ( [ eq : enanisotropic ] ) @xmath234 & = & \frac{\delta_{\rm z}(b_{\rm t})}{2}\int \frac{d^{2}r}{2\pi l_{b}^{2 } }   \times{\nonumber}\\ & & \left [ u_{\perp}\cos\varphi\left ( m_{\rm px}^{2}+m_{\rm py}^{2}\right)+u_{\rm z}\cos\varphi m_{\rm pz}^{2}-m_{\rm sz}\right],{\nonumber}\\\label{eq : enanisotropictilt}\end{aligned}\ ] ] where @xmath231 is the tilting angle from an upright direction , @xmath235 , and @xmath236 , @xmath237 are computed by eq . ( [ eq : ti_z ] ) , eq . ( [ eq : si_z ] ) respectively . such a constant @xmath220 guarantees a constant length scale @xmath44 and a constant energy scale @xmath238 . after the rescaling of length and energy to unit @xmath44 and @xmath238 , @xmath239 is replaced by @xmath240 , where @xmath241 is computed by eq . ( [ eq : delta_z - b ] ) with @xmath242 , i.e. in the absence of sample tilting . at this stage , we see that the sample tilting by an angle @xmath231 changes the anisotropic energy as if we simply replace @xmath243 , and @xmath83 in the model without tilting by @xmath244 , and @xmath245 .    to illustrate the effect of the tilt , we have plotted the evolution of the phase diagram in fig . [ fig : skyrmion - type - tilting ] for a value of @xmath246 , as well as that for the skyrmion energy and size along line a in the insets . one first notices that the tilt shifts the borders between the unentangled and entangled types of optimal fm states @xmath247 . generally the unentangled states are slightly favoured . this can easily be understood as the consequence of the relative increase of the zeeman energy with respect to the pseudospin - symmetry - breaking terms  the zeeman coupling favours indeed a full spin polarization and thus a maximal value of @xmath118 , which precisely means that the unentangled fm states are favoured . change of the skyrmion type diagram for sample tilting by angle @xmath248 . shifted borders are marked by blue dashed lines for different types of fm backgrounds , and red for different types of skyrmions , in contrast to the grey and black lines which mark the borders in fig . [ fig : skyrmion_type_diagram ] with @xmath249 . the insets show the evolution of the skyrmion energy ( still in units of @xmath250 ) and size across line a. ]    furthermore , the borders ( red lines in fig .  [ fig : skyrmion - type - tilting ] for @xmath246 as compared to black lines for @xmath251 ) between the pseudospin skyrmions in the vicinity of @xmath252 are shifted . indeed , the pseudospin skyrmions are favoured over a larger range as one may expect from the relative increase of the zeeman energy that renders larger the energy cost to create a spin skyrmion . this is clearly seen in the inset of fig . [ fig : skyrmion - type - tilting ]  while the energy of the pseudospin skyrmion , in the tilt - independent units of @xmath250 , remains the same as for @xmath253 , the @xmath254-independent energy of the spin skyrmion is increased and its size reduced such that the region of stable pseudospin skyrmions is increased . notice finally that the scaling relations ( [ scalingrb ] ) and ( [ scalingeb ] ) are also affected by the sample tilting . the scaling of the size of pseudospin skyrmions still obeys the @xmath255 law because the pseudospin anisotropy depends only on @xmath62 . we can also express such scaling in the total magnetic field and the tilting angle , namely @xmath256 by contrast , the spin skyrmion follows a scaling relation @xmath257 the exponents on @xmath258 and the magnetic field @xmath62 or @xmath64 are different because the coulomb energy @xmath166 depends on @xmath259 while the zeeman coupling depends on @xmath260 . if one perform the sample tilting at constant total magnetic field @xmath64 , spin and pseudospin skyrmions would show a different scaling with respect to the tilt angle .       change of the skyrmion type diagram for @xmath261 . the color code is the same as in fig . [ fig : skyrmion - type - tilting ] , in the absence of tilt ( @xmath253 ) . ]    our model for the anisotropic energy @xmath137 in eq . ( [ eq : enanisotropic ] ) [ also eq .  ( [ eq : enanisotropictilt ] ) ] has a u@xmath227 symmetry for the @xmath228- and @xmath229-component of the pseudospin . formally , this symmetry can be broken with the help of the parameter @xmath262 , which yields the modified energy @xmath234 & = & \frac{\delta_{\rm z}(b_{\rm t})}{2}\int\frac{d^{2}r}{2\pi l_{b}^{2}}\left[(u_{\perp } + \delta u_{\perp})\cos\varphi m_{\rm px}^{2}\right . { \nonumber}\\ & & \left . + ( u_{\perp}-\delta u_{\perp})\cos\varphi m_{\rm py}^{2 } + u_{\rm z}\cos\varphi m_{\rm pz}^{2}-m_{\rm sz}\right],{\nonumber}\\\label{eq : enanisotropicsubstrateaniso}\end{aligned}\ ] ] where we have also considered sample tilting . it is tempting to attribute this symmetry breaking to an @xmath31-anisotropy in the graphene sheet , e.g. when one applies uni - axial strain to it that breaks the @xmath263 rotation symmetry of the lattice . however , more microscopic calculations beyond the scope of our paper , would be required to corroborate this picture . the main effect of this symmetry breaking is to favour , in the case of an easy - plane pseudospin fm , an orientation of the pseudospin magnetization along the @xmath228-axis ( @xmath264 ) for @xmath265 or along the @xmath229-axis ( @xmath266 ) when @xmath267 , while no particular direction is favoured in the isotropic case ( @xmath268 ) . the relevant energy scale for the in - plane anisotropy is therefore reduced , and we can use the above model ( [ eq : enanisotropictilt ] ) also to treat the present case if one replaces the model parameters @xmath145 by @xmath269 .    the evolution of the phase diagram is shown in fig .  [ fig : skyrmion - type - substrate - anisotropy ] for a value @xmath270 . one first notices that the transition lines towards the easy - plane fm states is modified , whereas the easy - axis fm states are naturally not affected in energy by the change in @xmath82 . indeed , the transition line between the unentangled easy - plane and the easy - axis fm states is shifted downwards  the transition occurs at @xmath271 instead of @xmath188 as in the isotropic case . furthermore , the zeeman energy is again effectively increased with respect to the easy - plane fm energy scale ( for positive values of @xmath139 ) such that the unentangled easy - plane fm states are stabilized , whence the right shift of the transition between the unentangled and entangled easy - plane fm states in fig . [ fig : skyrmion - type - substrate - anisotropy ] . similarly one may understand the borders of the regions for the different skyrmion types . while the transition between the spin and the pseudospin skyrmion is unaffected in the case of an easy - axis fm background characterized by the unaltered energy scale @xmath83 , that between the spin an pseudospin skyrmions in an easy - plane fm background is shifted downwards ( red line in fig . [ fig : skyrmion - type - substrate - anisotropy ] . this can be understood from the evolution of the skyrmion size shown in the lower inset ( cut along line a ) . when approaching the transition between the two different underlying fm states from the right hand side ( by lowering the value of @xmath139 , the skyrmion size follows first the same evolution as in the isotropic case until the maximum is reached at the value of the underlying transition . notice , however , that due to the in - plane anisotropy the pseudospin @xmath8 symmetry is no longer restored so that the skyrmion size does not diverge . left to the transition , the skyrmion size is therefore generically smaller than in the isotropic case , and the skyrmion energy higher ( see upper inset of fig .  [ fig : skyrmion - type - substrate - anisotropy ] ) . the energy of the spin skyrmion is therefore reached at smaller values of @xmath272 such that the region where one encounters pseudospin skyrmions becomes smaller . furthermore , the transition between the region where one finds entanglement skyrmions and that of deflated pseudospin skyrmions in the region @xmath273 is strongly affected by the in - plane anisotropy . this can be attributed to the very small energy difference and thus delicate competition between entanglement and deflated pseudospin skyrmions that are also characterized by a certain degree of entanglement , as discussed in sec . [ subsec : deflated - pseudospin - skyrmion ] . change of the skyrmion type diagram for @xmath274 . the color code is the same as in fig . [ fig : skyrmion - type - tilting ] , in the absence of tilt ( @xmath253 ) . ]    the anisotropic energy eq . ( [ eq : enanisotropic ] ) can also be modified by adding the term @xmath275 , which is a pseudospin analogue of the zeeman coupling and describes the sublattice asymmetry . omitting the @xmath262 term discussed in the previous section but including the sample tilting , we have @xmath234 & = & \frac{\delta_{\rm z}(b_{\rm t})}{2}\int\frac{d^{2}r}{2\pi l_{b}^{2 } } \left [ u_{\perp}\cos\varphi\left ( m_{\rm px}^{2}+m_{\rm py}^{2}\right)\right . { \nonumber}\\ & & \left.+ u_{\rm z}\cos\varphi m_{\rm pz}^{2 } + \delta u_{\rm z}\cos\varphi m_{\rm pz } -m_{\rm sz}\right],{\nonumber}\\\label{eq : enanisotropicpspinzeeman}\end{aligned}\ ] ] where @xmath276 $ ] is the relative energy scale of the pseudospin analogue of the zeeman coupling . the original @xmath75 symmetry ( equivalence of a pseudospin orientation along the @xmath28 or @xmath76-axis ) is now broken , and pseudospin orientation along @xmath28 are favoured for @xmath277 and along @xmath76 for @xmath278 . the @xmath279 term modifies the optimal fm states @xmath247 for the anisotropic energy in a complementary manner as @xmath262  instead of the easy - plane fm states , the easy - axis fm states are affected by this type of anisotropy . this can indeed be seen in the phase diagram shown in fig . [ fig : skyrmion - type - pseudospin - zeeman ] for @xmath274 . first , it shifts the border between the regions for unentangled easy - axis fm and for unentangled easy - plane fm , as well as the border between the regions for unentangled easy - axis fm and entangled easy - axis fm . both are shifted upwards by @xmath280 . second , the slope of the border between the regions for entangled easy - axis fm and for entangled easy - plane fm , is increased from @xmath117 to @xmath281 . moreover , at the upper - left side of the sloped borders separating the easy - axis fm and easy - plane fm regions , the @xmath282 points close to the border corresponds to fm spinor @xmath86 with optimal value of @xmath132 less than @xmath283 . as @xmath282 moves away from these borders ( to the upper - left side ) , @xmath132 approaches to @xmath283 . these modifications can be understood by rewriting @xmath284 $ ] as @xmath234 & = & \frac{\delta_{\rm z}(b_{\rm t})}{2}\int\frac{d^{2}r}{2\pi l_{b}^{2}}\times\nonumber\\ & & \left[u_{\perp}\cos\varphi\left(m_{{\rm px}}^{2}+m_{{\rm py}}^{2}\right)+\left(u_{{\rm z}}+\frac{\delta u_{{\rm z}}}{2}\right)\cos\varphi m_{{\rm pz}}^{2}\right.\nonumber\\ & & \left.-\frac{\delta u_{{\rm z}}}{2}\cos\varphi(m_{{\rm pz}}-1)^{2}-m_{{\rm sz}}+\frac{\delta u_{{\rm z}}}{2}\cos\varphi\right ] \label{eq : enanisotropicpspinzeeman1}\end{aligned}\ ] ] the analysis is similar to those for the the @xmath262 term for the substrate anisotropy in the previous subsection . the upward shift of the transition between the easy - axis and easy - plane fm states is a consequence of the lowered energy of the former ( for @xmath285 ) that favours easy - axis states . similarly to the easy - plane anisotropy discussed above , there is no restoration of the @xmath8 pseudospin symmetry at the transition line @xmath286 because of the remaining term @xmath287 in eq . ( [ eq : enanisotropicpspinzeeman1 ] . as a consequence , the skyrmion size does not diverge at this value and its energy does not reach the minimal ( scale - invariant ) value @xmath288 ( see insets of fig . [ fig : skyrmion - type - pseudospin - zeeman ] ) . in conclusion , our analysis of the quarter - filled @xmath1 ll of graphene shows that the anisotropic terms in the energy functional , which break the spin and valley pseudospin symmetry of the leading coulomb energy , lead to four types of ferromagnetic ground state and various types of @xmath0-skyrmions . a skyrmion is embedded in the ferromagnetic background @xmath86 , and the center spinor @xmath198 gives rise to different types of skyrmions because the orthogonality condition between the spinors , @xmath289 , does not entirely fix the skyrmion type . the favored skyrmion type is then obtained by a minimization of the anisotropic energy terms . when the electron spin and pseudospin contribution to the anisotropic energy varies , a spin skyrmion can change to a pseudospin skyrmion while the ferromagnetic background remains unaltered . this is strikingly different from the two - fold degenerate ll split by electron spin , where a skyrmion simply disappears when the magnetic field , and hence the zeeman coupling increases to a critical value . a large spin skyrmion requires weak zeeman coupling and hence a weak applied magnetic field . however , in our four - component system where @xmath0-skyrmions are formed , one can tune the model parameters @xmath290 to approximately restore the @xmath8 symmetry of the valley pseudospin , so as to have a large pseudospin skyrmion , even when the anisotropic energy for both spin and valley pseudospin is high . by analogy , skyrmion also appears in graphene monolayer at @xmath291 when the valley pseudospin @xmath8 symmetry or the @xmath292 symmetry@xcite is approximately restored . in addition to these skyrmion types , one encounters more exotic skyrmions that are formed due to spin - pseudospin entanglement . a direct transposition of the concept of a skyrmion that entirely covers the spin or pseudospin bloch sphere yields the _ entanglement _ skyrmion , which covers the third _ entanglement _ bloch sphere . this bloch sphere is required , at @xmath293 , to fully characterize the skyrmion and fm states i.e. to account for two different angles @xmath114 and @xmath109 that naturally arise in our @xmath0 description of @xmath26 states . in the case of easy - axis fm states , entanglement manifests itself , rather counter - intuitively , in the form of locally anti - ferromagnetic or anti - ferrimagnetic patterns . moreover , we have identified another type of skyrmion , a deflated pseudospin skyrmions with partial entanglement . while all directions in the pseudospin bloch sphere are explored within this skyrmion type , similarly to an unentangled pseudospin skyrmion , the magnitude of the pseudospin magnetization is reduced in most places . this provides the deflated image of the representation on the bloch sphere , accompanied by an exploration of parts of the spin bloch sphere . in lattice - resolved images , deflated pseudospin skyrmions are manifest by a lower spin - pseudospin contrast and by local deviations of the spin magnetization from the @xmath28-axis . the lattice - resolved images we obtain in our study can in principle be used as a guide in a possible experimental identification of the various skyrmions in graphene , e.g. in spin - resolved stm / sts measurements . while the varying spin magnetization can be probed by a combination of magnetic tips , not only in the @xmath28-direction but also in the @xmath228- and @xmath229-directions , the @xmath28-component of the pseudospin magnetization can directly be read of from the relative electronic occupation of the different sublattices in graphene . in contrast to the spin magnetization , the @xmath228- and @xmath229-components of the pseudospin magnetization are unfortunately not accessible in this type of experiments that do not provide insight into the combination @xmath294 of the parameters in our parametrization of @xmath0 fields . our analysis also shows that the model parameters @xmath82 and @xmath83 that correspond to large optimal @xmath0-skyrmions reside in the vicinity of the borders between two different types of ferromagnetic backgrounds . indeed , the transitions are characterized by either ( i ) a partial symmetry restoration , e.g. of the @xmath8 pseudospin symmetry at @xmath295 , or ( ii ) a tendency of such a symmetry restoration , e.g. at @xmath187 for @xmath296 or @xmath297 for @xmath298 between unentangled and entangle fm backgrounds . in general , the energy to create a skyrmion is reduced due to a reduction of the anisotropy energy . in the case ( i ) , the partial symmetry restoration yields a divergent skyrmion size described by a power law @xmath299 , with a critical exponent @xmath199 close to @xmath200 , as expected from simple scaling arguments . at the same time , the skyrmion size  in @xmath225-independent units  scales in the vicinity of the transition as @xmath300 with the @xmath225-field and its energy as @xmath301 when compared to the energy @xmath302 of the non - linear sigma model . right at the transition , the skyrmion energy retrieves its scale - invariant value @xmath302 . if there is only a tendency towards symmetry restoration [ case ( ii ) ] , either in the above mentioned transition between unentangled and entangled fm backgrounds or in the presence of an anisotropy in the substrate that couples to the pseudospin , the divergence in the skyrmion size is truncated and follows approximately the law @xmath303 , where @xmath198 is some positive constant and @xmath304 is a generic coupling constant with critical value @xmath305 . again we find that the exponent @xmath199 is close to @xmath200 . several terms are not included in our model . the @xmath26 invariance of the non - linear sigma model energy @xmath306 $ ] can be broken by a term of the form@xcite @xmath307 $ ] , where @xmath308 is a superposition of the tensor product of pauli matrices , and the coefficient @xmath309 is of the same order as the input parameters @xmath82 , @xmath83 for the anisotropic energy in our model . at the fourth order of the gradient expansion , which may be relevant for small skyrmion sizes , we may have@xcite @xmath310 ^ 2 $ ] , where @xmath311 is the laplacian . the minus sign of such a term is on the same order as the coulomb energy @xmath312 $ ] in the gradient expansion , and it implies that our model has overestimated@xcite the energy of a skyrmion , or equivalently , underestimated the skyrmion size . generally speaking , the non - linear sigma model of skyrmions does not take account for the reduction of magnetization in the core of the texture caused by quantum fluctuations.@xcite we most gratefully acknowledge fruitful discussions and former collaboration with achim rosch . furthermore , we thank markus morgenstern for helpful experimental insights . yl is funded by a scholarship from the china scholarship council . in sec .  [ sec:4-types - skyrmion ] we represent the @xmath0 textures @xmath313 in lattice - scale profiles of the electron density @xmath158 and the @xmath28-component of the spin magnetization @xmath159 for the @xmath0-field @xmath313 . these profiles are computed with the help of eqs . ( [ eq : si_z ] ) and ( [ eq : ti_z ] ) , where the components of the @xmath0-field are convoluted with a form factor @xmath314 that mimics the amplitude square of the atomic wave function . here , @xmath315 is the position on the sublattice @xmath316 in the @xmath317-th unit cell . for illustration purposes , we have chosen a gaussian profile for @xmath318 that is normalized as @xmath319 . moreover , if we neglect the overlap between the atomic wave functions centred around different lattice site in the function @xmath320 , the expressions for @xmath158 and @xmath159 simply read @xmath321 the electronic occupation of the sublattices a and b in the @xmath317-th unit cell is given by @xmath322 , while the @xmath28-component of the spin magnetization is @xmath323 , with @xmath324z({{\bf r}}_{j}^{a , b}),\\ s_{\rm z}({{\bf r}}_{j}^{a , b } ) & = z^{\dagger}({{\bf r}}_{j}^{a , b})\left[\frac{\sigma_{z}\pm1}{2}\otimes\sigma_{z}\right]z({{\bf r}}_{j}^{a , b}).\end{aligned}\ ] ] in the above expression , the @xmath325-sign in the projector is chosen for a site on the @xmath326-sublattice and @xmath327 for a site on the @xmath225-sublattice . the second homotopy group@xcite @xmath329 of a topological manifold @xmath330 is defined as the equivalence class of the continuous mapping @xmath331 , where two mappings @xmath332 and @xmath333 are equivalent means that there exists a continuous mapping @xmath334\times s^2\rightarrow { \cal m}$ ] such that @xmath335 and @xmath336 . the multiplication in the group is the homotopic composition of the continuous mappings . the second homotopy group @xmath12 of the 2-sphere @xmath11 can be understood as `` wrapping a 2-sphere on another 2-sphere '' . the winding number ( brouwer degree ) of the mapping @xmath337 classifies all possible ways of the `` wrapping '' . since @xmath338 , we have @xmath339 . it can also be computed@xcite by the following relations between a simply connected lie group @xmath340 , one of its subgroups @xmath341 and the coset space @xmath342 : @xmath343 where @xmath344 is the fundamental group of a manifold defined by replacing @xmath11 in the definition of @xmath329 by @xmath13 . the manifold @xmath345 is isomorphic to @xmath346 , therefore @xmath347 where the last equation @xmath348 follows from the classification of mappings from a closed path to another closed path . the manifold @xmath328 is isomorphic to @xmath349 \cong \su{4}/[\su{3}\times { \rm u}(1)],\ ] ] therefore @xmath350 ) = \pi_1(\su{3}\times { \rm u}(1)).{\nonumber}\ ] ] using the fact that the homotopy group for product manifold factorizes , i.e. @xmath351 and the fact that any simple lie group @xmath340 has @xmath352 , we obtain @xmath353 this result can be generalized@xcite to compute @xmath354 for integer @xmath355 , because @xmath356 ) { \nonumber}\\ & = & \pi_1(\su{{\mathcal{n}}-1}\times { \rm u}(1)){\nonumber}\\ & = & \pi_1(\su{{\mathcal{n}}-1})\times \pi_1({\rm u}(1)){\nonumber}\\ & = & \mathbb{z}\end{aligned}\ ] ] this argument shows that any @xmath16-component qh system or , more generally , any @xmath16-component ferromagnet in two spatial dimensions with one filled component can principally host topological textures in the form of skyrmions that are characterized by an integer topological charge @xmath357 . the specificity of the quantum hall ferromagnet is that this topological charge is directly related to an electric charge . the @xmath0 space is a collection of normalized , four - component complex vectors @xmath358 . in addition , two vectors @xmath359 and @xmath360 are equivalent for arbitrary @xmath361 , in the sense that they correspond to the same matrix @xmath362 , which plays the role of order parameter in the theory of qhfm in ref .  [ ] . to be more precise , @xmath359 should be understood as a representative of the equivalent class @xmath363\in v/\sim$ ] , where @xmath364 denotes the set of normalized @xmath365 vectors and @xmath366 denotes the equivalence relation @xmath367 for @xmath368 and @xmath361 . in this paper , we call the normalized @xmath365 vector @xmath359 a `` @xmath0-spinor '' if it represents an element in the @xmath0 space . the name `` spinor '' here is inherited from the physics literature without mathematical rigour . the @xmath0 space can also be understood as a coset space ( see eq .  ( [ eq : group ] ) in app . [ subsec : second - homotopy ] ) @xmath369 \cong \su{4}/[\su{3}\times { \rm u}(1)],\ ] ] which means that the integer filling of one of the four landau sublevels is distinguished by removing the @xmath370 ( or @xmath371 ) rotations among three empty sublevels from the entire rotation group @xmath372 ( or @xmath26 ) among all four sublevels , followed by a removal of a global phase factor @xmath373 . the parametrization of a cp@xmath49-spinor is discussed in sec . [ subsec : qhfm - ground - states ] . there we construct the cp@xmath49-spinor from six angles @xmath374 , @xmath375 , @xmath132 , @xmath130 , @xmath114 , @xmath109 , and visualized them via a triplet of bloch spheres . the inverse problem  to obtain the six angles parametrizing a given cp@xmath49-spinor up to an overall phase factor @xmath376  is equally important for the complete understanding of the parametrization eq . ( [ eq : parametrizationz ] ) . given a cp@xmath49 -spinor @xmath96 , we have access to the parameter @xmath114 by computing the magnitude of spin and pseudospin magnetization via eqs . ( [ eq : si_f ] ) and ( [ eq : ti_f ] ) . there are two possible values : @xmath377 and @xmath378 . they correspond to the same cp@xmath49-spinor because of the following equivalence between two cp@xmath49-spinors parametrized in eq . ( [ eq : parametrizationz ] ) : @xmath379 the effect of changing @xmath114 into @xmath380 while keeping the same cp@xmath49-spinor is to reverse the direction of the spin and pseudospin magnetization . it reveals the fact that our parametrization is redundant for @xmath381 . in the discussion of qhfm states , we can always restrict @xmath114 in the range @xmath382 $ ] . under this restriction , we have @xmath383 and @xmath384 , in agreement with the schmidt decomposition theorem , which claims that we have non - negative real numbers as the coefficients in front of the direct - product - state basis obtained from the decomposition . notice , however , that this restriction is not the unique choice to avoid redundancies in the parametrization . instead one can also choose the restriction @xmath374 or @xmath132 in @xmath382 $ ] , such that @xmath114 and @xmath109 span the full bloch sphere , as in the discussion of entanglement skyrmions in sec . [ subsec : entanglement - skyrmion ] . the cp@xmath49-non - linear sigma model ( cp@xmath49-nlsm ) has the following form of energy : @xmath385=\int d^2r { { \bf d}}z^{\dagger}({{\bf r}})\cdot { { \bf d}}z({{\bf r}}),\label{eq : nlsm - general}\ ] ] where we defined @xmath386 z({{\bf r}})$ ] . the finite - energy configurations of @xmath157 satisfies the boundary condition at spatial infinity : @xmath387 where @xmath388 is a function of the polar angle @xmath231 of coordinate plane , and @xmath389 is a constant cp@xmath49-spinor . under such boundary condition , the base manifold of field @xmath157 can be extended to the riemann sphere @xmath390 via riemann stereographic projection,@xcite and the value of the cp@xmath49-spinor field at spatial infinity is @xmath391 . after the extension , each field @xmath157 can be considered as a mapping @xmath392 and can be classified by the degree of mapping @xmath393 & = \int\left(-\frac{i}{2\pi}\right)[{{\bf d}}z({{\bf r}})^{\dagger } \times { { \bf d}}z({{\bf r}})]_z d^{2}r\nonumber\\ & = \int\rho_{\rm topo}({{\bf r}})d^{2}r\end{aligned}\ ] ] because the second homotopy group @xmath394 of the cp@xmath49 space ( as target space of field @xmath157 ) is non - trivial . @xmath7 is called _ topological charge _ in the physics literature , and correspondingly , @xmath24 is called _ topological charge density_. it is impossible to continuously deform a field @xmath395 into another configuration @xmath396 of different topological charge @xmath397\neq{\cal q}[z_{a}]$ ] . therefore , our variational analysis on @xmath398 $ ] ( eq . ( [ eq : nlsm - general ] ) ) is limited within a subspace of field configurations @xmath399=m\in\mathbb{z}\right\ } .\ ] ] the minimal - energy configuration @xmath400}({{\bf r}})$ ] for @xmath398 $ ] in each @xmath401 serves as the starting point of our analysis of cp@xmath49-skyrmions . we observe an inequality @xmath402\pm4\pi{\cal q}[z]\ge 0\end{aligned}\ ] ] or , equivalently , @xmath385\ge2\pi\left|{\cal q}[z]\right|.\ ] ] it means that for each @xmath401 , the lower bound of @xmath403 $ ] is @xmath404\right|$ ] . comparing eq . ( [ eq : nlsm - general ] ) to eq . ( [ eq : ennlsm ] ) to restore the units of energy , we find that the right hand side in the above inequality gives @xmath405 as the lower bound of @xmath206 for charge @xmath406 skyrmions . moreover , the minimal - energy configuration which saturates the inequality is the solution of the following first order partial differential equations , which are called the bogomolnyi - prasad - sommerfield ( bps ) equations : @xmath407<0,\label{eq : bps_minus}\\ d_{i}z - i\epsilon_{ik}d_{k}z & = 0\;\mbox { for } { \cal q}[z]>0.\label{eq : bps_plus}\end{aligned}\ ] ] inserting @xmath408 with the _ unormalized _ complex vector @xmath409 and using the complex coordinate @xmath410 , the above equations are rewritten as @xmath411 where @xmath412 and @xmath413 .    the topological charge @xmath7 can also be expressed in @xmath414 with the help of stoke s theorem : @xmath415 where the contour @xmath416 encloses a simply connected region @xmath417 , and @xmath417 maximally covers the punctured complex plane where @xmath414 and @xmath418 are well defined . the generic solution of bps equation eq . ( [ eq : bps_plus ] ) for @xmath419 ( @xmath420 ) is a four - component vector field @xmath421 , in which the components @xmath422 are meromorphic ( anti - meromorphic ) functions .    in the main text we are interested in the solution of bps equation with topological charge @xmath203 . the generic form is @xmath423}(z ) = \left [ f_1 z - c_1 , f_2 z - c_2 , f_3 z - c_3 , f_4 z - c_4 \right].\ ] ] however , to demonstrate the use of the above formula in eq . ( [ eq : topou1charge ] ) , let us compute @xmath7 for the following complex vector field @xmath424,\label{eq : w - example}\ ] ] where the topological charge can not be simply read off . the contour @xmath425 consists three parts  an anticlockwise circle @xmath426 , a clockwise circle @xmath427 , and a pair of straight lines @xmath428 of opposite directions connecting the two circles . we have @xmath429 \frac{w^{\dagger}\boldsymbol{\nabla}w}{w^{\dagger}w}\cdot d\boldsymbol{l } { \nonumber}\\ & = & \frac{1}{2\pi i}\left [ \left(0 + 0 + \frac{2\pi i}{2 } + \frac{2\pi i}{2}\right ) - \left(0 - 2\pi i + 0 + 0\right ) + \left(0\right ) \right]{\nonumber}\\ & = & 2,{\nonumber}\end{aligned}\ ] ] where each term in the round bracket corresponds to the contribution from the components of @xmath430 in eq . ( [ eq : w - example ] ) accordingly . it is easy to verify the above result for the topological charge of @xmath430 in eq . ( [ eq : w - example ] ) by directly computation with eq . ( [ eq : topologicaldensity ] ) in the main text . in sec .  [ subsec : radial - deformation ] we discussed the radial deformed skyrmion @xmath212 . it has the form @xmath431}/\sqrt{\check{w}_{\rm [ 1]}^{\dagger}\check{w}_{\rm [ 1]}}\ ] ] with @xmath432}(x , y ) = ( x+iy)f-\lambda(r)c$ ] and @xmath433 . in the limit @xmath434 , the three components of the energy @xmath435 $ ] have the following behavior @xmath436 & \rightarrow &   4\pi\sqrt{\frac{\pi}{32 } } \frac{e^{2}}{4\pi\epsilon l_{b}},\\ e_{\rm c}[\check{z}_{\rm sk } ] & \rightarrow & \frac{3\pi}{64 } \frac{e^{2}}{4\pi\epsilon l_{b } } \left(\frac{\lambda_0}{l_b}\right)^{-1},\\ e_{\rm a}[\check{z}_{\rm sk } ] & \sim & \left ( a_{0}+a_{1}\left|\delta_{\rm z}\right|\log\kappa\right)\left(\frac{\lambda_0}{l_b}\right)^{-2}.\end{aligned}\ ] ] as a function of @xmath178 , @xmath437 and @xmath166 decrease monotonically . meanwhile , @xmath137 is a monotonic increase function of @xmath178 . [ fig : kappa_dependence ] displays their @xmath178 dependence . based on this analysis , the radial - deformation ansatz @xmath212 gives a finite anisotropic energy @xmath137 and a lowered total energy . @xmath178-dependence of the nlsm energy @xmath206 ( red line ) , the coulomb interaction energy @xmath166 ( blue line ) and the anisotropic energy @xmath137 ( green lines ) . ] the @xmath345-skyrmion ( or o(3)-skyrmion ) can be visualized by the image of the riemann sphere on the bloch sphere . the riemann sphere is obtained by the stereographic projection between the cartesian coordinates @xmath438 on the plane and the cartesian coordinates @xmath439 on the riemann sphere : @xmath440 it is illustrated in fig . [ fig : riemann_sphere ] . the `` hedgehog '' skyrmion is in fact an identity mapping from the riemann sphere to the bloch sphere .      as discussed in sec . [ subsec : qhfm - ground - states ] and appendix . [ subsec : cp3-spinor - and - nlsm ] , a cp@xmath49-spinor can be parametrized by six angles @xmath374 , @xmath375 , @xmath132 , @xmath130 , @xmath114 , @xmath109 , and therefore can be visualized by three bloch spheres , namely the spin bloch sphere ( for @xmath374 and @xmath375 ) , the pseudospin bloch sphere ( for @xmath132 and @xmath130 ) and the entanglement bloch sphere ( encodes @xmath114 and @xmath109 ) . hence a cp@xmath49-skyrmion can be visualized by plotting the image of the riemann sphere on the three bloch spheres via the parametrization of the cp@xmath49-spinor .    in practice , we draw the spin magnetization and pseudospin magnetization @xmath441 at points specified by longitudinal and latitudinal lines on the riemann sphere . these lines correspond to rays of constant angle @xmath48 and circles of constant radius @xmath172 circles on the @xmath31-plane where the skyrmion lives . in general , at a given point @xmath442 on the @xmath31-plane , @xmath443 and @xmath444 may have a magnitude less than @xmath117 , because @xmath445 can be different from @xmath446 or @xmath116 . as a consequence on the visualization , the spin and pseudospin magnetization lay _ inside _ the corresponding bloch spheres . this is the case for the cp@xmath49-entanglement skyrmion and the deflated cp@xmath49-pseudospin skyrmion . for example , in chapter 10 of l. h. ryder , _ quantum field theory _ ( second edition ) , cambridge university press 1996 ; chapter 6 of n. manton and p. sutcliffe , _ topological solitons _ , cambridge university press 2004 . s. mhlbauer , b. binz , f. jonietz , c. pfleiderer , a. rosch , a. neubauer , r. georgii , and p. bni , . n. romming , c. hanneken , m. menzel , j. e. bickel , b. wolter , k. von bergmann , a. kubetzka , and r. wiesendanger , .                                s. m. girvin and a. h. macdonald , chapter 5 in _ perspectives in quantum hall effects : novel quantum liquids in low - dimensional semiconductor structures _ , editor(s ) : sankar das sarma , aron pinczuk wiley - vch verlag gmbh & co. kgaa ( 2004 ) .
we model quantum hall skyrmions in graphene monolayer at quarter filling by a theory of @xmath0 fields and study the energy minimizing skyrmions in presence of valley pseudospin anisotropy and zeeman coupling . 
 we present a diagram of all types of skyrmions in a wide range of the anisotropy parameters . for each type of skyrmion 
 , we visualize it on three bloch spheres , and present the profiles of its texture on the graphene honeycomb lattice , thus providing references for the stm / sts imaging of spin - pseudospin textures in graphene monolayer in quantum hall regime . 
 besides the spin and pseudospin skyrmions for the corresponding degrees of freedom of an electron in the @xmath1 landau level , we discuss two unusual types  the `` entanglement skyrmion '' whose texture lies in the space of the entanglement of spin and pseudospin , as well as the `` deflated pseudospin skyrmion '' with partial entanglement . for all skyrmion types , 
 we study the dependence of the energy and the size of a skyrmion on the anisotropy parameters and perpendicular magnetic field . we also propose three ways to modify the anisotropy energy , namely the sample tilting , the substrate anisotropy and the valley pseudospin analogue of zeeman coupling .