Local perturbation in a Tomonaga-Luttinger liquid at g = 1/2: orthogonality catastrophe, Fermi-edge singularity, and local density of states A. Furusaki arXiv:cond-mat/9702195v1 [cond-mat.str-el] 21 Feb 1997 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-01, Japan (April 21, 2018) The orthogonality catastrophe in a Tomonaga-Luttinger liquid with an impurity is reexamined for the case when the interaction parameter or the dimensionless conductance is g = 1/2. By transforming bosons back to fermions, the Hamiltonian is reduced to a quadratic form, which allows for explicit calculation of the overlap integral and the local density of states at the defect site. The exponent of the orthogonality catastrophe due to a backward scattering center is found to be 1/8, in agreement with previous studies using different approaches. The time-dependence of the corehole Green’s function is computed numerically, which shows a clear crossover from a non-universal short-time behavior to a universal long-time behavior. The local density of states vanishes linearly in the low-energy limit at g = 1/2. 71.10.Pm,72.10.Fk a unitary transformation. The other exponent γB due to the backward scattering is believed to be independent of the strength of the potential and take a universal value, 1/8.14–17 In Refs. 14,15, and 17 the exponent γB is calculated by assuming that a backward scattering center can be replaced with an impenetrable potential barrier. Oreg and Finkel’stein,18 however, questioned the validity of the assumption and argued that the exponent of the Fermi-edge singularity due to a backward scattering center is zero, which implies γB = 0. On the other hand, Kane et al.16 used a renormalization-group equation which becomes exact in the limit of weak repulsive interaction between fermions. They could describe a crossover from the high-energy regime to the low-energy regime, and obtained the same exponent γB = 1/8 in the low-energy limit. The result of a recent direct numerical calculation of the overlap integral19 is also consistent with γB = 1/8. I. INTRODUCTION One-dimensional interacting fermion systems, Tomonaga-Luttinger (TL) liquids,1–3 have recently attracted much attention due to their anomalous response to local perturbations. Recent extensive studies4–9 on transport properties of TL liquids with an impurity revealed that repulsively interacting fermions have vanishing transmission probability through a potential barrier in the low-energy limit. This is because the interaction between fermions strongly enhances the backward scattering at the barrier. Thus, a single defect effectively cuts a TL liquid into two disconnected ones at zero temperature.4 This implies that the local density of states (LDOS) at the defect is reduced for low energy, and according to Kane and Fisher4 it shows a power-law energy dependence, 1 ρ(ω) ∝ ω g −1 , (1) It is known that, when the TL-liquid parameter g is 1/2, the bosonized Hamiltonian containing a nonlinear term representing the backward scattering can be transformed to a quadratic Hamiltonian of fermions.20 This is essentially the same technique as the Emery-Kivelson solution of the two-channel Kondo problem.22 The exact results on the conductance4 and non-equilibrium noise spectra21 were obtained using this refermionization technique. It is thus natural to expect that exact calculation should also be possible for the above-mentioned problems. The purpose of this paper is to show that this is indeed the case. where g is a parameter characterizing the TL liquid. This picture was, however, questioned recently by Oreg and Finkel’stein,10 who claimed based on a mapping to a Coulomb gas problem that the LDOS at the defect is enhanced, rather than suppressed, in the low-energy limit for weakly interacting fermions. This controversy motivates us to reexamine this issue. The orthogonality catastrophe11 in a TL liquid is another interesting subject which has been discussed by several authors.12–18 They showed that the overlap between the ground state of a pure TL liquid |p and that of a TL liquid with a single scatterer |s vanishes in the limit of large system size: | p|s |2 ∝ L−γF −γB , The structure of this paper is as follows. After introducing a model of interacting fermions in Sec. II, we discuss in Sec. III the exact low-energy behavior of the LDOS for g = 1/2. For g = 1/2 we show that Eq. (1) follows from the assumption that the phase field is pinned at the defect site. The importance of zero-modes is emphasized. In Sec. IV we calculate γB analytically for g = 1/2 without assuming the nature of the low-energy (2) where L is the length of the system. The exponent γF is due to the forward-scattering potential and depends on its strength.12,13 It can be calculated directly using 1 fixed point. We find γB = 1/8. The so-called core-hole Green’s function is then computed numerically in Sec. V, which shows a clear crossover from short-time to longtime regimes. We show in Sec. VI that the exponent of the Fermi-edge singularity due to backward scattering is also given by γB . We summarize the results in Sec. VII. H = ivF g4 + 2 ∞ † dx ψL (x) −∞ ∞ −∞ In this section we introduce a model of interacting spinless fermions and briefly explain the bosonization rule to fix the notation. The Hamiltonian of our model is given by ∞ d d † ψL (x) − ψR (x) ψR (x) + g2 dx dx † † : ψµ (x)ψµ (x) :: ψµ (x)ψµ (x) : dx II. MODEL −∞ +λF µ=L,R † † dx : ψL (x)ψL (x) : : ψR (x)ψR (x) : † † : ψµ (0)ψµ (0) : +λB eiθ ψL (0)ψR (0) + h.c. , (3) µ=L,R where ψL(R) describes left-going (right-going) fermions, : A : represents normal-ordered operator A, and λF (λB eiθ ) is the forward-scattering (backward-scattering) potential. Following the standard bosonization rule,23 we express fermions ψµ in terms of bosonic operators: 1 ηL e−iϕL (x) , ψL (x) = √ 2πα 1 ψR (x) = √ ηR eiϕR (x) , 2πα 1 d † : ψµ (x)ψµ (x) : = ϕµ (x), 2π dx We then introduce new bosonic fields as 1 φ(x) = √ [ϕR (x) + ϕL (x)], 4π 1 d [ϕR (x) − ϕL (x)], Π(x) = − √ 4π dx (5a) (5b) (4a) which obey [φ(x), Π(y)] = iδ(x − y). With these fields the Hamiltonian can be transformed to a bosonic form, (4b) H= (4c) where α is a short-distance cutoff. The bosonic fields satisfy the commutation relations [ϕL (x), ϕL (y)] = −iπsgn(x − y), [ϕR (x), ϕR (y)] = iπsgn(x − y), and [ϕL (x), ϕR (y)] = 0. The operator ηµ ’s are Majorana fermions corresponding to zero modes of bosons, which are needed to ensure the anticommutation relation be2 tween ψL and ψR . They satisfy {ηL , ηR } = 0 and ηµ = 1. v 2 2 1 dφ + gΠ2 g dx √ λF dφ(0) λB +√ +i ηL ηR sin 4πφ(0) + θ . π dx πα dx (6) The parameter g is related to g2 and g4 by g = [(1 + g4 − ˜ g2 )/(1 + g4 + g2 )]1/2 with gi = gi /2πvF . Since the in˜ ˜ ˜ ˜ teraction is repulsive, g is less than 1. The renormalized velocity is given by v = vF [(1 + g4 )2 − (˜2 )2 ]1/2 . ˜ g We then introduce another set of bosonic fields ϕ± (x):17 1 ϕ± (x) = √ 8 1 √ √ − g [ϕR (x) ± ϕL (−x)] + g 1 √ √ + g [ϕR (−x) ± ϕL (x)] . g (7) These fields satisfy [ϕ+ (x), ϕ+ (y)] = [ϕ− (x), ϕ− (y)] = −iπsgn(x − y) and [ϕ+ (x), ϕ− (y)] = 0. The advantage of using ϕ± is that we may separate the Hamiltonian into two commuting parts, H = HF + HB , where HF = HB = v 4π v 4π ∞ −∞ ∞ dϕ− dx 2 dx dϕ+ dx 2 dx −∞ + λF π +i g dϕ− (0) , 2 dx λB ηL ηR sin πα (8) 2gϕ+ (0) + θ . (9) The fermion field at x = 0 may be written as i 1 exp − √ ϕ− (0) ψ(0) = √ 2g 2πα ηL exp −i g ϕ+ (0) + ηR exp i 2 g ϕ+ (0) 2 III. LOCAL DENSITY OF STATES AT A SCATTERING CENTER In this section we calculate the following correlation function: 2 . (10) D(t) ≡ gθ |eiHt ψ † (0)e−iHt ψ(0)|gθ , (11) where |gθ is a ground state of H. The LDOS is given by ρ(ω) = (dω/2π)eiωt D(t). In general we expect D(t) ∝ e−i∆t t−ν for t → ∞. Since H has gapless excitations, we know that ∆ must be zero. Thus, we will not pay attention to ∆ and concentrate only on the exponent ν in the following discussion. 1 Since HF and HB commute, the correlation function is factorized into two parts as D(t) = 2πα DF (t)DB (t), where DF (t) = F|eiHF t eiΦ− e−iHF t e−iΦ− |F , DB (t) = B|eiHB t ηL eiΦ+ + ηR e−iΦ+ e−iHB t ηL e−iΦ+ + ηR eiΦ+ |B , (12a) (12b) √ Here Φ− = ϕ− (0)/ 2g, Φ+ = g/2ϕ+ (0), and |F (|B ) is a ground states of HF (HB ). The Hamiltonian HF is (0) related to a free Hamiltonian by a unitary transformation as U HF U † = HF + const, where (0) HF = v 4π ∞ dx −∞ dϕ− dx 2 (13) and U = exp −i g ϕ− (0) . 2 λF πv (14) (0) This means |F = U † |F0 with |F0 being the ground state of HF . We thus get DF (t) = F0 |e (0) (0) iHF t iΦ− −iHF t −iΦ− e e e |F0 = vt 1+i α 1 − 2g 1 ∼ t− 2g . (15) As pointed out in Ref. 10, the forward-scattering potential does not affect the LDOS. Next we rewrite Eq. (12b) as DB (t) = B|eiHB t eiΦ+ − ηL ηR e−iΦ+ e−iHB t e−iΦ+ + ηL ηR eiΦ+ |B , where HB ≡ ηL HB ηL = HB (λB → −λB ). Note that this sign change of the cosine term is a direct consequence of the anticommutation relation {ψL , ψR } = 0. At this point we may set ηL ηR = −i because only the terms involving even powers of ηL ηR will contribute to DB (t) when Eq. (16) is calculated perturbatively in powers of λB . We then shift ϕ+ (x) → ϕ+ (x) + √1 ( π − θ) and 2g 2 obtain H− = ∞ dx −∞ λ − cos πα dϕ+ −π dx 2 2gδ(x) 2gϕ+ (0) . (20) We first consider the case of g = 1/2. A crucial point in this case is that the cosine term becomes eiϕ+ (0) + e−iϕ+ (0) . Therefore, fermionizing the chiral boson ϕ+ as DB (t) = 2 +|eiH+ t eiΦ+ e−iH− t e−iΦ+ |+ +2 cos θ +|eiH+ t eiΦ+ e−iH− t eiΦ+ |+ , v 4π (16) eiϕ+ (x) √ = ηψ+ (x), 2πα (17) (21) where v H± ≡ 4π ∞ −∞ dx dϕ+ dx 2 λB ± cos πα we may transform Eq. (18) to20,24 2gϕ+ (0) (18) H± = iv and we have used the fact that the ground state of H+ , |+ , is invariant under ϕ+ → −ϕ+ . It is useful to transform Eq. (17) further to the form DB (t) = 2 +|eiH+ t e−iH− t |+ +2 cos θ +|e iH+ t −iH− t 2iΦ+ e e |+ , ∞ −∞ † dxψ+ (x) d ψ+ (x) dx λB † ηψ+ (0) + ψ+ (0)η , ±√ 2πα (22) where η is a Majorana fermion, satisfying η 2 = 1. This leads to a simple relation, ηH+ η = H− . Note that H+ is a quadratic Hamiltonian, which can be easily diagonalized:20 (19) where 3 ∞ H+ = = −∞ ∞ 0 dk ξk a† ak + k √ where ξk ≡ vk and ψ+ (x) = (dk/ 2π)e−ikx ak . For later convenience we write the transformation rule here:20 λB √ ηak + a† η k 2π α dkξk c† ck + d† dk + const, k k (23) 1 ak = √ ck + 2 Γ d +√ P 2 + Γ2 ) k 2π 2(ξk 1 a−k = √ c† − 2 k ξk Γ d† + √ P 2 + Γ2 ) k 2π 2(ξk η= λB π ∞ ξk 2 P α ∞ dq 0 1 2 ξq + Γ2 dq 0 ∞ dq 0 1 2 ξq + Γ2 1 2 ξq + Γ2 d† dq q − q−k q+k , (24a) d† dq q − q+k q−k , (24b) dq + d† , q (24c) where k > 0, Γ ≡ λ2 /(παv), and ck and dk satisfy the ordinary anticommutation relation. The ground state |+ is B the vacuum of ck and dk . Using Eq. (21), we rewrite Eq. (19) in a fermionic form, √ (25) DB (t) = 2 +|eiH+ t ηe−iH+ t η|+ + 2 2πα cos θ +|eiH+ t ηe−iH+ t ψ+ (0)|+ , where † H+ = H+ + πv : ψ+ (0)ψ+ (0) : +const. (26) From Eqs. (24a) and (24b), the second term becomes † πv : ψ+ (0)ψ+ (0) : = v 2 ∞ 0 ∞ dk dp 0 which is an irrelevant operator with scaling dimension 2. To find the long-time behavior of DB (t), we can thus treat Eq. (27) as a small perturbation. The lowest-order calculation then gives, for Γt ≫ 1, DB (t) = − √ λB ln(vt/α) 4i . + 2πα cos θ πΓt πv Γ2 t2 2ω π 2 vΓ (28) HM = ∞ 0 dk dp − d† , p (27) v 2 ∞ −∞ dx dφ dx 1 g 2 + gΠ2 + M [φ(0)]2 , (30) 2 where M should be a characteristic energy scale at which the cosine term becomes of the order of the band width (M = Γ for g = 1/2). It immediately follows from the scaling equation dλB /dl = (1 − g)λB that v M∝ α (29) for ω ≪ Γ. This is consistent with Eq. (1). We see that the single scatterer at x = 0 indeed depletes the low-energy excitations around it. φ= ck + c† k 2 ξp + Γ2 For g = 1/2 (0 < g < 1) we take a different approach. We assume from the outset that the phase field ϕ+ is pinned at x = 0 by the cosine potential in H+ (18), as in Refs. 14,15, and 17. We thus replace the cosine by a term which is easier to deal with. A convenient choice is Note that the 1/t-dependence of the first term comes from the correlator +|η(t)η(0)|+ , which also appeared in the two-channel Kondo problem.22 Combining Eqs. (15) and (28), we get D(t) = −2/(π 2 vΓt2 ) for Γt ≫ 1, which implies ρ(ω) = ξp λB v 1 1−g . (31) Since HM is a quadratic Hamiltonian, it is easily diago† nalized as HM = dkξk α† αk + βk βk with k g † sin(kx) αk + α† + cos(k|x| − δk ) βk + βk k 2πk (32) and Π = (1/gv)∂Π/∂t, where αk and βk satisfy the ordinary commutation relations of bosons. The phase shift is given by δk = tan−1 (gM/2vk). Note that δk → π/2 as k → 0. Let us denote the ground state of HM by |0M . We then find 0M |∂x ϕ+ (0, t) ∂x ϕ+ (0, 0)|0M = 2πg 0M |Π(0, t)Π(0, 0)|0M = 4 24 g 2 M 2 v 2 t4 (33) for M t ≫ 1, implying that ∂x ϕ+ (0) is an irrelevant operator with dimension 2. This is consistent with the observation made in Eq. (27). In fact, this is an expected result because ϕ+ is pinned at x = 0. We may thus use H− instead of H− to obtain the long-time asymptotic behavior of DB (t) in Eq. (19). It is also important to note that eiΦ+ is not fluctuating too much and can be regarded essentially as a constant because ϕ+ (0) is pinned. In fact, we find 0M |eiΦ+ |0M = 0M | exp i = eγ gαM IV. ORTHOGONALITY CATASTROPHE In this section we discuss the orthogonality catastrophe for the special case of g = 1/2. We calculate the overlap integral | p|s |2 = | F0 |F |2 × | 0|+ |2 , where |0 is the ground state of the Hamiltonian H0 ≡ H+ |λB =0 . It is almost trivial to find γF in Eq. (2) because F0 |F = F0 |U † |F0 . We get12,13 2π/gφ(0) |0M γF = 2g (34) 2v −iH+ t H0 = iv † DB (t) ∝ 0M |e Ve V |0M ≈ 0M |eiHM t V e−iHM t V † |0M , (35) . † dxψ+ (x) d ψ+ (x), dx 0 |+ = T exp −i −∞ dt1 · · · −∞ (38) eǫt H ′ (t)dt |0 , (39) where ǫ is positive infinitesimal and H ′ (t) = eiH0 t (H+ − H0 )e−iH0 t . Using the linked-cluster theorem, we can write the overlap integral as 0|+ = exp[Gc (0, −∞)], (40) where Gc (0, −∞) is a sum of connected ring diagrams, 2n 0 0 (37) and |0 is the filled Fermi sea. Then the ground state of H+ can be written as We therefore conclude D(t) ∝ t−1/g , from which Eq. (1) follows. We emphasize that the above calculation should give the exact value of the exponent, although the amplitude may not be correct. λ2n Gc (0, −∞) = − 2n n=1 ∞ −∞ where V is a unitary operator which shifts φ(x) → √ φ(x) + 2π . The rhs of Eq. (35) is known to decay as ∼ t−1/2g .25 This result can be easily obtained using the following representation for V : √ π dxΠ(x) V = exp i 2 ∞ e−αk † (36) = exp − dk √ sin δk βk − βk . 2gk 0 ∞ 2 Hence our problem is reduced to calculate the overlap 0|+ . In the fermion language, H0 is for αM ≪ v, where γ = 0.577 . . . is Euler’s constant. Note that, at g = 1/2, we get +|eiΦ+ |+ = −(λB /πv) ln(v/αΓ), which is consistent with Eqs. (31) and (34). Hence, from Eq. (19), we get iH+ t λF 2πv dt2n s0 (t1 − t2 )g0 (t2 − t3 ) · · · s0 (t2n−1 − t2n )g0 (t2n − t1 ) exp −∞ ǫti . (41) i=1 √ Here λ ≡ λB / 2πα and the propagators s0 (t) and g0 (t) are given by s0 (t) = 0|Tη(t)η(0)|0 = sgn(t), (42a) i † † g0 (t) = 0|T[ψ+ (x = 0, t) − ψ+ (0, t)][ψ+ (0, 0) − ψ+ (0, 0)]|0 = , πv[t − iεsgn(t)] (42b) where ε is positive infinitesimal. Differentiating Eq. (41) with respect to λ, we obtain Gc (0, −∞) = − v 4 Γ 0 0 dΓ 0 dt1 −∞ −∞ dt2 eǫ(t1 +t2 ) s0 (t1 − t2 )g(t2 , t1 ), (43) where g(t1 , t2 ) is a solution of a Dyson equation, g(t1 , t2 ) = g0 (t1 − t2 ) − Γ P 2πi 0 0 dt3 −∞ dt4 −∞ eǫ(t3 +t4 ) sgn(t3 − t4 )g(t4 , t2 ). t1 − t3 (44) Since Eq. (44) contains double integral, working in real time is not so convenient as it is in the Fermi-liquid case.26 On the other hand, the Fourier transform of Eq. (44) contains only a single integral: 5 g(ω, t2 ) = − ˜ ∞ eiωt2 iΓ sgn(ω) + v |ω| −∞ 1 1 dν . g (ν, t2 ) ˜ − 2πi ν − ω + 2iǫ 2(ν + iǫ) (45) This equation can be solved in the limit ǫ → 0 in the standard way.27 We first introduce functions g± by ˜ ∞ g± (ω) = ˜ −∞ dν 1 1 . g (ν, t2 ) ˜ − 2πi ν − ω ∓ 2iǫ 2(ν + iǫ) (46) We can then express Eq. (45) as g+ (ω) − 1 − i ˜ Γ |ω| g− (ω) = − ˜ eiωt2 sgn(ω). v (47) A solution of this equation with correct analytic properties is g± (ω) = − ˜ ∞ dν eiνt2 sgn(ν) X± (ω) , −∞ 2πi ν − ω ∓ iδ X+ (ν) 1 v (48) where δ is positive infinitesimal and  Γ dν ln 1 − i |ν|  . X± (ω) = exp −∞ 2πi ν − ω ∓ iδ  With this solution Eq. (43) becomes 1 8π 2 i 1 = 2 8π Gc (0, −∞) = Γ 0 t1 ∞ ∞ ∞ (49) sgn(ν) X− (ω) ν − ω + iδ X+ (ν) −∞ −∞ 0 −∞ −∞ ∞ Γ 0 ∞ X− (ω) i(ν−iǫ)τ sgn(ν) , dω dΓ dτ dν e X+ (ν) −∞ (ω − ν + 2iǫ)(ν − ω + iδ) 0 −∞ −∞ dt2 dt1 dΓ dω dν e−iωt1 +iνt2 +ǫ(t1 +t2 ) (50) where τ = t2 − t1 and we have integrated over (t1 + t2 )/2. As pointed out by Hamann,28 in the next step in which we perform the ω integral, it is important to keep ǫ finite while taking the limit δ → +0: Gc (0, −∞) = − =− i 4π 1 8πǫ 0 Γ dτ dΓ Γ dΓ 0 dνei(ν−iǫ)τ −∞ −∞ 0 ∞ 0 dτ −∞ ∞ dν −∞ sgn(ν) X− (ν − 2iǫ) 2iǫ X+ (ν) −1 After replacing ν/(|ν| − iΓ) by ν[(|ν| − iΓ) over τ to obtain −1 − (ν − iΓ) Γ 0 1 1 ν − 2 dΓ dν 2 4πǫi 0 ν + Γ2 2π −∞ i Γ 1 Λ Γ = ln +1 − ln 2ǫ 2π Γ 16 EL Gc (0, −∞) = Γ 1 νei(ν−iǫ)τ + 2 |ν| − iΓ 8π 0 dΓ 0 −∞ ∞ dΓ 0 0 , ∞ dν1 Γ i |ν| dν2 0 dν1 −∞ ] and ln 1 − Γ ∞ dτ ∞ dν2 −∞ Γ ν1 ei(ν1 −iǫ)τ ln 1 − i |ν2 | . |ν1 | − iΓ (ν2 − ν1 + iδ)2 Γ by ln 1 − i |ν| / 1 + i Γ , we integrate ν ν1 1 tan−1 2 (ν1 + ν2 )2 ν1 + Γ2 Γ ν2 (51) where we have introduced the high-energy cutoff Λ ∼ v/α and the low-energy cutoff EL ∼ v/L. From Eqs. (40) and (51) we get γB = 1/8 in agreement with the previous studies.14–17,19 Note that the quantity E0 ≡ −(Γ/2π)[ln(Λ/Γ)+1] appearing in the first term is equal to the difference between the ground state energies of H+ and H0 .20 Since δ(E) ≡ tan−1 (Γ/E) in Eq. (51) is the phase shift for fictitious chiral fermions due to the coupling λB in Eq. (22), the above calculation implies that γB = 1 [δ(0)/π]2 , in contrast to the Fermi-liquid result11,26 γFermi = 2 [δ(0)/π]2 . The extra factor 1/2 in our result can be traced back to the peculiar form of the scattering term in † † Eq. (22). Only the combination ψ+ − ψ+ interacts with η, and the other combination ψ+ + ψ+ is decoupled. Hence only half of the degrees of freedom have the phase shift (δ(0) = π/2), giving the factor 1/2. As pointed out by Matveev,20 the Hamiltonian (22) is equivalent to the effective Hamiltonian of the two-channel Kondo model in the Toulouse limit,22 where the Majorana fermion η corresponds to the xy-component of the impurity spin. Thus our calculation also applies to the orthogonality catastrophe in the two-channel Kondo problem in which J⊥ is turned on and off while Jz kept constant. 6 V. CORE-HOLE GREEN’S FUNCTION Next we calculate the core-hole Green’s function, G(t) = 0|eiH0 t e−iH+ t |0 (52) for g = 1/2. Using the linked-cluster theorem again, we get G(t) = exp[Gc (t, 0)], where Gc (t, 0) is Gc (t, 0) = − ∞ λ2n 2n n=1 t t dt1 · · · 0 0 dt2n s0 (t1 − t2 )g0 (t2 − t3 ) · · · s0 (t2n−1 − t2n )g0 (t2n − t1 ). 0 This time we differentiate Eq. (53) with respect to t to get d Gc (t, 0) = λ2 dt 10 t 0 −1 dt1 g0 (t − t1 )s(t1 ), (54) −Re[dGc /d Γt] − where s(t1 ) is defined for 0 ≤ t1 ≤ t and is a solution of a Dyson equation, Γ P s(t1 ) = −1 − 2πi t dt3 0 −2 10 −3 sgn(t1 − t3 ) s(t4 ). dt4 t3 − t4 0 Γ Γ d Gc (t, 0) = − dt 4 2πi 10 10 t (55) −1 t dt1 0 s(t1 ) . t1 0 10 1 10 10 2 10 Γt From this equation we can easily show that s(t1 ) = s(t − t1 ) and s(+0) = −1. Thus Eq. (54) becomes − (53) FIG. 1. Time evolution of the core-hole Green’s function. There is a clear crossover at Γt ∼ 1. The dashed line represents Re[dGc /dΓt] = −1/(8Γt). (56) VI. FERMI-EDGE SINGULARITY Here the first term comes from the real part of g0 in Eq. (42b). In this section we briefly discuss the Fermi-edge singularity for g < 1 to show that the exponents can be easily obtained from the analysis of Secs. IV and V. Here we are concerned with the correlation function For short times Γt ≪ 1, we can solve Eq. (55) perturbatively. Up to order (Γt)2 we obtain Gc (t, 0) = i t Γt ln 2π tc 1 1 − 1 − Γt + (Γt)2 , 4 24 (0) I(t) = g0 |ei(HF (57) +H0 )t ψ(0)e−i(HF +HB )t ψ † (0)|g0 , (58) where |g0 ≡ |F0 ⊗ |0 . Following the same path as in 1 Sec. III, we write the correlator as I(t) = 2πα IF (t)IB (t), where12,13 where tc is a short-time cutoff ∼ 1/Λ. This expansion, however, starts to fail around Γt ∼ 1. From the analysis in Sec. IV, for Γt ≫ 1 we expect Gc (t, 0) to approach 1 −iE0 t − 8 ln(Γt).14–17 (0) (0) IF (t) = F0 |eiHF t e−iΦ− U e−iHF t U † eiΦ− |F0 ∼ t−νF The crossover from the short-time to the long-time regimes can be seen most conveniently by solving Eq. (55) numerically and putting the solution into Eq. (56). Note that the integral in Eq. (56) is well-defined because Ims(t1 ) ∼ t1 | ln t1 | for t1 → 0. Figure 1 shows the tdependence of the real part of (d/dt)Gc (t, 0) computed in this way. It clearly exhibits the crossover at Γt ∼ 1 from the short-time behavior, Eq. (56), to the long-time asymptote, Re[dGc (t, 0)/dt] = −1/8t. with νF = √1 2g + λF √ 2πv 2g 2 (59) and IB (t) = 2 0|eiH0 t e−iH− t |0 +2 cos θ 0|eiH0 t e−iH− t e2iΦ+ |0 . (60) We expect that IB (t) should decay as IB (t) ∝ t−νB in the long-time limit. We now notice that the first term 7 1 in Eq. (60) is similar to the core-hole Green’s function discussed in Sec. V. As we saw in Fig. 1, it should decay γ as ∼ t−˜ with γ being the exponent of the orthogonal˜ ity catastrophe between |0 and the ground state of H− : γ | 0|− |2 ∝ L−˜ . The latter state has a finite overlap with the ground state of H− , because ∂x ϕ(0) [∝ (H− − H− )] is an irrelevant operator around the fixed point of H− . This means γ = γB = 1/8. Since the second term in ˜ Eq. (60) contains extra factor, e2iΦ+ , at least it is not larger than the first term. Hence we conclude νB = 1/8, in agreement with Refs. 15 and 17. The fact that νB equals γB is a direct consequence of the pinning of ϕ+ at x = 0. 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The characteristic, anomalous low-energy (long-time) properties were obtained by exact calculations for g = 1/2 by mapping the bosonized Hamiltonian back to a fermionic quadratic Hamiltonian. This method has allowed us to describe the crossover from the weakcoupling (short-time) to the strong-coupling (long-time) regimes. The exact results obtained for g = 1/2 agree with the previous studies based on the assumption that the phase fields are completely pinned at the impurity site in the low-energy limit. The agreement implies that, to describe the low-energy physics, it is sufficient to use an effective model which incorporates the perfect reflection by the local potential. We conclude that γB = 1/8 1 and ρ(ω) ∝ ω g −1 for g < 1. It seems that the mapping to a Coulomb gas problem used in Refs. 10 and 18 makes it difficult to capture the Majorana fermions which have played an essential role in this paper. After completion of this work the author became aware that Fabrizio and Gogolin29 obtained a similar result on the low-energy behavior of the LDOS, Eq. (1). ACKNOWLEDGMENTS The author would like to thank N. Kawakami, N. Nagaosa, and V. Ponomarenko for helpful discussions. The numerical computation was supported by the Yukawa Institute for Theoretical Physics and also done in part on VPP500 at the Institute for Solid State Physics, University of Tokyo. 8 29 M. Fabrizio and A. O. Gogolin, cond-mat/9702080. 9