J Supercond Nov Magn (2015) 28:1331–1335
DOI 10.1007/s10948-014-2914-4

ORIGINAL PAPER

Generic Features of an Electron Injected
into the Luttinger Liquid
Hideaki Maebashi · Yasutami Takada

Received: 31 July 2014 / Accepted: 12 November 2014 / Published online: 14 December 2014
© Springer Science+Business Media New York 2014

Abstract An extra electron injected into the Luttinger liquid (LL) is not completely separated into spin and charge
excitations, but it provides a peak of an electron-like excitation in addition to the two prominent spin- and chargeexcitation peaks in the one-electron spectral function of aninteracting one-dimensional (1D) system. By a scaling argument based on a renormalization-group scheme, we obtain
universal features of this electron-like excitation for low
energies and in the vicinity of the Fermi point, which areentirely characterized by the Luttinger parameter Kc andthe renormalized Fermi velocity vF at the LL fixed point.
The latter quantity vF is shown to be given by the slope ofthe lowest excitation spectrum near the Fermi point in thesystem with large but finite size. Some of exact results arepresented for the 1D Hubbard model at n = 0.59 filling.
Keywords One-dimensional electron systems ·Luttinger liquid · Spin-charge separation · One-electron
spectral function

1 Introduction
A Luttinger liquid (LL) is the most typical example of nonFermi liquids that are characterized by the absence of adiscontinuity in the momentum distribution function n(k)at temperature T = 0. It describes universal features of aninteracting one-dimensional (1D) electron system for low
energies, long wavelengths, and/or in the vicinity of the

H. Maebashi () · Y. Takada
Institute for Solid State Physics, University of Tokyo,
Kashiwa, Chiba 277-8581, Japane-mail: maebashi@issp.u-tokyo.ac.jp

Fermi point kF [1–3]. One of those features is a powerlaw singularity. For example, the behavior of n(k) with
momentum k near kF is specified by the Luttinger parameter
Kc (= 1) as n(k) ∝ const. − |k − kF |θ sgn(k − kF ) with apower-law exponent θ = (Kc − 1)2 /4Kc > 0. Anotherimportant feature is spin-charge separation, due to which
the one-electron spectral function A(k, ω) at T = 0 has
two power-law divergences at energies ω = us (k −kF ) anduc (k−kF ), where us and uc are the spin and charge velocities,
respectively [4–6].
The central idea of the LL theory is the equivalence
between the low-energy spectrum of the 1D system and thatof the Luttinger model, which is a 1D solvable model ofinteracting electrons with linear dispersion [3]. The spincharge separation is then precisely defined by the fact thatthe bosonized Hamiltonian for the Luttinger model fully
separates into spin and charge parts. However, it does notmean that an extra electron added to the ground state of the
Luttinger model is completely fractionalized into spin andcharge excitations. Instead, the extra electron leaves behinditself an electron-like excitation with a velocity vF in the
spectral function. This was noted by Voit about two decades
ago who considered a toy model with θ = 0 but us = uc ,which is not a typical LL with θ > 0, though [7]. Recently, ageneral proof of it has been given by us, including the clarification of the electron-like excitation for high energies andaway from the Fermi point beyond the LL theory [8, 9].
It is, however, noted that the spectral function A(k, ω)of a 1D microscopic model (e.g., the 1D Hubbard model)
generally differs from that of the Luttinger model even if
the low-energy spectra of those models are equivalent toeach other. For low energies and in the vicinity of kF , thedifference in A(k, ω) is ascribed to the renormalization factor in a renormalization-group scheme. In the same scheme,
the velocity vF of an electron-like particle (renormalized

1332

J Supercond Nov Magn (2015) 28:1331–1335

electron or pseudoelectron) is not equal to the noninteracting Fermi velocity v0 of the microscopic model but is, ingeneral, renormalized.
In this paper, carefully discussing on the above renormalization effects, we study generic features of A(k, ω) of aninteracting 1D electron system at T = 0 for low energies
and in the vicinity of kF . Even with the inclusion of thoseeffects A(k, ω) is shown to have a singularity related to thepseudoelectron at ω = vF (k − kF ) as a function of ω withdifferent prefactors above and below vF (k −kF ). We obtain
universal values for the power-law exponent and the prefactor ratio, which are uniquely determined by Kc . We find thatthe pseudoelectron velocity vF is, in fact, given by the renormalized Fermi velocity at the LL fixed point and is equal
to the slope of the lowest excitation spectrum near kF in thesystem with large but finite size.
To illustrate our claim, we present some of exact results
for the 1D Hubbard model at n = 0.59 filling. It is notedthat the overall structure of A(k, ω) for the 1D Hubbard
model at this filling (or n = 0.6) and U = 4.9 t has been
intensively studied by various methods in relation to theangle-resolved photoemission spectroscopy of the quasi1D organic conductor TTF-TCNQ [10–15], although thosestudies are not concerned with the detailed structure ofA(k, ω) for low energies and in the vicinity of kF . Here, weconcentrate on it within the LL theory.

It is, here, noted that (1) and (2) also hold in threedimensional (3D) systems with suitable changes of thenotation. In 3D, the fixed point is a Fermi liquid, where ZF
is finite for  → 0 and Ã+ (kF + q, ω) corresponds to thequasiparticle’s spectral function given by a Lorentzian. In
1D, on the other hand, the fixed point is a LL, where ZF
vanishes for  → 0 as ZF ∝ (/kF )θ and Ã+ (kF +q, ω) has
singularities as a function of ω.
If we consider the case of q = 0, we can take the limitof  → 0. Then, the dependence of ZF on  is canceledby Ã+ (kF , ω) ∝ (v0 )−θ |ω|−1+θ in A(kF , ω). In this paper,
however, we are interested in the structure of low-energy
excitations in A(kF + q, ω) and therefore we have to consider the case of a finite q. For a finite q, the limit of  → 0
cannot be taken for A(kF+q, ω) because of the limitation of|q| < , so that  must be scaled to |q|. This scaling argument leads to ZF ∝ (|q|/kF )θ , which is regarded as a finite
number. By introducing a dimensionless function f (x, y)through Ã+ (kF +q, ω) = f (ω/v0 q, |q|/) /v0 |q|, we canwrite (1) with a finite q asA(kF +q, ω) ∝ |q|−1+θ f (ω/v0 q, |q|/) /v0 kFθ .

(3)

Then, by examining the characteristics of the function
f (x, y) irrespective of the argument y, we can find theuniversal behaviors of A(kF +q, ω) (see Section 4).

3 Finite-Size Spectrum
2 Renormalization-Group Scheme
Let us consider the spectral function A(kF +q, ω) for a 1D†
, ck,σ ] (σ = ± labels the
microscopic Hamiltonian H [ck,σspin) and introduce a bandwidth cutoff . By eliminating“fast” modes with |k ∓ kF | >  in a renormalization-groupscheme, the spectral function can be written in the formA(kF +q, ω) = ZF Ã+ (kF +q, ω)+A(kF +q, ω),(1)where the momentum q is limited by |q| <  but the energyω is allowed to take all values [16, 17]. Here, Ã+ (kF +q, ω)is the spectral function for the fixed-point Hamiltonian†, c̃k,α,σ ] (α = ± distinguishes the right- and leftH̃ [c̃k,α,σmoving renormalized electrons) and ZF is a renormalization
factor that is introduced
to satisfy
 the fermion anticom
†mutation relation c̃k,α,σ , c̃k  ,α  ,σ  = δk,k  δα,α  δσ,σ  in the
fixed-point Hamiltonian. We then have the sum rule onÃ+ (kF +q, ω) as
 ∞


Ã+ (kF+q, ω)dω = c̃kF +q,+,σ , c̃k†F +q,+,σ = 1, (2)−∞for any |q| < . The last term A(kF +q, ω) represents an“incoherent” part in A(kF +q, ω). We do not expect that itgives an important contribution for low energies. Therefore,
we do not consider it in the following.

†
The LL fixed-point Hamiltonian H̃ [c̃k,α,σ, c̃k,α,σ ] is composed of two branches of linear dispersion with a bandwidth
cutoff  and is entirely characterized by a renormalized
Fermi velocity vF , an intra-branch coupling g4 , and an interbranch coupling g2 . On the other hand, the Luttinger model
has an infinite bandwidth and momentum-dependent couplings with a momentum transfer cutoff . There are fourkinds of those couplings in general, which are conventionally denoted by g4 (q), g4⊥ (q), g2 (q), and g2⊥ (q). In thispaper, we take g4 (q) = 0 and g2 (q) = g2⊥ (q) for all
q together with g4⊥ (0) = g4 and g2 (0) = g2⊥ (0) =g2 , so that the Luttinger model becomes equivalent to†H̃ [c̃k,α,σ, c̃k,α,σ ] except a cutoff scheme. The difference
between two cutoff schemes for the bandwidth and themomentum transfer does not change physical results for low
energies, but a mathematically rigorous treatment such as
bosonization is possible for the latter scheme. Therefore,†, c̃k,α,σ ] from
we can derive physical results for H̃ [c̃k,α,σmathematically rigorous ones for the Luttinger model.
By using the above equivalence and the correspondence
between the low-energy spectra of the fixed-point Hamilto††
nian H̃ [c̃k,α,σ, c̃k,α,σ ] and the bare one H [ck,σ, ck,σ ], we seethat the spectrum of low-energy excitations in a 1D interacting electron system with large but finite size L (the number

J Supercond Nov Magn (2015) 28:1331–1335

1333

G
of electrons in the ground state NG = 2×odd and kF ≡ πN
2L )is given by


2π 
−
+
−
−2
uν +
ν +ν +nν +nν +O(L ),L ν=s,c2π   ++− −−P = kF Jc +
ν +nν −nν ,L ν=s,c ν
E =

(4)
(5)

(δNc ±Jc Kc )(δNs ±Js )where ±
, ±
, n±
c =
s =
ν is a nonneg16Kc
16
ative integer, δNc , δNs , Jc , and Js are, respectively, quantum
numbers related to total charge added into the system, spin,
charge current, and spin current [3, 18, 19].
In (4) and (5), us , uc , and Kc are given in terms of the LL
fixed-point parameters vF , g4 , and g2 as us = vF − g4 /2π ,1/2, and Kc = [(vF +uc = (vF +g4 /2π )2 −(g2 /π )2
g4 /2π−g2 /π )/(vF +g4 /2π+g2 /π )]1/2 . We can solve thoseequations for vF , g4 , and g2 . In particular, we find that therenormalized Fermi velocity vF at the LL fixed point isgiven by2Δ㻱㻌ε㻜㻔㼗㻕㻌

㻙㼗㻲

㻜

㼗㻲

㼗

2

vF = [us +(1+2θ)uc ]/2,

(6)
2

c)
as alreadywhere θ is related to Kc by θ = (1−K
4Kc
mentioned in Section 1.
To elucidate the physical meaning of (6), let us consider the spectrum in adding an electron with up spin in
the vicinity of kF . The corresponding quantum numbers are:
δNc = Jc = δNs = Js = 1. By substituting those quantum
numbers into (4) and (5) and using (6), we obtain


−
−2
uν n+
E = (π/L)vF +(2π/L)ν +nν +O(L ), (7)Fig. 1 Sketch of a process of adding an extra electron into theground state of a noninteracting 1D system with finite size and thecorresponding low-energy spectrum (color online)

4 One-Electron Spectral Function
The exact result of the Green’s function of the Luttinger
model is well known and it takes a simple form in real
space and time [20, 21]. By using the physical equiva†, c̃k,α,σ ]lence between the Luttinger model and H̃ [c̃k,α,σ
discussed in the last section, we obtain the Green’s function†
for the LL fixed-point Hamiltonian H̃ [c̃k,α,σ, c̃k,α,σ ] as
G̃+ (x, t) =−θ x −vF t +i/(t)[x −us t +i/(t)]−1/22π x −vF t +iη(t)× [x −uc t +i/(t)]−(1+θ )/2
× [x +uc t −i/(t)]−θ/2 ,ν=s,cP = kF +π/L+(2π/L)

 

−
n+
ν −nν .

(8)

ν=s,c

Then, we see that vF corresponds to the slope of the lowest excitation spectrum near kF as limL→∞ E/(P −kF ).
Remember how the Fermi velocity v0 is defined for a noninteracting system with a dispersion ε0 (k). The lowest energy
excitation near kF is given by adding an electron to the unoccupied state with momentum kF+π/L (kF L/π is an odd integer), as sketched in Fig. 1. Then, the noninteracting Fermi
velocity v0 can be defined as limL→∞ E/(P − kF ) =limL→∞ [ε0 (kF +π/L)−ε0 (kF )]/(π/L) = ∂ε0 (k)/∂k|k=kF .
Because an interacting system has rigid Fermi points as welland the process of adding an extra electron into the system
also obeys the Fermi statistics, we understand that (6) is anatural extension to the interacting case of v0 .
In Fig. 2, we have plotted vF /v0 as a function of the interaction U in units of 2π v0 together with us /v0 , uc /v0 , andθ obtained by using the Bethe ansatz method for the 1D
Hubbard model with the filling n = 0.59. Those values for
U = 4.9 t will be used in obtaining the result in Fig. 3.

Δ㻼㻌

㼗㻲

㻜㻚㻜
㻜㻚㻝㻜

㼁㻩㻠㻚㻥㼠
㻜㻚㻠
㻜㻚㻢

㻜㻚㻞

(9)

㻜㻚㻤

㻝㻚㻜

θ

㻜㻚㻜㻡
㻜㻚㻜㻜
㻝㻚㻡

㼡㼏㻛㼢㻜

㻝㻚㻜

㼢㻲㻛㼢㻜

㼡㼟 㻛㼢㻜

㻜㻚㻡㼢㻜㻌㻩 㻞㼠 㼟㼕㼚㻔π㼚㻛㻞㻕

㻜㻚㻜
㻜㻚㻜

㻜㻚㻞

㻜㻚㻠

㻜㻚㻢

㻜㻚㻤

㻝㻚㻜

㼁㻌㻔㼡㼚㼕㼠㼟㻌㼛㼒㻌㻞π㼢㻜㻌㻕Fig. 2 Exact results of vF /v0 , us /v0 , uc /v0 , and θ as a function of theinteraction U in units of 2πv0 ≈ 10 t for the 1D Hubbard model withn = 0.59. The corresponding weak-coupling results are also shown by
the broken curves (color online)

J Supercond Nov Magn (2015) 28:1331–1335

㼪 㻗㻔㼗㻌㻗㼝㻘ω㻕㻌㻔㼡㼚㼕㼠㼟㻌㼛㼒㻌㻝㻛㼢㻭㻜㻌㼨㼝㼨㻕㻌
㻲

1334

㼡㼏㻌㼝 㼢㻲㻌㼝 㼡㼟㻌㼝㻠㼝㻩㻙㻜㻚㻝Λ

㻟
㻞
㻝
㻜
㻙㻞㻚㻜

㻙㻝㻚㻡

㻙㻝㻚㻜

㻙㻜㻚㻡

ω㻌㻔㼡㼚㼕㼠㼟㻌㼛㼒㻌㼢㻜㻌㼨㼝㼨㻕㻜㻚㻜

Fig. 3 Fixed-point spectral function Ã+ (kF +q, ω) with q = −0.1
for the 1D Hubbard model with n = 0.59 and U = 4.9 t (color online)

where η(t) = η sgn(t) with a positive infinitesimal η and(t) =  sgn(t). This Green’s function has the same form
as that of the Luttinger model, but vF , us , uc , and θ are notparameters independent of one another. They are quantities
that is derived from a 1D microscopic Hamiltonian as inFig. 2.
The retarded Green’s function G̃R+ (x, t) is derived from(9) by using G̃R
(x,
t)
=
G̃
(x,
t)
+
G̃∗+ (−x, −t) for t >++0. The spectral function Ã+ (k, ω) ≡ −(1/π )ImG̃R+ (k, ω)is then obtained by performing a double Fourier transform∞∞Riωt−i(k−kF )x .
G̃R
+ (k, ω) = 0 dt −∞ dx G̃+ (x, t)e
By using the above definition of Ã+ (k, ω), we can easily
verify that the sum rule, (2), actually holds for (9). This is+because (9) necessarily leads to G̃R+ (x, t = 0 ) = −iδ(x),for which the presence of the second factor including vF is
essential. If we replace  by 1/η in (9), we have an approximate result of the Green’s function without this factor,
which is often cited in the literature as the Green’s function
obtained by the bosonization method. It is, however, notedthat in the Luttinger model, taking  = 1/η → ∞ corresponds to neglecting the momentum dependence of thecouplings, for which a mathematically rigorous formulationof bosonization to satisfy the fermion anticommutation relation cannot be possible unless θ = 0 [3]. In fact, a simple+
calculation leads to G̃R+ (x, t = 0 )|=1/η = −iw(θ)δ(x)−1/2
with w(θ) = π (1/2+θ/2) /  (1+θ/2). Becausew(θ) < 1 for θ > 0, the replacement of  by 1/η results in a
violation of the sum rule, (2), unless θ = 0.
Exact calculations of the spectral function of the Luttinger model with θ > 0 were first made with satisfying thesum rule by Schönhammer and Meden [22]. But their results
are limited only for vF = us . This is not the case in general
for an interacting 1D system as clearly seen in Fig. 2. The
calculations for a generic case were made by us by developing a new scheme to perform the double Fourier transformwith use of the Feynman parameter trick, leading to a proofof the existence of an additional peak structure called thepseudoelectron excitation [8].
In Fig. 3, by using the same scheme, we have plotted thefixed-point spectral function Ã+ (kF+q, ω) with q = −0.1
for the 1D Hubbard model with n = 0.59 and U = 4.9 t.
The peak-like cusp observed near the center is the pseudoelectron excitation. Because it is located at ω = vF q, thepseudoelectron velocity is given by vF , i.e., the renormalized Fermi velocity at the LL fixed point. It is, here, notedthat since the function f (x, y) introduced in Section 2 isdefined by Ã+ (kF + q, ω) = f (ω/v0 q, |q|/) /v0 |q|, theplot in Fig. 3 corresponds to that of f (−x, y) for y = 0.1
as a function of x. We can show that for any value of y as
well as y = 0.1, the function f (x, y) has divergences at
x = us /v0 , uc /v0 and a peak-like cusp at x = vF /v0 . On
the other hand, a detailed description of those singularities
is given by Eq. (4) in Ref. [8].
By using the above knowledge of f (x, y) for (3), we canderive the universal behaviors of A(kF +q, ω) ≈ ZF Ã+ (kF +q, ω) for low energies and in the vicinity of the Fermi point.
For |ω−uν q| uν |q| with ν = s, c, we have|q|
A(kF +q, ω) ≈ aν±−1+θ

uν kFθ

ω−uν q −μν
,
uν q
LL

(10)

where the upper and lower signs of aν± are taken for |ω| >uν |q| and |ω| < uν |q|, respectively. The exponents are
given by μLL= 1/2 − θ and μLL= (1 − θ)/2, as+ is asc−±positive constant, as = 0, and ac are positive constants
with ac+ /ac− = sin π θ/2, reproducing well-known results[4–6]. Our new result is a description of the pseudoelectron
excitation including the renormalization effects. Namely,
for |ω−vF q| vF |q|, we findω−vF q θ|q|−1+θ
A(kF +q, ω) ≈b0 −b±,(11)vF q
vF kFθ
where b0 and b± are positive constants with b+ /b− =cos π θ , and the upper and lower signs of b± are taken for|ω| > vF |q| and |ω| < vF |q|, respectively. Therefore, thepseudoelectron excitation is characterized by the power-law
exponent θ , the prefactor ratio b+ /b− = cos π θ , and thevelocity vF = [us +(1+2θ)uc ]/2.

5 Summary and Conclusion
We have studied generic features of the spectral function
A(k, ω) of an interacting 1D electron system at T = 0 for
low energies and in the vicinity of the Fermi point. By ascaling argument based on a renormalization-group scheme,
we have shown that the pseudoelectron excitation in A(k, ω)
is described by (11) to find that the universal exponent is

J Supercond Nov Magn (2015) 28:1331–1335

equal to θ and the universal ratio of the prefactors is given by
b+ /b− = cos π θ with θ = (Kc−1)2 /4Kc . Then, in addition
to the two power-law divergences at ω = us (k − kF ) andω = uc (k −kF ), A(k, ω) has an asymmetric peak-like cusp
characterized by those universal quantities at ω = vF (k−kF ).
The pseudoelectron corresponds to the renormalized
electron in the renormalization-group language and its
velocity is given by the renormalized Fermi velocity at
the LL fixed point, an important quantity to determine theposition of the peak-like cusp. We have found that the pseudoelectron velocity is given by vF = [us +(1+2θ)uc ]/2 andis equal to the slope of the lowest excitation spectrum near
kF in the system with large but finite size. For the 1D Hubbard model at n = 0.59 filling, we have obtained explicit
values of vF as a function of the interaction U .
Acknowledgments This work is partially supported by Innovative
Area “Materials Design through Computics: Complex Correlation and
Non-Equilibrium Dynamics” (No. 22104011) from MEXT, Japan.

References
1. Tomonaga, S.: Prog. Theor. Phys. 5, 544 (1950)
2. Luttinger, J.M.: J. Math. Phys. 4, 1154 (1963)

1335
3. Haldane, F.D.M.: J. Phys. C 14, 2585 (1981)
4. Meden, V., Schönhammer, K.: Phys. Rev. B 46, 15753 (1992)
5. Voit, J.: Phys. Rev. B 47, 6740 (1993)
6. Orignac, E., Tsuchiizu, M., Suzumura, Y.: Phys. Rev. B 84,165128 (2011)
7. Voit, J.: J. Phys.: Condens. Matter 5, 8305 (1993)
8. Maebashi, H., Takada, Y.: Phys. Rev. B 89, 201109(R) (2014)
9. For studies beyond the Luttinger liquid theory, see A. Imambekov,
T. L. Schmidt, and L. I. Glazman, Rev. Mod. Phys. 84 1253
(2012)
10. Claessen, R. et al.: Phys. Rev. Lett. 88, 096402 (2002)
11. Sing, M. et al.: Phys. Rev. B 68, 125111 (2003)
12. Benthien, H., Gebhard, F., Jeckelmann, E.: Phys. Rev. Lett. 92,
256401 (2004)
13. Carmelo, J.M.P. et al.: J. Phys.: Condens. Matter 18, 5191 (2006)
14. Carmelo, J.M.P., Bozi, D., Penc, K.: J. Phys.: Condens. Matter 20,
415103 (2008)
15. Essler, F.H.L.: Phys. Rev. B 81, 205120 (2010)
16. Shankar, R.: Rev. Mod. Phys. 66, 129 (1994)
17. Metzner, W., Castellani, C., Castro, C.D.: Adv. Phys. 47, 317(1998)
18. Sólyom, J.: Fundamentals of the Physics of Solids, vol. 3.
Springer, Berlin, Heidelberg (2010)
19. Woynarovich, F.: J. Phys. A 22, 4243 (1989)
20. Sólyom, J.: Adv. Phys. 28, 201 (1979)
21. Suzumura, Y.: Prog. Theor. Phys. 63, 51 (1980)
22. Schönhammer, K., Meden, V.: Phys. Rev. B 47, 16205 (1993); 48,
11390 (1993)