- Email: [email protected]
Instanton method for the electron propagator
Physica E 12 (2002) 84 – 87
www.elsevier.com/locate/physe
Instanton method for the electron propagator Michael R. Geller ∗ Department of Physics and Astronomy, University of Georgia, Athens, GA 30602-2451, USA
Abstract A nonperturbative theory of the electron propagator is developed and used to calculate the one-particle Green’s function and tunneling density of states in strongly correlated electron systems. The method, which is based on a Hubbard–Stratonovich decoupling of the electron–electron interaction combined with a cumulant expansion of the resulting noninteracting propagator, provides a possible generalization of the instanton technique to the calculation of the electron propagator in a many-body system. Application to the one-dimensional electron gas with short-range interaction is discussed. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.43.Jn; 71.10.Pm; 71.27.+a Keywords: Tunneling; Strongly correlated electron systems; Instantons
1. Introduction In a large class of one-dimensional systems, electron–electron interaction leads to the breakdown of Landau’s Fermi liquid theory and to the formation of a highly correlated Tomonaga–Luttinger liquid phase. In 1990, Wen [1] proposed that an edge state in the fractional quantum Hall e>ect (FQHE) regime of the two-dimensional electron gas is a chiral Luttinger liquid (CLL), a chiral counterpart to this non-Fermi-liquid state. Theoretical work by Wen [2] and by Kane and Fisher [3] showed that the transport and spectral properties of edge states in the FQHE regime should be strikingly di>erent than edge states in the integral regime, and several of these properties have been observed experimentally [4]. Recently, however, an important experiment by Grayson et al. [5] on the tunneling spectra of ∗ Corresponding author. Tel.: +1-706-542-2834; fax: +1-706542-2492. E-mail address: [email protected] (M.R. Geller).
quantum Hall edges found three surprising results: First, power-law tunneling characteristics were observed over a range of Flling factors , irrespective of whether a quantized Hall state existed at that Flling factor. Second, within a given FQHE plateau, the tunneling exponent was found to vary with the Flling factor and was not Fxed when the Hall conductance was quantized. Measurements on some samples, however, do show a density of states (DOS) plateau [6]. And third, the experiment shows that the tunneling DOS predicted by the multicomponent CLL theory for the hierarchy states is incorrect. This experiment was motivated by an earlier experiment by Chang et al. [7], who observed CLL-like tunneling characteristics at = 12 . Although some aspects of Ref. [5] are beginning to be understood [8], especially in connection with the third point above, the most fundamental aspects are not. In this paper I will outline a new theoretical method to calculate the one-particle Green’s function and tunneling DOS in a strongly correlated electron system. The method, discussed in detail in Ref. [9], is based on
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 2 4 8 - X
M.R. Geller / Physica E 12 (2002) 84 – 87
a Hubbard–Stratonovich decoupling of the electron– electron interaction combined with a cumulant expansion of the resulting noninteracting propagator. It provides a potential generalization of the instanton technique [10] to the calculation of the electron propagator in a many-body system [11], and a variant of the intuitive but phenomenological “charge spreading” picture [12] emerges automatically. Underlying the present method is the assumption that the physical processes most important in determining the low-energy spectral properties of a strongly correlated conductor are the infrared catastrophes associated with the response of a Fermi system to a sudden perturbation, in this case the introduction of a new electron into the conductor. After describing the general formalism I will apply it to spinless fermions with short-range interaction. In future work the method will be applied to the sharp and smooth edges of FQHE systems. 2. Outline of the method I consider a D-dimensional interacting electron system, possibly in an external magnetic Feld. For simplicity I will suppress spin indices and assume the system to be translationally invariant with density n0 . The grand-canonical Hamiltonian is H = H0 + V , where 1 d D r d D r n(r)U (r − r )n(r ); V ≡ 2 n(r) ≡
†
(r) (r) − n0 :
(1)
H0 is the noninteracting Hamiltonian. We want to calculate the Euclidean propagator G(rf ; ri ; 0 ) ≡ −T
O
H (rf ; 0 ) H (ri ; 0)H ;
(2)
which can be written (in the interaction representation with respect to H0 ) as G(rf ; ri ; 0 ) = −
T (rf ; 0 ) O (ri ; 0)e− T e
−
0
d V ( )
0
0
d V ( )
0
: (3)
A Hubbard–Stratonovich transformation then leads to −1 De−(1=2) U g(rf ; ri ; 0 |) ; G(rf ; ri ; 0 ) = N −1 De−(1=2) U (4)
85
where g(rf ; ri ; 0 |) ≡ −T (rf ; 0 ) O (ri ; 0)ei
0
d
d D r(r; )n(r; )
0
(5)
is a noninteracting correlation function, and N ≡ T exp(− 0 d V )−1 0 is a constant. So far everything is exact. What remains is to Fnd an appropriate approximation for g(r; r ; |) and to do the resulting functional integral. I evaluate Eq. (5) with a second-order cumulant expansion g(rf ; ri ; 0 |) = G0 (rf ; ri ; 0 ) ×e
C1 (r; )(r; )+ C2 (r ; r )(r; )(r ; )
: (6)
This form correctly accounts for the infrared divergences (such as the orthogonality catastrophe) caused by the tunneling electron; see Ref. [9] for details. The Cn are known in terms of noninteracting equilibrium Green’s functions. For example G0 (rf ; r; 0 − )G0 (r; ri ; ) C1 (r; ) = −i : (7) G0 (rf ; ri ; 0 ) The functional integral in Eq. (4) can be done exactly, leading to [9] G(rf ; ri ; 0 ) = A( 0 )G0 (rf ; ri ; 0 )e−S( 0 ) ; where A ≡ N[Det(1 − 2C2 U )] determinant and 1 S≡ d d d D r d D r 2 0
−1=2
(8)
is a Puctuation
×(r; )Ue> (r ; r )(r ; ):
(9)
Here (r; ) ≡ − iC1 (r ) and Ue> (r ; r ) ≡ (U −1 − 2C2 )−1 r ; r . I interpret Eq. (8) as follows: S is the Euclidean action for a time-dependent charge distribution (r; ) whose dynamics, governed by H0 , describes the charge density associated with an electron inserted into position r = ri at time = 0 and removed from rf at
0 . The charge interacts via an e>ective interaction Ue> that accounts for the modiFcation of the electron– electron interaction by dynamic screening. My interpretation of −iC1 (r ) as the charge density associated with a tunneling electron follows from extensive numerical D studies and from the exact (at T = 0) identity d r (r; ) = ( )( 0 − ).
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M.R. Geller / Physica E 12 (2002) 84 – 87
3. Application to one-dimensional spinless fermions
Acknowledgements
Here I assume a short-range interaction of the form U (x − x ) = U0 Q(x − x ), where Q(x) is a broadened delta function with range , and show that Eq. (8) correctly predicts a power-law DOS. First consider the “tree-level” instanton approximation, obtained by keeping the Frst cumulant C1 only. In this case the action is ∞ U0 S= d d x [(x; )]2 (10) 2 0 −∞
This work was supported by the National Science Foundation under CAREER Grant No. DMR0093217, and by a Research Innovation Award and Cottrell Scholars Award from the Research Corporation. It is a pleasure to thank Claudio Chamon, Yong Baek Kim, Allan MacDonald, Gerry Mahan, David Thouless, Giovanni Vignale, and Ziqiang Wang for useful discussions.
and we can choose xi = xf = 0. In polar coordinates x = R cos and vF = R sin , the low-energy noninteracting propagator can be written as sin(kF R cos − ) ; (11) G0 (R; ) = R which shows that G0 falls o> as 1=R in the Euclidean plane ˜R = (x; vF ). Because (x; ) has 1=R “singularities” 1 at ˜R = (0; 0) and (0; vF 0 ), the action diverges logarithmically, the infrared divergence from the tail of one singularity being cut o> by the position of the other, a distance vF 0 away. S therefore diverges logarithmically in 0 , leading to the well-known power-law DOS in one dimension. In the limit kF−1 it can be shown that 3 U0 0 ; (12) S= ln 8 vF a where a is a microscopic cuto> length, leading to a DOS 3 U0 N (") = const × " with = : (13) 8 vF Including the second cumulant C2 in does not qualitatively change this picture. However, it does a>ect the value of the DOS exponent . The goal of this work is to develop a method that is asymptotically exact in the low energy and=or weak tunneling limit. Although the ultimate limitations of this method are not understood at present, application of the second cumulant (or “one-loop”) analysis to the Tomonaga–Luttinger model leads to an exponent in exact agreement with the bosonization result [9]. 1 These singularities arise from using the low-energy approximation (11) to the noninteracting propagator; they are not present in the exact expression. However, the 0 -dependent part of S comes from the 1=R tail of G0 , which is correctly described by the low-energy form.
References [1] X.G. Wen, Phys. Rev. B 41 (1990) 12 838. [2] X.G. Wen, Int. J. Mod. Phys. B 6 (1992) 1711; X.G. Wen, Adv. Phys. 44 (1995) 405. [3] C.L. Kane, M.P.A. Fisher, in: S. Das Sarma, A. Pinczuk (Eds.), Perspectives in Quantum Hall E>ects, Wiley, New York, 1997. [4] F.P. Milliken, C.P. Umbach, R.A. Webb, Solid State Commun. 97 (1996) 309; A.M. Chang, L.N. Pfei>er, K.W. West, Phys. Rev. Lett. 77 (1996) 2538; L. Saminadayar, D.C. Glattli, Y. Jin, B. Etienne, Phys. Rev. Lett. 79 (1997) 2526; R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, D. Mahalu, Nature 389 (1997) 162; M. Reznikov, R. de-Picciotto, T.G. GriUths, M. Heiblum, V. Umansky, Nature 399 (1999) 238; M. Grayson, D.C. Tsui, L.N. Pfei>er, K.W. West, A.M. Chang, Phys. Rev. Lett. 86 (2001) 2645. [5] M. Grayson, D.C. Tsui, L.N. Pfei>er, K.W. West, A.M. Chang, Phys. Rev. Lett. 80 (1998) 1062. [6] A.M. Chang, M.K. Wu, C.C. Chi, L.N. Pfei>er, K.W. West, Phys. Rev. Lett. 86 (2001) 143. [7] A.M. Chang, L.N. Pfei>er, K.W. West, Physica B 249 (1998) 383. [8] A.V. Shytov, L.S. Levitov, B.I. Halperin, Phys. Rev. Lett. 80 (1998) 141; S. Conti, G. Vignale, J. Phys.: Condens. Matter 10 (1998) L779; D.H. Lee, X.G. Wen, cond-mat=9809160; K. Imura, Europhys. Lett. 47 (1999) 233; A. Lopez, E. Fradkin, Phys. Rev. B 59 (1999) 15 323; U. Zulicke, A.H. MacDonald, Phys. Rev. B 60 (1999) 1837; I. D’Amico, G. Vignale, Phys. Rev. B 60 (1999) 2084; A.M.M. Pruisken, B. Skoric, M.A. Baranov, Phys. Rev. B 60 (1999) 16 838; B. Skoric, A.M.M. Pruisken, Nucl. Phys. B 559 (1999) 637; D.V. Khveshchenko, Solid State Commun. 111 (1999) 501; A. Alekseev, V. Cheianov, A.P. Dmitriev, V.Yu. Kachorovskii, JETP Lett. 72 (2000) 333; D.V. Khveshchenko, Phys. Rev. B 61 (2000) 5610;
M.R. Geller / Physica E 12 (2002) 84 – 87 L.S. Levitov, A.V. Shytov, B.I. Halperin, Phys. Rev. B 64 (2001) 75322. [9] M.R. Geller, to be published. [10] J.S. Langer, Ann. Phys. 41 (1967) 108; S. Coleman, Phys. Rev. D 15 (1977) 2929; C.G. Callen, S. Coleman, Phys. Rev. D 16 (1977) 1762; J.W. Negele, Rev. Mod. Phys. 54 (1982) 913.
[11] Yu.V. Nazarov, Sov. Phys. JETP 68 (1989) 561; Y.B. Kim, X.G. Wen, Phys. Rev. B 50 (1994) 8078; Z. Wang, S. Xiong, Phys. Rev. Lett. 83 (1999) 828; A. Kamenev, A. Andreev, Phys. Rev. B 60 (1999) 2218. [12] B.Z. Spivak, (unpublished); L.S. Levitov, A.V. Shytov, JETP Lett. 66 (1997) 214.
87
www.elsevier.com/locate/physe
Instanton method for the electron propagator Michael R. Geller ∗ Department of Physics and Astronomy, University of Georgia, Athens, GA 30602-2451, USA
Abstract A nonperturbative theory of the electron propagator is developed and used to calculate the one-particle Green’s function and tunneling density of states in strongly correlated electron systems. The method, which is based on a Hubbard–Stratonovich decoupling of the electron–electron interaction combined with a cumulant expansion of the resulting noninteracting propagator, provides a possible generalization of the instanton technique to the calculation of the electron propagator in a many-body system. Application to the one-dimensional electron gas with short-range interaction is discussed. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.43.Jn; 71.10.Pm; 71.27.+a Keywords: Tunneling; Strongly correlated electron systems; Instantons
1. Introduction In a large class of one-dimensional systems, electron–electron interaction leads to the breakdown of Landau’s Fermi liquid theory and to the formation of a highly correlated Tomonaga–Luttinger liquid phase. In 1990, Wen [1] proposed that an edge state in the fractional quantum Hall e>ect (FQHE) regime of the two-dimensional electron gas is a chiral Luttinger liquid (CLL), a chiral counterpart to this non-Fermi-liquid state. Theoretical work by Wen [2] and by Kane and Fisher [3] showed that the transport and spectral properties of edge states in the FQHE regime should be strikingly di>erent than edge states in the integral regime, and several of these properties have been observed experimentally [4]. Recently, however, an important experiment by Grayson et al. [5] on the tunneling spectra of ∗ Corresponding author. Tel.: +1-706-542-2834; fax: +1-706542-2492. E-mail address: [email protected] (M.R. Geller).
quantum Hall edges found three surprising results: First, power-law tunneling characteristics were observed over a range of Flling factors , irrespective of whether a quantized Hall state existed at that Flling factor. Second, within a given FQHE plateau, the tunneling exponent was found to vary with the Flling factor and was not Fxed when the Hall conductance was quantized. Measurements on some samples, however, do show a density of states (DOS) plateau [6]. And third, the experiment shows that the tunneling DOS predicted by the multicomponent CLL theory for the hierarchy states is incorrect. This experiment was motivated by an earlier experiment by Chang et al. [7], who observed CLL-like tunneling characteristics at = 12 . Although some aspects of Ref. [5] are beginning to be understood [8], especially in connection with the third point above, the most fundamental aspects are not. In this paper I will outline a new theoretical method to calculate the one-particle Green’s function and tunneling DOS in a strongly correlated electron system. The method, discussed in detail in Ref. [9], is based on
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 2 4 8 - X
M.R. Geller / Physica E 12 (2002) 84 – 87
a Hubbard–Stratonovich decoupling of the electron– electron interaction combined with a cumulant expansion of the resulting noninteracting propagator. It provides a potential generalization of the instanton technique [10] to the calculation of the electron propagator in a many-body system [11], and a variant of the intuitive but phenomenological “charge spreading” picture [12] emerges automatically. Underlying the present method is the assumption that the physical processes most important in determining the low-energy spectral properties of a strongly correlated conductor are the infrared catastrophes associated with the response of a Fermi system to a sudden perturbation, in this case the introduction of a new electron into the conductor. After describing the general formalism I will apply it to spinless fermions with short-range interaction. In future work the method will be applied to the sharp and smooth edges of FQHE systems. 2. Outline of the method I consider a D-dimensional interacting electron system, possibly in an external magnetic Feld. For simplicity I will suppress spin indices and assume the system to be translationally invariant with density n0 . The grand-canonical Hamiltonian is H = H0 + V , where 1 d D r d D r n(r)U (r − r )n(r ); V ≡ 2 n(r) ≡
†
(r) (r) − n0 :
(1)
H0 is the noninteracting Hamiltonian. We want to calculate the Euclidean propagator G(rf ; ri ; 0 ) ≡ −T
O
H (rf ; 0 ) H (ri ; 0)H ;
(2)
which can be written (in the interaction representation with respect to H0 ) as G(rf ; ri ; 0 ) = −
T (rf ; 0 ) O (ri ; 0)e− T e
−
0
d V ( )
0
0
d V ( )
0
: (3)
A Hubbard–Stratonovich transformation then leads to −1 De−(1=2) U g(rf ; ri ; 0 |) ; G(rf ; ri ; 0 ) = N −1 De−(1=2) U (4)
85
where g(rf ; ri ; 0 |) ≡ −T (rf ; 0 ) O (ri ; 0)ei
0
d
d D r(r; )n(r; )
0
(5)
is a noninteracting correlation function, and N ≡ T exp(− 0 d V )−1 0 is a constant. So far everything is exact. What remains is to Fnd an appropriate approximation for g(r; r ; |) and to do the resulting functional integral. I evaluate Eq. (5) with a second-order cumulant expansion g(rf ; ri ; 0 |) = G0 (rf ; ri ; 0 ) ×e
C1 (r; )(r; )+ C2 (r ; r )(r; )(r ; )
: (6)
This form correctly accounts for the infrared divergences (such as the orthogonality catastrophe) caused by the tunneling electron; see Ref. [9] for details. The Cn are known in terms of noninteracting equilibrium Green’s functions. For example G0 (rf ; r; 0 − )G0 (r; ri ; ) C1 (r; ) = −i : (7) G0 (rf ; ri ; 0 ) The functional integral in Eq. (4) can be done exactly, leading to [9] G(rf ; ri ; 0 ) = A( 0 )G0 (rf ; ri ; 0 )e−S( 0 ) ; where A ≡ N[Det(1 − 2C2 U )] determinant and 1 S≡ d d d D r d D r 2 0
−1=2
(8)
is a Puctuation
×(r; )Ue> (r ; r )(r ; ):
(9)
Here (r; ) ≡ − iC1 (r ) and Ue> (r ; r ) ≡ (U −1 − 2C2 )−1 r ; r . I interpret Eq. (8) as follows: S is the Euclidean action for a time-dependent charge distribution (r; ) whose dynamics, governed by H0 , describes the charge density associated with an electron inserted into position r = ri at time = 0 and removed from rf at
0 . The charge interacts via an e>ective interaction Ue> that accounts for the modiFcation of the electron– electron interaction by dynamic screening. My interpretation of −iC1 (r ) as the charge density associated with a tunneling electron follows from extensive numerical D studies and from the exact (at T = 0) identity d r (r; ) = ( )( 0 − ).
86
M.R. Geller / Physica E 12 (2002) 84 – 87
3. Application to one-dimensional spinless fermions
Acknowledgements
Here I assume a short-range interaction of the form U (x − x ) = U0 Q(x − x ), where Q(x) is a broadened delta function with range , and show that Eq. (8) correctly predicts a power-law DOS. First consider the “tree-level” instanton approximation, obtained by keeping the Frst cumulant C1 only. In this case the action is ∞ U0 S= d d x [(x; )]2 (10) 2 0 −∞
This work was supported by the National Science Foundation under CAREER Grant No. DMR0093217, and by a Research Innovation Award and Cottrell Scholars Award from the Research Corporation. It is a pleasure to thank Claudio Chamon, Yong Baek Kim, Allan MacDonald, Gerry Mahan, David Thouless, Giovanni Vignale, and Ziqiang Wang for useful discussions.
and we can choose xi = xf = 0. In polar coordinates x = R cos and vF = R sin , the low-energy noninteracting propagator can be written as sin(kF R cos − ) ; (11) G0 (R; ) = R which shows that G0 falls o> as 1=R in the Euclidean plane ˜R = (x; vF ). Because (x; ) has 1=R “singularities” 1 at ˜R = (0; 0) and (0; vF 0 ), the action diverges logarithmically, the infrared divergence from the tail of one singularity being cut o> by the position of the other, a distance vF 0 away. S therefore diverges logarithmically in 0 , leading to the well-known power-law DOS in one dimension. In the limit kF−1 it can be shown that 3 U0 0 ; (12) S= ln 8 vF a where a is a microscopic cuto> length, leading to a DOS 3 U0 N (") = const × " with = : (13) 8 vF Including the second cumulant C2 in does not qualitatively change this picture. However, it does a>ect the value of the DOS exponent . The goal of this work is to develop a method that is asymptotically exact in the low energy and=or weak tunneling limit. Although the ultimate limitations of this method are not understood at present, application of the second cumulant (or “one-loop”) analysis to the Tomonaga–Luttinger model leads to an exponent in exact agreement with the bosonization result [9]. 1 These singularities arise from using the low-energy approximation (11) to the noninteracting propagator; they are not present in the exact expression. However, the 0 -dependent part of S comes from the 1=R tail of G0 , which is correctly described by the low-energy form.
References [1] X.G. Wen, Phys. Rev. B 41 (1990) 12 838. [2] X.G. Wen, Int. J. Mod. Phys. B 6 (1992) 1711; X.G. Wen, Adv. Phys. 44 (1995) 405. [3] C.L. Kane, M.P.A. Fisher, in: S. Das Sarma, A. Pinczuk (Eds.), Perspectives in Quantum Hall E>ects, Wiley, New York, 1997. [4] F.P. Milliken, C.P. Umbach, R.A. Webb, Solid State Commun. 97 (1996) 309; A.M. Chang, L.N. Pfei>er, K.W. West, Phys. Rev. Lett. 77 (1996) 2538; L. Saminadayar, D.C. Glattli, Y. Jin, B. Etienne, Phys. Rev. Lett. 79 (1997) 2526; R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, D. Mahalu, Nature 389 (1997) 162; M. Reznikov, R. de-Picciotto, T.G. GriUths, M. Heiblum, V. Umansky, Nature 399 (1999) 238; M. Grayson, D.C. Tsui, L.N. Pfei>er, K.W. West, A.M. Chang, Phys. Rev. Lett. 86 (2001) 2645. [5] M. Grayson, D.C. Tsui, L.N. Pfei>er, K.W. West, A.M. Chang, Phys. Rev. Lett. 80 (1998) 1062. [6] A.M. Chang, M.K. Wu, C.C. Chi, L.N. Pfei>er, K.W. West, Phys. Rev. Lett. 86 (2001) 143. [7] A.M. Chang, L.N. Pfei>er, K.W. West, Physica B 249 (1998) 383. [8] A.V. Shytov, L.S. Levitov, B.I. Halperin, Phys. Rev. Lett. 80 (1998) 141; S. Conti, G. Vignale, J. Phys.: Condens. Matter 10 (1998) L779; D.H. Lee, X.G. Wen, cond-mat=9809160; K. Imura, Europhys. Lett. 47 (1999) 233; A. Lopez, E. Fradkin, Phys. Rev. B 59 (1999) 15 323; U. Zulicke, A.H. MacDonald, Phys. Rev. B 60 (1999) 1837; I. D’Amico, G. Vignale, Phys. Rev. B 60 (1999) 2084; A.M.M. Pruisken, B. Skoric, M.A. Baranov, Phys. Rev. B 60 (1999) 16 838; B. Skoric, A.M.M. Pruisken, Nucl. Phys. B 559 (1999) 637; D.V. Khveshchenko, Solid State Commun. 111 (1999) 501; A. Alekseev, V. Cheianov, A.P. Dmitriev, V.Yu. Kachorovskii, JETP Lett. 72 (2000) 333; D.V. Khveshchenko, Phys. Rev. B 61 (2000) 5610;
M.R. Geller / Physica E 12 (2002) 84 – 87 L.S. Levitov, A.V. Shytov, B.I. Halperin, Phys. Rev. B 64 (2001) 75322. [9] M.R. Geller, to be published. [10] J.S. Langer, Ann. Phys. 41 (1967) 108; S. Coleman, Phys. Rev. D 15 (1977) 2929; C.G. Callen, S. Coleman, Phys. Rev. D 16 (1977) 1762; J.W. Negele, Rev. Mod. Phys. 54 (1982) 913.
[11] Yu.V. Nazarov, Sov. Phys. JETP 68 (1989) 561; Y.B. Kim, X.G. Wen, Phys. Rev. B 50 (1994) 8078; Z. Wang, S. Xiong, Phys. Rev. Lett. 83 (1999) 828; A. Kamenev, A. Andreev, Phys. Rev. B 60 (1999) 2218. [12] B.Z. Spivak, (unpublished); L.S. Levitov, A.V. Shytov, JETP Lett. 66 (1997) 214.
87