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Ground state structure and excitations in strongly correlated Fermi systems
Volume 144, number 4,5
PHYSICS LETTERS A
5 March 1990
G R O U N D STATE S T R U C T U R E A N D E X C I T A T I O N S IN STRONGLY CORRELATED FERMI SYSTEMS A.P. P R O T O G E N O V Institute of Applied Physics, Academy of Sciences of the USSR, 46 Uljanov Street, 603600 Gortcy, USSR
Received 13 January 1989; revised manuscript received 15 December 1989; accepted for publication 27 December 1989 Communicated by A.A. Maradudin
The ground state structure and some excitation properties of the (2 + 1)-dimensional strongly correlated Fermi systems are discussed. Quasi-classical configurations of the fields with finite energy and fractional charge of quasi-particles are described. It is shown that a holon and a spinon are different states of an anyon.
A deep insight into the properties o f new phases [ 1-5] in strongly correlated Fermi systems stimulates the unification of various types of their elementary excitations. The 2 + 1 dimensions and the delocalization of excitations in a m o m e n t u m space produced by a strong coupling mostly determine the soliton-like excitations. The study of quasi-particles enables one to get information on the v a c u u m structure in the low-energy region. The present paper makes an attempt to investigate possible types of elementary excitations. I consider, mainly, the excitations at temperatures exceeding the superconducting transition one. The details will be published elsewhere. The configurations o f the field contributing to the partition function are studied in the quasi-classical approximation in a continuum limit taking into account topological characteristics o f the system. Strongly interacting Fermi particles are described with a well-known Hubbard model with strong (compared to a hop t) Coulomb repulsion U on the lattice site. The Hubbard Hamiltonian projected on the states of the low zone can be written as a Heisenberg Hamiltonian with additional terms describing the hole transport [ 1-3 ] and the nontrivial topology [4,5]. In the absence of the external electromagnetic field the action in the low-energy limit has the form [5]
S= f dvd~x( ~=½Jl(Ou+iAu)zkl2
i.
- 4n e~"XAuFvz
)"
(1)
The complex doublet z = (~) is related to the antiferromagnetic order parameter n (x, y, r) as follows: n = z +az. Here cr is the Pauli matrix and z is the Matsubara time. The restriction n2= 1 is equivalent to the condition IZl I2 71_122 I2 = 1. The statistical gauge field A~ compensates the arbitrary local choice z ~ z ' = z e x p [ i A (x) ] o f the additional freedom degrees of the z-field compared to the n-field. The gauge field of the CP ~representation at small scales is composite (Au = - iz + 0 uz). We shall consider the longwavelength limit and the states [ 5 ] where the gauge potential is an independent variable. In the longwavelength case the terms of the self-energy type F ~ are essential at smaller scales [6]. The term responsible for the hole transport in eq. ( 1 ) is omitted since the doping is assumed to be small (t~ << t 2/U). The temporal component of the statistical gauge potential is usually employed to take into account [7 ] spin density fluctuations near half-filling o f the site while the spatial components describe the space structure of frustrations. The parameter v in eq. ( 1 ) is the filling factor, i.e. the dimensionless density of frustrations. The last term in eq. (1) is the U( 1 ) Chern-Si269
V o l u m e 144, n u m b e r 4,5
PHYSICS LETTERS A
mons term reflecting the Bohm-Aharonov interaction. Its nonzero value in the continuous limit characterizes the so-called [7 ] Chiral spin state. If the ground state is an antiferromagnetic one, the ChernSimons term is absent [8 ]. Such a term occurs due to doping [ 9 ], is produced by a random lattice [ 10 ] and reflects [7,9] three-particle correlations initiated by the next-nearest-neighbor coupling. It should be noted that the Chern-Simons term is odd with respect to 2D parity P and time reversal T. This macroscopic violation of discrete symmetries is concealed by multilayer 3D systems where the signs of v in the neighboring layers may alternate in the equilibrium state. Let us consider the long-wavelength space structure of elementary excitations of the model ( 1 ). We imagine that we have broken the local antiferromagnetic order by removing a particle (with its spin and charge) from the pair. This means that in the test spin location n 2 # 1, or [z[ 2 ¢ 1. The violation of the condition [z[ 2= 1 can be taken into account by introducing a spin source in the form ofipAo into eq. ( 1 ), with the Lagrange multiplier iA0 and the density p = [z[ 2_ 1 describing the deviation from the local antiferromagnetic order. The functional (1) with such a source for the scalar order parameter at the quasi-classical level was discussed in relation with a fractional quantum Hall effect [ 11,12 ]. The equations of motion in the static case (0k+iAk)2Z~=0,
(2)
1/
4~ (VXA)± = IZl [2+ Iz212- 1,
(3)
yield the following field configurations •
Zk(r, ~0)=O(r--rm) exp0m~o)~ sin 2gr A~,(r) = - - u
m 2gr '
r
'
,
(4) (5a)
m
2nr'
r > rm,
(5b)
with zero action and the fractional excitation "charge" e* = ~. The expression (3) means the neutralization condition of the spin density Iz l 2_ 1 by the vacuum density (u/4n) ( V × A ) . . Let us stress again that for the sake of simplicity the external elec270
5 M a r c h 1990
tromagnetic gauge field is assumed to be absent (cf. ref. [7] ). In the expression (4) O(r-rm) is the step function, rm 2 = vrnlm 2 lH=H -- 1/2 is the magnetic length and Hz= ( V × A ) l is the statistical magnetic field. The expressions (4), (5) describe the vortex with a long-range Bohm-Aharonov potential (5b) outside the elementary vortex and the ferromagnetic order at r < rm,
n= (O2(r--rm) sin 2a, 0, O2(r--rm)
cos 2o~) .
(6)
Thus the expression (6) describes the ferromagnetic polaron induced by the test spin (doping). At the surface density n~1013-1014 cm -2 the statistical magnetic field is H ~ n¢~o~ 106_ 10 v e where ~o is the flux quantum. The vortex scale l~i~ 10-7 cm and the distance n-1/2 between vortices are of the same order. The fractional charge e * = v , where v=O/2== q/p= n × 2rd 2 is the filling factor of the degenerate ground state, is well known in relation to the fractional quantum Hall effect. Here 0 is the vacuum angle, q is the number of particles, the odd denominator p = 2 ( l + ½) [ 13 ] is the number of flux quanta, l is the orbital momentum, and n is the surface density. In our case the filling factor v = 0/4~ =p/2q, where p is the number of flux quanta of the statistical magnetic field and q is the number of particles. The multiplier ½ indicates that two orientations are possible. To avoid misunderstanding we note that our vacuum angle 0 coincides with the angle O~ in ref. [6]. The vacuum angles 0p and 0~ in the vortex picture (expressed in the z-field terms) and in the n-field picture (6), respectively, are dually conjugate [ 6,14 ]: 0 ~ = - ~ 2 / 0 ~ . As shown in refs. [15,16] 0 = 2 n , i.e. p = q= 1 and the filling factor u = ½ (cf. refs. [ 17,18 ] ). The 4n periodicity in the vacuum angle 0 is due to spontaneous time reversal violations in the chiral spin state [7 ]. The gauge configurations (5b) were introduced by Wilczek [ 19 ] and are called anyons. An anyon, as a particle, represents a vortex with several flux quanta and charges rotating around it. In the singular gauge when the potential outside the vortex is zero and the two anyons are interchanged the wave function acquires a phase: ~((2, ~1) = exp(iO~)~(~l, ¢2),
0~ =
nq
2p"
(7)
Volume 144, number 4,5
PHYSICSLETTERSA
The ratio 0 ~ = - ~ 2 / 0 a is used here [6,14]. This means that at p/q= 1 an anyon, as a quasi-particle, is a half-fermion [7,17] obeying the fractional statistics [ 19,20 ]. Therefore the picture of excitations with a collective long-range gauge Bohm-Aharonov interaction is equivalent [19,20] to the picture of free excitations with intermediate statistics in the singular gauge. When dealing with anyon-antianyon pairing one should bear in mind that the vector potential of the anyon pair written in relative variables is equivalent to the charge interacting with a double flux in the single-flux field of the anyon [ 20 ]. This fact yields the necessary 2e-rule for Fermi confinement of anyons in the "normal" state and Bose condensation of anyon pairs in the superconducting state. In fact, when measuring the number of quanta in units ch/2e the equality 2p= 1 is valid and at q = 1 an anyon-antianyon pair is a spinon while at q = 2 the anyon pair becomes a holon. On the other hand, the value of q yields the number of bands of a Hofstadter particle [21 ] for the rational value p/2q of the flux in the statistical magnetic field. It is known [22] from the quantum Hall effect study that different bands make different sign contributions to the Hall coefficient. Thus, when the temperature decreases and the chemical potential passes through the center of the spectrum the hole-like Hall coefficient changes to the electron-like one. This property may be related to the results of the recent Hall measurements [23] where the Hall coefficient changed its hole-like sign for the electron-like one at the temperature To usually called the onset of the superconducting state transition. According to (7) the q jump from q= 1 to q = 2 at this temperature corresponds to the phase transition with change of the anyon pair statistics from the Fermi case to the Bose one. In other words, at T = To there occurs a phase transition into a nonuniform ground state with a smaller number of cells with a nonzero flux of the statistical magnetic field on the underlying lattice. In the singular gauge of interest the action at large distances is logarithmically large, R
S ~ ~ rdr [ (Oz/Or)2-k - IzlZ(Ao+m/2nr) a] 0
~mZlnR,
(8)
5 March 1990
i.e. a temperature decrease may cause a KosterlitzThouless transition from the Coulomb (plasma) phase of single vortices (Fermi-anyon pairs and fluctuation Bose-anyon pairs) existing in the Tc < T < To region into the superconducting state where elementary excitations are the Kosterlitz-Thouless pairs of vortices (perhaps Bose-anyon fours). The Kosterlitz-Thouless phase transition accompanied with a Kosterlitz-Nelson jump has been observed in ref. [23]. In the mean statistical magnetic field H~-nOo the energy of stationary states is equal to
eN=hO)c(N+ 1 ) = n J ( 2 N + 1 ) = nkaT( ZN+ l ) .
(9)
The values of the m a s s M ~ h Z / J a 2 and density na z~- 1 are used here, where a is the lattice constant. The characteristic temperature Tc = 3"/kB = 102 K in the spectrum of the Matsubara frequencies is, evidently, the temperature of Bose condensation of anyon-antianyon vortices. The relation (9) Tc ~ n / M is yielded by the recent experimental studies [25 ] of local magnetic fields by means of ~t-muon spin relaxation. Further ordering produced by the temperature decrease below Tc can be due to the Wigner crystallization of the vortex structure. The complete scenario of phase transitions is presented in ref. [26 ]. The energy o f quasi-particle motion td increases with doping and at t(~>~t 2 / U ~ J condensate anyons will effectively decay into spin-roton excitations [27,28]. These collective modes in the anyon medium possess a roton part of the spectrum at a finite wavelength of the order of lH and a gap of the order of J as well as a finite gap (larger than J) existing in the long-wavelength region due to the incompressibility condition. A reduced number of condensate Bose anyons decreases the critical temperature with the doping increase. What is the relation between the phenomena considered and the experimental results obtained? I compare only some of them. (1) When studying tunneling in SIN-contacts violation of charge antisymmetry of the C - V characteristics at the substitution V~ - V was found [ 29 ]. (2) It has been found that at helium temperatures a lattice of vortices is present even in weak magnetic fields but at nitrogen temperatures it is absent [30]. (3) Two types of closely located phase transitions were discovered 271
Volume 144, number 4,5
PHYSICS LETTERS A
w h e n s t u d y i n g the h e a t c a p a c i t y [ 3 1 ] . (4) I n v e s t i gations o f n o n e r g o d i c b e h a v i o u r w i t h l o n g - t e r m rel a x a t i o n o f the m a g n e t i c m o m e n t u m [32] i n d i c a t e the existence o f m e t a s t a b l e a n y o n - s p i n o n a n d any o n - h o l o n states d u e to cusps in the c o r r e l a t i o n energy o f the a n y o n i n t e r a c t i o n w i t h the f o r m a t i o n o f a gap system [33 ] a n d a h i e r a r c h y o f relaxation t i m e s in a strong statistical m a g n e t i c field. I a m t h a n k f u l to G.E. V o l o v i c k for r e f e r e n c e to the papers [ 17,18 ] a n d useful c o m m e n t s . I also wish to t h a n k P.B. W i e g m a n n , L.V. K e l d y s h , D.A. K i r z h n i t s a n d A.M. P o l y a k o v for helpful d i s c u s s i o n a n d A.V. G a p o n o v - G r e k h o v for s t i m u l a t i n g interest.
References [ 1 ] P.W. Anderson, Science 235 (1987) 1196; G. Baskaran, Z. Zou and P.W. Anderson, Solid State Commun. 63 (1987) 973. [2] S. Kivelson, J. Sethna and D. Rokhsar, Phys. Rev. B 35 (1987) 8865. [3 ] A. Ruckenstein, P. Hirschfeld and J. Appel, Phys. Rev. B 36 (1987) 857. [4] I.E. Dzyaloshinskii, A.M. Polyakov and P.B. Wiegmann, Phys. Lett. A 137 (1988) 112. [5] P.B. Wiegmann, Phys. Rev. Lett. 60 (1988) 821. [6 ] X.G. Wen and A. Zee, Phys. Rev. Lett. 62 ( 1989 ) 1937. [7] X.G. Wen, F. Wilczek and A. Zee, Phys. Rev. B 39 (1989) 11413. [8] L.B. Ioffe and A.I. Larkin, preprint ( 1988); X.G. Wen and A. Zee, Phys. Rev. Lett. 61 ( 1988 ) 1025; F.D.M. Haldane, Phys. Rev. Lett. 61 (1988) 1029; E. Fradkin and M. Stone, Phys. Rev. B 38 ( 1988 ) 7215; T. Dombre and N. Read, Phys. Rev. B 38 ( 1988 ) 7181.
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[9] I.J.R. Aitchison and N.E. Mavromatos, Phys. Rev. B 39 (1989) 6544. [ 10 ] S.M. Catterall and J.F. Wheater, Phys. Lett. B 213 (1989) 186. [ 11 ] S.M. Girvin, in: The quantum Hall effect, eds. R.E. Prange and S.M. Girvin (Springer, Berlin, 1986). [12]S.M. Girvin and A.H. MacDonald, Phys. Rev. Lett. 58 (1987) 1252. [13]M.H. Friedman, J.B. Sokoloff, A. Widom and Y.N. Srivastava, Phys. Rev. Lett. 52 (1984) 1587. [ 14 ] A.M. Polyakov, Mod. Phys. Lett. A 3 ( 1988 ) 325. [ 15 ] P.B. Wiegmann, Landau Institute preprint ( 1988 ). [ 16 ] D.V. Khveshchenko, Y.I. Kogan and P.B. Wiegmann, to be published. [ 17 ] R.B. Laughlin, Phys. Rev. Lett. 60 ( 1988 ) 2677. [ 18 ] R.B. Laughlin, Science 242 ( 1988 ) 525. [ 19] F. Wilczek, Phys. Rev. Lett. 48 (1982) 1144; F. Wilczek and A. Zee, Phys. Rev. Lett. 51 ( 1983 ) 2250. [20] Y.S. Wu, Phys. Rev. Lett. 53 (1984) 111. [21 ] D. Hofstadter, Phys. Rev. B 14 (1976) 2239. [22] M. Kohmoto, Phys. Rev. B 39 (1989) 1193. [23] S.N. Artemenko, I.G. Gorlova and Yu. I. Latyshev, Phys. Lett. A 138 (1989) 428. [24] V.L. Berezinskii, Zh. Eksp. Teor. Fiz. 61 (1971 ) 1144; J.M. Kosterlitz and D.J. Thouless, J. Phys. C 5 (1972) L124. [25] Y.J. Uemura et al., Phys. Rev. Lett. 62 (1989) 2317. [26] A.P. Protogenov, Phys. Lett. A 142 (1989 ) 285. [27] V. Kalmeyer and R.B. Laughlin, Phys. Rev. B 39 (1989) 11879. [28] P. Chandra, P. Coleman and A.I. Larkin, preprint RU-8918. [29] M. Naito, D.P.E. Smith, M.D. Kirk, B. Oh, M.R. Hahn, K. Char, D.B. Mitzi, J.Z. Sun, D.J. Webb, M.R. Beasley, O. Fischer, T.H. Geballe, R.H. Hammond, A. Kapitulnik and C.F. Quate, Phys. Rev. B 35 (1987) 7228. [30] P.L. Gammel et al., Phys. Rev. Lett. 59 (1982) 2592. [31 ] R.A. Butera, Phys. Rev. B 37 (1988) 5909. [32] V.P. Khavronin, I.D. Luzyanin and S.L. Ginzburg, Phys. Lett. A 129 (1988) 399. [ 33 ] B.I. Halperin, Phys. Rev. Lett. 52 (1984) 1583; Helv. Phys. Acta 56 (1983) 75.
PHYSICS LETTERS A
5 March 1990
G R O U N D STATE S T R U C T U R E A N D E X C I T A T I O N S IN STRONGLY CORRELATED FERMI SYSTEMS A.P. P R O T O G E N O V Institute of Applied Physics, Academy of Sciences of the USSR, 46 Uljanov Street, 603600 Gortcy, USSR
Received 13 January 1989; revised manuscript received 15 December 1989; accepted for publication 27 December 1989 Communicated by A.A. Maradudin
The ground state structure and some excitation properties of the (2 + 1)-dimensional strongly correlated Fermi systems are discussed. Quasi-classical configurations of the fields with finite energy and fractional charge of quasi-particles are described. It is shown that a holon and a spinon are different states of an anyon.
A deep insight into the properties o f new phases [ 1-5] in strongly correlated Fermi systems stimulates the unification of various types of their elementary excitations. The 2 + 1 dimensions and the delocalization of excitations in a m o m e n t u m space produced by a strong coupling mostly determine the soliton-like excitations. The study of quasi-particles enables one to get information on the v a c u u m structure in the low-energy region. The present paper makes an attempt to investigate possible types of elementary excitations. I consider, mainly, the excitations at temperatures exceeding the superconducting transition one. The details will be published elsewhere. The configurations o f the field contributing to the partition function are studied in the quasi-classical approximation in a continuum limit taking into account topological characteristics o f the system. Strongly interacting Fermi particles are described with a well-known Hubbard model with strong (compared to a hop t) Coulomb repulsion U on the lattice site. The Hubbard Hamiltonian projected on the states of the low zone can be written as a Heisenberg Hamiltonian with additional terms describing the hole transport [ 1-3 ] and the nontrivial topology [4,5]. In the absence of the external electromagnetic field the action in the low-energy limit has the form [5]
S= f dvd~x( ~=½Jl(Ou+iAu)zkl2
i.
- 4n e~"XAuFvz
)"
(1)
The complex doublet z = (~) is related to the antiferromagnetic order parameter n (x, y, r) as follows: n = z +az. Here cr is the Pauli matrix and z is the Matsubara time. The restriction n2= 1 is equivalent to the condition IZl I2 71_122 I2 = 1. The statistical gauge field A~ compensates the arbitrary local choice z ~ z ' = z e x p [ i A (x) ] o f the additional freedom degrees of the z-field compared to the n-field. The gauge field of the CP ~representation at small scales is composite (Au = - iz + 0 uz). We shall consider the longwavelength limit and the states [ 5 ] where the gauge potential is an independent variable. In the longwavelength case the terms of the self-energy type F ~ are essential at smaller scales [6]. The term responsible for the hole transport in eq. ( 1 ) is omitted since the doping is assumed to be small (t~ << t 2/U). The temporal component of the statistical gauge potential is usually employed to take into account [7 ] spin density fluctuations near half-filling o f the site while the spatial components describe the space structure of frustrations. The parameter v in eq. ( 1 ) is the filling factor, i.e. the dimensionless density of frustrations. The last term in eq. (1) is the U( 1 ) Chern-Si269
V o l u m e 144, n u m b e r 4,5
PHYSICS LETTERS A
mons term reflecting the Bohm-Aharonov interaction. Its nonzero value in the continuous limit characterizes the so-called [7 ] Chiral spin state. If the ground state is an antiferromagnetic one, the ChernSimons term is absent [8 ]. Such a term occurs due to doping [ 9 ], is produced by a random lattice [ 10 ] and reflects [7,9] three-particle correlations initiated by the next-nearest-neighbor coupling. It should be noted that the Chern-Simons term is odd with respect to 2D parity P and time reversal T. This macroscopic violation of discrete symmetries is concealed by multilayer 3D systems where the signs of v in the neighboring layers may alternate in the equilibrium state. Let us consider the long-wavelength space structure of elementary excitations of the model ( 1 ). We imagine that we have broken the local antiferromagnetic order by removing a particle (with its spin and charge) from the pair. This means that in the test spin location n 2 # 1, or [z[ 2 ¢ 1. The violation of the condition [z[ 2= 1 can be taken into account by introducing a spin source in the form ofipAo into eq. ( 1 ), with the Lagrange multiplier iA0 and the density p = [z[ 2_ 1 describing the deviation from the local antiferromagnetic order. The functional (1) with such a source for the scalar order parameter at the quasi-classical level was discussed in relation with a fractional quantum Hall effect [ 11,12 ]. The equations of motion in the static case (0k+iAk)2Z~=0,
(2)
1/
4~ (VXA)± = IZl [2+ Iz212- 1,
(3)
yield the following field configurations •
Zk(r, ~0)=O(r--rm) exp0m~o)~ sin 2gr A~,(r) = - - u
m 2gr '
r
'
,
(4) (5a)
m
2nr'
r > rm,
(5b)
with zero action and the fractional excitation "charge" e* = ~. The expression (3) means the neutralization condition of the spin density Iz l 2_ 1 by the vacuum density (u/4n) ( V × A ) . . Let us stress again that for the sake of simplicity the external elec270
5 M a r c h 1990
tromagnetic gauge field is assumed to be absent (cf. ref. [7] ). In the expression (4) O(r-rm) is the step function, rm 2 = vrnlm 2 lH=H -- 1/2 is the magnetic length and Hz= ( V × A ) l is the statistical magnetic field. The expressions (4), (5) describe the vortex with a long-range Bohm-Aharonov potential (5b) outside the elementary vortex and the ferromagnetic order at r < rm,
n= (O2(r--rm) sin 2a, 0, O2(r--rm)
cos 2o~) .
(6)
Thus the expression (6) describes the ferromagnetic polaron induced by the test spin (doping). At the surface density n~1013-1014 cm -2 the statistical magnetic field is H ~ n¢~o~ 106_ 10 v e where ~o is the flux quantum. The vortex scale l~i~ 10-7 cm and the distance n-1/2 between vortices are of the same order. The fractional charge e * = v , where v=O/2== q/p= n × 2rd 2 is the filling factor of the degenerate ground state, is well known in relation to the fractional quantum Hall effect. Here 0 is the vacuum angle, q is the number of particles, the odd denominator p = 2 ( l + ½) [ 13 ] is the number of flux quanta, l is the orbital momentum, and n is the surface density. In our case the filling factor v = 0/4~ =p/2q, where p is the number of flux quanta of the statistical magnetic field and q is the number of particles. The multiplier ½ indicates that two orientations are possible. To avoid misunderstanding we note that our vacuum angle 0 coincides with the angle O~ in ref. [6]. The vacuum angles 0p and 0~ in the vortex picture (expressed in the z-field terms) and in the n-field picture (6), respectively, are dually conjugate [ 6,14 ]: 0 ~ = - ~ 2 / 0 ~ . As shown in refs. [15,16] 0 = 2 n , i.e. p = q= 1 and the filling factor u = ½ (cf. refs. [ 17,18 ] ). The 4n periodicity in the vacuum angle 0 is due to spontaneous time reversal violations in the chiral spin state [7 ]. The gauge configurations (5b) were introduced by Wilczek [ 19 ] and are called anyons. An anyon, as a particle, represents a vortex with several flux quanta and charges rotating around it. In the singular gauge when the potential outside the vortex is zero and the two anyons are interchanged the wave function acquires a phase: ~((2, ~1) = exp(iO~)~(~l, ¢2),
0~ =
nq
2p"
(7)
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PHYSICSLETTERSA
The ratio 0 ~ = - ~ 2 / 0 a is used here [6,14]. This means that at p/q= 1 an anyon, as a quasi-particle, is a half-fermion [7,17] obeying the fractional statistics [ 19,20 ]. Therefore the picture of excitations with a collective long-range gauge Bohm-Aharonov interaction is equivalent [19,20] to the picture of free excitations with intermediate statistics in the singular gauge. When dealing with anyon-antianyon pairing one should bear in mind that the vector potential of the anyon pair written in relative variables is equivalent to the charge interacting with a double flux in the single-flux field of the anyon [ 20 ]. This fact yields the necessary 2e-rule for Fermi confinement of anyons in the "normal" state and Bose condensation of anyon pairs in the superconducting state. In fact, when measuring the number of quanta in units ch/2e the equality 2p= 1 is valid and at q = 1 an anyon-antianyon pair is a spinon while at q = 2 the anyon pair becomes a holon. On the other hand, the value of q yields the number of bands of a Hofstadter particle [21 ] for the rational value p/2q of the flux in the statistical magnetic field. It is known [22] from the quantum Hall effect study that different bands make different sign contributions to the Hall coefficient. Thus, when the temperature decreases and the chemical potential passes through the center of the spectrum the hole-like Hall coefficient changes to the electron-like one. This property may be related to the results of the recent Hall measurements [23] where the Hall coefficient changed its hole-like sign for the electron-like one at the temperature To usually called the onset of the superconducting state transition. According to (7) the q jump from q= 1 to q = 2 at this temperature corresponds to the phase transition with change of the anyon pair statistics from the Fermi case to the Bose one. In other words, at T = To there occurs a phase transition into a nonuniform ground state with a smaller number of cells with a nonzero flux of the statistical magnetic field on the underlying lattice. In the singular gauge of interest the action at large distances is logarithmically large, R
S ~ ~ rdr [ (Oz/Or)2-k - IzlZ(Ao+m/2nr) a] 0
~mZlnR,
(8)
5 March 1990
i.e. a temperature decrease may cause a KosterlitzThouless transition from the Coulomb (plasma) phase of single vortices (Fermi-anyon pairs and fluctuation Bose-anyon pairs) existing in the Tc < T < To region into the superconducting state where elementary excitations are the Kosterlitz-Thouless pairs of vortices (perhaps Bose-anyon fours). The Kosterlitz-Thouless phase transition accompanied with a Kosterlitz-Nelson jump has been observed in ref. [23]. In the mean statistical magnetic field H~-nOo the energy of stationary states is equal to
eN=hO)c(N+ 1 ) = n J ( 2 N + 1 ) = nkaT( ZN+ l ) .
(9)
The values of the m a s s M ~ h Z / J a 2 and density na z~- 1 are used here, where a is the lattice constant. The characteristic temperature Tc = 3"/kB = 102 K in the spectrum of the Matsubara frequencies is, evidently, the temperature of Bose condensation of anyon-antianyon vortices. The relation (9) Tc ~ n / M is yielded by the recent experimental studies [25 ] of local magnetic fields by means of ~t-muon spin relaxation. Further ordering produced by the temperature decrease below Tc can be due to the Wigner crystallization of the vortex structure. The complete scenario of phase transitions is presented in ref. [26 ]. The energy o f quasi-particle motion td increases with doping and at t(~>~t 2 / U ~ J condensate anyons will effectively decay into spin-roton excitations [27,28]. These collective modes in the anyon medium possess a roton part of the spectrum at a finite wavelength of the order of lH and a gap of the order of J as well as a finite gap (larger than J) existing in the long-wavelength region due to the incompressibility condition. A reduced number of condensate Bose anyons decreases the critical temperature with the doping increase. What is the relation between the phenomena considered and the experimental results obtained? I compare only some of them. (1) When studying tunneling in SIN-contacts violation of charge antisymmetry of the C - V characteristics at the substitution V~ - V was found [ 29 ]. (2) It has been found that at helium temperatures a lattice of vortices is present even in weak magnetic fields but at nitrogen temperatures it is absent [30]. (3) Two types of closely located phase transitions were discovered 271
Volume 144, number 4,5
PHYSICS LETTERS A
w h e n s t u d y i n g the h e a t c a p a c i t y [ 3 1 ] . (4) I n v e s t i gations o f n o n e r g o d i c b e h a v i o u r w i t h l o n g - t e r m rel a x a t i o n o f the m a g n e t i c m o m e n t u m [32] i n d i c a t e the existence o f m e t a s t a b l e a n y o n - s p i n o n a n d any o n - h o l o n states d u e to cusps in the c o r r e l a t i o n energy o f the a n y o n i n t e r a c t i o n w i t h the f o r m a t i o n o f a gap system [33 ] a n d a h i e r a r c h y o f relaxation t i m e s in a strong statistical m a g n e t i c field. I a m t h a n k f u l to G.E. V o l o v i c k for r e f e r e n c e to the papers [ 17,18 ] a n d useful c o m m e n t s . I also wish to t h a n k P.B. W i e g m a n n , L.V. K e l d y s h , D.A. K i r z h n i t s a n d A.M. P o l y a k o v for helpful d i s c u s s i o n a n d A.V. G a p o n o v - G r e k h o v for s t i m u l a t i n g interest.
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