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Pair distribution function and off-diagonal long range order parameter of one-dimensional spinless fermion system
Physics Letters A 305 (2002) 289–297 www.elsevier.com/locate/pla
Pair distribution function and off-diagonal long range order parameter of one-dimensional spinless fermion system Yu-Liang Liu Center for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China Received 5 July 2002; received in revised form 14 October 2002; accepted 15 October 2002 Communicated by R. Wu
Abstract Using the eigen-functional theory, we derive the exact functional expressions of pair distribution function, off-diagonal long range order parameter and fermion Green’s function of a one-dimensional spinless fermion system. With the linear spectrum approximation and for weak fermion interaction, we calculate these functions and the ground state wave function of the system. Moreover, under this approximation we show that the commutation relations of the fermion density operators are not influenced by the fermion interaction, and explain why operators with the same conformal dimensions at V = 0 have different conformal dimensions when fermion interaction is turned on (V = 0). 2002 Elsevier Science B.V. All rights reserved.
Strongly correlated system is one of the most interesting and hardest problem [1–10] in condensed matter physics nowadays, of which the most prominent character is the strong particle correlation that completely controls its low energy behavior. This strong particle correlation is controlled by the particle interactions, thus the key point in treating this kind of problem is how to treat the particle interaction (potential) terms accurately. Applying usual perturbation methods [11,12] to strongly correlated systems meet many serious problems. Because of the strong correlation among particles, there is no a small suitable quantity to be used as a perturbation expansion parameter. It is natural that one uses other ways to treat strongly correlated systems, for example, treating the interaction (potential) terms exactly and then exactly or perturbatively treating the kinetic energy term of the particles. This is the strategy of bosonization method in treating 1D interacting fermion systems [13–19]. The eigen-functional theory [20] provides a new way to treat general strongly correlated systems, in which it ends in solving equation for the phase fields which are determined by the particle interaction, and reflects the particle correlation of the systems directly. There are two key steps in eigen-functional theory, one is that under the path integral formulation, we exactly map a D-dimensional strongly correlated system into an effective (D + 1)dimensional time-dependent “single-particle” problem. By solving the resulting eigen-functional equation of the propagator operator of the particles, we can obtain the formally exact functional expressions of the action and a variety of correlation functions of the system, respectively. By taking the functional average over Lagrangian
E-mail address: [email protected] (Y.-L. Liu). 0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 1 4 2 2 - 6
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multiplier fields, we obtain the corresponding physical quantities. The most prominent character of the eigenfunctional theory is that in the eigen-functional representation, the particle operators are non-interacting, and the particle correlation is determined by the phase fields. In this Letter, we study a one-dimensional spinless fermion system with repulsive interaction with the eigenfunctional theory. We first derive the formally exact functional expressions of off-diagonal long range order parameter, pair distribution function, fermion Green’s function and ground state wave function of the system, then calculate these functions under the linear spectrum approximation and for weak fermion interaction. Moreover, we prove that the commutation relations of the fermion density operators are not influenced by the fermion interaction, and clarify some ambiguities in usual bosonization method. We now consider a one-dimensional spinless fermion system with repulsive interaction, † † µcˆi cˆi + V nˆ i nˆ i+1 , cˆi cˆj + cˆj† cˆi − H = −t (1) ij
iσ
i
where ij indicates the summation over the nearest neighbor sites, V is the repulsive interaction strength of the fermions, and µ is the chemical potential. This system has a metal-Mott insulator transition at half-filling as V t. It is obvious that in this limit and at half-filling the fermions cannot be hopping, because each hopping at least needs the energy V , which is formidable in low energy region. At half-filling, as taking the Jordan–Wigner inverse-transformation, the system becomes the Heisenberg model at V = 2t, which can be exactly solved by the Bethe ansatz [21]. Here we only consider the case of less than half-filling of the fermions, in which the system is in metal state even if V t. We introduce a Lagrangian multiplier field φi (t) which enforces nˆ i = cˆi† cˆi as a constraint condition on the system (when integrating out the density field, it is usual Hubbard–Stratonovich boson field). Using the Hamiltonian (1), we have the following action [20] (choosing h¯ = 1), dt ci† (t)Mij (t)cj (t) + dt ni (t)φi (t) − V ni (t)ni+1 (t) , S[φ, n] = (2) ij
i
where Mij (t) = δij [i∂t + µ − φi (t)] + t (γij + γj i ) is a N × N matrix, where γij = 1 for j = i ± 1, and γij = 0 for other j . Now the fermion fields, density field and the Lagrangian multiplier field are independent variables, and the first term describes the fermions moving in the time-dependent “potential” φi (t). We note that with the introduction of the Lagrangian multiplier field, the original Hilbert space of the fermions is enlarged by the new fields ni (t) and φi (t), and there appear some unphysical states in this enlarged Hilbert space. To remove these unphysical states, all physical quantities of the system must be the functional average of those corresponding quantities obtained by the operators cˆi† (t)(cˆi (t)) and nˆ i (t) over the Lagrangian multiplier field. The operators cˆi† (t)(cˆi (t)) and nˆ i (t) are not the same as the original operators cˆi† (cˆi ) and nˆ i that after introducing the Lagrangian multiplier field, the translation symmetry of the system appears to be broken, and the fermions become “non-interaction”. All the correlation effects of the fermions are produced by this “potential” φi (t) which plays a role of a time-depending random potential on the fermions. However, after taking the functional average over this “potential” φi (t), the translation symmetry of the system is restored, and all physical quantities are translating invariant. The action (2) only includes quadratic form of the fermion fields and can be integrated out easily. We obtain the resulting action [22] θ (−Ek [φ])Ek [φ] + S[φ, n] = − (3) dt ni (t)φi (t) − V ni (t)ni+1 (t) , k
i
where the eigenvalues Ek [φ] are determined by the eigen-functional equation of the fermion propagator operator Mij (t). Thus the eigen-functional equation of the fermion propagator operator Mij (t) plays a key role in the eigenfunctional theory. It reads explicitly [23,24] M(t)Ψkω (t, [φ]) = (ω − Ek [φ])Ψkω (t, [φ]).
(4)
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With the Hellmann–Feynman theorem the eigen-values can be written as Ek [φ] = εk + Σk [φ], where εk = 1 † −2t cos(ak) − µ, Σk [φ] = a 0 dξ dt Ψkω (t, [φ])φ(t)Ψkω (t, [φ]), and φ(t) = (φi (t)δij ) is a N × N matrix. We have used the momentum k and frequency ω to denote the states of the equation. The quantum numbers k and ω are well-defined in the case of φi (t) = 0, i.e., V = 0, where they are good quantum numbers. After turning on the “potential” φi (t), the translation symmetry of the system is broken, and they are not good quantum numbers, but they can be still used to label the states of the fermions, as what is usually being done in impurity and/or defect scattering problems where the systems do not have the translation symmetry. The physical meaning of the states labelled by the momentum k and frequency ω are clear. They represent the eigenstates of the fermions moving in the random potential φi (t). These states are adiabatically connected to the corresponding unperturbed states where k and ω are good quantum numbers. According to the expression of the eigen-values, the eigen-functionals Ψkω (t, [φ]) have the following general expression, ikx1 Qk (x1 ,t,[φ]) e e 1 A .. k −iωt , Ψkω (t, [φ]) = √ Ψk (t, [φ])e (5) , Ψk (t, [φ]) = √ . T L eikxN eQk (xN ,t,[φ]) where Ak is a normalization constant, and the phase field Qk (xi , t) satisfies the differential equation i∂t Qk (xi , t, [φ]) − φi (t) − γk 1 − e−Qk (xi ,t,[φ])+Qk (xi+δ ,t,[φ]) − γk∗ 1 − e−Qk (xi ,t,[φ])+Qk (xi−δ ,t,[φ]) = −Σk [φ],
(6) Ak ikxi Qk (xi ,t,[φ]) √ e e L
are the eigen wave functions of the here γk = teiak . The eigen-functionals Ψk (xi , t, [φ]) = fermions for a given boson field φi (t) with which we can calculate a variety of correlation functions of the system (see below). After integrating out the density field and neglecting a constant term, we obtain the formally exact expression of the action of the system 2 1 1 φ(q, Ω) , S[φ] = − dt F (xi , t) + 2T L v(q) q,Ω
i
R φ(xi , t)e2Qk (xi ,t,[ξ φ]) F (xi , t) = T θ −Σk [φ] − ε(k) dξ , 2QR k (xi ,t,[ξ φ]) k i dte
1
(7)
0
where v(q) = aV cos(aq), and QR k (xi , t, [φ]) is the real part of the phase field Qk (xi , t, [φ]). In the eigen-functional presentations Ψkω (t, [φ]), the fermions are non-interacting, and the
fermion operators
can be written as √ √ cˆi (t) = a Ψkω (xi , t, [φ])cˆkω = a Ψk (xi , t, [φ])cˆk (t), √ cˆi† (t) = a
k,ω
k
Ψk∗ (xi , t, [φ])cˆk† (t),
(8)
k
where cˆk (t) and cˆk† (t) are the fermion annihilation and creation operators with momentum k at time t, respectively, and satisfy the anticommutation relation {cˆk (t), cˆk† (t )} = δkk δ(t − t ) and/or {cˆkω , cˆk† ω } = δkk δωω . It is worth noting that the eigen-functionals Ψkω (xi , t, [φ]) are the eigen-wave functions of the fermions for a given Lagrangian multiplier field φi (t), thus the fermion operators cˆi (t) and cˆi† (t) are the functional of the Lagrangian multiplier field
† iωt −iωt and cˆ† (t) = √1 φi (t), while the fermion operators cˆk (t) = √1 are not the functional ω cˆkω e ω cˆkω e k T T of the Lagrangian multiplier field φi (t), and they are only representing the fermions unoccupied (annihilation operator) and occupied (creation operator) the states in the Hilbert space constructed by the eigen-functionals Ψkω (xi , t, [φ]), just like in usual periodic potential the electron operators can be represented by Bloch wave
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functions, here the role of the eigen-functionals Ψkω (xi , t, [φ]) is the same as that of the Bloch wave functions. However, at zero temperature, whether the state represented by the eigen-functional Ψkω (xi , t, [φ]) is occupied depends on the corresponding eigen-value Ek [φ]. By the orthogonality and completeness of the eigen-functionals, it can be easily proved that the fermion operators cˆi (t) and cˆi† (t) satisfy usual anti-commutation relations. Eq. (8) can be seen as some kind of bosonization representation of the fermion operators, in which the fermion operators cˆk (t) and cˆk† (t) represent non-interacting fermions. Our approach is different from usual bosonization representation, where the fermion operators are represented by their density operators which are taken as the basic ingredients of the system [13–15], while at present the basic ingredient are the phase field which represents the fermion correlation effect. With the expressions of the fermion operators cˆi (t) and cˆi† (t), the off-diagonal long-range order (ODLRO) parameter can be written as R(xi − xj ) =
1 † 1 ∗ Ψk (xi , t, [φ])Ψk (xj , t, [φ]) φ , cˆi (t)cˆj (t) c,φ = an n
(9)
k
where n = N/L is the fermion density, and · · ·c,φ means the average on the ground state of the free fermions cˆk (t) and cˆk† (t) and the functional average over the Lagrangian multiplier field φi (t), respectively. It can be easily proved that the ODLRO parameter R(x) satisfies the relation, R(0) = 1. While the fermion Green’s function reads i T cˆi (t)cˆj† (t ) c,φ a Ψk (xi , t, [φ])Ψ ∗ (xj , t , [φ]) Ψk (xi , t, [φ])Ψ ∗ (xj , t , [φ])φ k k ∼ = , = ω − Ek [φ] ω − E k [φ]φ φ
G(xi − xj , t − t ) = −
k,ω
(10)
k,ω
where we have used the approximations A{Σk [φ]}n φ Aφ {Σk [φ]}n φ , and {Σk [φ]}n φ {Σk [φ]φ }n . These approximations are reasonable because the correlations among the self-energy Σk [φ] and between Ψk (xi , t, [φ])Ψk∗ (xj , t , [φ]) and Σk [φ] are high order (1/kF ) corrections, and the low energy behavior of the fermion Green’s function is determined by Ψk (xi , t, [φ])Ψk∗ (xj , t , [φ])φ . The pair distribution function of the system is 1 nˆ i (t)nˆ j (t)c,φ − δij nˆ i (t)c,φ a 2 n2 1 |Ψk (xi , t, [φ])Ψk (xj , t, [φ])|2 φ = 2 n k,k ∗ − Ψk (xi , t, [φ])Ψk (xj , t, [φ])Ψk∗ (xj , t, [φ])Ψk (xi , t, [φ]) φ
g(xi − xj ) =
which satisfies the usual normalization integral an (g(xi − xj ) − 1) = −1,
(11)
(12)
j
where a is the lattice constant, and g(0) = 0. These relations are useful in numerical calculations, and can be used as criterions for taking some approximations, because they are exact, and independent of the fermion filling and the fermion (repulsive) interaction strength. The above relations are formally exact, in which the most basic ingredients are the phase field Qk (xi , t, [φ]) with the corresponding differential equation it satisfies. However, to analytically study the low energy behavior of the system, we have to take some approximations. In contrast to previous perturbation theory [11,12] where one directly takes a series expansion of the interaction potential or the Lagrangian multiplier field, the phase field is assumed to be “small” and is taken as an expansion parameter. Under linearization approximation (i.e., only taking
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the linear terms in solving the differential equation of the phase field), we obtain the solution of the phase field, φ(q, Ω)eiqxi −iΩt 1 , Qk (xi , t, [φ]) = (13) TL Ω − 2t sin(ak) sin(aq) + ξk [1 − cos(aq)] q,Ω
where ξk = −2t cos(ak). In the low energy region, all physical properties of the system are determined by the fermion states near the Fermi level of the system. In one dimension, the Fermi level of the system is composed of two points defined by the Fermi momentum kF , one is at kF , and another one is at −kF . The phase field can be divided correspondingly into two parts corresponding to these Fermi level +kF and −kF , φR (q, Ω)eiqxi −iΩt 1 , QRk (xi , t, [φ]) = Qk (xi , t, [φ]) k∼kF = TL Ω − 2t sin(a(kF + k)) sin(aq) + ξkF +k [1 − cos(aq)] q,Ω
φL (q, Ω)eiqxi −iΩt 1 , QLk (xi , t, [φ]) = Qk (xi , t, [φ]) k∼−kF = TL Ω − 2t sin(a(−kF + k)) sin(aq) + ξ−kF +k [1 − cos(aq)] q,Ω (14) where the momentums k and q only take the values −D k, q D, and D kF is a momentum cut-off. Correspondingly, the Lagrangian multiplier field has to be separated into R (L) parts near the point +kF (−kF ). Eq. (8) can be rewritten formally as √ ΨRk (xi , t, [φ])cˆRk (t) + ΨLk (xi , t, [φ])cˆLk (t) , cˆi (t) = a √ cˆi† (t) = a
k
† † ∗ ∗ ΨRk (xi , t, [φ])cˆRk (t) + ΨLk (xi , t, [φ])cˆLk (t) ,
(15)
k Ak ±ikxi QR(L)k (xi ,t,[φ]) where ΨR(L)k (xi , t, [φ]) = √ e e , and Ψk (xi , t, [φ]) = ΨRk (xi , t, [φ]) + ΨLk (xi , t, [φ]). Notice L that the fermions are divided into two parts, the right-moving and the left-moving fermions. The above approximations are good only for weak interaction V /t < 1, while for strong interaction V /t > 1 they are invalid, and we must use Eq. (13) to calculate the low energy effective action and correlation functions of the system. The reason is that the above approximation is equivalent to the linearized dispersion approximation in bosonization method, where the modification of fermion velocities away from the Fermi points is omitted. We should point out that in our formulation the fermions move in the time-depending random potential produced by the Lagrangian multiplier field which represents the influence of other fermions due to interaction in original Hilbert space, and it alters the Fermi velocity of the system. After taking functional average over the Lagrangian multiplier field, the Fermi velocity of the system depends upon the fermion interaction potential. However, under the above approximation the Fermi velocity is invariant, which is correct only for weak interaction V /t < 1. Note that the fermion interaction strength V appears in the action of the system as the parameter v(q) = aV cos(aq). Thus strong interaction V /t > 1 does not mean a large v(q) due to the factor cos(aq). For strong interaction, we need to consider all the momenta in the range of −2kF q 2kF . While, for the weak interaction, it is enough to only consider the momenta in the range of −D q D, and take approximately v(q) aV . Substituting Eq. (14) into Eq. (7), we have the low energy effective action of the system, 1 1 φR (−q, −Ω)φL(q, Ω) , Seff [φ] = AR (q, Ω)|φR (q, Ω)|2 + AL (q, Ω)|φL(q, Ω)|2 + TL 2aV q,Ω
1 1 − cos(aq) 1 (16) and vF = 2at sin(akF). 4π sin(aq) vF sin(aq) ∓ aΩ The last term originates from the fermion interactions under the above approximations, † † † † † cˆk cˆk+q cˆk† cˆk −q ˆLk −q + cˆLk cˆLk+q cˆRk ˆRk −q = nˆ Rq nˆ L−q + nˆ Lq nˆ R−q . nˆ q nˆ −q = cˆRk cˆRk+q cˆLk c c where AR(L)(q, Ω) =
k,k
k,k
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q 1 It is obvious that in the long wavelength region aq 1, we have AR(L)(q, Ω) = 4π vF q∓Ω , and the low energy effective action (16) reduces into that of the Tomonaga–Luttinger model [23]. Moreover, the phase fields are independent of the fermion momentum k:
QRk (xi , t, [φ]) = QR (xi , t, [φ]) =
1 φR (q, Ω)eiqxi −iΩt , TL Ω − vF q q,Ω
1 φL (q, Ω)eiqxi −iΩt . QLk (xi , t, [φ]) = QL (xi , t, [φ]) = TL Ω + vF q
(17)
q,Ω
Notice that the phase fields are pure imaginary fields Q∗R(L)k (xi , t, [φ]) = −QR(L)k (xi , t, [φ]), this property of the phase field is a result of considering only the linear spectrum of the fermions. If we take the scaling √ † √ (xi , t) and cˆi (t) = a ψ(x i , t), under the above approximations Eq. (15) can be rewritten as cˆi† (t) = a ψ 1 ikxi QR (xi ,t,[φ]) (xi , t) = √ ψ e e cˆRk (t) + eikxi eQL (xi ,t,[φ]) cˆLk (t) , 2L k 1 −ikxi −QR (xi ,t,[φ]) † † † (xi , t) = √ e e cˆRk (t) + e−ikxi e−QL (xi ,t,[φ]) cˆLk (t) . ψ 2L k
(18)
For a real system, the right-moving fermions take the momenta in the range 0 k π/a, and the left-moving fermions take the momenta in the range −π/a k 0. However, after taking the above approximations where (xi , t) in (18) cannot † (xi , t) and ψ the phase fields are independent of the momentum k, the fermion operators ψ exactly satisfy usual anti-commutation relations. If we choose k = ±kF + k (+ for right-moving fermions, and − for left-moving fermions), where −D k D, and formally take D → ∞ in calculating the integration of the † (xi , t) momentum k as in usual bosonization method [13–15], it can be seen easily that the fermion operators ψ i , t) satisfy the anti-commutation relations, and Eq. (18) can be seen as the bosonization representation and ψ(x of the fermions, which is equivalent to usual bosonization method. This equivalence between them becomes more clear if we rewrite the fermion operator as two parts, one is the right-moving fermion operator, and another one is the left-moving fermion operator: 1 R (xi , t) + ψ L (xi , t) , (xi , t) = √ ψ ψ 2 1 R(L)(xi , t) = √ eQR(L)(xi ,t,[φ]) (0) (xi , t), where ψ eikxi cˆR(L)k (t) = eQR(L)(xi ,t,[φ]) ψ R(L) L k and ψ R(L) (xi , t) are the corresponding “non-interacting” fermion operators, and define the right- and left-moving fermion density operators (0)
† (xi , t)ψ R (xi , t) = ρR (xi , t) = ψ R
1 iqxi e ρRq (t), L q
† (xi , t)ψ L (xi , t) = ρL (xi , t) = ψ L
1 iqxi e ρLq (t), L q
† (t)cˆR(L)k+q (t). The right- and left-moving fermion density operators do not depend where ρR(L)q (t) = k cˆR(L)k on the phase field, and are the same as that of the corresponding “non-interacting” fermions. This character of the right and left-moving fermion density operators comes from the linear spectrum approximation for the fermions, in which the phase field is a pure imaginary quantity. With the fermion density operators ρR(L)q (t), we can obtain
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their well-known commutation relations which are the cornerstone of usual bosonization method [13–15], qL ρiq (t), ρj −q (t) = ± δij δqq , 2π
i, j = R, L,
where ‘+’ for the right-moving fermions, and ‘−’ for the left-moving fermions. Under the linear spectrum approximation, these commutation relations of the density operators are not influenced by the fermion interaction. R(L) (xi , t), we can also easily explain why the field operators with same conformal Using the fermion operators ψ L(R) (xi , t) and ρR(L) (xi , t), have the different conformal dimensions † (xi , t)ψ dimensions at V = 0, such as ψ R(L) when turning on the fermion interaction. The fermion density operators ρR(L) (xi , t) do not include the phase field, L(R) (xi , t) include the phase field, † (xi , t)ψ and their conformal dimension is keeping as 1, while the operators ψ R(L) and their conformal dimension is changed into g (see below for definition) due to the fermion correlation. Thus we see that under the framework of the eigen-functional theory, the assumptions and validity of usual bosonization theory become very clear. Using the expressions (18) and the low energy effective action (16), we can obtain the fermion Green’s functions (t > 0), R(L) (xi , t)ψ † (0, 0) GR(L)(xi , t) = −i ψ R(L) c,φ (g+1/g−2)/4 1 e±ikF xi δ2 =± , (19) 2π xi ∓ αt ± iδ (xi + αt − iδ)(xi − αt + iδ) 1−γ −1 ∼ D is a momentum cut-off constant. This result is the same where α = 1 − γ 2 , γ = π2aV , g = hv 1+γ , and δ ¯ F as that obtained by usual bosonization methods [13–15,23]. The ODLRO parameter has a simple form (g+1/g−2)/4 1 sin(kF (xi − xj )) δ2 R(xi − xj ) = , πn xi − xj (xi − xj − iδ)(xi − xj + iδ)
(20)
|x|→∞
and it is short-range R(x) ∼ x −(g+1/g)/2 sin(kF x). The relation R(0) = 1 is retained where n = kF /π . Based on Eq. (11), we can obtain the pair distribution function of the system, (g−1)/2 k x 1 2 sin2 ( F2 ) δ2 g(x) = 1 − 2 2 (21) 1 + cos(kF x) . π n x2 (x − iδ)(x + iδ) For a non-interacting fermion system (V = 0, and g = 1), the pair distribution function (21) reduces to the well2 know form [11] g(x) = 1 − π 21n2 sin x(k2 F x) , and the relation g(0) = 0 is retained. Moreover, from Eq. (18), the ground state wave function of the system can be approximately written as a simple form Ψg (x1 , . . . , xN ) = C deteiki xj · (22) |xi − xj |1−1/g , i>j
where C is the normalization constant, and the factor deteiki xj denote the corresponding free right- and leftmoving fermions. The last factor is contributed by the phase field, and represents the fermion correlation. This ground state wave function clearly demonstrates that even weak fermion interaction can produce the long or quasilong range correlation among the fermions. This expression of the ground state wave function of the system is correct under the linear spectrum approximation for the fermions and weak interaction. However, for strong fermion interaction, although we can still formally divide the fermions into the right and left-moving fermions, the phase field would depend on the momentum k of the corresponding “non-interacting” fermions, and the momentum q in Eq. (14) will take the values in the range of −π q π , which make the problem difficult to handle. In general,
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we expect that the ground state wave function of the system can be written as |xi − xj |β , Ψg (x1 , . . . , xN ) = C deteiki xj ·
(23)
i>j
where C is the normalization constant, and the exponent β < 0 depends on the Fermi momentum kF and the fermion interaction strength V . The appearance of the last factor is reasonable because the ground state wave function can be approximately written as Ψg (x1 , . . . , xN ) ∼ deteiki xj +Qki (xj ,t,[φ]) φ ik x i j · exp QRkF (xi , t, [φ]) + QL−kF (xi , t, [φ]) ∼ det e . i
φ
With the low energy effective action (16) in which replacing aV by v(q), the last functional average factor over the Lagrangian multiplier field will contribute a factor i>j |xi − xj |β to the ground state wave function of the system. The factor i>j |xi − xj |β means that there is a long or quasi-long range fermion correlation for any no zero fermion interaction strength V , where the exponent β has the property, β = 0 at V = 0. Notice that in the Hilbert space constructed by the eigen-functionals Ψk (xi , t, [φ]), the system is reduced into a single-particle system, and the fermions are “non-interacting”, thus in this Hilbert space we can construct a “ground state” which is the functional of the Lagrangian multiplier field φi (t). After taking the functional average over the field φi (t), we obtain the physical ground state wave function (22) and (23) of the system. In summary, using the eigen-functional theory, we have studied the one-dimensional spinless fermion system, and calculated the ODLRO parameter, the pair distribution function, the fermion Green’s function and the ground state wave function of the system. Under the linear spectrum approximation, we have proved the commutation of the fermion density operators are not influenced by the fermion interaction, and explained why operators with the same conformal dimensions at V = 0 may have different conformal dimensions when turning on the fermion interaction, thus clarifying some ambiguities in the assumptions and validity of usual bosonization method.
Acknowledgement This work is supported by the Beiren foundation of the Tsinghua University and NSFC.
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A.C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge Univ. Press, 1993. P.W. Anderson, Science 273 (1987) 1196. F.C. Zhang, T.M. Rice, Phys. Rev. B 37 (1988) 3759. C.M. Varma, et al., Phys. Rev. Lett. 63 (1989) 1986. G. Baskaran, P.W. Anderson, Phys. Rev. B 37 (1988) 580; L. Ioffe, A. Larkin, Phys. Rev. B 39 (1989) 8988; P.A. Lee, N. Nagaosa, Phys. Rev. B 46 (1992) 5621. R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. R.F. Prange, S.M. Girvin (Eds.), The Quantum Hall Effect, Springer-Verlag, 1987. H.G. Keiss, Conjugated Conducting Polymers, Springer-Verlag, 1990; M.S. Dresselhaus, G. Dresselhaus, P.C. Ekluud, Science of Fullerences and Carbon Nanotubes, Academic Press, San Diego, CA, 1996. P. Nozières, Theory of Interacting Fermi System, Addison-Wesley, 1974. P.W. Anderson, The Theory of Superconductivity in the High-Tc Cuprates, Princeton Univ. Press, Princeton, NJ, 1997. G.D. Mahan, Many-Particle Physics, Plenum, New York, 1981. J.W. Negele, H. Orland, Quantum Many-Particle Systems, Addison-Wesley, 1988.
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[13] A. Luther, I. Peschel, Phys. Rev. B 9 (1974) 2911. [14] V.J. Emery, in: J.T. Devreese (Ed.), Highly Conducting One-Dimensional Solids, Plenum, New York, 1979; J. Sólyom, Adv. Phys. 28 (1979) 201. [15] F.D.M. Haldane, J. Phys. C 14 (1981) 2585. [16] A. Houghton, H.J. Kwon, J.B. Marston, Phys. Rev. B 50 (1994) 1351. [17] P. Kopietz, J. Hermisson, K. Schoenhammer, Phys. Rev. B 52 (1995) 10877. [18] A.H. Castro Neto, E. Fradkin, Phys. Rev. B 51 (1995) 4084. [19] P. Kopietz, Bosonization of Interacting Fermions in Arbitrary Dimensions, Springer-Verlag, 1996. [20] Y.L. Liu, Int. J. Mod. Phys. B 16 (2002) 773; Y.L. Liu, Int. J. Mod. Phys. B 16 (2002) 2201. [21] H.B. Thacker, Rev. Mod. Phys. 53 (1981) 253. [22] Y.L. Liu, Application of eigen-functional theory on quantum many-particle systems, Int. J. Mod. Phys. B, in press. [23] K.K. Lee, Y. Chen, J. Phys. A 21 (1988) 4155; Y.L. Liu, Phys. Rev. B 59 (1999) 7937. [24] T.K. Ng, cond-mat/9706033; Y.L. Liu, T.K. Ng, Phys. Rev. Lett. 83 (1999) 5539.
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Pair distribution function and off-diagonal long range order parameter of one-dimensional spinless fermion system Yu-Liang Liu Center for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China Received 5 July 2002; received in revised form 14 October 2002; accepted 15 October 2002 Communicated by R. Wu
Abstract Using the eigen-functional theory, we derive the exact functional expressions of pair distribution function, off-diagonal long range order parameter and fermion Green’s function of a one-dimensional spinless fermion system. With the linear spectrum approximation and for weak fermion interaction, we calculate these functions and the ground state wave function of the system. Moreover, under this approximation we show that the commutation relations of the fermion density operators are not influenced by the fermion interaction, and explain why operators with the same conformal dimensions at V = 0 have different conformal dimensions when fermion interaction is turned on (V = 0). 2002 Elsevier Science B.V. All rights reserved.
Strongly correlated system is one of the most interesting and hardest problem [1–10] in condensed matter physics nowadays, of which the most prominent character is the strong particle correlation that completely controls its low energy behavior. This strong particle correlation is controlled by the particle interactions, thus the key point in treating this kind of problem is how to treat the particle interaction (potential) terms accurately. Applying usual perturbation methods [11,12] to strongly correlated systems meet many serious problems. Because of the strong correlation among particles, there is no a small suitable quantity to be used as a perturbation expansion parameter. It is natural that one uses other ways to treat strongly correlated systems, for example, treating the interaction (potential) terms exactly and then exactly or perturbatively treating the kinetic energy term of the particles. This is the strategy of bosonization method in treating 1D interacting fermion systems [13–19]. The eigen-functional theory [20] provides a new way to treat general strongly correlated systems, in which it ends in solving equation for the phase fields which are determined by the particle interaction, and reflects the particle correlation of the systems directly. There are two key steps in eigen-functional theory, one is that under the path integral formulation, we exactly map a D-dimensional strongly correlated system into an effective (D + 1)dimensional time-dependent “single-particle” problem. By solving the resulting eigen-functional equation of the propagator operator of the particles, we can obtain the formally exact functional expressions of the action and a variety of correlation functions of the system, respectively. By taking the functional average over Lagrangian
E-mail address: [email protected] (Y.-L. Liu). 0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 1 4 2 2 - 6
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multiplier fields, we obtain the corresponding physical quantities. The most prominent character of the eigenfunctional theory is that in the eigen-functional representation, the particle operators are non-interacting, and the particle correlation is determined by the phase fields. In this Letter, we study a one-dimensional spinless fermion system with repulsive interaction with the eigenfunctional theory. We first derive the formally exact functional expressions of off-diagonal long range order parameter, pair distribution function, fermion Green’s function and ground state wave function of the system, then calculate these functions under the linear spectrum approximation and for weak fermion interaction. Moreover, we prove that the commutation relations of the fermion density operators are not influenced by the fermion interaction, and clarify some ambiguities in usual bosonization method. We now consider a one-dimensional spinless fermion system with repulsive interaction, † † µcˆi cˆi + V nˆ i nˆ i+1 , cˆi cˆj + cˆj† cˆi − H = −t (1) ij
iσ
i
where ij indicates the summation over the nearest neighbor sites, V is the repulsive interaction strength of the fermions, and µ is the chemical potential. This system has a metal-Mott insulator transition at half-filling as V t. It is obvious that in this limit and at half-filling the fermions cannot be hopping, because each hopping at least needs the energy V , which is formidable in low energy region. At half-filling, as taking the Jordan–Wigner inverse-transformation, the system becomes the Heisenberg model at V = 2t, which can be exactly solved by the Bethe ansatz [21]. Here we only consider the case of less than half-filling of the fermions, in which the system is in metal state even if V t. We introduce a Lagrangian multiplier field φi (t) which enforces nˆ i = cˆi† cˆi as a constraint condition on the system (when integrating out the density field, it is usual Hubbard–Stratonovich boson field). Using the Hamiltonian (1), we have the following action [20] (choosing h¯ = 1), dt ci† (t)Mij (t)cj (t) + dt ni (t)φi (t) − V ni (t)ni+1 (t) , S[φ, n] = (2) ij
i
where Mij (t) = δij [i∂t + µ − φi (t)] + t (γij + γj i ) is a N × N matrix, where γij = 1 for j = i ± 1, and γij = 0 for other j . Now the fermion fields, density field and the Lagrangian multiplier field are independent variables, and the first term describes the fermions moving in the time-dependent “potential” φi (t). We note that with the introduction of the Lagrangian multiplier field, the original Hilbert space of the fermions is enlarged by the new fields ni (t) and φi (t), and there appear some unphysical states in this enlarged Hilbert space. To remove these unphysical states, all physical quantities of the system must be the functional average of those corresponding quantities obtained by the operators cˆi† (t)(cˆi (t)) and nˆ i (t) over the Lagrangian multiplier field. The operators cˆi† (t)(cˆi (t)) and nˆ i (t) are not the same as the original operators cˆi† (cˆi ) and nˆ i that after introducing the Lagrangian multiplier field, the translation symmetry of the system appears to be broken, and the fermions become “non-interaction”. All the correlation effects of the fermions are produced by this “potential” φi (t) which plays a role of a time-depending random potential on the fermions. However, after taking the functional average over this “potential” φi (t), the translation symmetry of the system is restored, and all physical quantities are translating invariant. The action (2) only includes quadratic form of the fermion fields and can be integrated out easily. We obtain the resulting action [22] θ (−Ek [φ])Ek [φ] + S[φ, n] = − (3) dt ni (t)φi (t) − V ni (t)ni+1 (t) , k
i
where the eigenvalues Ek [φ] are determined by the eigen-functional equation of the fermion propagator operator Mij (t). Thus the eigen-functional equation of the fermion propagator operator Mij (t) plays a key role in the eigenfunctional theory. It reads explicitly [23,24] M(t)Ψkω (t, [φ]) = (ω − Ek [φ])Ψkω (t, [φ]).
(4)
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With the Hellmann–Feynman theorem the eigen-values can be written as Ek [φ] = εk + Σk [φ], where εk = 1 † −2t cos(ak) − µ, Σk [φ] = a 0 dξ dt Ψkω (t, [φ])φ(t)Ψkω (t, [φ]), and φ(t) = (φi (t)δij ) is a N × N matrix. We have used the momentum k and frequency ω to denote the states of the equation. The quantum numbers k and ω are well-defined in the case of φi (t) = 0, i.e., V = 0, where they are good quantum numbers. After turning on the “potential” φi (t), the translation symmetry of the system is broken, and they are not good quantum numbers, but they can be still used to label the states of the fermions, as what is usually being done in impurity and/or defect scattering problems where the systems do not have the translation symmetry. The physical meaning of the states labelled by the momentum k and frequency ω are clear. They represent the eigenstates of the fermions moving in the random potential φi (t). These states are adiabatically connected to the corresponding unperturbed states where k and ω are good quantum numbers. According to the expression of the eigen-values, the eigen-functionals Ψkω (t, [φ]) have the following general expression, ikx1 Qk (x1 ,t,[φ]) e e 1 A .. k −iωt , Ψkω (t, [φ]) = √ Ψk (t, [φ])e (5) , Ψk (t, [φ]) = √ . T L eikxN eQk (xN ,t,[φ]) where Ak is a normalization constant, and the phase field Qk (xi , t) satisfies the differential equation i∂t Qk (xi , t, [φ]) − φi (t) − γk 1 − e−Qk (xi ,t,[φ])+Qk (xi+δ ,t,[φ]) − γk∗ 1 − e−Qk (xi ,t,[φ])+Qk (xi−δ ,t,[φ]) = −Σk [φ],
(6) Ak ikxi Qk (xi ,t,[φ]) √ e e L
are the eigen wave functions of the here γk = teiak . The eigen-functionals Ψk (xi , t, [φ]) = fermions for a given boson field φi (t) with which we can calculate a variety of correlation functions of the system (see below). After integrating out the density field and neglecting a constant term, we obtain the formally exact expression of the action of the system 2 1 1 φ(q, Ω) , S[φ] = − dt F (xi , t) + 2T L v(q) q,Ω
i
R φ(xi , t)e2Qk (xi ,t,[ξ φ]) F (xi , t) = T θ −Σk [φ] − ε(k) dξ , 2QR k (xi ,t,[ξ φ]) k i dte
1
(7)
0
where v(q) = aV cos(aq), and QR k (xi , t, [φ]) is the real part of the phase field Qk (xi , t, [φ]). In the eigen-functional presentations Ψkω (t, [φ]), the fermions are non-interacting, and the
fermion operators
can be written as √ √ cˆi (t) = a Ψkω (xi , t, [φ])cˆkω = a Ψk (xi , t, [φ])cˆk (t), √ cˆi† (t) = a
k,ω
k
Ψk∗ (xi , t, [φ])cˆk† (t),
(8)
k
where cˆk (t) and cˆk† (t) are the fermion annihilation and creation operators with momentum k at time t, respectively, and satisfy the anticommutation relation {cˆk (t), cˆk† (t )} = δkk δ(t − t ) and/or {cˆkω , cˆk† ω } = δkk δωω . It is worth noting that the eigen-functionals Ψkω (xi , t, [φ]) are the eigen-wave functions of the fermions for a given Lagrangian multiplier field φi (t), thus the fermion operators cˆi (t) and cˆi† (t) are the functional of the Lagrangian multiplier field
† iωt −iωt and cˆ† (t) = √1 φi (t), while the fermion operators cˆk (t) = √1 are not the functional ω cˆkω e ω cˆkω e k T T of the Lagrangian multiplier field φi (t), and they are only representing the fermions unoccupied (annihilation operator) and occupied (creation operator) the states in the Hilbert space constructed by the eigen-functionals Ψkω (xi , t, [φ]), just like in usual periodic potential the electron operators can be represented by Bloch wave
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functions, here the role of the eigen-functionals Ψkω (xi , t, [φ]) is the same as that of the Bloch wave functions. However, at zero temperature, whether the state represented by the eigen-functional Ψkω (xi , t, [φ]) is occupied depends on the corresponding eigen-value Ek [φ]. By the orthogonality and completeness of the eigen-functionals, it can be easily proved that the fermion operators cˆi (t) and cˆi† (t) satisfy usual anti-commutation relations. Eq. (8) can be seen as some kind of bosonization representation of the fermion operators, in which the fermion operators cˆk (t) and cˆk† (t) represent non-interacting fermions. Our approach is different from usual bosonization representation, where the fermion operators are represented by their density operators which are taken as the basic ingredients of the system [13–15], while at present the basic ingredient are the phase field which represents the fermion correlation effect. With the expressions of the fermion operators cˆi (t) and cˆi† (t), the off-diagonal long-range order (ODLRO) parameter can be written as R(xi − xj ) =
1 † 1 ∗ Ψk (xi , t, [φ])Ψk (xj , t, [φ]) φ , cˆi (t)cˆj (t) c,φ = an n
(9)
k
where n = N/L is the fermion density, and · · ·c,φ means the average on the ground state of the free fermions cˆk (t) and cˆk† (t) and the functional average over the Lagrangian multiplier field φi (t), respectively. It can be easily proved that the ODLRO parameter R(x) satisfies the relation, R(0) = 1. While the fermion Green’s function reads i T cˆi (t)cˆj† (t ) c,φ a Ψk (xi , t, [φ])Ψ ∗ (xj , t , [φ]) Ψk (xi , t, [φ])Ψ ∗ (xj , t , [φ])φ k k ∼ = , = ω − Ek [φ] ω − E k [φ]φ φ
G(xi − xj , t − t ) = −
k,ω
(10)
k,ω
where we have used the approximations A{Σk [φ]}n φ Aφ {Σk [φ]}n φ , and {Σk [φ]}n φ {Σk [φ]φ }n . These approximations are reasonable because the correlations among the self-energy Σk [φ] and between Ψk (xi , t, [φ])Ψk∗ (xj , t , [φ]) and Σk [φ] are high order (1/kF ) corrections, and the low energy behavior of the fermion Green’s function is determined by Ψk (xi , t, [φ])Ψk∗ (xj , t , [φ])φ . The pair distribution function of the system is 1 nˆ i (t)nˆ j (t)c,φ − δij nˆ i (t)c,φ a 2 n2 1 |Ψk (xi , t, [φ])Ψk (xj , t, [φ])|2 φ = 2 n k,k ∗ − Ψk (xi , t, [φ])Ψk (xj , t, [φ])Ψk∗ (xj , t, [φ])Ψk (xi , t, [φ]) φ
g(xi − xj ) =
which satisfies the usual normalization integral an (g(xi − xj ) − 1) = −1,
(11)
(12)
j
where a is the lattice constant, and g(0) = 0. These relations are useful in numerical calculations, and can be used as criterions for taking some approximations, because they are exact, and independent of the fermion filling and the fermion (repulsive) interaction strength. The above relations are formally exact, in which the most basic ingredients are the phase field Qk (xi , t, [φ]) with the corresponding differential equation it satisfies. However, to analytically study the low energy behavior of the system, we have to take some approximations. In contrast to previous perturbation theory [11,12] where one directly takes a series expansion of the interaction potential or the Lagrangian multiplier field, the phase field is assumed to be “small” and is taken as an expansion parameter. Under linearization approximation (i.e., only taking
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the linear terms in solving the differential equation of the phase field), we obtain the solution of the phase field, φ(q, Ω)eiqxi −iΩt 1 , Qk (xi , t, [φ]) = (13) TL Ω − 2t sin(ak) sin(aq) + ξk [1 − cos(aq)] q,Ω
where ξk = −2t cos(ak). In the low energy region, all physical properties of the system are determined by the fermion states near the Fermi level of the system. In one dimension, the Fermi level of the system is composed of two points defined by the Fermi momentum kF , one is at kF , and another one is at −kF . The phase field can be divided correspondingly into two parts corresponding to these Fermi level +kF and −kF , φR (q, Ω)eiqxi −iΩt 1 , QRk (xi , t, [φ]) = Qk (xi , t, [φ]) k∼kF = TL Ω − 2t sin(a(kF + k)) sin(aq) + ξkF +k [1 − cos(aq)] q,Ω
φL (q, Ω)eiqxi −iΩt 1 , QLk (xi , t, [φ]) = Qk (xi , t, [φ]) k∼−kF = TL Ω − 2t sin(a(−kF + k)) sin(aq) + ξ−kF +k [1 − cos(aq)] q,Ω (14) where the momentums k and q only take the values −D k, q D, and D kF is a momentum cut-off. Correspondingly, the Lagrangian multiplier field has to be separated into R (L) parts near the point +kF (−kF ). Eq. (8) can be rewritten formally as √ ΨRk (xi , t, [φ])cˆRk (t) + ΨLk (xi , t, [φ])cˆLk (t) , cˆi (t) = a √ cˆi† (t) = a
k
† † ∗ ∗ ΨRk (xi , t, [φ])cˆRk (t) + ΨLk (xi , t, [φ])cˆLk (t) ,
(15)
k Ak ±ikxi QR(L)k (xi ,t,[φ]) where ΨR(L)k (xi , t, [φ]) = √ e e , and Ψk (xi , t, [φ]) = ΨRk (xi , t, [φ]) + ΨLk (xi , t, [φ]). Notice L that the fermions are divided into two parts, the right-moving and the left-moving fermions. The above approximations are good only for weak interaction V /t < 1, while for strong interaction V /t > 1 they are invalid, and we must use Eq. (13) to calculate the low energy effective action and correlation functions of the system. The reason is that the above approximation is equivalent to the linearized dispersion approximation in bosonization method, where the modification of fermion velocities away from the Fermi points is omitted. We should point out that in our formulation the fermions move in the time-depending random potential produced by the Lagrangian multiplier field which represents the influence of other fermions due to interaction in original Hilbert space, and it alters the Fermi velocity of the system. After taking functional average over the Lagrangian multiplier field, the Fermi velocity of the system depends upon the fermion interaction potential. However, under the above approximation the Fermi velocity is invariant, which is correct only for weak interaction V /t < 1. Note that the fermion interaction strength V appears in the action of the system as the parameter v(q) = aV cos(aq). Thus strong interaction V /t > 1 does not mean a large v(q) due to the factor cos(aq). For strong interaction, we need to consider all the momenta in the range of −2kF q 2kF . While, for the weak interaction, it is enough to only consider the momenta in the range of −D q D, and take approximately v(q) aV . Substituting Eq. (14) into Eq. (7), we have the low energy effective action of the system, 1 1 φR (−q, −Ω)φL(q, Ω) , Seff [φ] = AR (q, Ω)|φR (q, Ω)|2 + AL (q, Ω)|φL(q, Ω)|2 + TL 2aV q,Ω
1 1 − cos(aq) 1 (16) and vF = 2at sin(akF). 4π sin(aq) vF sin(aq) ∓ aΩ The last term originates from the fermion interactions under the above approximations, † † † † † cˆk cˆk+q cˆk† cˆk −q ˆLk −q + cˆLk cˆLk+q cˆRk ˆRk −q = nˆ Rq nˆ L−q + nˆ Lq nˆ R−q . nˆ q nˆ −q = cˆRk cˆRk+q cˆLk c c where AR(L)(q, Ω) =
k,k
k,k
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q 1 It is obvious that in the long wavelength region aq 1, we have AR(L)(q, Ω) = 4π vF q∓Ω , and the low energy effective action (16) reduces into that of the Tomonaga–Luttinger model [23]. Moreover, the phase fields are independent of the fermion momentum k:
QRk (xi , t, [φ]) = QR (xi , t, [φ]) =
1 φR (q, Ω)eiqxi −iΩt , TL Ω − vF q q,Ω
1 φL (q, Ω)eiqxi −iΩt . QLk (xi , t, [φ]) = QL (xi , t, [φ]) = TL Ω + vF q
(17)
q,Ω
Notice that the phase fields are pure imaginary fields Q∗R(L)k (xi , t, [φ]) = −QR(L)k (xi , t, [φ]), this property of the phase field is a result of considering only the linear spectrum of the fermions. If we take the scaling √ † √ (xi , t) and cˆi (t) = a ψ(x i , t), under the above approximations Eq. (15) can be rewritten as cˆi† (t) = a ψ 1 ikxi QR (xi ,t,[φ]) (xi , t) = √ ψ e e cˆRk (t) + eikxi eQL (xi ,t,[φ]) cˆLk (t) , 2L k 1 −ikxi −QR (xi ,t,[φ]) † † † (xi , t) = √ e e cˆRk (t) + e−ikxi e−QL (xi ,t,[φ]) cˆLk (t) . ψ 2L k
(18)
For a real system, the right-moving fermions take the momenta in the range 0 k π/a, and the left-moving fermions take the momenta in the range −π/a k 0. However, after taking the above approximations where (xi , t) in (18) cannot † (xi , t) and ψ the phase fields are independent of the momentum k, the fermion operators ψ exactly satisfy usual anti-commutation relations. If we choose k = ±kF + k (+ for right-moving fermions, and − for left-moving fermions), where −D k D, and formally take D → ∞ in calculating the integration of the † (xi , t) momentum k as in usual bosonization method [13–15], it can be seen easily that the fermion operators ψ i , t) satisfy the anti-commutation relations, and Eq. (18) can be seen as the bosonization representation and ψ(x of the fermions, which is equivalent to usual bosonization method. This equivalence between them becomes more clear if we rewrite the fermion operator as two parts, one is the right-moving fermion operator, and another one is the left-moving fermion operator: 1 R (xi , t) + ψ L (xi , t) , (xi , t) = √ ψ ψ 2 1 R(L)(xi , t) = √ eQR(L)(xi ,t,[φ]) (0) (xi , t), where ψ eikxi cˆR(L)k (t) = eQR(L)(xi ,t,[φ]) ψ R(L) L k and ψ R(L) (xi , t) are the corresponding “non-interacting” fermion operators, and define the right- and left-moving fermion density operators (0)
† (xi , t)ψ R (xi , t) = ρR (xi , t) = ψ R
1 iqxi e ρRq (t), L q
† (xi , t)ψ L (xi , t) = ρL (xi , t) = ψ L
1 iqxi e ρLq (t), L q
† (t)cˆR(L)k+q (t). The right- and left-moving fermion density operators do not depend where ρR(L)q (t) = k cˆR(L)k on the phase field, and are the same as that of the corresponding “non-interacting” fermions. This character of the right and left-moving fermion density operators comes from the linear spectrum approximation for the fermions, in which the phase field is a pure imaginary quantity. With the fermion density operators ρR(L)q (t), we can obtain
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their well-known commutation relations which are the cornerstone of usual bosonization method [13–15], qL ρiq (t), ρj −q (t) = ± δij δqq , 2π
i, j = R, L,
where ‘+’ for the right-moving fermions, and ‘−’ for the left-moving fermions. Under the linear spectrum approximation, these commutation relations of the density operators are not influenced by the fermion interaction. R(L) (xi , t), we can also easily explain why the field operators with same conformal Using the fermion operators ψ L(R) (xi , t) and ρR(L) (xi , t), have the different conformal dimensions † (xi , t)ψ dimensions at V = 0, such as ψ R(L) when turning on the fermion interaction. The fermion density operators ρR(L) (xi , t) do not include the phase field, L(R) (xi , t) include the phase field, † (xi , t)ψ and their conformal dimension is keeping as 1, while the operators ψ R(L) and their conformal dimension is changed into g (see below for definition) due to the fermion correlation. Thus we see that under the framework of the eigen-functional theory, the assumptions and validity of usual bosonization theory become very clear. Using the expressions (18) and the low energy effective action (16), we can obtain the fermion Green’s functions (t > 0), R(L) (xi , t)ψ † (0, 0) GR(L)(xi , t) = −i ψ R(L) c,φ (g+1/g−2)/4 1 e±ikF xi δ2 =± , (19) 2π xi ∓ αt ± iδ (xi + αt − iδ)(xi − αt + iδ) 1−γ −1 ∼ D is a momentum cut-off constant. This result is the same where α = 1 − γ 2 , γ = π2aV , g = hv 1+γ , and δ ¯ F as that obtained by usual bosonization methods [13–15,23]. The ODLRO parameter has a simple form (g+1/g−2)/4 1 sin(kF (xi − xj )) δ2 R(xi − xj ) = , πn xi − xj (xi − xj − iδ)(xi − xj + iδ)
(20)
|x|→∞
and it is short-range R(x) ∼ x −(g+1/g)/2 sin(kF x). The relation R(0) = 1 is retained where n = kF /π . Based on Eq. (11), we can obtain the pair distribution function of the system, (g−1)/2 k x 1 2 sin2 ( F2 ) δ2 g(x) = 1 − 2 2 (21) 1 + cos(kF x) . π n x2 (x − iδ)(x + iδ) For a non-interacting fermion system (V = 0, and g = 1), the pair distribution function (21) reduces to the well2 know form [11] g(x) = 1 − π 21n2 sin x(k2 F x) , and the relation g(0) = 0 is retained. Moreover, from Eq. (18), the ground state wave function of the system can be approximately written as a simple form Ψg (x1 , . . . , xN ) = C deteiki xj · (22) |xi − xj |1−1/g , i>j
where C is the normalization constant, and the factor deteiki xj denote the corresponding free right- and leftmoving fermions. The last factor is contributed by the phase field, and represents the fermion correlation. This ground state wave function clearly demonstrates that even weak fermion interaction can produce the long or quasilong range correlation among the fermions. This expression of the ground state wave function of the system is correct under the linear spectrum approximation for the fermions and weak interaction. However, for strong fermion interaction, although we can still formally divide the fermions into the right and left-moving fermions, the phase field would depend on the momentum k of the corresponding “non-interacting” fermions, and the momentum q in Eq. (14) will take the values in the range of −π q π , which make the problem difficult to handle. In general,
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we expect that the ground state wave function of the system can be written as |xi − xj |β , Ψg (x1 , . . . , xN ) = C deteiki xj ·
(23)
i>j
where C is the normalization constant, and the exponent β < 0 depends on the Fermi momentum kF and the fermion interaction strength V . The appearance of the last factor is reasonable because the ground state wave function can be approximately written as Ψg (x1 , . . . , xN ) ∼ deteiki xj +Qki (xj ,t,[φ]) φ ik x i j · exp QRkF (xi , t, [φ]) + QL−kF (xi , t, [φ]) ∼ det e . i
φ
With the low energy effective action (16) in which replacing aV by v(q), the last functional average factor over the Lagrangian multiplier field will contribute a factor i>j |xi − xj |β to the ground state wave function of the system. The factor i>j |xi − xj |β means that there is a long or quasi-long range fermion correlation for any no zero fermion interaction strength V , where the exponent β has the property, β = 0 at V = 0. Notice that in the Hilbert space constructed by the eigen-functionals Ψk (xi , t, [φ]), the system is reduced into a single-particle system, and the fermions are “non-interacting”, thus in this Hilbert space we can construct a “ground state” which is the functional of the Lagrangian multiplier field φi (t). After taking the functional average over the field φi (t), we obtain the physical ground state wave function (22) and (23) of the system. In summary, using the eigen-functional theory, we have studied the one-dimensional spinless fermion system, and calculated the ODLRO parameter, the pair distribution function, the fermion Green’s function and the ground state wave function of the system. Under the linear spectrum approximation, we have proved the commutation of the fermion density operators are not influenced by the fermion interaction, and explained why operators with the same conformal dimensions at V = 0 may have different conformal dimensions when turning on the fermion interaction, thus clarifying some ambiguities in the assumptions and validity of usual bosonization method.
Acknowledgement This work is supported by the Beiren foundation of the Tsinghua University and NSFC.
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