- Email: [email protected]
Evidence for a chiral anomaly in solid state physics
Volume 113A, number 6
PHYSICS LETTERS
30 December 1985
E V I D E N C E F O R A C H I R A L A N O M A L Y IN S O L I D S T A T E P H Y S I C S I.V. K R I V E and A.S. R O Z H A V S K Y Physical-Technical Institute of Low Temperatures, Ukrainian Academy of Sciences, 47, Lenin Avenue, Kharkov 310164, USSR Received 5 October 1985; accepted for publication 21 October 1985
The role of chiral anomaly in solid state physics is considered. Using the field theoretical model of the Peierls insulator we show that a well-known experimentallyverified phenomenon, the charge density wave in quasi-one-dimensionalconductors, is closely connected with the chiral anomaly in two-dimensional space-time. A particular consequence of this fact which is related to the effect of charge fractionalization in condensed matter is also discussed.
One of the most interesting phenomena in quantum field theory, the chiral anomaly [ 1], has had an appreciable influence on the modern development of high energy physics (see, for example, ref. [2]). Recently in ref. [3] the role of the chiral anomaly in solid state physics was examined for the first time. In particular, the current state in gapless semiconductors which is the analog of the Adler-Bell-Jackiw anomaly was predicted. It is shown below that there exists a well-known experimentally confirmed phenomenon in solid state physics, the charge density wave in quasi-one-dimensional conductors [4,5], which is closely connected with chiral anomaly in two-dimensional space-time
[6,71. It is well known that a one-dimensional metal chain is unstable at low temperatures respective to structure (Peierls) transition which changes an original lattice period and forms an energy gap on the Fermi level in the conduction electron spectrum. The lattice reconstruction involves the redistribution of the electron density. The charge density modulation with the new superstructure period is called Fr6nlich charge density wave (CDW). The microscopic description of the Peierls phase transition usually starts from the Fr6nlich electronphonon hamiltonian
k,o + N -1/2
q ~
ga~+qoaka(b q + b + q ) .
(1)
k,q ,a H e re a k+ (ak) and bq+ (bq) are the electron and phonon creation (destruction) operators; ek and 6Oq the electron and phonon dispersion laws; o the spin projection index;g the electron-phonon coupling constant; N the total number of sites in the chain of length L ((7 = L / N is the intersite distance). The traditional investigation of the Peierls transition based on the thermodynamics of hamiltonian (1) is presented for example in refs. [4,5]. The modern treatment of the Peierls state uses the powerful methods of soliton theory and of quantum field theory (QFT) and predicts such unusual phenomena as the existence of solitons, nonlinear dynamics of CDW in electromagnetic field, effects of charge fractionalization and so on (for a review of experimental data, see refs. [8,9]). There is a deep analogy between the one-dimensional models of QFT and the continuum model of PI [ 10-12]. The hamiltonian (1) is reduced to this level of description after averaging over atomic distances and linearizing of the electron spectrum in the vicinity of the Fermi level. Remind that PI is characterized by the complex order parameter [5] A = IAI exp(i~0) ~ N -1/2 (b2kv). 313
Volume 113A, number 6
PHYSICS LETTERS
Its modulus is the gap on the Fermi level in the electron spectrum (in the forthcoming analysis we shall put the Fermi energy equal to zero). This gap separates the occupied valence band from the empty conduction band and the phase ~ogoverns the large-scale variations of the chemical potential. The mechanisms of collective CDW-conductivity are connected with phase dynamics. If the dimensionless coupling constant ~ ---g2/ rr~ VF < 1 the Peierls gap is much less than the band width IAI ,~ W; the electron spectrum may be linearised near k ~- k F :
E(k) = -+(e2 + [AI2) 1/2 =*'E(p)=+(p2 +lA[2)l/2,
p=k-k
F.
(2)
Here ek = - W cos ka is the energy of bare (metallic) electrons and we use the units ti = 1, VF = Wa = 1. After change (2) the electron sector of the PI model takes a form typical for one-dimensional models of relativistic QFT:
£e = CI'a i'Yuauada •
(3)
Here a,, ( ~ = ~ + 7 0 ) is the two-component relativistic spinor in 1 + 1-dimensional space-time describing two degrees of freedom (particle-hole) and the spin of the real electron is accounted via "isotopic" doubling of components o = 1, 2. The interaction of electrons with a complex order parameter leads to a mass term in the Dirac lagrangian (3). It is easy to construct the symmetry-consistent form of interactions of Dirac electrons • with the order parameter: £eA = - A 1 ~c,q*c, - iA2~075~1'.,
(5)
AS a result the lagrangian of PI with so called incommensurate CDW is (see also ref. [10])
314
" 2 /g 2 GO2k 2 F --IAI2/g 2 ,~ = Ia{
+ C~a{iTua u - A 1 - i75A2}q%.
(6)
Without the first (kinetic) term the lagrangian (6) coincides with the quasiclassical lagrangian of the U(1) × U(1) chiral-invariant Gross-Neveu model (GNM) [13]. Lagrangian (6) has two time scales: an electronic one t e ~ [A0[-1 [A 0 is the equilibrium gap, A 0 ~ W e x p ( - 1 / k ) ] and the lattice one t L (gW2kF)-1. The small ratio te/t L ~ g W 2 k . / A o "< I guarantees the adiabatic approximation (or PI (the classical nature of field A). Now we rewrite the lagrangian (6) in terms of modulus and phase and introduce the electromagnetic field Fur = BuA u - OvA u in a standard way Ou -+ D u
= a u - ieAu: £ = IAI2/g2W2kF -- IAI2/g 2 + IAI2~o2/g2¢.,O2kF + ~aiTuDuxI'a - IAl~aei'rs s°xlso .
(7)
The effective lagrangian of the model depending only on the order parameter and electromagnetic field can be obtained from (7) after integration over Fermivariables:
d2x £eff= - i ln f D~oDxI'a exp(i f d2x £ ) .
(8)
We shall show that the phase-dependent effective lagrangian (the CDW lagrangian) is easily extracted from (8) in an adiabatic approximation. For this purpose we introduce new variables (chiral transformation), ~ o = exp[-½i 75~(x, t)]~o ,
(4)
where A 1 = IAI COS ~, A 2 = IAI sin ~0 are real and imaginary parts of the order parameter, 75 = "Y071The kinetic and potential energy of the Peierls lattice in harmonic approximation can easily be derived from the phonon terms in (1) leaving only contributions with q = 2k F (phonon condensate),
£A = IAl2/g2c°2k F --IAI2/g 2 •
30 December 1985
Xa exp [-71 ')'5~0(X, t)] .
(9)
Then the fermionic part of lagrangian (7) takes the form
+ ½"~o3,uT~aOu~o.
(10)
The last term in (10) does not contribute to the effective lagrangian (8) if the gradients of ~0are small compared to A. Indeed in 1 + 1 dimensions the matrices 7 u obey the relation 7u~,5 = eU~'%, (e uv is an antisymmetric tensor). Therefore the axial-vector coupling in
Volume 113A, number 6
PHYSICS LETTERS
(1 0) can be presented as j u A (5) where Ju is the ordinary fermion current and A (5) = ½'e uO0¢ thevector field. After the supplementary change of variables in the path integral (8),~ o = exp(i f dx~ A(5)Xo , the currentfield interaction ]UA(u5) takes the form
i"A(u5)=~o'rUXof a.A~5)dxv (U~v),
(11)
i.e, the vector fields are presented only by their gradients. For weakly inhomogeneous ~0the expansion (11) starts from ~0",~ and its contribution to the effective lagrangian can be omitted comparatively to the contribution of the chiral afiomaly (see later). So the effective lagrangian is
£x = [AI2/gEW2kF -- IAl2/g2 +Xo(iTuOu - IAI)×° + [Al2(o2/g2~O2kv.
(12)
However, it is well known that the functional measure in (8) d/z - D~oD,I, o is not invariant under chiral rotations (9) (the chiral anomaly) [ 14]. The jacobian
is [6] J = exp(i
fd2x[(1/41r)(au~o)2-(e/2~r)eUVFuv~o])" (13)
30 December 1985
£(~) = (IAI2/g2eO~kF) (~b2 - c02(~')2) - (e/rr)E~, (16) where c o =gW2kF/2nl/21AI is the CDW velocity [15]. Within the phenomenological approach the lagrangian (16) was studied in ref. [16]. In ref. [17] the CDW lagrangian (in the absence of E) was also obtained within the path integral method but in a perturbation series over the last term in (10). As we have shown (see also ref. [18]) in our approach this interaction contributes to the effective lagrangian only terms proportional to higher order derivatives: (¢")2 and so on. We point once more that both the spatial phase gradients and the interaction of phase with the electric field appear in £(~) owing to chiral anomaly. Due to the topological origin of the chiral anomaly the coefficients attached to these terms in (17) are universal. This important property explains the unusual (topological) form of CDW current [1 5]. Indeed from the last term in (16) one can see that the CDW charge density is PCDW = -(e/Tr)~' and the current density is correspondingly/'CDW = (e/zr~. Thus the density of phase current has the form jCDWu =
The expression in the exponent of (15) is twice the standard one, because of the additional summation over electron spin (isotopic spin in QFT-model). Taking into account the chiral anomaly we obtain from (8), (12), £eff = £(e~l) + £(~t, where
-(e/n) euv~v¢ ,
typical for topologically conserved currents aujCDW -----0 and the net CDW charge is independent of the local phase variations (the topological charge), oo
QCDW = f
fd x
--f
= IAI 2 ¢ 2 / g 2 ~ k v
dx OCDW =-(e/zr)A¢,
--oo
A~p = ~(~) _ ~(_oo).
- i In Det(iTuDu - IAD ,
~t
(17)
(14)
-- (1/4.) (~,)2 _ (e/.)E~,
(15) where E is the electric field. We have omitted the "fermion-origin" time derivatives of phase in (15) because they are less in 6O2kF/Ao ~ 1 times of analogous terms of the lattice origin. The effective lagrangian (14) describing the Peierls insulator in an electromagnetic field was carefully investigated for the case of a homogeneous electric field in ref. [ 12]. Eq. (15) defines the lagrangian of incommensurate CDW in an electric field,
(18)
The direct connection of eqs. (17), (18) with chiral anomaly is evident from the following considerations. In accordance with the general formulae of QFT the charge induced in the vacuum by the external field A~)5) = ~¢ 1 , [see lagrangian (10)] is t~o
Q=f
dx ~Seff/SA~ 5) .
(19)
--oo
Here S e fr is the total effective action (8) including the chiral anomaly (13). It is easy to see that the charge (19) can be broken into parts Q = Qpol + QtOp, where Qpol is the charge arising from the vacuum polarization by an external electric field, and QtOp the topological charge connected with chiral anomaly (1 3). 315
Volume 113A, number 6
PHYSICS LETTERS
For weak inhomogeneous phase fluctuations, I~'1 ,~ A0, polarization effects are small as we have seen and the total charge appearing due to the interaction of order parameter A = A0ei¢ with occupied band ("vacuum" electrons) is governed by global phase variations o0
Q=f
dx ~Seff/~A~ 5) ~_ a t o p = -(e/Tr)Atp.
(20)
--o0
This vacuum charge is also the charge of the charge density wave (CDW) in quasi-one-dimensional conductors. In the continuum model of the Peierls insulator without taking into account the effects of lattice periodicity, the CDW is the Goldstone mode arising from the spontaneous U(1) symmetry breaking. This actually is the case if the period of CDW is incommensurate with the lattice period and the other effects of CDW pinning are negligible. Then the moving of CDW is activationless and the phase difference A¢ is not fixed. Really there is always pinning of the CDW. In some quasi-one-dimensional conductors (NbSe 3, TaS 3, etc.) the correlation of CDW period with the lattice period a is the main reason of the pinning. When the concen"tration of conduction electrons is kFa = 7rl/M (l < M, l and M are the reciprocals, M > 2, the commensurability index of CDW), the lagrangian of CDW must be supplemented by an extra term, the energy of commensurability [ 15] : 6£ c(~) = (/~/rrX) A 2 cosM~o orn //~ (A/W~- 2 ~ 1 .
(21)
When E = 0 the lagranglan (16), (21) is the wellknown sine-Gordon lagrangian. The topologically stable soliton solutions can fix the phase difference A 9 = 27riM. Thus we tend to the conclusion that there can exist collective excitations in CDW carrying fractional charge [Qs[ = 2e]M [16]. It is necessary to make one significant note here. The CDW in consideration is the collective degree of freedom describing the dynamics of filled band electrons. Therefore naively the charge of CDW solitons is delocalized because it is composed from the charge of all the electrons of the occupied valence band. The nontriviality of the situation consists in the following; despite of its collective origin the interaction of CDW 316
30 December 1985
with an electric field is local due to chiral anomaly (13) and the CDW solitons behave in an electric field like ordinary particles having a charge form factor. In other words in the fields with inhomogeneity scale more than the soliton size d the solitons can be treated as point-like particles. It is of special interest to consider chiral anomaly in PI with double commensurability. An example of such a system is the conducting linear polymer transpolyacetylene [trans-(CH)x ] (see the detailed discussion in ref. [8]). The CDW withM = 2 is described b) GNM [13] (see also refs. [11,12]) with real order parameter -+A (the commensurability M = 2 fixes the phase ¢ = 0, 7r). That is why the local consequence of chiral anomaly, the phase dynamics in trans-(CH)x, is absent. However the relation (20) survives because of its global (topological) nature. So the question arises what is the object whose charge is defined by the expression (20)? In the GNM with real order parameter two types of inhomogeneous static solutions exist: topologically stable solitons (kinks) &s(X ~ -+~) = +A 0 and polarons Ap(X -~ +co) = A 0 [19]. The kink fixes the global phase difference A¢ = 7r (antikink, A9 = --r0. That is, according to (20), a kink has topological charge Qs = - e and an antikink, Qs = e. At the same time the spin of charge solitons is zero because we have summarised over index o when deriving (15), (20) (for spinless "electrons" the well known Jackiw-Rebbi result Qs = -+e/2 [20] is valid). These topological charges are formed by all the electrons of the occupied valence band and thus they are formally speaking delocalized (see details in ref. [10]). The kink extracts one localized level in the centre of the energy gap (i.e. with zero energy) from the continuum of the Dirac "sea" levels. From the electroneutrality condition it follows that the collective (topological) charge has to be compensated by the charge of the electron extracted from the occupied band to local level. Such compensation on the distance 1 > d (d ~ A~ 1 is the size of kink) is possible because of the local coupling of topological charge density to the electric field. Thus, the kink turns to be neutral but nonzero spin 1/2 appears [21]. One can occupy the zero-energy local level by the additional charge-electron (hole) with the opposite spin-projection when doping the polymer. Then kink becomes charged but spinless [21]. The unusual spin-charge relation in trans-polyacetylene is experi-
Volume 113A, number 6
PHYSICS LETTERS
mentally confirmed when measuring magnetic susceptibility and dc-conductivity in the doping process
[81. The same considerations state that for polarons with zero topological charge the spin-charge relation is normal: polarons are charged (+e) and have spin 1/2. Thus we have shown that the CDW itself, its electromagnetic properties and the unusual quantum numbers in Peierls-Fr6nlich systems are of the chiral anomaly origin. This fact demonstrated once more the unexpected and interesting connections between solid state physics and modern quantum field theory. The authors thank P.I. Fomin and V.A. Miransky for the valuable discussions.
References [1] J. Schwinger, Phys. Rev. 82 (1951) 664; S.L. Adler, Phys. Rev. 177 (1969) 2426; T.S. Bell and R. Jackiw, Nuovo Cimento A60 (1969) 47. [2] R. Jackiw, in: Lectures on current algebra and its applications, eds. S. Treiman, R. Jackiw and D. Gross (Princeton Univ. Press, Princeton, 1972). [3] H.B. Nielson and M. Ninomiya, Phys. Lett. 130B (1983) 389. [4] L.N. Bulaevskii, Usp. Fiz. Nauk 115 (1975) 261. [5] G.A. Toombs, Phys. Rep. 40 (1978) 182. [6] K. Jonson, Phys. Lett. 5 (1963) 253.
30 December 1985
[7] R. Roskies and F. Shaposnik, Phys. Rev. D23 (1981) 558. [8] A.J. Heeger and A.G. MacDiarmd, Chem. Scr. 17 (1981) 115. [9] G. Grinner,Comm. Solid State Phys. 10 (1983) 183. [10] S.A. Brazovsky, Zh. Eksp. Teor. Fiz. 78 (1980) 677; S.A. Brazovsky and N.N. Kirova, Chem. Scr. 17 (1981). [11] R. Jackiw and J.S. Schrieffer, Nucl. Phys. B190 (1981) 253. [12] I.V. Krive and A.S. Rozhavsky, Solid State Commun. 39 (1981) 427; Zh. Eksp. Teor. Fiz. 85 (1983) 1122. [13] D.J. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235. [14] K. Fujikawa, Phys. Rev. D21 (1980) 2848. [15] P.A. Lee, T.M. Rice and P.W. Anderson, Solid State Commun. 14 (1974) 703. [16] M.J. Rice, A.R. Bishop, J.A. Krumhansl and S.E. TruUinger,Phys. Rev. Lett. 36 (1976)432; K. Maki, Phys. Rev. Lett. 39 (1977) 46; I.V. Krive and A.S. Rozhavsky, Fiz. Niz. Temp (Soy. Phys. J. Low Temp.) 6 (1980) 1272. [17] S.A. Brazovsky and I.E. Dzylashisky, Zh. Eksp. Teor. Fiz. 71 (1976) 2338. [18] W.A. Bardeen, S. Elitzur, Y. Frishman and E. Rabinovici, Nucl..Phys. B218 (1983) 445; A.Z. Capri, R. Ferrari and L.E. Picasso, Phys. Rev. D30 (1984) 2136. [19] R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D13 (1976) 3398. [20] R. Jackiw and C. Rebbi, Phys. Rev. D13 (1976) 3398; G. Goldstone and F. Wilczek, Phys. Rev. Lett. 48 (1981) 986. [21 ] W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698; M.J. Rice, Phys. Lett. 71A (1979) 152.
317
PHYSICS LETTERS
30 December 1985
E V I D E N C E F O R A C H I R A L A N O M A L Y IN S O L I D S T A T E P H Y S I C S I.V. K R I V E and A.S. R O Z H A V S K Y Physical-Technical Institute of Low Temperatures, Ukrainian Academy of Sciences, 47, Lenin Avenue, Kharkov 310164, USSR Received 5 October 1985; accepted for publication 21 October 1985
The role of chiral anomaly in solid state physics is considered. Using the field theoretical model of the Peierls insulator we show that a well-known experimentallyverified phenomenon, the charge density wave in quasi-one-dimensionalconductors, is closely connected with the chiral anomaly in two-dimensional space-time. A particular consequence of this fact which is related to the effect of charge fractionalization in condensed matter is also discussed.
One of the most interesting phenomena in quantum field theory, the chiral anomaly [ 1], has had an appreciable influence on the modern development of high energy physics (see, for example, ref. [2]). Recently in ref. [3] the role of the chiral anomaly in solid state physics was examined for the first time. In particular, the current state in gapless semiconductors which is the analog of the Adler-Bell-Jackiw anomaly was predicted. It is shown below that there exists a well-known experimentally confirmed phenomenon in solid state physics, the charge density wave in quasi-one-dimensional conductors [4,5], which is closely connected with chiral anomaly in two-dimensional space-time
[6,71. It is well known that a one-dimensional metal chain is unstable at low temperatures respective to structure (Peierls) transition which changes an original lattice period and forms an energy gap on the Fermi level in the conduction electron spectrum. The lattice reconstruction involves the redistribution of the electron density. The charge density modulation with the new superstructure period is called Fr6nlich charge density wave (CDW). The microscopic description of the Peierls phase transition usually starts from the Fr6nlich electronphonon hamiltonian
k,o + N -1/2
q ~
ga~+qoaka(b q + b + q ) .
(1)
k,q ,a H e re a k+ (ak) and bq+ (bq) are the electron and phonon creation (destruction) operators; ek and 6Oq the electron and phonon dispersion laws; o the spin projection index;g the electron-phonon coupling constant; N the total number of sites in the chain of length L ((7 = L / N is the intersite distance). The traditional investigation of the Peierls transition based on the thermodynamics of hamiltonian (1) is presented for example in refs. [4,5]. The modern treatment of the Peierls state uses the powerful methods of soliton theory and of quantum field theory (QFT) and predicts such unusual phenomena as the existence of solitons, nonlinear dynamics of CDW in electromagnetic field, effects of charge fractionalization and so on (for a review of experimental data, see refs. [8,9]). There is a deep analogy between the one-dimensional models of QFT and the continuum model of PI [ 10-12]. The hamiltonian (1) is reduced to this level of description after averaging over atomic distances and linearizing of the electron spectrum in the vicinity of the Fermi level. Remind that PI is characterized by the complex order parameter [5] A = IAI exp(i~0) ~ N -1/2 (b2kv). 313
Volume 113A, number 6
PHYSICS LETTERS
Its modulus is the gap on the Fermi level in the electron spectrum (in the forthcoming analysis we shall put the Fermi energy equal to zero). This gap separates the occupied valence band from the empty conduction band and the phase ~ogoverns the large-scale variations of the chemical potential. The mechanisms of collective CDW-conductivity are connected with phase dynamics. If the dimensionless coupling constant ~ ---g2/ rr~ VF < 1 the Peierls gap is much less than the band width IAI ,~ W; the electron spectrum may be linearised near k ~- k F :
E(k) = -+(e2 + [AI2) 1/2 =*'E(p)=+(p2 +lA[2)l/2,
p=k-k
F.
(2)
Here ek = - W cos ka is the energy of bare (metallic) electrons and we use the units ti = 1, VF = Wa = 1. After change (2) the electron sector of the PI model takes a form typical for one-dimensional models of relativistic QFT:
£e = CI'a i'Yuauada •
(3)
Here a,, ( ~ = ~ + 7 0 ) is the two-component relativistic spinor in 1 + 1-dimensional space-time describing two degrees of freedom (particle-hole) and the spin of the real electron is accounted via "isotopic" doubling of components o = 1, 2. The interaction of electrons with a complex order parameter leads to a mass term in the Dirac lagrangian (3). It is easy to construct the symmetry-consistent form of interactions of Dirac electrons • with the order parameter: £eA = - A 1 ~c,q*c, - iA2~075~1'.,
(5)
AS a result the lagrangian of PI with so called incommensurate CDW is (see also ref. [10])
314
" 2 /g 2 GO2k 2 F --IAI2/g 2 ,~ = Ia{
+ C~a{iTua u - A 1 - i75A2}q%.
(6)
Without the first (kinetic) term the lagrangian (6) coincides with the quasiclassical lagrangian of the U(1) × U(1) chiral-invariant Gross-Neveu model (GNM) [13]. Lagrangian (6) has two time scales: an electronic one t e ~ [A0[-1 [A 0 is the equilibrium gap, A 0 ~ W e x p ( - 1 / k ) ] and the lattice one t L (gW2kF)-1. The small ratio te/t L ~ g W 2 k . / A o "< I guarantees the adiabatic approximation (or PI (the classical nature of field A). Now we rewrite the lagrangian (6) in terms of modulus and phase and introduce the electromagnetic field Fur = BuA u - OvA u in a standard way Ou -+ D u
= a u - ieAu: £ = IAI2/g2W2kF -- IAI2/g 2 + IAI2~o2/g2¢.,O2kF + ~aiTuDuxI'a - IAl~aei'rs s°xlso .
(7)
The effective lagrangian of the model depending only on the order parameter and electromagnetic field can be obtained from (7) after integration over Fermivariables:
d2x £eff= - i ln f D~oDxI'a exp(i f d2x £ ) .
(8)
We shall show that the phase-dependent effective lagrangian (the CDW lagrangian) is easily extracted from (8) in an adiabatic approximation. For this purpose we introduce new variables (chiral transformation), ~ o = exp[-½i 75~(x, t)]~o ,
(4)
where A 1 = IAI COS ~, A 2 = IAI sin ~0 are real and imaginary parts of the order parameter, 75 = "Y071The kinetic and potential energy of the Peierls lattice in harmonic approximation can easily be derived from the phonon terms in (1) leaving only contributions with q = 2k F (phonon condensate),
£A = IAl2/g2c°2k F --IAI2/g 2 •
30 December 1985
Xa exp [-71 ')'5~0(X, t)] .
(9)
Then the fermionic part of lagrangian (7) takes the form
+ ½"~o3,uT~aOu~o.
(10)
The last term in (10) does not contribute to the effective lagrangian (8) if the gradients of ~0are small compared to A. Indeed in 1 + 1 dimensions the matrices 7 u obey the relation 7u~,5 = eU~'%, (e uv is an antisymmetric tensor). Therefore the axial-vector coupling in
Volume 113A, number 6
PHYSICS LETTERS
(1 0) can be presented as j u A (5) where Ju is the ordinary fermion current and A (5) = ½'e uO0¢ thevector field. After the supplementary change of variables in the path integral (8),~ o = exp(i f dx~ A(5)Xo , the currentfield interaction ]UA(u5) takes the form
i"A(u5)=~o'rUXof a.A~5)dxv (U~v),
(11)
i.e, the vector fields are presented only by their gradients. For weakly inhomogeneous ~0the expansion (11) starts from ~0",~ and its contribution to the effective lagrangian can be omitted comparatively to the contribution of the chiral afiomaly (see later). So the effective lagrangian is
£x = [AI2/gEW2kF -- IAl2/g2 +Xo(iTuOu - IAI)×° + [Al2(o2/g2~O2kv.
(12)
However, it is well known that the functional measure in (8) d/z - D~oD,I, o is not invariant under chiral rotations (9) (the chiral anomaly) [ 14]. The jacobian
is [6] J = exp(i
fd2x[(1/41r)(au~o)2-(e/2~r)eUVFuv~o])" (13)
30 December 1985
£(~) = (IAI2/g2eO~kF) (~b2 - c02(~')2) - (e/rr)E~, (16) where c o =gW2kF/2nl/21AI is the CDW velocity [15]. Within the phenomenological approach the lagrangian (16) was studied in ref. [16]. In ref. [17] the CDW lagrangian (in the absence of E) was also obtained within the path integral method but in a perturbation series over the last term in (10). As we have shown (see also ref. [18]) in our approach this interaction contributes to the effective lagrangian only terms proportional to higher order derivatives: (¢")2 and so on. We point once more that both the spatial phase gradients and the interaction of phase with the electric field appear in £(~) owing to chiral anomaly. Due to the topological origin of the chiral anomaly the coefficients attached to these terms in (17) are universal. This important property explains the unusual (topological) form of CDW current [1 5]. Indeed from the last term in (16) one can see that the CDW charge density is PCDW = -(e/Tr)~' and the current density is correspondingly/'CDW = (e/zr~. Thus the density of phase current has the form jCDWu =
The expression in the exponent of (15) is twice the standard one, because of the additional summation over electron spin (isotopic spin in QFT-model). Taking into account the chiral anomaly we obtain from (8), (12), £eff = £(e~l) + £(~t, where
-(e/n) euv~v¢ ,
typical for topologically conserved currents aujCDW -----0 and the net CDW charge is independent of the local phase variations (the topological charge), oo
QCDW = f
fd x
--f
= IAI 2 ¢ 2 / g 2 ~ k v
dx OCDW =-(e/zr)A¢,
--oo
A~p = ~(~) _ ~(_oo).
- i In Det(iTuDu - IAD ,
~t
(17)
(14)
-- (1/4.) (~,)2 _ (e/.)E~,
(15) where E is the electric field. We have omitted the "fermion-origin" time derivatives of phase in (15) because they are less in 6O2kF/Ao ~ 1 times of analogous terms of the lattice origin. The effective lagrangian (14) describing the Peierls insulator in an electromagnetic field was carefully investigated for the case of a homogeneous electric field in ref. [ 12]. Eq. (15) defines the lagrangian of incommensurate CDW in an electric field,
(18)
The direct connection of eqs. (17), (18) with chiral anomaly is evident from the following considerations. In accordance with the general formulae of QFT the charge induced in the vacuum by the external field A~)5) = ~¢ 1 , [see lagrangian (10)] is t~o
Q=f
dx ~Seff/SA~ 5) .
(19)
--oo
Here S e fr is the total effective action (8) including the chiral anomaly (13). It is easy to see that the charge (19) can be broken into parts Q = Qpol + QtOp, where Qpol is the charge arising from the vacuum polarization by an external electric field, and QtOp the topological charge connected with chiral anomaly (1 3). 315
Volume 113A, number 6
PHYSICS LETTERS
For weak inhomogeneous phase fluctuations, I~'1 ,~ A0, polarization effects are small as we have seen and the total charge appearing due to the interaction of order parameter A = A0ei¢ with occupied band ("vacuum" electrons) is governed by global phase variations o0
Q=f
dx ~Seff/~A~ 5) ~_ a t o p = -(e/Tr)Atp.
(20)
--o0
This vacuum charge is also the charge of the charge density wave (CDW) in quasi-one-dimensional conductors. In the continuum model of the Peierls insulator without taking into account the effects of lattice periodicity, the CDW is the Goldstone mode arising from the spontaneous U(1) symmetry breaking. This actually is the case if the period of CDW is incommensurate with the lattice period and the other effects of CDW pinning are negligible. Then the moving of CDW is activationless and the phase difference A¢ is not fixed. Really there is always pinning of the CDW. In some quasi-one-dimensional conductors (NbSe 3, TaS 3, etc.) the correlation of CDW period with the lattice period a is the main reason of the pinning. When the concen"tration of conduction electrons is kFa = 7rl/M (l < M, l and M are the reciprocals, M > 2, the commensurability index of CDW), the lagrangian of CDW must be supplemented by an extra term, the energy of commensurability [ 15] : 6£ c(~) = (/~/rrX) A 2 cosM~o orn //~ (A/W~- 2 ~ 1 .
(21)
When E = 0 the lagranglan (16), (21) is the wellknown sine-Gordon lagrangian. The topologically stable soliton solutions can fix the phase difference A 9 = 27riM. Thus we tend to the conclusion that there can exist collective excitations in CDW carrying fractional charge [Qs[ = 2e]M [16]. It is necessary to make one significant note here. The CDW in consideration is the collective degree of freedom describing the dynamics of filled band electrons. Therefore naively the charge of CDW solitons is delocalized because it is composed from the charge of all the electrons of the occupied valence band. The nontriviality of the situation consists in the following; despite of its collective origin the interaction of CDW 316
30 December 1985
with an electric field is local due to chiral anomaly (13) and the CDW solitons behave in an electric field like ordinary particles having a charge form factor. In other words in the fields with inhomogeneity scale more than the soliton size d the solitons can be treated as point-like particles. It is of special interest to consider chiral anomaly in PI with double commensurability. An example of such a system is the conducting linear polymer transpolyacetylene [trans-(CH)x ] (see the detailed discussion in ref. [8]). The CDW withM = 2 is described b) GNM [13] (see also refs. [11,12]) with real order parameter -+A (the commensurability M = 2 fixes the phase ¢ = 0, 7r). That is why the local consequence of chiral anomaly, the phase dynamics in trans-(CH)x, is absent. However the relation (20) survives because of its global (topological) nature. So the question arises what is the object whose charge is defined by the expression (20)? In the GNM with real order parameter two types of inhomogeneous static solutions exist: topologically stable solitons (kinks) &s(X ~ -+~) = +A 0 and polarons Ap(X -~ +co) = A 0 [19]. The kink fixes the global phase difference A¢ = 7r (antikink, A9 = --r0. That is, according to (20), a kink has topological charge Qs = - e and an antikink, Qs = e. At the same time the spin of charge solitons is zero because we have summarised over index o when deriving (15), (20) (for spinless "electrons" the well known Jackiw-Rebbi result Qs = -+e/2 [20] is valid). These topological charges are formed by all the electrons of the occupied valence band and thus they are formally speaking delocalized (see details in ref. [10]). The kink extracts one localized level in the centre of the energy gap (i.e. with zero energy) from the continuum of the Dirac "sea" levels. From the electroneutrality condition it follows that the collective (topological) charge has to be compensated by the charge of the electron extracted from the occupied band to local level. Such compensation on the distance 1 > d (d ~ A~ 1 is the size of kink) is possible because of the local coupling of topological charge density to the electric field. Thus, the kink turns to be neutral but nonzero spin 1/2 appears [21]. One can occupy the zero-energy local level by the additional charge-electron (hole) with the opposite spin-projection when doping the polymer. Then kink becomes charged but spinless [21]. The unusual spin-charge relation in trans-polyacetylene is experi-
Volume 113A, number 6
PHYSICS LETTERS
mentally confirmed when measuring magnetic susceptibility and dc-conductivity in the doping process
[81. The same considerations state that for polarons with zero topological charge the spin-charge relation is normal: polarons are charged (+e) and have spin 1/2. Thus we have shown that the CDW itself, its electromagnetic properties and the unusual quantum numbers in Peierls-Fr6nlich systems are of the chiral anomaly origin. This fact demonstrated once more the unexpected and interesting connections between solid state physics and modern quantum field theory. The authors thank P.I. Fomin and V.A. Miransky for the valuable discussions.
References [1] J. Schwinger, Phys. Rev. 82 (1951) 664; S.L. Adler, Phys. Rev. 177 (1969) 2426; T.S. Bell and R. Jackiw, Nuovo Cimento A60 (1969) 47. [2] R. Jackiw, in: Lectures on current algebra and its applications, eds. S. Treiman, R. Jackiw and D. Gross (Princeton Univ. Press, Princeton, 1972). [3] H.B. Nielson and M. Ninomiya, Phys. Lett. 130B (1983) 389. [4] L.N. Bulaevskii, Usp. Fiz. Nauk 115 (1975) 261. [5] G.A. Toombs, Phys. Rep. 40 (1978) 182. [6] K. Jonson, Phys. Lett. 5 (1963) 253.
30 December 1985
[7] R. Roskies and F. Shaposnik, Phys. Rev. D23 (1981) 558. [8] A.J. Heeger and A.G. MacDiarmd, Chem. Scr. 17 (1981) 115. [9] G. Grinner,Comm. Solid State Phys. 10 (1983) 183. [10] S.A. Brazovsky, Zh. Eksp. Teor. Fiz. 78 (1980) 677; S.A. Brazovsky and N.N. Kirova, Chem. Scr. 17 (1981). [11] R. Jackiw and J.S. Schrieffer, Nucl. Phys. B190 (1981) 253. [12] I.V. Krive and A.S. Rozhavsky, Solid State Commun. 39 (1981) 427; Zh. Eksp. Teor. Fiz. 85 (1983) 1122. [13] D.J. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235. [14] K. Fujikawa, Phys. Rev. D21 (1980) 2848. [15] P.A. Lee, T.M. Rice and P.W. Anderson, Solid State Commun. 14 (1974) 703. [16] M.J. Rice, A.R. Bishop, J.A. Krumhansl and S.E. TruUinger,Phys. Rev. Lett. 36 (1976)432; K. Maki, Phys. Rev. Lett. 39 (1977) 46; I.V. Krive and A.S. Rozhavsky, Fiz. Niz. Temp (Soy. Phys. J. Low Temp.) 6 (1980) 1272. [17] S.A. Brazovsky and I.E. Dzylashisky, Zh. Eksp. Teor. Fiz. 71 (1976) 2338. [18] W.A. Bardeen, S. Elitzur, Y. Frishman and E. Rabinovici, Nucl..Phys. B218 (1983) 445; A.Z. Capri, R. Ferrari and L.E. Picasso, Phys. Rev. D30 (1984) 2136. [19] R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D13 (1976) 3398. [20] R. Jackiw and C. Rebbi, Phys. Rev. D13 (1976) 3398; G. Goldstone and F. Wilczek, Phys. Rev. Lett. 48 (1981) 986. [21 ] W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698; M.J. Rice, Phys. Lett. 71A (1979) 152.
317