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Topological character of excitations in strongly correlated electronic systems: confinement and dimensional crossover
Synthetic Metals 133–134 (2003) 41–43
Topological character of excitations in strongly correlated electronic systems: confinement and dimensional crossover S. Brazovskiib,*, N. Kirovaa,c a
LPTMS, Baˆt. 100, Universite´ Paris-Sud, 91405 Orsay Cedex, France b Landau Institute for Theoretical Physics, Moscow, Russia c POMA, Universite´ Angers, 49045 Angers Cedex 01, France
Abstract Topologically nontrivial states, the solitons, emerge as elementary excitations in 1D electronic systems. In a quasi 1D material the topological requirements originate the spin- or charge-roton like configurations with charge- or spin-kinks localized in the core. The rotons possess semi-integer winding numbers which results in the spin-charge recombination due to confinement and the combined symmetry. Practically important is the case of the spinon functioning as the single electronic p-junction in a quasi 1D superconducting material. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Solitons; Confinement; Spinon; Holon
1. Introduction to solitons
2. Confinement and combined excitations
Topologically nontrivial states are common in symmetry broken phases at macroscopic scales. Low dimensional systems bring them to a microscopic level where solitons emerge as single particles (see reviews [1,2]). The earliest and latest applications are conducting polymers and spinPeierls chains. We shall discuss topological aspects of elementary excitations, especially the confinement and the dimensional D crossover. To extend physics of solitons to the higher D world, the most important problem one faces is the effect of confinement (Brazovskii, 1980): as topological objects connecting degenerate vacuums, the solitons at D > 1 acquire an infinite energy unless they reduce or compensate their topological charges. The problem is generic to solitons, but it becomes particularly interesting at the single electronic level where the spin-charge reconfinement appears as the result of topological constraints. The topological effects of D > 1 ordering reconfine the charge and the spin locally while still with essentially different distributions. Nevertheless, integrally one of the two is screened again, being transferred to the collective mode, so that in kinetics the massive particles carry only either charge or spin as in 1D. Details can be found in [3–5].
2.1. The classical commensurate CDW: confinement of phase solitons and of kinks
* Corresponding author. Fax: þ33-1-69-15-65-25. E-mail address: [email protected] (S. Brazovskii).
Being a case of spontaneous symmetry breaking, the CDW order parameter OCDW D cos ½Qr þ j possesses a manifold of degenerate ground states. For the M-fold commensurate CDW the energy cos ½Mj reduces the allowed positions to multiples of 2p=M, M > 1. Connecting trajectories j ! j 2p=M are phase solitons. Particularly important is the case M ¼ 2 for which solitons are clearly observed, e.g. in polyacethylene or in organic Mott insulators (see articles by Nad and Brazovskii in this volume). Above Tc , the symmetry is not broken and solitons are allowed to exist as elementary particles. But in the symmetry broken phase at T < Tc , any local deformation must return the configuration to the same (modulo 2p for the phase) state. Otherwise the interchain interaction energy (with the linear density F hD0 Dn ð1 cos ½j0 jn Þi is lost when the effective phase j0 þ p signðD0 Þ at the soliton bearing chain n ¼ 0 acquires a finite (and 6 2pÞ increment with respect to the neighboring chain values jn . The 1D allowed solitons do not satisfy this condition which originates a constant confinement force F between them, hence the infinitely growing confinement energy Fjxj. For example, M ¼ 2, the kinks should be bound in pairs or aggregate into macroscopic complexes with a particular role plaid by Coulomb interactions [6].
0379-6779/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 9 - 6 7 7 9 ( 0 2 ) 0 0 4 1 6 - 2
42
S. Brazovskii, N. Kirova / Synthetic Metals 133–134 (2003) 41–43
Especially interesting is the more complicated case of coexisting discrete and continuous symmetries. As a result of their interference the topological charge of solitons originated by the discrete symmetry can be compensated by gapless degrees of freedom originated by the continuous one. This scenario we shall discuss through the rest of the article. 2.2. The incommensurate CDW: confinement of amplitude solitons with phase wings In the incommensurate CDW (ICDW) any additional pair of electrons or holes is accommodated to the extended ground states for which the overall phase difference becomes 2p. Phase increments are produced by phase slips which provide the spectral flow from the upper þD0 to the lower D0 rims of the single particle gap. The phase slip requires the amplitude Dðx; tÞ to pass through zero, at which moment the complex order parameter has a shape of the amplitude soliton (AS), the kink, Dðx ¼ 1Þ $
Dðx ¼ 1). Curiously, this instantaneous configuration becomes the stationary ground state for the case when only one electron is added to the system or when the total spin polarization is controlled to be nonzero. The AS carries the singly occupied mid-gap state, thus having a spin 1=2, but its charge is compensated to zero by local dilatation of singlet vacuum states [1,2]. As a nontrivial topological object (OCDW does not map onto itself) the pure AS is prohibited in D > 1 environment. Nevertheless, it becomes allowed even their if it acquires phase tails with the total increment dj ¼ p. The length of these tails xj is determined by the weak interchain coupling, thus xj x0 . As in 1D, the sign of D changes within the scale x0 but further on, at the scale xj , the factor cos ½Qx þ j also changes the sign thus leaving the product in OCDW to be invariant. As a result, the 3D allowed particle is formed with the AS core x0 carrying the spin and the two wings stretched over xj, where the wing phase is twisting by p=2, each carrying the charge e=2. 2.3. Spin-gap cases: the quantum CDW and the superconductivity Consider firstly systems with attraction which originates the gap and leads to the discrete degeneracy in the spin channel. Within the boson representation the variables can be chosen as j which is the analog of the CDW phase and y which is the angle of the SU2 spin rotation. For the incommensurate electronic systems the Lagrangian can be written as (the velocity v ¼ 1) Latr fC1 ð@yÞ2 þ V cos ð2yÞg þ C2 ð@jÞ2 where C1 ; C2 are constants and V is the backward exchange scattering and ð@f Þ2 ¼ ð@ t f Þ2 ð@ x f Þ2 . Elementary excitations at 1D are the spinon as a soliton y ¼ 0 ! y ¼ p,
hence carrying the spin 1=2, and the gapless charge sound in j. An alternative description invokes the standard gauge phase w of the superconductivity which is conjugated to j. ~ 2 ð@wÞ2 with C ~ 2 1=C2 . The term C2 ð@jÞ2 is duel to C The CDW order parameter is OCDW exp½iðQx þ jÞ cos y. The spin operator cos y stands for what was the amplitude D in the quasi-classical description and at presence of the spinon it also changes the sign. Hence for the CDW ordered phase the allowed configuration must be composed with two components: the spin soliton y ! y p and the phase wings j ! j p, where the charge e ¼ 1 is concentrated. Beyond D ¼ 1, a general view is: the spinon as a soliton bound to the semi-integer dislocation loop. For the singlet superconductivity the order parameter is Oss exp½iw cos y. In D > 1 the elementary spin excitation is composed with the spin soliton y ! y þ p supplied with current wings w ! w þ p. The quasi 1D interpretation is that the spinon works as a Josephson p-junction in the superconducting wire. The 2D view is a pair of superconducting p-vortices sharing the common core where the unpaired spin is localized. The 3D view is a half flux vortex loop which collapse is prevented by the spin confined in its center. The solitonic nature of the spinon in the quasi 1D picture corresponds to the string of reversed sign of the order parameter left behind in the course of the spinon motion. The spin soliton becomes an elementary fragment of the stripe pattern near the pair breaking limit (FFLO phase). The p wings along the line of the spinon motion become the persistent current (Yakovenko, private communication). For this combined particle the electronic quantum numbers are reconfined (while with different scales of localization). But integrally over the cross-section the on-chain electric current induced by the spinon is compensated exactly by the back-flows at distant chains. This is a general property of the vortex dipole configuration constructed above. Finally, the soliton as a state of the coherent media will not carry a current and itself will not be driven by a homogeneous electric field. 2.3.1. A hole in the AFM environment Consider the quasi 1D system with repulsion at a nearly half filled band which is the spin density wave (SDW) rout to a general doped antiferromagnetic (AFM) Mott–Hubbard insulator (see [4] for more details). The 1D bosonized Hamiltonian (Luther and Emery) can be written schematically as Hrep fC2 ð@jÞ2 U cos ð2jÞg þ ðC1 @yÞ2 where j is the analog of the CDW displacement phase and y is the spin rotation angle. Here U is the Umklapp scattering amplitude (or g3 of Dzyaloshinskii and Larkin) which is supposed to be small as in Bechgaard salts. In 1D the excitations are the (anti)holon as the p soliton in j, and the spin sound in y which are decoupled. At D > 1 below the SDW ordering transition TSDW , the order parameter
S. Brazovskii, N. Kirova / Synthetic Metals 133–134 (2003) 41–43
Fig. 1. Motion of the topologically combined excitation in a spin-gap (charge gap) media. The string of the amplitude reversal of the order parameter created by the spinon (holon) is cured by the semi-vortex pair (the loop in 3D) of the phase circulation. For the CDW case the curls are displacements contours for the half-integer dislocation pair. For the superconductivity the curls are lines of electric currents circulating through the normal core carrying the unpaired spin. For the SDW, AFM cases the curls are directions of the staggered magnetization.
is OSDW cos j exp fiðQx þ yÞg. To survive in D > 1, the p-soliton in j ( cos j ! cos j) should enforce a p rotation in y, then the sign changes of the two factors composing OSDW cancel each other. Extending to isotropic AFMs like the CuO2 planes, the SDW becomes a staggered magnetization, the soliton becomes a hole moving within the AFM environment, the p wings become the magnetic semi-vortices (see [5] for a macroscopic illustration: the chimera). The resulting configuration is a half-integer vortex ring of the staggered magnetization (a semiroton) with the holon confined in its center similar to Fig. 1 (in 2D the vortex ring is reduced to the pair of vortices). The solitonic nature of the holon in quasi 1D corresponds to the string of reversed staggered magnetization left by the moving hole.
43
ton–spinon complex. This construction can be processed from another side considering a vortex configuration bound to an unpaired electron. Without assistance of the quasi onedimensionality, a short coherence length is required to leave only a small number of intragap levels in the vortex core. Thus, the existence of complex spin excitations is ultimately related to robustness of the FFLO phase at finite spin polarization. It must withstand a fragmentation due to quantum or thermal melting. Then any termination point of one stripe within the regular pattern (the dislocation) will be accompanied by the phase semiroton in accordance with the quasi 1D picture. The same concerns holons in the AFM: the combined state becomes an element of the melted charged stripe. Exploiting an analogy to quarks and gluons in QCD, separate spinons and holons are not observed directly because of the topological confinement. The topological effects of D > 1 ordering reconfine the charge and the spin locally while still with essentially different distributions. Nevertheless, integrally one of the two is screened again, being transferred to the collective mode, so that in kinetics the massive particles carry only either charge or spin as in 1D. For more details a view of spinons as p-junctions can be found in [3], while [4] is more accented on holons in the AFM. Brazovskii [4] suggests also the introduction in history of microscopic solitons, while a broader view of solitons in the framework of theory of strongly correlated systems can be found in [7]. Kirova [5] exploits the picture of combined topological defects (the chimers) for sliding ISDW.
References 3. Conclusions Our conclusions have been derived for weakly interacting chains. Since the results are symmetrically and topologically characterized, they can be extrapolated to isotropic systems with strong coupling where a clear microscopic derivation is not available. Here the hypothesis is that, instead of normal carries excited or injected above the gap, the lowest states are the symmetry broken ones described above as semiro-
[1] S. Brazovskii, N. Kirova, in: I.M. Khalatnikov (Ed.), vol. A5, Harwood Academic Publishers, 1984, Sov. Sci. Rev. [2] S. Brazovskii, in: Modern Problems in Condensed Matter Science, vol. 25, Elsevier, Amsterdam, 1990, p. 425. [3] S. Brazovskii, cond-mat/0204147. [4] S. Brazovskii, cond-mat/0006355. [5] N. Kirova, cond-mat/0004313. [6] S. Teber, B.P. Stojkovic, S.A. Brazovskii, A.R. Bishop, J. Phys. Condens. Matter. 13 (2001) 4015. [7] S. Brazovskii, J. Phys. IV 10 (2000) 3–169.
Topological character of excitations in strongly correlated electronic systems: confinement and dimensional crossover S. Brazovskiib,*, N. Kirovaa,c a
LPTMS, Baˆt. 100, Universite´ Paris-Sud, 91405 Orsay Cedex, France b Landau Institute for Theoretical Physics, Moscow, Russia c POMA, Universite´ Angers, 49045 Angers Cedex 01, France
Abstract Topologically nontrivial states, the solitons, emerge as elementary excitations in 1D electronic systems. In a quasi 1D material the topological requirements originate the spin- or charge-roton like configurations with charge- or spin-kinks localized in the core. The rotons possess semi-integer winding numbers which results in the spin-charge recombination due to confinement and the combined symmetry. Practically important is the case of the spinon functioning as the single electronic p-junction in a quasi 1D superconducting material. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Solitons; Confinement; Spinon; Holon
1. Introduction to solitons
2. Confinement and combined excitations
Topologically nontrivial states are common in symmetry broken phases at macroscopic scales. Low dimensional systems bring them to a microscopic level where solitons emerge as single particles (see reviews [1,2]). The earliest and latest applications are conducting polymers and spinPeierls chains. We shall discuss topological aspects of elementary excitations, especially the confinement and the dimensional D crossover. To extend physics of solitons to the higher D world, the most important problem one faces is the effect of confinement (Brazovskii, 1980): as topological objects connecting degenerate vacuums, the solitons at D > 1 acquire an infinite energy unless they reduce or compensate their topological charges. The problem is generic to solitons, but it becomes particularly interesting at the single electronic level where the spin-charge reconfinement appears as the result of topological constraints. The topological effects of D > 1 ordering reconfine the charge and the spin locally while still with essentially different distributions. Nevertheless, integrally one of the two is screened again, being transferred to the collective mode, so that in kinetics the massive particles carry only either charge or spin as in 1D. Details can be found in [3–5].
2.1. The classical commensurate CDW: confinement of phase solitons and of kinks
* Corresponding author. Fax: þ33-1-69-15-65-25. E-mail address: [email protected] (S. Brazovskii).
Being a case of spontaneous symmetry breaking, the CDW order parameter OCDW D cos ½Qr þ j possesses a manifold of degenerate ground states. For the M-fold commensurate CDW the energy cos ½Mj reduces the allowed positions to multiples of 2p=M, M > 1. Connecting trajectories j ! j 2p=M are phase solitons. Particularly important is the case M ¼ 2 for which solitons are clearly observed, e.g. in polyacethylene or in organic Mott insulators (see articles by Nad and Brazovskii in this volume). Above Tc , the symmetry is not broken and solitons are allowed to exist as elementary particles. But in the symmetry broken phase at T < Tc , any local deformation must return the configuration to the same (modulo 2p for the phase) state. Otherwise the interchain interaction energy (with the linear density F hD0 Dn ð1 cos ½j0 jn Þi is lost when the effective phase j0 þ p signðD0 Þ at the soliton bearing chain n ¼ 0 acquires a finite (and 6 2pÞ increment with respect to the neighboring chain values jn . The 1D allowed solitons do not satisfy this condition which originates a constant confinement force F between them, hence the infinitely growing confinement energy Fjxj. For example, M ¼ 2, the kinks should be bound in pairs or aggregate into macroscopic complexes with a particular role plaid by Coulomb interactions [6].
0379-6779/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 9 - 6 7 7 9 ( 0 2 ) 0 0 4 1 6 - 2
42
S. Brazovskii, N. Kirova / Synthetic Metals 133–134 (2003) 41–43
Especially interesting is the more complicated case of coexisting discrete and continuous symmetries. As a result of their interference the topological charge of solitons originated by the discrete symmetry can be compensated by gapless degrees of freedom originated by the continuous one. This scenario we shall discuss through the rest of the article. 2.2. The incommensurate CDW: confinement of amplitude solitons with phase wings In the incommensurate CDW (ICDW) any additional pair of electrons or holes is accommodated to the extended ground states for which the overall phase difference becomes 2p. Phase increments are produced by phase slips which provide the spectral flow from the upper þD0 to the lower D0 rims of the single particle gap. The phase slip requires the amplitude Dðx; tÞ to pass through zero, at which moment the complex order parameter has a shape of the amplitude soliton (AS), the kink, Dðx ¼ 1Þ $
Dðx ¼ 1). Curiously, this instantaneous configuration becomes the stationary ground state for the case when only one electron is added to the system or when the total spin polarization is controlled to be nonzero. The AS carries the singly occupied mid-gap state, thus having a spin 1=2, but its charge is compensated to zero by local dilatation of singlet vacuum states [1,2]. As a nontrivial topological object (OCDW does not map onto itself) the pure AS is prohibited in D > 1 environment. Nevertheless, it becomes allowed even their if it acquires phase tails with the total increment dj ¼ p. The length of these tails xj is determined by the weak interchain coupling, thus xj x0 . As in 1D, the sign of D changes within the scale x0 but further on, at the scale xj , the factor cos ½Qx þ j also changes the sign thus leaving the product in OCDW to be invariant. As a result, the 3D allowed particle is formed with the AS core x0 carrying the spin and the two wings stretched over xj, where the wing phase is twisting by p=2, each carrying the charge e=2. 2.3. Spin-gap cases: the quantum CDW and the superconductivity Consider firstly systems with attraction which originates the gap and leads to the discrete degeneracy in the spin channel. Within the boson representation the variables can be chosen as j which is the analog of the CDW phase and y which is the angle of the SU2 spin rotation. For the incommensurate electronic systems the Lagrangian can be written as (the velocity v ¼ 1) Latr fC1 ð@yÞ2 þ V cos ð2yÞg þ C2 ð@jÞ2 where C1 ; C2 are constants and V is the backward exchange scattering and ð@f Þ2 ¼ ð@ t f Þ2 ð@ x f Þ2 . Elementary excitations at 1D are the spinon as a soliton y ¼ 0 ! y ¼ p,
hence carrying the spin 1=2, and the gapless charge sound in j. An alternative description invokes the standard gauge phase w of the superconductivity which is conjugated to j. ~ 2 ð@wÞ2 with C ~ 2 1=C2 . The term C2 ð@jÞ2 is duel to C The CDW order parameter is OCDW exp½iðQx þ jÞ cos y. The spin operator cos y stands for what was the amplitude D in the quasi-classical description and at presence of the spinon it also changes the sign. Hence for the CDW ordered phase the allowed configuration must be composed with two components: the spin soliton y ! y p and the phase wings j ! j p, where the charge e ¼ 1 is concentrated. Beyond D ¼ 1, a general view is: the spinon as a soliton bound to the semi-integer dislocation loop. For the singlet superconductivity the order parameter is Oss exp½iw cos y. In D > 1 the elementary spin excitation is composed with the spin soliton y ! y þ p supplied with current wings w ! w þ p. The quasi 1D interpretation is that the spinon works as a Josephson p-junction in the superconducting wire. The 2D view is a pair of superconducting p-vortices sharing the common core where the unpaired spin is localized. The 3D view is a half flux vortex loop which collapse is prevented by the spin confined in its center. The solitonic nature of the spinon in the quasi 1D picture corresponds to the string of reversed sign of the order parameter left behind in the course of the spinon motion. The spin soliton becomes an elementary fragment of the stripe pattern near the pair breaking limit (FFLO phase). The p wings along the line of the spinon motion become the persistent current (Yakovenko, private communication). For this combined particle the electronic quantum numbers are reconfined (while with different scales of localization). But integrally over the cross-section the on-chain electric current induced by the spinon is compensated exactly by the back-flows at distant chains. This is a general property of the vortex dipole configuration constructed above. Finally, the soliton as a state of the coherent media will not carry a current and itself will not be driven by a homogeneous electric field. 2.3.1. A hole in the AFM environment Consider the quasi 1D system with repulsion at a nearly half filled band which is the spin density wave (SDW) rout to a general doped antiferromagnetic (AFM) Mott–Hubbard insulator (see [4] for more details). The 1D bosonized Hamiltonian (Luther and Emery) can be written schematically as Hrep fC2 ð@jÞ2 U cos ð2jÞg þ ðC1 @yÞ2 where j is the analog of the CDW displacement phase and y is the spin rotation angle. Here U is the Umklapp scattering amplitude (or g3 of Dzyaloshinskii and Larkin) which is supposed to be small as in Bechgaard salts. In 1D the excitations are the (anti)holon as the p soliton in j, and the spin sound in y which are decoupled. At D > 1 below the SDW ordering transition TSDW , the order parameter
S. Brazovskii, N. Kirova / Synthetic Metals 133–134 (2003) 41–43
Fig. 1. Motion of the topologically combined excitation in a spin-gap (charge gap) media. The string of the amplitude reversal of the order parameter created by the spinon (holon) is cured by the semi-vortex pair (the loop in 3D) of the phase circulation. For the CDW case the curls are displacements contours for the half-integer dislocation pair. For the superconductivity the curls are lines of electric currents circulating through the normal core carrying the unpaired spin. For the SDW, AFM cases the curls are directions of the staggered magnetization.
is OSDW cos j exp fiðQx þ yÞg. To survive in D > 1, the p-soliton in j ( cos j ! cos j) should enforce a p rotation in y, then the sign changes of the two factors composing OSDW cancel each other. Extending to isotropic AFMs like the CuO2 planes, the SDW becomes a staggered magnetization, the soliton becomes a hole moving within the AFM environment, the p wings become the magnetic semi-vortices (see [5] for a macroscopic illustration: the chimera). The resulting configuration is a half-integer vortex ring of the staggered magnetization (a semiroton) with the holon confined in its center similar to Fig. 1 (in 2D the vortex ring is reduced to the pair of vortices). The solitonic nature of the holon in quasi 1D corresponds to the string of reversed staggered magnetization left by the moving hole.
43
ton–spinon complex. This construction can be processed from another side considering a vortex configuration bound to an unpaired electron. Without assistance of the quasi onedimensionality, a short coherence length is required to leave only a small number of intragap levels in the vortex core. Thus, the existence of complex spin excitations is ultimately related to robustness of the FFLO phase at finite spin polarization. It must withstand a fragmentation due to quantum or thermal melting. Then any termination point of one stripe within the regular pattern (the dislocation) will be accompanied by the phase semiroton in accordance with the quasi 1D picture. The same concerns holons in the AFM: the combined state becomes an element of the melted charged stripe. Exploiting an analogy to quarks and gluons in QCD, separate spinons and holons are not observed directly because of the topological confinement. The topological effects of D > 1 ordering reconfine the charge and the spin locally while still with essentially different distributions. Nevertheless, integrally one of the two is screened again, being transferred to the collective mode, so that in kinetics the massive particles carry only either charge or spin as in 1D. For more details a view of spinons as p-junctions can be found in [3], while [4] is more accented on holons in the AFM. Brazovskii [4] suggests also the introduction in history of microscopic solitons, while a broader view of solitons in the framework of theory of strongly correlated systems can be found in [7]. Kirova [5] exploits the picture of combined topological defects (the chimers) for sliding ISDW.
References 3. Conclusions Our conclusions have been derived for weakly interacting chains. Since the results are symmetrically and topologically characterized, they can be extrapolated to isotropic systems with strong coupling where a clear microscopic derivation is not available. Here the hypothesis is that, instead of normal carries excited or injected above the gap, the lowest states are the symmetry broken ones described above as semiro-
[1] S. Brazovskii, N. Kirova, in: I.M. Khalatnikov (Ed.), vol. A5, Harwood Academic Publishers, 1984, Sov. Sci. Rev. [2] S. Brazovskii, in: Modern Problems in Condensed Matter Science, vol. 25, Elsevier, Amsterdam, 1990, p. 425. [3] S. Brazovskii, cond-mat/0204147. [4] S. Brazovskii, cond-mat/0006355. [5] N. Kirova, cond-mat/0004313. [6] S. Teber, B.P. Stojkovic, S.A. Brazovskii, A.R. Bishop, J. Phys. Condens. Matter. 13 (2001) 4015. [7] S. Brazovskii, J. Phys. IV 10 (2000) 3–169.