- Email: [email protected]
Potential scattering in a Luttinger liquid: weak versus strong correlations
PERGAMON
Solid State Communications 113 (2000) 263–266 www.elsevier.com/locate/ssc
Potential scattering in a Luttinger liquid: weak versus strong correlations Debanand Sa a,b,1 a
Department of Physics, Indian Institute of Technology, Kanpur-208 016, India b Institute of Physics, Sachivalaya Marg, Bhubaneswar-751 005, India Received 4 August 1999; accepted 13 October 1999 by H. Eschrig
Abstract The effect of a scalar impurity in Luttinger liquid is analysed. In the weak Coulomb correlation limit, the scenario presented by Oreg and Finkel’stein seems to be correct, where the problem resembles the physics of Kondo resonance. On the contrary, we propose an emergence of new physics when the correlation energy is larger than the Kondo scale, i.e. the resonance breaks, but still either the left or the right moving electron might get localised near the impurity site, giving rise to a local moment and hence a Curie-like impurity susceptibility in the system. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Metals; D. Electron–electron interactions; D. Electronic states (localised); D. Kondo effects
The study of one-dimensional (1D) interacting fermion systems has attracted renewed attention recently due to the rapid advancement in the sub-micron technology [1]. It is well known from the exact [2] as well as the approximate solutions [3–5] that the electron–electron interaction in 1D electron gas, when away from the density-wave or the superconducting instabilities, leads to Luttinger liquid behaviour. The properties of these 1D systems are expected to be unusual in the sense that one does not have well defined single particle excitations in this state, i.e. the occupation number at the Fermi energy has a singularity rather than a jump, their density of states vanishes at the Fermi surface in a power-law way, etc. The true coherent excitations, i.e. the eigen modes of the systems are charge- and spin-density fluctuations [6,7], which are by nature bosonic, dynamically independent and in general propagate with different velocities (the so called spin–charge separation). The striking non-Fermi liquid behaviour of 1D electrons can clearly be exhibited in transport properties, especially in the presence of impurities or barriers [8,9]. For a repulsive electron–electron interaction, it has been predicted that at zero temperature even a single weak backward potential
scatterer eventually causes the conductance to vanish. It is widely accepted that the low energy physics of this system can be described by two semi-infinite chains, connected by a weak link junction [10,11]. But recently, this problem has been reconsidered by Oreg and Finkel’stein [12,13] questioning the earlier works. They addressed the problem of Fermi-edge singularity in 1D and concluded that the low energy physics in such a system together with the electron– electron repulsion resembles the physics of Kondo resonance. While this result seems to be valid in the weak Coulomb correlation limit, the strong correlation case is quite different [14]. Starting from the Kondo resonance physics of Oreg and Finkel’stein, we propose an emergence of new physics in the case of strong Coulomb correlation, that is, the resonance breaks and either the left or the right moving electron gets localised near the impurity site, giving rise to a local moment in the system. We give an estimate of the critical Coulomb correlation energy beyond which the physics of local moment could take place. Furthermore, we discuss the role of temperature and also a new phase diagram in the U 2 T plane, where U is the Coulomb correlation energy and T, the temperature. The minimal model which gives rise to Luttinger liquid behaviour is the 1D single band Hubbard model, given by
1
Present Address: Institut fu¨r Physik, Universita¨t Dortmund, 44221 Dortmund, Germany, Fax: 149-231-755 3569. E-mail address: [email protected] (D. Sa)
H 2t
X i;s
0038-1098/00/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00487-1
Ci;†s Ci11;s 1 h:c: 1
1 X U n n 2 i;s is i2s
1
264
D. Sa / Solid State Communications 113 (2000) 263–266
Ci;†s Ci;s
where is the electron operator which creates (annihilates) an electron of spin s at site i and ni;s Ci;†s Ci;s is the electron number, t and U are the hopping integral and the Coulomb correlation energy, respectively. The hopping term in the Hamiltonian can be solved exactly and the excitation spectrum is obtained as, ek 22tcosk. In the ground state, all the Fermions with j k j# kF are filled which creates a Fermi surface at the Fermi wave vector kF ^p=2. The low energy modes of this theory are given by the excitations near the Fermi points. Thus, one can linearise the dispersion near the two Fermi points, which of course, gives rise to left and right moving Fermions. Hence, the Fermion operators can be written as
interaction drives the system to a non-Fermi liquid fixed point, i.e. Luttinger liquid, the low energy excitations being the collective spin- and charge-density fluctuations. In such situations, the presence of a potential scatterer might have profound consequences. This has already been pointed out by many authors [10–14] but the problem remains unclear to date. For completeness, the effect of electron– electron interaction on a potential scatterer in 1D is analysed through Anderson’s poor man’s scaling approach [15]. In the weak coupling limit, i.e. U=pvF p 1; Vf =pvF p 1 and Vb =pvF p 1, the scaling equations for the forward as well as the backward impurity scattering up to the order of UV are written as
Ci;s e2ikF ri ai;s 1 eikF ri bi;s
dVf 0 dl
6
dVb 1r0 UVb dl
7
2
The operators a=b, which are the left/right branch Fermion operators, do not contain high momentum modes. One presumes here that these fields have slowly varying matrix elements on the scale of the lattice. While this is valid in the non-interacting theory, it is also assumed to be so even when the interactions are turned on. Thus, by neglecting the rapidly oscillatory terms, one can express the Hubbard model in terms of the left/right moving branch Fermion operators. Now, on Fourier transforming and retaining the inter-branch density density interaction term which is sufficient to yield Luttinger liquid behaviour, the Hamiltonian can be written as, H H0 1 HI , where X H0 vF k 2 kF a†k;s ak;s 1 2k 2 kF b†k;s bk;s 3 k;s
HI U
where dl 2d lnEc with Ec being the band width cut-off and r0 1=2pvF is the free 1D density of states. It is obvious from the above differential equation that the forward impurity potential becomes a marginal perturbation whereas the backward becomes a relevant one for the case of a repulsive electron–electron interaction. Thus, the backward impurity potential which flows to the strong coupling fixed point implies that the system has a cross-over scale, analogous to the Kondo scale, given by kB Tk , Ec
X 0
k;k ;q;s
a†k;s b†k 0 ;2s bk 0 1q;2s ak2q;s
1 s $ 2s
4
where vF is the Fermi velocity. Since the exchange term is irrelevant in the renormalisation group sense for the repulsive electron–electron interaction, it is neglected here. In terms of of the left/right branch Fermion operators, the scattering of the conduction band electrons by a scalar impurity can be written as X † Himp Vf ak;s ak 0 s 1 a ! b 0
k;k ;s
1 Vb
X 0
k;k s
a†k;s bk 0 s 1 a $ b
5
where Vf Vf q , 0 is the forward impurity scattering amplitude and Vb Vb q , 2kF is the backward one. For simplicity, we take these to be momentum independent. In the usual potential scattering problems in three or higher dimensions, the effect of electron–electron interaction is always neglected, since the interacting electron systems can be mapped to a nearly non-interacting systems of quasi-particles, which is the basis of Landau Fermi liquid and the problem of a scalar impurity in this case can be handled in an usual perturbation theory. The situation in 1D is completely different, where the repulsive electron–electron
V U
1 r0 U
8
The flow diagram which is rather complex is discussed extensively by Oreg and Finkel’stein in a later paper [13]. Since there are several marginal parameters characterising the impurity, each of the fixed point describing the backward impurity problem corresponds to a manifold of fixed points which can be represented by lines in a two-dimensional plot. In the limit of strong potential barrier of the impurity, the low temperature behaviour is controlled by the line of fixed points corresponding to two disconnected semi-infinite wires which is proposed by Kane et al., and many others [10]. This is because in the course of renormalisation, the strength of the large bare backward impurity potential increases further and at the final stage it transforms into a strong barrier which ultimately breaks the chain. However, a mapping of the weak impurity problem onto a Coulomb gas theory [16], and hence to the Kondo problem [17] apparently contradicts this simple intuitive picture. The scenario of Ref. [10] is based on the assumption that no other fixed points intervene in the scaling from the repulsive fixed point of a weakly scattering defect to the attractive fixed point corresponding to a tunneling junction of two half-wires. By mapping the impurity problem onto a spin1/2 Heisenberg chain, it has been shown [13] that the line of fixed points describing the limit Vb ! ∞ is unstable. This implies that there exists an another line of fixed points which
D. Sa / Solid State Communications 113 (2000) 263–266
is attractive at an intermediate value of the parameter Vb . Oreg and Finkel’stein pointed out that this is the fixed point (i.e. the weak impurity limit flows to a point where Vb is finite and Vf is small, equal to its bare value) which controls the low energy physics of the system when the bare potential of the impurity is weak. This has been utilised in the mapping of the impurity problem onto the Coulomb gas and hence to the Kondo problem where the localised mixture of left and right moving electrons act as the role of Kondo singlet. In the derivation of the scaling equations, it has been presumed that the interaction parameter U is not renormalised. This is due to the fact that U describes the electron– electron interaction in the bulk electron liquid, so the presence of a single impurity cannot influence this parameter. The appearance of a cross-over scale in the preceeding discussion suggests that the Kondo resonance physics proposed by Oreg and Finkel’stein is valid only below this scale. Thus, U also has to be less than this scale. Furthermore, in obtaining the Kondo resonance physics, a weak Coulomb correlation energy U has been used. However, the cross-over energy scale is nothing but the binding energy of the left and right moving electrons. Thus, in the present problem, there is a competition between two energy scales, the Coulomb correlation energy and the cross-over energy scale. Since the Kondo resonance physics of Oreg and Finkel’stein is valid for a weak U, the obvious question arises at this point is, what happens when the Coulomb correlation energy is larger than the cross-over scale? We try to address precisely this question in the present letter. Since U describes the Coulomb repulsion between the left and the right moving electrons in the electron liquid, strong correlation implies that the resonance is broken and hence there will be a breakdown of the Kondo resonance physics. We propose that even though the resonance breaks, still either the left or the right moving electron might get localised near the impurity site, which ultimately corresponds to a local moment in the system. Qualitatively, one can understand this in a following way: In the large U Hubbard model, the spin sector has BCS like condensate of pair of spinons. The introduction of a vacancy implies removal of a spinon from the vacant site which converts its partner from the condensate into a free spinon and it acts as a local moment in the system. This is in accordance with the earlier work on the formation of local moment near a static vacancy in the 1D large U Hubbard model [14]. Moreover, we can give an estimate of the critical Coulomb correlation Ucr , beyond which the physics of local moment will be valid. Ucr can be obtained by solving the following equation for U, U Ec
V U
1 r0 U
UU
9 cr
The appearance of a Ucr in the present formulation provides us a prescription to distinguish between the weak as well as the strong Coulomb correlation regime. When
265
U , Ucr , it is the case of weak Coulomb correlation where the problem of the scalar impurity in the Luttinger liquid maps to the problem of Kondo resonance proposed by Oreg and Finkel’stein. On the other hand, the strong correlation case corresponds to U . Ucr , where the resonance breaks and there is a local moment formation in the system. Therefore, as usual, one is expected to get a Curie-like impurity susceptibility in the system. Of course, the local moment will interact with the other excitations in the system, which will ultimately lead to a problem similar to that of a magnetic impurity in a Luttinger liquid [18,19]. But in the present case, since the local moment appears at zero temperature, one might ask, what happens to the moment at high temperature phase? We believe that the moment might survive to high temperature also, since the phenomenon of local moment occurs here is only due to the strong bulk Coulomb repulsion. This might be an important difference in the usual problem of a magnetic impurity in a Luttinger liquid and the present case. The stability of the local moment and the related phenomenon for U . Ucr with respect to the variation in temperature is currently under investigation [20]. Now, let us consider the interplay of temperature in the impurity problem explicitely. The Kondo resonance physics of Oreg and Finkel’stein which is valid in the weak Coulomb correlation limit, i.e. U , Ucr , will break down again when the temperature (T) is larger than the Kondolike temperature (Tk ) (Tk is analogous to the Kondo temperature, given in Eq. (8)). But in this case, the high temperature phase is just the problem of a potential scatterer in a Luttinger liquid which can well be described by a usual perturbation theory. This prompts us to propose a novel phase diagram in the U 2 T plane. For U , Ucr , one has two regimes corresponding to T . Tk which is the regime of usual potential scattering problem in a Luttinger liquid where perturbation theory is valid and for T , Tk , it is the Kondo resonance regime of Oreg and Finkel’stein. On the other hand for U . Ucr , the physics of local moment appears which might persist to high temperature also. This shows that even though the pure model in both the weak as well as the strong Coulomb correlation limit yields Luttinger liquid behaviour [2,21], the presence of a weak potential scatterer makes both the limit distinct from each other. In recent times, since there is rapid advancement in the fabrication of nano-materials such as quantum dots, semiconducting quantum wires etc., the present model study can be tested through a straightforward substitutional study which will definitely help in our understanding on the properties of low-dimensional materials. In conclusion, we briefly summarise the main results of the paper. The physics of a scalar impurity in an 1D interacting electron gas is analysed in both the weak as well as the strong bulk Coulomb correlation limit. In the weak correlation limit, i.e. when U , Ucr , the problem maps to the Kondo resonance physics proposed by Oreg and Finkel’stein whereas in the strong Coulomb correlation
266
D. Sa / Solid State Communications 113 (2000) 263–266
case, i.e. for U . Ucr , the theory breaks down. In the latter case, we propose that even if the resonance breaks, still either the left or the right moving electron might get localised giving rise to a local moment in the system. This is in accordance with our earlier work on the formation of a local moment near a static vacancy in 1D large U Hubbard model. We give an estimate of the critical Coulomb correlation from which one can distinguish the weak as well as the strong correlation regime. Furthermore, we discuss the interplay of temperature as well as the U 2 T phase diagram in the impurity problem. This indicates that even though the pure model in both the weak as well as the strong Coulomb correlation limit yields Luttinger liquid behaviour, the ground state in presence of a weak scalar impurity in both the cases are quite different from each other.
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Acknowledgements The author would like to thank Prof. G. Baskaran and Prof. S.N. Behera for useful discussions. References [1] D.V. Averin, K.K. Likharev, Mesoscopic Phenomena in
[14] [15] [16] [17] [18] [19] [20] [21]
Solids, in B.L. Altshuler et al., (Ed.), North-Holland, Elsevier Science Publishers B.V., Amsterdam, 1991. E.H. Lieb, F.Y. Wu, Phys. Rev. Lett. 20 (1968) 1445. V.J. Emery, Highly Conducting One Dimensional Solids, in: J.T. Devreese et al., (Ed.), Plenum, New York, 1979. J. Solyom, Adv. Phys. 28 (1979) 201. F.D.M. Haldane, J. Phys. C 14 (1981) 2585. J. Voit, Rep. Prog. Phys. 58 (1995) 977. H.J. Schulz et al., Cond-mat/9807366 v2, 4 Aug. (1998). C.L. Kane, M.P.A. Fisher, Phys. Rev. Lett. 68 (1992) 1220. K.A. Matveev, D. Yue, L.I. Glazman, Phys. Rev. Lett. 71 (1993) 3351. C.L. Kane, M.P.A. Fisher, Phys. Rev. B 46 (1992) 15233 and references therein. M. Fabrizio, A.O. Gogolin, Phys. Rev. B 51 (1995) 17827. Y. Oreg, A.M. Finkel’stein, Phys. Rev. 53 (1996) 10928 and references therein. Y. Oreg, A.M. Finkel’stein, Philo. Mag. B 77 (1998) 1145 and references therein. Debanand Sa, Europhys. Lett. 42 (1998) 289. P.W. Anderson, J. Phys. C 3 (1970) 2436. P. Minnhagen, Rev. Mod. Phys. 59 (1987) 1001. A.C. Hewson, The Kondo Problems to Heavy Fermions, Cambridge University Press, Cambridge, 1993. D.H. Lee, J. Toner, Phys. Rev. Lett. 69 (1992) 3378. A. Furusaki, N. Nagaosa, Phys. Rev. Lett. 72 (1994) 892. Debanand Sa, in preparation. M. Ogata, H. Shiba, Phys. Rev. B 41 (1990) 2326.
Solid State Communications 113 (2000) 263–266 www.elsevier.com/locate/ssc
Potential scattering in a Luttinger liquid: weak versus strong correlations Debanand Sa a,b,1 a
Department of Physics, Indian Institute of Technology, Kanpur-208 016, India b Institute of Physics, Sachivalaya Marg, Bhubaneswar-751 005, India Received 4 August 1999; accepted 13 October 1999 by H. Eschrig
Abstract The effect of a scalar impurity in Luttinger liquid is analysed. In the weak Coulomb correlation limit, the scenario presented by Oreg and Finkel’stein seems to be correct, where the problem resembles the physics of Kondo resonance. On the contrary, we propose an emergence of new physics when the correlation energy is larger than the Kondo scale, i.e. the resonance breaks, but still either the left or the right moving electron might get localised near the impurity site, giving rise to a local moment and hence a Curie-like impurity susceptibility in the system. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Metals; D. Electron–electron interactions; D. Electronic states (localised); D. Kondo effects
The study of one-dimensional (1D) interacting fermion systems has attracted renewed attention recently due to the rapid advancement in the sub-micron technology [1]. It is well known from the exact [2] as well as the approximate solutions [3–5] that the electron–electron interaction in 1D electron gas, when away from the density-wave or the superconducting instabilities, leads to Luttinger liquid behaviour. The properties of these 1D systems are expected to be unusual in the sense that one does not have well defined single particle excitations in this state, i.e. the occupation number at the Fermi energy has a singularity rather than a jump, their density of states vanishes at the Fermi surface in a power-law way, etc. The true coherent excitations, i.e. the eigen modes of the systems are charge- and spin-density fluctuations [6,7], which are by nature bosonic, dynamically independent and in general propagate with different velocities (the so called spin–charge separation). The striking non-Fermi liquid behaviour of 1D electrons can clearly be exhibited in transport properties, especially in the presence of impurities or barriers [8,9]. For a repulsive electron–electron interaction, it has been predicted that at zero temperature even a single weak backward potential
scatterer eventually causes the conductance to vanish. It is widely accepted that the low energy physics of this system can be described by two semi-infinite chains, connected by a weak link junction [10,11]. But recently, this problem has been reconsidered by Oreg and Finkel’stein [12,13] questioning the earlier works. They addressed the problem of Fermi-edge singularity in 1D and concluded that the low energy physics in such a system together with the electron– electron repulsion resembles the physics of Kondo resonance. While this result seems to be valid in the weak Coulomb correlation limit, the strong correlation case is quite different [14]. Starting from the Kondo resonance physics of Oreg and Finkel’stein, we propose an emergence of new physics in the case of strong Coulomb correlation, that is, the resonance breaks and either the left or the right moving electron gets localised near the impurity site, giving rise to a local moment in the system. We give an estimate of the critical Coulomb correlation energy beyond which the physics of local moment could take place. Furthermore, we discuss the role of temperature and also a new phase diagram in the U 2 T plane, where U is the Coulomb correlation energy and T, the temperature. The minimal model which gives rise to Luttinger liquid behaviour is the 1D single band Hubbard model, given by
1
Present Address: Institut fu¨r Physik, Universita¨t Dortmund, 44221 Dortmund, Germany, Fax: 149-231-755 3569. E-mail address: [email protected] (D. Sa)
H 2t
X i;s
0038-1098/00/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00487-1
Ci;†s Ci11;s 1 h:c: 1
1 X U n n 2 i;s is i2s
1
264
D. Sa / Solid State Communications 113 (2000) 263–266
Ci;†s Ci;s
where is the electron operator which creates (annihilates) an electron of spin s at site i and ni;s Ci;†s Ci;s is the electron number, t and U are the hopping integral and the Coulomb correlation energy, respectively. The hopping term in the Hamiltonian can be solved exactly and the excitation spectrum is obtained as, ek 22tcosk. In the ground state, all the Fermions with j k j# kF are filled which creates a Fermi surface at the Fermi wave vector kF ^p=2. The low energy modes of this theory are given by the excitations near the Fermi points. Thus, one can linearise the dispersion near the two Fermi points, which of course, gives rise to left and right moving Fermions. Hence, the Fermion operators can be written as
interaction drives the system to a non-Fermi liquid fixed point, i.e. Luttinger liquid, the low energy excitations being the collective spin- and charge-density fluctuations. In such situations, the presence of a potential scatterer might have profound consequences. This has already been pointed out by many authors [10–14] but the problem remains unclear to date. For completeness, the effect of electron– electron interaction on a potential scatterer in 1D is analysed through Anderson’s poor man’s scaling approach [15]. In the weak coupling limit, i.e. U=pvF p 1; Vf =pvF p 1 and Vb =pvF p 1, the scaling equations for the forward as well as the backward impurity scattering up to the order of UV are written as
Ci;s e2ikF ri ai;s 1 eikF ri bi;s
dVf 0 dl
6
dVb 1r0 UVb dl
7
2
The operators a=b, which are the left/right branch Fermion operators, do not contain high momentum modes. One presumes here that these fields have slowly varying matrix elements on the scale of the lattice. While this is valid in the non-interacting theory, it is also assumed to be so even when the interactions are turned on. Thus, by neglecting the rapidly oscillatory terms, one can express the Hubbard model in terms of the left/right moving branch Fermion operators. Now, on Fourier transforming and retaining the inter-branch density density interaction term which is sufficient to yield Luttinger liquid behaviour, the Hamiltonian can be written as, H H0 1 HI , where X H0 vF k 2 kF a†k;s ak;s 1 2k 2 kF b†k;s bk;s 3 k;s
HI U
where dl 2d lnEc with Ec being the band width cut-off and r0 1=2pvF is the free 1D density of states. It is obvious from the above differential equation that the forward impurity potential becomes a marginal perturbation whereas the backward becomes a relevant one for the case of a repulsive electron–electron interaction. Thus, the backward impurity potential which flows to the strong coupling fixed point implies that the system has a cross-over scale, analogous to the Kondo scale, given by kB Tk , Ec
X 0
k;k ;q;s
a†k;s b†k 0 ;2s bk 0 1q;2s ak2q;s
1 s $ 2s
4
where vF is the Fermi velocity. Since the exchange term is irrelevant in the renormalisation group sense for the repulsive electron–electron interaction, it is neglected here. In terms of of the left/right branch Fermion operators, the scattering of the conduction band electrons by a scalar impurity can be written as X † Himp Vf ak;s ak 0 s 1 a ! b 0
k;k ;s
1 Vb
X 0
k;k s
a†k;s bk 0 s 1 a $ b
5
where Vf Vf q , 0 is the forward impurity scattering amplitude and Vb Vb q , 2kF is the backward one. For simplicity, we take these to be momentum independent. In the usual potential scattering problems in three or higher dimensions, the effect of electron–electron interaction is always neglected, since the interacting electron systems can be mapped to a nearly non-interacting systems of quasi-particles, which is the basis of Landau Fermi liquid and the problem of a scalar impurity in this case can be handled in an usual perturbation theory. The situation in 1D is completely different, where the repulsive electron–electron
V U
1 r0 U
8
The flow diagram which is rather complex is discussed extensively by Oreg and Finkel’stein in a later paper [13]. Since there are several marginal parameters characterising the impurity, each of the fixed point describing the backward impurity problem corresponds to a manifold of fixed points which can be represented by lines in a two-dimensional plot. In the limit of strong potential barrier of the impurity, the low temperature behaviour is controlled by the line of fixed points corresponding to two disconnected semi-infinite wires which is proposed by Kane et al., and many others [10]. This is because in the course of renormalisation, the strength of the large bare backward impurity potential increases further and at the final stage it transforms into a strong barrier which ultimately breaks the chain. However, a mapping of the weak impurity problem onto a Coulomb gas theory [16], and hence to the Kondo problem [17] apparently contradicts this simple intuitive picture. The scenario of Ref. [10] is based on the assumption that no other fixed points intervene in the scaling from the repulsive fixed point of a weakly scattering defect to the attractive fixed point corresponding to a tunneling junction of two half-wires. By mapping the impurity problem onto a spin1/2 Heisenberg chain, it has been shown [13] that the line of fixed points describing the limit Vb ! ∞ is unstable. This implies that there exists an another line of fixed points which
D. Sa / Solid State Communications 113 (2000) 263–266
is attractive at an intermediate value of the parameter Vb . Oreg and Finkel’stein pointed out that this is the fixed point (i.e. the weak impurity limit flows to a point where Vb is finite and Vf is small, equal to its bare value) which controls the low energy physics of the system when the bare potential of the impurity is weak. This has been utilised in the mapping of the impurity problem onto the Coulomb gas and hence to the Kondo problem where the localised mixture of left and right moving electrons act as the role of Kondo singlet. In the derivation of the scaling equations, it has been presumed that the interaction parameter U is not renormalised. This is due to the fact that U describes the electron– electron interaction in the bulk electron liquid, so the presence of a single impurity cannot influence this parameter. The appearance of a cross-over scale in the preceeding discussion suggests that the Kondo resonance physics proposed by Oreg and Finkel’stein is valid only below this scale. Thus, U also has to be less than this scale. Furthermore, in obtaining the Kondo resonance physics, a weak Coulomb correlation energy U has been used. However, the cross-over energy scale is nothing but the binding energy of the left and right moving electrons. Thus, in the present problem, there is a competition between two energy scales, the Coulomb correlation energy and the cross-over energy scale. Since the Kondo resonance physics of Oreg and Finkel’stein is valid for a weak U, the obvious question arises at this point is, what happens when the Coulomb correlation energy is larger than the cross-over scale? We try to address precisely this question in the present letter. Since U describes the Coulomb repulsion between the left and the right moving electrons in the electron liquid, strong correlation implies that the resonance is broken and hence there will be a breakdown of the Kondo resonance physics. We propose that even though the resonance breaks, still either the left or the right moving electron might get localised near the impurity site, which ultimately corresponds to a local moment in the system. Qualitatively, one can understand this in a following way: In the large U Hubbard model, the spin sector has BCS like condensate of pair of spinons. The introduction of a vacancy implies removal of a spinon from the vacant site which converts its partner from the condensate into a free spinon and it acts as a local moment in the system. This is in accordance with the earlier work on the formation of local moment near a static vacancy in the 1D large U Hubbard model [14]. Moreover, we can give an estimate of the critical Coulomb correlation Ucr , beyond which the physics of local moment will be valid. Ucr can be obtained by solving the following equation for U, U Ec
V U
1 r0 U
UU
9 cr
The appearance of a Ucr in the present formulation provides us a prescription to distinguish between the weak as well as the strong Coulomb correlation regime. When
265
U , Ucr , it is the case of weak Coulomb correlation where the problem of the scalar impurity in the Luttinger liquid maps to the problem of Kondo resonance proposed by Oreg and Finkel’stein. On the other hand, the strong correlation case corresponds to U . Ucr , where the resonance breaks and there is a local moment formation in the system. Therefore, as usual, one is expected to get a Curie-like impurity susceptibility in the system. Of course, the local moment will interact with the other excitations in the system, which will ultimately lead to a problem similar to that of a magnetic impurity in a Luttinger liquid [18,19]. But in the present case, since the local moment appears at zero temperature, one might ask, what happens to the moment at high temperature phase? We believe that the moment might survive to high temperature also, since the phenomenon of local moment occurs here is only due to the strong bulk Coulomb repulsion. This might be an important difference in the usual problem of a magnetic impurity in a Luttinger liquid and the present case. The stability of the local moment and the related phenomenon for U . Ucr with respect to the variation in temperature is currently under investigation [20]. Now, let us consider the interplay of temperature in the impurity problem explicitely. The Kondo resonance physics of Oreg and Finkel’stein which is valid in the weak Coulomb correlation limit, i.e. U , Ucr , will break down again when the temperature (T) is larger than the Kondolike temperature (Tk ) (Tk is analogous to the Kondo temperature, given in Eq. (8)). But in this case, the high temperature phase is just the problem of a potential scatterer in a Luttinger liquid which can well be described by a usual perturbation theory. This prompts us to propose a novel phase diagram in the U 2 T plane. For U , Ucr , one has two regimes corresponding to T . Tk which is the regime of usual potential scattering problem in a Luttinger liquid where perturbation theory is valid and for T , Tk , it is the Kondo resonance regime of Oreg and Finkel’stein. On the other hand for U . Ucr , the physics of local moment appears which might persist to high temperature also. This shows that even though the pure model in both the weak as well as the strong Coulomb correlation limit yields Luttinger liquid behaviour [2,21], the presence of a weak potential scatterer makes both the limit distinct from each other. In recent times, since there is rapid advancement in the fabrication of nano-materials such as quantum dots, semiconducting quantum wires etc., the present model study can be tested through a straightforward substitutional study which will definitely help in our understanding on the properties of low-dimensional materials. In conclusion, we briefly summarise the main results of the paper. The physics of a scalar impurity in an 1D interacting electron gas is analysed in both the weak as well as the strong bulk Coulomb correlation limit. In the weak correlation limit, i.e. when U , Ucr , the problem maps to the Kondo resonance physics proposed by Oreg and Finkel’stein whereas in the strong Coulomb correlation
266
D. Sa / Solid State Communications 113 (2000) 263–266
case, i.e. for U . Ucr , the theory breaks down. In the latter case, we propose that even if the resonance breaks, still either the left or the right moving electron might get localised giving rise to a local moment in the system. This is in accordance with our earlier work on the formation of a local moment near a static vacancy in 1D large U Hubbard model. We give an estimate of the critical Coulomb correlation from which one can distinguish the weak as well as the strong correlation regime. Furthermore, we discuss the interplay of temperature as well as the U 2 T phase diagram in the impurity problem. This indicates that even though the pure model in both the weak as well as the strong Coulomb correlation limit yields Luttinger liquid behaviour, the ground state in presence of a weak scalar impurity in both the cases are quite different from each other.
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