- Email: [email protected]
The independent-pair model for one-dimensional fermions interacting through a repulsive delta-potential
20 January
&A
I997
PHYSICS
LETTERS
A
I
Physics Letters A 224 ( 1997) 367-37
The independent-pair model for one-dimensional fermions interacting through a repulsive delta-potential ED. Buzatu I Department Received
23 August
of Theoretical Physics, Institute of Atomic Physics, Bucharesr MC-6 1996; revised manuscript received 21 October 1996; accepted Communicated by V.M. Agranovich
R-76900, Romania
for publication
23 October
1996
Abstract The single-particle energies and the ground-state energy for a one-dimensional system of interacting fermions via a repulsive S-function potential are self-consistently calculated using the Brueckner and Gammel method. The breakdown of the usual quasi-particle description is signaled in this approach by the occurrence of a negative gap at SF in the excitation spectrum. A new ground-state configuration of the system emerges: a reduced Fermi sea and an assembly of bosons at fkr. The ground-state energy computed within this scheme is in remarkable agreement with the Bethe-ansatz results at all coupling constants.
It is well known that the Landau’s theory of the Fermi liquids breaks down in one dimension (ID) (see, e.g., Section 2.2 from Ref. [ l] ; the breakdown of the Landau quasi-particle description is also signaled by a singular forward scattering, as it was pointed out (for the 2D Hubbard model) in Ref. [ 21) . An alternative theory, based on an exactly soluble model [ 31 by the bosonization technique [ 41, leads to the concept of the Luttinger liquid [ 51, with distinct properties [ 11. Other methods, like the renormalization group [ 61 or Bethe-ansatz (BA) [ 71 ones, provide a quite rich picture of the 1D systems. Nevertheless, the traditional methods of the many-body theory can give a much more intuitive image of the specific phenomena. The repulsive &function interaction between the fermions suggested us to use a well-known method introduced in nuclear physics many years ago in order to over-
the Brueckner theory (BT) [ 81. We shall apply here the Brueckner and Gammel (BG) method [ 91 to determine self-consistently the single-particle energies (SPEs) and the ground-state energy (GSE) for the 1D repulsive delta model (RDM), i.e. a system of fermions interacting via a repulsive S-function potential, its Bethe-ansatz solution [IO] providing a good test to check our considerations. The Hamiltonian for the 1D RDM in the second quantization (momentum representation) reads
come the difficulties
where, in the chosen units,
associated
with the hard-core
’ E-mail: [email protected].
in-
teractions:
kn +
’
c c6,+k,.,c~~-kl,o’Ck~.~‘Ck,.rr. kl e-1 .u.d
(2)
03759601/97/$17.00
0 1997 Elsevier Science
I
368
F.D. Buzutu / Physics Letters A 224
is the SPE of the free fermions, C > 0 is the (dimensionless) coupling constant and ci rr (~k,~) denotes the usual Fermi creation (annihilation) operator of a one-particle state with a given momentum k (in units of the Fermi momentum kF) and spin g = &l/2. It was assumed in ( 1) that the system is enclosed in a large box of finite length but we shall consider in fact only the thermodynamic limit. In the BA approach, the GSE of the ID RDM can be found by solving an integral equation; the numerical results [ 111 have been compared with other approximate schemes, the T-matrix approximation (TMA) [ 121 giving the best agreement with the BA solution. The idea of the BT can be easily understood by comparing it with the Hat-tree-Fock method: if the last considers the motion of a particle in the average field produced by the others (independent-particle model), in the Brueckner approach the interaction of any two particles is treated exactly and the effect of the rest of the particles on the interacting pair is replaced by an average (independent-pair pair model). According to the BT, the effective interaction between two particles is described by the reaction matrix K, the solution of an integral equation depending on the bare potential and on the still undetermined SPE (in our terminology, the T-matrix corresponds to the solution of the same integral equation but with the self-consistent single-particle energies replaced by the kinetic energies) ; these energies are assumed to have the HartreeFock form, with the one-particle potential given by the diagonal elements of the K-matrix. Formally, the problem reduces to finding a solution for the SPE from these two coupled equations; however, while there is a unique prescription to calculate the energy E of a particle in the ground state, i.e. its on-energy-shell (OnES) propagation, the energy E of a particle in an excited state cannot be uniquely defined when the effects of the off-energy-shell (OffES) propagation are taken into account. BG [ 91 assumed a particular form for E, depending parametrically on the total energy 0 of the initial two-particle state ( OnES) under consideration (E depends also on some average excitation energy A, but it enters as a parameter in the whole calculation of both E and E, a dependence which will not be explicitly specified). In the particular case of a constant interaction in the k-space, the K-matrix equation can be solved immediately independently of the SPE form. The resulting
f 19961367-371
integral equations BG method are
for E and E for the ID RDM in the
121 < 1,
40 =q)(O + V(I), e(h;W
=&g(h)
+V(h;rz),
IhI > 1,
(3)
where the one-particle potential V (V) experienced by a particle in an unexcited (excited) state, as a result of the interaction with all the other particles, is given by
K = If
I’,
K=l+h, =~[K/2fq;~(l,l’)l
e(q;l,I’)
+~[K/2-q;fl(l,l’)l K=l+l’, e(q;K,O)
-L?(l,l’),
n( I, I’) = &(I) + E( I’) =e(K/2+q;f2)
+e(K/2-q;fl)
-fi+ll, K=l+h, fl=
parameter E [2&(0),2&(l)].
The range VD, appearing defined by %
= (--00,~]K1/2-
(5)
in the integrals in Eqs. (4))
1) U (JKl/2+
l,+co)
(6)
and consistent with the form of the denominators (5), restricts all the intermediate scatterings to the states above the Fermi level. The definition (6) is a consequence of two requirements: (i) only the principal part of the q-integrals (in general, over all possible values of q) has to be taken; (ii) the exclusion principle is satisfied in the intermediate unexcited states. However, let us note that the definition (6) follows from (i) and (ii) only if all the energies of the excited states are higher than those of the unexcited ones, an implicit assumption for a normal Fermi liquid.
F.D. Bu:atu/Physics
Letters A 224 (1996) 367-371
369
The parameter A has been introduced as an average excitation energy of a particle. In the nuclear matter calculations it was considered an arbitrary quantity taking values from 0 to E( 1) - E( 0) corresponding to the energy difference of particles lying respectively at the top and the bottom of the Fermi sea; the results depend only weakly on the values of A [ 91, justifying, in the opinion of Brueckner and Gammel, the introduced approximations. Let us note that the above range of A assumes implicitly that there is no gap at the Fermi level between the OnES and OffES propagations (in fact it occurs for the nuclear matter [ 131) ; by allowing such a possibility, we should define the range as AE [e(l;&,,)
-E(l),~(l;&J
-E(O)l,
(7)
where fi,,, denotes the maximum value of R (we shall use also ami,, with a similar meaning). However, the definition of A is still ambiguous: the limits of the range (7) depend themselves on the initial value of A used in the determination of E and E. In a full self-consistent treatment A should be uniquely determined from its definition; this can be done by averaging the momentum dependence of E. We defined A as the arithmetical mean of the two limits of its range (7)> A=e(l;&,)-$(O)+E(~)].
(8)
This choice corresponds to a minimum average excitation energy (assuming E a convex function of its variable, as it follows from the numerical computations). The condition (8) determines self-consistently the parameter A. The integral coupled equations (3) can be solved numerically by iteration, starting with the free expression (2) for the SPE and with A = 0; the first iteration results (Th4A) in the C = cc case are represented in Fig. 1 by dashed lines. For the OffES propagation only two curves have been drawn: the lower (upper) one corresponds to fi = J.&,, ( O,,,i,). For k 2 3 the OffES curves join the free dispersion law E~( k), a consequence of the assumption that in the intermediate states IKI < 2. The same property occurred in the BG results for [kl 2 2.6k, [9] and reflects the fact that the excitations with the momentum far from the Fermi level obey the same dispersion law as in the free case. But a more important remark is the discontinuity at the Fermi momentum (1 in our units) between
Uomcntum Fig. 1. The SPE versus the momentum (in units of k,) RDM with an infinite coupling constant C in the TMA
-A for the ID and KMA.
F( 1) and E( 1; L&,) corresponding to a negative gap (NG) and not to a positive one as in the nuclear matter calculations [ 131. We could try to avoid the occurrence of such a NG at each iteration step by increasing the value of A; the convergence is reached after a few iterations, but the value of A required to overcome the NG will be much greater (at least for strong couplings) than the value indicated by the right side of Eq. (8). It follows that the NG is unavoidable. As a result, the denominators e given by Eqs. (5) will vanish for some values of the q-variable, contradicting thus the condition (i) mentioned above. Consequently, only the principal part of the q-integrals from Eqs. (4) must be taken; in our case, the definition (6) of the range V, does not guarantee this fact. The selfconsistent solution of the system (3) and (8), i.e. in the K-matrix approximation (KMA) corresponds for C = co to A = 0; the results for the SPE are represented in Fig. 1 by continuous lines. The final value A of the NG is approximately -0.6. For smaller values of the interaction constant C, both E and E get closer to the free dispersion law Ed, concomitantly with a reduction of the NG and an increase of A; in the C + 0 limit, A --+ 0 and A -+ 0.5. From Fig. 1 it follows also that the main effect of the interaction on the OnES propagation is to shift the values of E,, by a constant quantity, i.e. the average potential V experienced by a fermion in the ground-state depends very weakly on its momentum; this is of course a consequence of the k-independence of the bare potential in the momentum representation.
370
F.D. Buzcrtu/Phy.ysic.s Letters A 224 (1996) 367-371 I
I
[~g([)+fV(l)]dl=$+l
oxlI
r. -1
V(l) dl, b
(10)
jo., -
5
__:
___iik --.
0.1
Intcractlon
constant
C
Fig. 2. The GSE of the ID RDM in the TMA and F’BPas function
of the interaction constant.
The occurrence of a NG at the Fermi surface, a consequence of the kinetic restrictions imposed by the momentum conservation on the two-body scatterings, signals the breakdown of the usual quasi-particle picture in 1D: it is related, through the diagonal elements of the reaction matrix entering in the calculation of the SPE, with a singular forward scattering of two particles [2]. The NG determines an instability of the Fermi surface, and we shall assume that at the top of the unstable Fermi sea, the particles will fall down to their lowest energetic positions from the excitation spectrum (corresponding to k = rtl) ; this process ends when the level of the remaining Fermi sea is as high as the upper level of the filled vertical edges at the Fermi momentum, i.e. when
F(q)
=E[1;4Kp)l,
(9)
a condition that defines the effective Fermi momentum K~. The particles at k = &l can be viewed either as fermions, if one employs an R-type label for their states, or as bosons, in the momentum representation; they can be considered as a ID effect induced by the necessity of taking into account diagrams in the linked cluster expansion contradicting the exclusion principle [ 81. Consequently, the ground-state configuration in this fermion-boson picture (FBP) consists of a reduced sea of core fermions for Ikl < K~ and a certain number of shell bosons at k = f 1 (a fermion-boson duality has been previously emphasized in Ref. [ 141). In the BT, the GSE is computed from
where E represents the density of the GSE in units of kFeO( kF) . The values of E for C < 6 in the TMA are represented in Fig. 2 by dashed line; they coincide with the results obtained previously [ 121. The maximum (relative) deviation from the BA values [ 1 l] is realized when C = co: it decreases from 20% in the TMA to 9% in the KMA (without taking into account the NG problem). In calculating the GSE in the FBP, we divided the last integral from Eq. ( 10) in two parts: from 0 to K~, computed by using the KMA results for V, and from K~ to 1. The contribution of the second term was assumed to change with some parameter t E [0, l] describing the continuous transformationfrom{1,V(Z)}at~=Oto{1,1/[1;2~(I)]} att=l;therange[KF,l]ismappedforanyt # 1 in another one [ K~, 11, with K~ = K~ and lim,,, K, = 1, while the transformed values of V(I) are finite at any 1. It follows that the second term goes to zero, i.e. the contribution of the shell bosons to the GSE is the same as that corresponding to the free particles above the K,-level. The values of E calculated in this manner coincide numerically with the BA results [ 1I] at any interaction constant: in the C = cc case we obtained K~ = 0.876 and E = 0.849, while the BA value is E = 8/37r; for C < 6, the results in the FBP are along the continuous curve from Fig. 2 and they are very close to the KMA ones (in the weak coupling regime, K~ N 1 and the NG problem becomes irrelevant for the GSE) . Within the FBP, the lowest-energy excitations starting from the interacting ground-state correspond to those of the shell bosons: a spin exchange between them across the two edges gives rise to a 2k,-spin density wave and a similar transfer of a boson (with spin 0 and momentum 2k,) corresponds to a 4kF-charge density wave, indicating the charge-spin separation that occurs in ID systems. A detailed discussion of the excitation spectrum will be given elsewhere. The occurrence of a NG in the excitation spectrum of a Fermi liquid with pure repulsive interactions is, in our opinion, strictly related to the dimensionality of the system; it has been also found in the 1D Hubbard model [ 151. The existence of such a gap could be
F.D. Bu:atu/Physics
relevant in the dispute about a possible breakdown of the usual Fermi liquid behavior in 2D. Let us shortly discuss the relation between the BT, based on the Goldstone expansion, and other approximate schemes from the Feynman-Dyson (timedependent) perturbation theory. In the Goldstone expansion the particles in intermediate states have physical energies (SPE) depending on their momentum and the virtual nature of the intermediate states is summarized in the energy denominators; in contrast, in the Feynman-Dyson expansion the intermediate particles can propagate with any frequency, independent of the momentum. Consequently, the breakdown of the Fermi liquid theory in 1D has different aspects: while in terms of the Green functions the Peierls instability prevents the quasiparticle formation, within the BT the “particles” propagate with the determined SPE but they are unstable (at least around the Fermi level). In studying the response of a 1D system to an external perturbation with the aid of the Green functions, the competition between the Peierls and Cooper instabilities requires the inclusion of both ladder and ring diagrams in a consistent manner, as it is realized within the parquet approximation [ 161. On the other hand, for systems in the low-density limit (small concentrations or/and short-range interactions) and for properties like GSE (where only averages over the frequency are required), the contributions of the ladder diagrams dominate not only in 3D but in ID too, as the comparison with exact results indicates. The simplest BT (no OffES effects) could be viewed as a self-consistent, frequency averaged version of the ladder approximation [ 17,181. In conclusion, one might say that the BA results for the GSE of a 1D system of fermions interacting through a repulsive S-potential can be reproduced within the approximate BG method with a certain, natural self-consistency condition imposed on the parameter n of this theory and interpreting, in an appropriate manner, the instability induced by the occurrence of the NG at the Fermi level. One might have expected perhaps such a result, as the system under considera-
Letters A 224 (1996) 367-371
371
tion is an extremely dilute fermion gas. In addition, this independent-pair approximation suggests some possible connections with well known results from the BA and bosonization theory. I am grateful to M. Apostol, R. Hayn, V.I. Inozemtsev, F. Mancini, D. Villani and J. Voit for helpful suggestions, remarks and discussions on this subject.
References ] I ] J. Voit, Rep. Prog. Phys. 58 (1995) 977. [2] PW. Anderson, Phys. Rev. Lett. 64 ( 1990) 1839; 65 ( 1990) 2306; P.W. Anderson and D. Khveshchenko, Phys. Rev. B 52 (1995) 16415. [3] J.M. Luttinger, J. Math. Phys. 4 (1963) 1154; DC. Mattis and E.H. Lieb, J. Math. Phys. 6 (1965) 304. [4] V.J. Emery, in: Highly conducting one-dimensional solids, eds. J.T. Devreese, R.E. Evrard and V.E. van Doren (Plenum, New York, 1979). [S] F.D.M. Haldane, J. Phys. C 14 (1981) 2585. [6] J. S6lyom. Adv. Phys. 28 ( 1979) 201. [7] B. Sutherland, in: Lecture notes in physics, Vol. 242. Exactly solvable problems in condensed matter and relativistic field theory (Springer, Berlin, 1985) p. 1. [ 8 I B.D. Day, Rev. Mod. Phys. 39 ( 1967) 719, and references therein. 19 K.A. Brueckner and J.L. Gammel, Phys. Rev. 109 (1958) 1023. 110 M. Gaudin, Phys. Lett. A 24 (1967) 55; C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. W.I. Friesen and B. Bergersen, J. Phys. C 13 (1980) 6627. [ll 112 1 S. Nagano and K.S. Singwi, Phys. Rev. B 27 ( 1983) 6732. ]13 K.A. Brueckner and D.T. Goldman, Phys. Rev. 117 ( 1960) 207. [ 141 M. Apostol, V. Btirsan and C. Mantea, Rev. Roum. Phys. 29 (1984) 673. [ 151 ED. Buzatu, R. Hayn and V. Oudovenko. in preparation. [ 161 Yu.A. Bychkov, L.P Gor’kov and I.E. Dzyaloshinskii, Zh. Eksp. Tear. Fiz. 50 ( 1966) 738 [Sov. Phys. JETP 23 ( 1966) 4891. [ 171 V.M. Galitskii, Zh. Eksp. Teor. Fiz. 34 (1958) 151 [Sov. Phys. JETP 7 (1958) 1041. [ IS] A.L. Fetter and J.D. Walecka, Quantum theory of manyparticle systems (McGraw-Hill, New York, 1971) Sect, 42.
&A
I997
PHYSICS
LETTERS
A
I
Physics Letters A 224 ( 1997) 367-37
The independent-pair model for one-dimensional fermions interacting through a repulsive delta-potential ED. Buzatu I Department Received
23 August
of Theoretical Physics, Institute of Atomic Physics, Bucharesr MC-6 1996; revised manuscript received 21 October 1996; accepted Communicated by V.M. Agranovich
R-76900, Romania
for publication
23 October
1996
Abstract The single-particle energies and the ground-state energy for a one-dimensional system of interacting fermions via a repulsive S-function potential are self-consistently calculated using the Brueckner and Gammel method. The breakdown of the usual quasi-particle description is signaled in this approach by the occurrence of a negative gap at SF in the excitation spectrum. A new ground-state configuration of the system emerges: a reduced Fermi sea and an assembly of bosons at fkr. The ground-state energy computed within this scheme is in remarkable agreement with the Bethe-ansatz results at all coupling constants.
It is well known that the Landau’s theory of the Fermi liquids breaks down in one dimension (ID) (see, e.g., Section 2.2 from Ref. [ l] ; the breakdown of the Landau quasi-particle description is also signaled by a singular forward scattering, as it was pointed out (for the 2D Hubbard model) in Ref. [ 21) . An alternative theory, based on an exactly soluble model [ 31 by the bosonization technique [ 41, leads to the concept of the Luttinger liquid [ 51, with distinct properties [ 11. Other methods, like the renormalization group [ 61 or Bethe-ansatz (BA) [ 71 ones, provide a quite rich picture of the 1D systems. Nevertheless, the traditional methods of the many-body theory can give a much more intuitive image of the specific phenomena. The repulsive &function interaction between the fermions suggested us to use a well-known method introduced in nuclear physics many years ago in order to over-
the Brueckner theory (BT) [ 81. We shall apply here the Brueckner and Gammel (BG) method [ 91 to determine self-consistently the single-particle energies (SPEs) and the ground-state energy (GSE) for the 1D repulsive delta model (RDM), i.e. a system of fermions interacting via a repulsive S-function potential, its Bethe-ansatz solution [IO] providing a good test to check our considerations. The Hamiltonian for the 1D RDM in the second quantization (momentum representation) reads
come the difficulties
where, in the chosen units,
associated
with the hard-core
’ E-mail: [email protected].
in-
teractions:
kn +
’
c c6,+k,.,c~~-kl,o’Ck~.~‘Ck,.rr. kl e-1 .u.d
(2)
03759601/97/$17.00
0 1997 Elsevier Science
I
368
F.D. Buzutu / Physics Letters A 224
is the SPE of the free fermions, C > 0 is the (dimensionless) coupling constant and ci rr (~k,~) denotes the usual Fermi creation (annihilation) operator of a one-particle state with a given momentum k (in units of the Fermi momentum kF) and spin g = &l/2. It was assumed in ( 1) that the system is enclosed in a large box of finite length but we shall consider in fact only the thermodynamic limit. In the BA approach, the GSE of the ID RDM can be found by solving an integral equation; the numerical results [ 111 have been compared with other approximate schemes, the T-matrix approximation (TMA) [ 121 giving the best agreement with the BA solution. The idea of the BT can be easily understood by comparing it with the Hat-tree-Fock method: if the last considers the motion of a particle in the average field produced by the others (independent-particle model), in the Brueckner approach the interaction of any two particles is treated exactly and the effect of the rest of the particles on the interacting pair is replaced by an average (independent-pair pair model). According to the BT, the effective interaction between two particles is described by the reaction matrix K, the solution of an integral equation depending on the bare potential and on the still undetermined SPE (in our terminology, the T-matrix corresponds to the solution of the same integral equation but with the self-consistent single-particle energies replaced by the kinetic energies) ; these energies are assumed to have the HartreeFock form, with the one-particle potential given by the diagonal elements of the K-matrix. Formally, the problem reduces to finding a solution for the SPE from these two coupled equations; however, while there is a unique prescription to calculate the energy E of a particle in the ground state, i.e. its on-energy-shell (OnES) propagation, the energy E of a particle in an excited state cannot be uniquely defined when the effects of the off-energy-shell (OffES) propagation are taken into account. BG [ 91 assumed a particular form for E, depending parametrically on the total energy 0 of the initial two-particle state ( OnES) under consideration (E depends also on some average excitation energy A, but it enters as a parameter in the whole calculation of both E and E, a dependence which will not be explicitly specified). In the particular case of a constant interaction in the k-space, the K-matrix equation can be solved immediately independently of the SPE form. The resulting
f 19961367-371
integral equations BG method are
for E and E for the ID RDM in the
121 < 1,
40 =q)(O + V(I), e(h;W
=&g(h)
+V(h;rz),
IhI > 1,
(3)
where the one-particle potential V (V) experienced by a particle in an unexcited (excited) state, as a result of the interaction with all the other particles, is given by
K = If
I’,
K=l+h, =~[K/2fq;~(l,l’)l
e(q;l,I’)
+~[K/2-q;fl(l,l’)l K=l+l’, e(q;K,O)
-L?(l,l’),
n( I, I’) = &(I) + E( I’) =e(K/2+q;f2)
+e(K/2-q;fl)
-fi+ll, K=l+h, fl=
parameter E [2&(0),2&(l)].
The range VD, appearing defined by %
= (--00,~]K1/2-
(5)
in the integrals in Eqs. (4))
1) U (JKl/2+
l,+co)
(6)
and consistent with the form of the denominators (5), restricts all the intermediate scatterings to the states above the Fermi level. The definition (6) is a consequence of two requirements: (i) only the principal part of the q-integrals (in general, over all possible values of q) has to be taken; (ii) the exclusion principle is satisfied in the intermediate unexcited states. However, let us note that the definition (6) follows from (i) and (ii) only if all the energies of the excited states are higher than those of the unexcited ones, an implicit assumption for a normal Fermi liquid.
F.D. Bu:atu/Physics
Letters A 224 (1996) 367-371
369
The parameter A has been introduced as an average excitation energy of a particle. In the nuclear matter calculations it was considered an arbitrary quantity taking values from 0 to E( 1) - E( 0) corresponding to the energy difference of particles lying respectively at the top and the bottom of the Fermi sea; the results depend only weakly on the values of A [ 91, justifying, in the opinion of Brueckner and Gammel, the introduced approximations. Let us note that the above range of A assumes implicitly that there is no gap at the Fermi level between the OnES and OffES propagations (in fact it occurs for the nuclear matter [ 131) ; by allowing such a possibility, we should define the range as AE [e(l;&,,)
-E(l),~(l;&J
-E(O)l,
(7)
where fi,,, denotes the maximum value of R (we shall use also ami,, with a similar meaning). However, the definition of A is still ambiguous: the limits of the range (7) depend themselves on the initial value of A used in the determination of E and E. In a full self-consistent treatment A should be uniquely determined from its definition; this can be done by averaging the momentum dependence of E. We defined A as the arithmetical mean of the two limits of its range (7)> A=e(l;&,)-$(O)+E(~)].
(8)
This choice corresponds to a minimum average excitation energy (assuming E a convex function of its variable, as it follows from the numerical computations). The condition (8) determines self-consistently the parameter A. The integral coupled equations (3) can be solved numerically by iteration, starting with the free expression (2) for the SPE and with A = 0; the first iteration results (Th4A) in the C = cc case are represented in Fig. 1 by dashed lines. For the OffES propagation only two curves have been drawn: the lower (upper) one corresponds to fi = J.&,, ( O,,,i,). For k 2 3 the OffES curves join the free dispersion law E~( k), a consequence of the assumption that in the intermediate states IKI < 2. The same property occurred in the BG results for [kl 2 2.6k, [9] and reflects the fact that the excitations with the momentum far from the Fermi level obey the same dispersion law as in the free case. But a more important remark is the discontinuity at the Fermi momentum (1 in our units) between
Uomcntum Fig. 1. The SPE versus the momentum (in units of k,) RDM with an infinite coupling constant C in the TMA
-A for the ID and KMA.
F( 1) and E( 1; L&,) corresponding to a negative gap (NG) and not to a positive one as in the nuclear matter calculations [ 131. We could try to avoid the occurrence of such a NG at each iteration step by increasing the value of A; the convergence is reached after a few iterations, but the value of A required to overcome the NG will be much greater (at least for strong couplings) than the value indicated by the right side of Eq. (8). It follows that the NG is unavoidable. As a result, the denominators e given by Eqs. (5) will vanish for some values of the q-variable, contradicting thus the condition (i) mentioned above. Consequently, only the principal part of the q-integrals from Eqs. (4) must be taken; in our case, the definition (6) of the range V, does not guarantee this fact. The selfconsistent solution of the system (3) and (8), i.e. in the K-matrix approximation (KMA) corresponds for C = co to A = 0; the results for the SPE are represented in Fig. 1 by continuous lines. The final value A of the NG is approximately -0.6. For smaller values of the interaction constant C, both E and E get closer to the free dispersion law Ed, concomitantly with a reduction of the NG and an increase of A; in the C + 0 limit, A --+ 0 and A -+ 0.5. From Fig. 1 it follows also that the main effect of the interaction on the OnES propagation is to shift the values of E,, by a constant quantity, i.e. the average potential V experienced by a fermion in the ground-state depends very weakly on its momentum; this is of course a consequence of the k-independence of the bare potential in the momentum representation.
370
F.D. Buzcrtu/Phy.ysic.s Letters A 224 (1996) 367-371 I
I
[~g([)+fV(l)]dl=$+l
oxlI
r. -1
V(l) dl, b
(10)
jo., -
5
__:
___iik --.
0.1
Intcractlon
constant
C
Fig. 2. The GSE of the ID RDM in the TMA and F’BPas function
of the interaction constant.
The occurrence of a NG at the Fermi surface, a consequence of the kinetic restrictions imposed by the momentum conservation on the two-body scatterings, signals the breakdown of the usual quasi-particle picture in 1D: it is related, through the diagonal elements of the reaction matrix entering in the calculation of the SPE, with a singular forward scattering of two particles [2]. The NG determines an instability of the Fermi surface, and we shall assume that at the top of the unstable Fermi sea, the particles will fall down to their lowest energetic positions from the excitation spectrum (corresponding to k = rtl) ; this process ends when the level of the remaining Fermi sea is as high as the upper level of the filled vertical edges at the Fermi momentum, i.e. when
F(q)
=E[1;4Kp)l,
(9)
a condition that defines the effective Fermi momentum K~. The particles at k = &l can be viewed either as fermions, if one employs an R-type label for their states, or as bosons, in the momentum representation; they can be considered as a ID effect induced by the necessity of taking into account diagrams in the linked cluster expansion contradicting the exclusion principle [ 81. Consequently, the ground-state configuration in this fermion-boson picture (FBP) consists of a reduced sea of core fermions for Ikl < K~ and a certain number of shell bosons at k = f 1 (a fermion-boson duality has been previously emphasized in Ref. [ 141). In the BT, the GSE is computed from
where E represents the density of the GSE in units of kFeO( kF) . The values of E for C < 6 in the TMA are represented in Fig. 2 by dashed line; they coincide with the results obtained previously [ 121. The maximum (relative) deviation from the BA values [ 1 l] is realized when C = co: it decreases from 20% in the TMA to 9% in the KMA (without taking into account the NG problem). In calculating the GSE in the FBP, we divided the last integral from Eq. ( 10) in two parts: from 0 to K~, computed by using the KMA results for V, and from K~ to 1. The contribution of the second term was assumed to change with some parameter t E [0, l] describing the continuous transformationfrom{1,V(Z)}at~=Oto{1,1/[1;2~(I)]} att=l;therange[KF,l]ismappedforanyt # 1 in another one [ K~, 11, with K~ = K~ and lim,,, K, = 1, while the transformed values of V(I) are finite at any 1. It follows that the second term goes to zero, i.e. the contribution of the shell bosons to the GSE is the same as that corresponding to the free particles above the K,-level. The values of E calculated in this manner coincide numerically with the BA results [ 1I] at any interaction constant: in the C = cc case we obtained K~ = 0.876 and E = 0.849, while the BA value is E = 8/37r; for C < 6, the results in the FBP are along the continuous curve from Fig. 2 and they are very close to the KMA ones (in the weak coupling regime, K~ N 1 and the NG problem becomes irrelevant for the GSE) . Within the FBP, the lowest-energy excitations starting from the interacting ground-state correspond to those of the shell bosons: a spin exchange between them across the two edges gives rise to a 2k,-spin density wave and a similar transfer of a boson (with spin 0 and momentum 2k,) corresponds to a 4kF-charge density wave, indicating the charge-spin separation that occurs in ID systems. A detailed discussion of the excitation spectrum will be given elsewhere. The occurrence of a NG in the excitation spectrum of a Fermi liquid with pure repulsive interactions is, in our opinion, strictly related to the dimensionality of the system; it has been also found in the 1D Hubbard model [ 151. The existence of such a gap could be
F.D. Bu:atu/Physics
relevant in the dispute about a possible breakdown of the usual Fermi liquid behavior in 2D. Let us shortly discuss the relation between the BT, based on the Goldstone expansion, and other approximate schemes from the Feynman-Dyson (timedependent) perturbation theory. In the Goldstone expansion the particles in intermediate states have physical energies (SPE) depending on their momentum and the virtual nature of the intermediate states is summarized in the energy denominators; in contrast, in the Feynman-Dyson expansion the intermediate particles can propagate with any frequency, independent of the momentum. Consequently, the breakdown of the Fermi liquid theory in 1D has different aspects: while in terms of the Green functions the Peierls instability prevents the quasiparticle formation, within the BT the “particles” propagate with the determined SPE but they are unstable (at least around the Fermi level). In studying the response of a 1D system to an external perturbation with the aid of the Green functions, the competition between the Peierls and Cooper instabilities requires the inclusion of both ladder and ring diagrams in a consistent manner, as it is realized within the parquet approximation [ 161. On the other hand, for systems in the low-density limit (small concentrations or/and short-range interactions) and for properties like GSE (where only averages over the frequency are required), the contributions of the ladder diagrams dominate not only in 3D but in ID too, as the comparison with exact results indicates. The simplest BT (no OffES effects) could be viewed as a self-consistent, frequency averaged version of the ladder approximation [ 17,181. In conclusion, one might say that the BA results for the GSE of a 1D system of fermions interacting through a repulsive S-potential can be reproduced within the approximate BG method with a certain, natural self-consistency condition imposed on the parameter n of this theory and interpreting, in an appropriate manner, the instability induced by the occurrence of the NG at the Fermi level. One might have expected perhaps such a result, as the system under considera-
Letters A 224 (1996) 367-371
371
tion is an extremely dilute fermion gas. In addition, this independent-pair approximation suggests some possible connections with well known results from the BA and bosonization theory. I am grateful to M. Apostol, R. Hayn, V.I. Inozemtsev, F. Mancini, D. Villani and J. Voit for helpful suggestions, remarks and discussions on this subject.
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