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Skyrmion-like states and quasi-molecular excitations in the doped cuprates
Physica B 259—261 (1999) 476—478
Skyrmion-like states and quasi-molecular excitations in the doped cuprates A.S. Moskvin*, A.S. Ovchinnikov Department of Theoretical Physics, Ural State University, Lenin ave. 51, 620083 Ekaterinburg, Russia
Abstract Within a framework of the pseudo-spin formalism we consider the skyrmion-like topological modes for the quasi-twodimensional quantum lattice bose gas. A skyrmion—skyrmion chiral pseudo-spin coupling results in a pseudo-spin-flip mode splitting to the bonding (ferro) and antibonding (antiferro) quasi-molecular modes with different momentum q-dependence of the effective pseudospin susceptibility. It is conjectured that CuO planes in the doped cuprates can be considered as a system of the skyrmion-like states pinned on the charge inhomogeneity centers. 1999 Elsevier Science B.V. All rights reserved. Keywords: Cuprate; Chirality; Skyrmion; Superconductivity
Numerous experimental data convincingly evidence an unconventional non-Landau—Fermi-liquid behavior of the electron-fluid realized in cuprate superconductors with a puzzling metal-dielectric duality, which must be associated with some quite unusual order parameter(s). Up to now various fermionic, bosonic, or boson—fermionic scenarios have been elaborated with the purpose of elucidation of the HTSC problem. Below, we would like to emphasize unconventional skyrmion-like states of the quantum lattice bose gas to be the probable candidates for the description of specific inhomogeneous states in the CuO planes of the cuprates with the non-isovalent substitution. The classical skyrmions represent the stationary non-uniform states of the vector field S(r) on 2D-plane with the energy H " J( S?(r))dx ? (a"x, y, z), which corresponds to the Hamiltonian of the classical isotropic 2D Heisenberg model [1—3]. The spatial distribution of the order parameter for the conventional skyrmion localized at the point r"0 obeys the equation *S(r)"0 and has the form S " V
* Corresponding author. Fax: #7-3432-615-978; e-mail: [email protected].
sin h(r)cos(lu#t), S "sin h(r)sin(lu#t), S "cos h(r), W X sin 0(r)"2(jr) J /(j J #r J ) where l"$1,$2,2 is a topological charge with the chirality defined by the sign ‘$’, the j is a skyrmion radius, and the angle t defines a global orientation of the transversal xy-components of the spin S(r). The skyrmion energy E "8p"l"J does not J depend on the j and t. Note, the topological charge distinguishes the irreducible representations of the plane symmetry group C and in a sense corresponds to an orbital azimuthal quantum number m"$l, so below one make use of the atomic terminology: p, d, f,2 skyrmions when l"$1, $2, $3, etc. An isolated skyrmion could be described by an effective chiral pseudospin s". The principal elements of the classical field ‘skyrmionology’ could be used for the description of the stationary non-uniform states of various quantum (effective) spin systems. Below, one considers a quantum lattice bose gas where Hamiltonian is equivalent to that of the anisotropic Heisenberg magnet (s") with the temperature dependent external magnetic field [4]: HK "k SK (m)# [t(mn)(SK (m)SK (n)#SK (m)SK (n)) X V V W W K KL #»(mn)SK (m)SK (n)], X X
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 9 2 9 - 6
A.S. Moskvin, A.S. Ovchinnikov / Physica B 259—261 (1999) 476—478
where the spin operators are associated with the local boson creation (annihilation) operators, the local boson transfer integral t(mn) and the boson—boson repulsion energy »(mn) are effective exchange integrals; the renormalized boson chemical potential k plays the role of an external field and is determined by the constraint: 1SK 2"N\ 1SK (m)2"(!1nˆ 2). Here, N is a latX K X tice site number and 1nˆ 2 an average on-site boson concentration. Along with the uniform non-ordered metallic (NO), charge ordering (CO), superconducting bose-superfluid (BS), mixed (CO#BS) phases for the local boson ordering, one should introduce various nonuniform topological configurations with either topological charge or chirality. One should emphasize a difference between quasi-skyrmions and classical spin skyrmions: 1. The quasi-skyrmions represent an MFA portrait of the appropriate quantum states distinguished by topological quantum numbers, namely, the summary spin S and its projection M . So, one might consider 1 the non-uniform states the M "0 or n " (neutral 1 quasi-skyrmions), and M "$1 when a single boson 1 is created (annihilated) (singly charged quasi-skyrmions). 2. Taking account of the real tetragonal symmetry within the CuO plane permits to associate the quasi skyrmions with appropriate irreducible representations of the tetragonal symmetry group C : !" A , A , B , B , E . So, the "l""1 (p-skyr mion) is associated with the doublet E representation, while the "l""2 (d-skyrmion) splits to the B (d ) V\W and B (d ) skyrmions. In general, it should be noted VW that lowering the symmetry results in the chirality mixing effect, so that the ‘tetragonal classification’ operates most appropriately. The p(E ) skyrmion as sociates with the current state, whereas the d(B , B ) skyrmions do with the currentless states because of the ‘orbital quenching’. 3. Taking account of different physical nature of the z and x, y components of the boson pseudospin one should introduce as a minimum three types of the quasi-skyrmion solutions in accordance with the different boundary conditions at rPR. For the COtype quasi-skyrmions S " "0, "S " " "1, for V W P X P the BS-type S " "0, S " "1, for the X W P V P (CO#BS)-type S " O0, S " O0, respectively. , P , P The skyrmion-like distribution of the superconducting order parameter corresponds to the pure p(E ) or d(B , B )-symmetry for the topological charge value "l""1 or "l""2, respectively, only for the pure COtype quasi-skyrmions, while for the mixed (CO#BS), or for the pure BS-type quasi-skyrmions, we come to the mixed (s(A )#p(E )) symmetry at "l""1, or S (s(A )#d(B , B )) symmetry at "l""2 with an s fraction determined mainly by the outskirts of the quasi-skyrmion.
477
The above considered quasi-skyrmionic model could be successfully applied to the real doped cuprates in a framework of the scenario developed in papers [5—8], where the CuO planes are considered to be a system of the local singlet bosons moving in a lattice of the pseudo-Jahn—Teller (PJT) centers which is characterized by additional vibronic pseudo-spin vector parameters. The purely charge quasi-skyrmions in the PJT lattice are unstable with respect to the formation of a bound hybrid charge-vibronic quasi-skyrmion (large polaron). The singly charged hybrid quasi-skyrmions as peculiar polaron-like pseudo-particles can be pinned near such centers of charge inhomogeneity as Sr> in La Sr CuO , or \V V well isolated vacancies in YBa Cu O (x+1). This >V model approach, which considers the doped cuprates as a system of coupled quasi-skyrmions, permits to explain many of their puzzling properties, particularly the lowenergy structure of a spectrum. A skyrmion—skyrmion interaction results in the appearance of the quasi-molecular excitations in pairs of the quasi-skyrmions, which could be revealed by the inelastic neutron scattering. Such excitations are associated with the transitions between different singlet and triplet (bonding—antibonding) pseudo-spin states which correspond to different magnitudes of the summary pseudo-spin RK "pˆ #pˆ of the pair. An effective pseudo-spin Hamiltonian for the skyrmion—skyrmion coupling within a CuO plane could be written as the sum of four contributions: HK " HK #HK #HK #HK , where HK describes an initial pseudo-spin b !b splitting for the d-skyrmions, HK "I (R )pˆ pˆ is an effective current—current coup , X X ling with the parameter I depending on the inter, skyrmionic separation R ; HK "I (R )
[(pˆ pˆ !pˆ pˆ )cos8u!(pˆ pˆ #pˆ pˆ )sin8u] is V V W W V W W V a multipole coupling with u being the azimuthal angle of the R , and HK "I (R )(pˆ pˆ #pˆ pˆ ) is a purely , V V W W quantum tunnel Hamiltonian, providing a quantum transfer of the pseudo-spin density between the quasiskyrmions resulting in a singlet—triplet splitting. The imaginary part of effective dynamical pseudo-spin susceptibility for skyrmion pair includes the contributions of the triplet—triplet and singlet—triplet transitions and could be written as follows:
s (q, u)J e SR[cos ?@
#sin
qR 1RK (0)RK (t)2 ? @ 2
qR 1¹K (0)¹K (t)2 ] dt, ? @ 2
where ¹K "pˆ !pˆ is the pseudo-spin multiplicity mixing operator. An occurrence of the singlet—triplet splitting and specific angular dependence for the two contributions could provide a reliable detection of both modes.
478
A.S. Moskvin, A.S. Ovchinnikov / Physica B 259—261 (1999) 476—478
References [1] A.A. Belavin, A.M. Polyakov, JETP Lett. 22 (1975) 245. [2] B.I. Shraiman, E.D. Siggia, Phys. Rev. Lett. 64 (1988) 467. [3] R.J. Gooding, Phys. Rev. Lett. 66 (1991) 2266.
[4] R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod. Phys. 62 (1990) 113. [5] A.S. Moskvin, JETP Lett. 58 (1993) 342. [6] A.S. Moskvin, Physica C 282—287 (1997) 1807. [7] A.S. Moskvin, Physica B 252 (1998) 186. [8] A.S. Moskvin, N.N. Loshkareva, Yu.P. Sukhorukov et al., JETP 105 (1994) 967.
Skyrmion-like states and quasi-molecular excitations in the doped cuprates A.S. Moskvin*, A.S. Ovchinnikov Department of Theoretical Physics, Ural State University, Lenin ave. 51, 620083 Ekaterinburg, Russia
Abstract Within a framework of the pseudo-spin formalism we consider the skyrmion-like topological modes for the quasi-twodimensional quantum lattice bose gas. A skyrmion—skyrmion chiral pseudo-spin coupling results in a pseudo-spin-flip mode splitting to the bonding (ferro) and antibonding (antiferro) quasi-molecular modes with different momentum q-dependence of the effective pseudospin susceptibility. It is conjectured that CuO planes in the doped cuprates can be considered as a system of the skyrmion-like states pinned on the charge inhomogeneity centers. 1999 Elsevier Science B.V. All rights reserved. Keywords: Cuprate; Chirality; Skyrmion; Superconductivity
Numerous experimental data convincingly evidence an unconventional non-Landau—Fermi-liquid behavior of the electron-fluid realized in cuprate superconductors with a puzzling metal-dielectric duality, which must be associated with some quite unusual order parameter(s). Up to now various fermionic, bosonic, or boson—fermionic scenarios have been elaborated with the purpose of elucidation of the HTSC problem. Below, we would like to emphasize unconventional skyrmion-like states of the quantum lattice bose gas to be the probable candidates for the description of specific inhomogeneous states in the CuO planes of the cuprates with the non-isovalent substitution. The classical skyrmions represent the stationary non-uniform states of the vector field S(r) on 2D-plane with the energy H " J( S?(r))dx ? (a"x, y, z), which corresponds to the Hamiltonian of the classical isotropic 2D Heisenberg model [1—3]. The spatial distribution of the order parameter for the conventional skyrmion localized at the point r"0 obeys the equation *S(r)"0 and has the form S " V
* Corresponding author. Fax: #7-3432-615-978; e-mail: [email protected].
sin h(r)cos(lu#t), S "sin h(r)sin(lu#t), S "cos h(r), W X sin 0(r)"2(jr) J /(j J #r J ) where l"$1,$2,2 is a topological charge with the chirality defined by the sign ‘$’, the j is a skyrmion radius, and the angle t defines a global orientation of the transversal xy-components of the spin S(r). The skyrmion energy E "8p"l"J does not J depend on the j and t. Note, the topological charge distinguishes the irreducible representations of the plane symmetry group C and in a sense corresponds to an orbital azimuthal quantum number m"$l, so below one make use of the atomic terminology: p, d, f,2 skyrmions when l"$1, $2, $3, etc. An isolated skyrmion could be described by an effective chiral pseudospin s". The principal elements of the classical field ‘skyrmionology’ could be used for the description of the stationary non-uniform states of various quantum (effective) spin systems. Below, one considers a quantum lattice bose gas where Hamiltonian is equivalent to that of the anisotropic Heisenberg magnet (s") with the temperature dependent external magnetic field [4]: HK "k SK (m)# [t(mn)(SK (m)SK (n)#SK (m)SK (n)) X V V W W K KL #»(mn)SK (m)SK (n)], X X
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 9 2 9 - 6
A.S. Moskvin, A.S. Ovchinnikov / Physica B 259—261 (1999) 476—478
where the spin operators are associated with the local boson creation (annihilation) operators, the local boson transfer integral t(mn) and the boson—boson repulsion energy »(mn) are effective exchange integrals; the renormalized boson chemical potential k plays the role of an external field and is determined by the constraint: 1SK 2"N\ 1SK (m)2"(!1nˆ 2). Here, N is a latX K X tice site number and 1nˆ 2 an average on-site boson concentration. Along with the uniform non-ordered metallic (NO), charge ordering (CO), superconducting bose-superfluid (BS), mixed (CO#BS) phases for the local boson ordering, one should introduce various nonuniform topological configurations with either topological charge or chirality. One should emphasize a difference between quasi-skyrmions and classical spin skyrmions: 1. The quasi-skyrmions represent an MFA portrait of the appropriate quantum states distinguished by topological quantum numbers, namely, the summary spin S and its projection M . So, one might consider 1 the non-uniform states the M "0 or n " (neutral 1 quasi-skyrmions), and M "$1 when a single boson 1 is created (annihilated) (singly charged quasi-skyrmions). 2. Taking account of the real tetragonal symmetry within the CuO plane permits to associate the quasi skyrmions with appropriate irreducible representations of the tetragonal symmetry group C : !" A , A , B , B , E . So, the "l""1 (p-skyr mion) is associated with the doublet E representation, while the "l""2 (d-skyrmion) splits to the B (d ) V\W and B (d ) skyrmions. In general, it should be noted VW that lowering the symmetry results in the chirality mixing effect, so that the ‘tetragonal classification’ operates most appropriately. The p(E ) skyrmion as sociates with the current state, whereas the d(B , B ) skyrmions do with the currentless states because of the ‘orbital quenching’. 3. Taking account of different physical nature of the z and x, y components of the boson pseudospin one should introduce as a minimum three types of the quasi-skyrmion solutions in accordance with the different boundary conditions at rPR. For the COtype quasi-skyrmions S " "0, "S " " "1, for V W P X P the BS-type S " "0, S " "1, for the X W P V P (CO#BS)-type S " O0, S " O0, respectively. , P , P The skyrmion-like distribution of the superconducting order parameter corresponds to the pure p(E ) or d(B , B )-symmetry for the topological charge value "l""1 or "l""2, respectively, only for the pure COtype quasi-skyrmions, while for the mixed (CO#BS), or for the pure BS-type quasi-skyrmions, we come to the mixed (s(A )#p(E )) symmetry at "l""1, or S (s(A )#d(B , B )) symmetry at "l""2 with an s fraction determined mainly by the outskirts of the quasi-skyrmion.
477
The above considered quasi-skyrmionic model could be successfully applied to the real doped cuprates in a framework of the scenario developed in papers [5—8], where the CuO planes are considered to be a system of the local singlet bosons moving in a lattice of the pseudo-Jahn—Teller (PJT) centers which is characterized by additional vibronic pseudo-spin vector parameters. The purely charge quasi-skyrmions in the PJT lattice are unstable with respect to the formation of a bound hybrid charge-vibronic quasi-skyrmion (large polaron). The singly charged hybrid quasi-skyrmions as peculiar polaron-like pseudo-particles can be pinned near such centers of charge inhomogeneity as Sr> in La Sr CuO , or \V V well isolated vacancies in YBa Cu O (x+1). This >V model approach, which considers the doped cuprates as a system of coupled quasi-skyrmions, permits to explain many of their puzzling properties, particularly the lowenergy structure of a spectrum. A skyrmion—skyrmion interaction results in the appearance of the quasi-molecular excitations in pairs of the quasi-skyrmions, which could be revealed by the inelastic neutron scattering. Such excitations are associated with the transitions between different singlet and triplet (bonding—antibonding) pseudo-spin states which correspond to different magnitudes of the summary pseudo-spin RK "pˆ #pˆ of the pair. An effective pseudo-spin Hamiltonian for the skyrmion—skyrmion coupling within a CuO plane could be written as the sum of four contributions: HK " HK #HK #HK #HK , where HK describes an initial pseudo-spin b !b splitting for the d-skyrmions, HK "I (R )pˆ pˆ is an effective current—current coup , X X ling with the parameter I depending on the inter, skyrmionic separation R ; HK "I (R )
[(pˆ pˆ !pˆ pˆ )cos8u!(pˆ pˆ #pˆ pˆ )sin8u] is V V W W V W W V a multipole coupling with u being the azimuthal angle of the R , and HK "I (R )(pˆ pˆ #pˆ pˆ ) is a purely , V V W W quantum tunnel Hamiltonian, providing a quantum transfer of the pseudo-spin density between the quasiskyrmions resulting in a singlet—triplet splitting. The imaginary part of effective dynamical pseudo-spin susceptibility for skyrmion pair includes the contributions of the triplet—triplet and singlet—triplet transitions and could be written as follows:
s (q, u)J e SR[cos ?@
#sin
qR 1RK (0)RK (t)2 ? @ 2
qR 1¹K (0)¹K (t)2 ] dt, ? @ 2
where ¹K "pˆ !pˆ is the pseudo-spin multiplicity mixing operator. An occurrence of the singlet—triplet splitting and specific angular dependence for the two contributions could provide a reliable detection of both modes.
478
A.S. Moskvin, A.S. Ovchinnikov / Physica B 259—261 (1999) 476—478
References [1] A.A. Belavin, A.M. Polyakov, JETP Lett. 22 (1975) 245. [2] B.I. Shraiman, E.D. Siggia, Phys. Rev. Lett. 64 (1988) 467. [3] R.J. Gooding, Phys. Rev. Lett. 66 (1991) 2266.
[4] R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod. Phys. 62 (1990) 113. [5] A.S. Moskvin, JETP Lett. 58 (1993) 342. [6] A.S. Moskvin, Physica C 282—287 (1997) 1807. [7] A.S. Moskvin, Physica B 252 (1998) 186. [8] A.S. Moskvin, N.N. Loshkareva, Yu.P. Sukhorukov et al., JETP 105 (1994) 967.