RAPID COMMUNICATIONSPHYSICAL REVIEW B 89, 201109(R) (2014)

Structural evolution of the one-dimensional spectral function from the low- to the high-energy limit
Hideaki Maebashi and Yasutami Takada
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan(Received 5 July 2013; revised manuscript received 18 April 2014; published 14 May 2014)
By exactly analyzing the spin- 21 Luttinger liquid (LL) and numerically solving a model of a mobile impurity
electron in the LL, we obtain the one-electron spectral function A(p,ω) in a one-dimensional metal in an entire
range of p at zero temperature. For |p| near the Fermi point pF , A(p,ω) is featured by two prominent peaks ofspinon and (anti)holon representing spin-charge separation, but we also find an additional cusp structure between
them. For |p|  pF , this structure evolves as a main peak in A(p,ω) by swallowing the antiholon mode and its
dispersion relation approaches that of a free electron, implying the existence of an electron excitation in the whole
region, but not quite a quasiparticle in the Fermi liquid due to ever existing power-law decay of the excitation.
DOI: 10.1103/PhysRevB.89.201109

PACS number(s): 71.10.Pm, 72.15.Nj, 79.60.−i

The concept of spin-charge separation plays a central
role in describing low-energy physics near Fermi points ina one-dimensional (1D) interacting electron gas, a typicalexample of the spin- 12 Luttinger liquid (LL) [1]. This concept
may be confirmed in real materials by various experiments[2], including the recent high-resolution angular resolved
photoemission spectroscopy in which the one-electron spectral
function A(p,ω) can be directly measured in the wide rangeof momentum p and energy ω.
If |p| is not restricted to the region near the Fermi
momentum pF , the linear spectrum approximation, usuallyadopted in the LL theory, is not sufficient in appropriately
obtaining A(p,ω). In fact, the effect of the nonlinear spectrumon A(p,ω) has been intensively studied in recent years [3–8].
According to those studies on integrable systems, A(p,ω)
has singularities for arbitrary p as A(p,ω) ∝ |ω−ν (p)|−μν (p)with ν = s and c, where s (p) and c (p) are energies of spinand charge collective excitations, respectively. In the usual
LL theory, the exponent μν (p) is independent of p, but thenonlinearity in the electron dispersion makes it depend on
p [5–9]. Since the edge of support of A(p,ω) is locatedat ω = s (p), μs (p) determines the power of the thresholdsingularity in A(p,ω) and its actual value has been given fromthe finite-size spectrum obtained by the Bethe-ansatz method[6,8]. For nonintegrable systems, this threshold singularity
remains intact, but the singularity at ω = c (p) is smeared intoa broad peak [7].
In those preceding works, only the singularities at ω = s (p)and c (p) are discussed on the belief that the electron nature
will not sustain in the spin-charge separated system. For |p| far
away from pF , however, the effect of interactions becomes soweak that we would naively expect that the nature of an injected
electron to measure A(p,ω) manifests itself as a main peak inA(p,ω). Then a natural question arises: Does an electronlike
excitation mode actually exist in the 1D interacting electron
gas for |p|  pF ? If yes, a related and more intriguing questionis, How does the electronlike mode reconcile with the physics
of spin-charge separation for |p| ≈ pF ?
In this Rapid Communication, we have carefully studied the
1D one-electron Green’s function G(p,t) in momentum space
and time and found that for p ≈ pF , its long-time asymptotic
form is composed of three independent modes of power-law
decay. Two of them correspond to well-known spinon and(anti)holon excitations, but the remaining one describes the1098-0121/2014/89(20)/201109(5)mode of an electronlike particle (pseudoelectron) which may
be regarded as an electron dressed with a “cloud” of low-lying
spin and charge collective excitations. This pseudoelectron
does not appear as a main structure in A(p,ω) and never leads
to a finite jump in the momentum distribution function n(p).
As p goes away from pF , the pseudoelectron structure getsbroader, but with the further increase of p, it becomes lessbroad and eventually for p  pF , it evolves as a main anddivergent peak in A(p,ω) by swallowing the antiholon mode.
Concomitantly, its dispersion relation approaches that of afree electron, allowing us to regard the pseudoelectron as afree electron; but actually it is not quite, nor the Landau’s
quasiparticle, basically because this excitation is accompaniedby power-law decay. Those results clarify the generic featureof A(p,ω) in a 1D metal and answer the aforementioned twoquestions. In the following, we shall substantiate our claim.
Let us consider the spin- 12 Luttinger model, for whichG(p,t) with p = pF +k is well known and is given by [10,11]
 ∞
e−ikx
1 − i(x − vF t)dx
iG(pF + k,t) =√1 − i(x − us t)−∞ 2π η − i(x − vF t)
×[1 − i(x − uc t)]−(1+θ)/2
×[1 + i(x + uc t)]−θ/2 ,(1)

for t > 0 with η a positive infinitesimal and  a finite
momentum-transfer cutoff. Here us and uc are the velocities ofspinon and (anti)holon, respectively, and θ denotes the powerof singularity in n(p). They are related to the velocity of anelectron vF and the interactions between electrons g4 and g2 in
the original Hamiltonian through us = vF −g4 /2π , uc = [(vF +g4 /2π )2 −(g2 /π )2 ]1/2 , and θ = (vF +g4 /2π −uc )/2uc [12]. By
eliminating g4 from those equations, we obtain vF = [(1+2θ )uc +us ]/2, and thus vF as well as us , uc , and θ can be
determined by the Bethe-ansatz method for integrable systems.
By regarding the integrand in Eq. (1) as an analytic functionof x to deform the integral path along the real axis into thelower-half complex x plane and assuming us < vF < uc , wecan evaluate iG(pF +k,t) for k > 0 in the long-time limit oft → ∞ as   1/2−γν e−iuν kt+iπφνLL /2−k/
iG(pF + k,t) = e LLk
LL t 1−μνν=s,cν201109-1+(F t)−1+μF −ivF kt+iπφF /2

e

.

(2)

©2014 American Physical Society

RAPID COMMUNICATIONSPHYSICAL REVIEW B 89, 201109(R) (2014)=s,c,e∞and evaluating A(p,ω) by π −1 Re 0 dt eiωt iG(p,t) for ω > 0,
we see that A(p,ω) has a singularity (peak or shoulder) atω =  (p) for |μ (p)| < 1 in the way ofA(p,ω) = const + C ± (p)|ω −  (p)|−μ (p) ,

(4)

where
C ± (p) = π −1 (p)−1+μ (p) [μ (p)]cos(π [μ (p)±φ (p)]/2) with the upper (lower) sign for ω larger (smaller)
than  (p).
As for the term = e in Eq. (3) in which e (pF +k) = vF k,
μe (pF ) = μF , and φe (pF ) = φF with e (pF ) = F for the spin- 21
Luttinger model, we easily see that ∂A(pF +k,ω)/∂ω diverges
k for 0 < θ  1/8 even at p → pF . By using the identity
at∞ω =μ vF±it
Ftedt = (μF +1)e±iπ(μF+1)/2 , we can also verify that0A(p,ω) behaves in accord with Eq. (4), exhibiting a structureof the pseudoelectron.
In Fig. 1, we have explicitly shown the generic feature ofA(p,ω) in which a peak with a cusp exists at the pseudoelectronmode [ω = e (p)] in addition to the well-known double
divergent peaks for p near pF . In plotting A(p,ω), we need toknow concrete values for us , uc , and θ ; we have determinedthem by adopting the Yang-Gaudin model [13], i.e., the 1D
electron gas with a δ-function interaction, described by theN∂2
1
Hamiltonian H = − 2m
i=1 ∂xi2 +V0
i<j δ(xi −xj ), in theintermediate-coupling region of λ0 ≡ mV0 /2πpF = 1.0.
Two comments are in order on the result in Fig. 1: (i) Irrespective of , the limit of η → 0+ is required in Eq. (1) to keep†
the correct electron anticommutation relation {cpσ ,cp σ } =∞δpp δσ σ so as to satisfy the sum rule −∞ A(p,ω)dω =†{cpσ ,cpσ } = 1 [14]. Thus the proper limiting procedure in
Eq. (1) is to make η zero first with  being finite, leading to
the result in Fig. 1. If η is replaced by −1 as was previously
the case, the pole contribution disappears, because the residue
vanishes with that choice of η, constituting the reason forthe omission of the pseudoelectron contribution in precedingstudies [15,16]. That choice is physically incorrect, because theanticommutation relation is not globally satisfied, violating
the sum rule [17]. (ii) If we take θ = 0 in Eq. (1), essentialλΛωThe terms specified by ν = s,c represent the contributions
from contours along the two branch cuts (which we takeparallel to the imaginary axis) associated with the branch
points at x = us t −i/ and uc t −i/, showing the well-known
time correlation specific to spinon and (anti)holon in the
LL
LL with the exponents μLL
s and μc given, respectively, as
LL
LL
LL
μs = 1+φs = 1/2−θ and μc = φcLL = (1−θ )/2.
√ Here γs =LL
θ/20, γc = θ/2,
(uc −us )(1+θ)/2 π ]2/(1+2θ) ,s = [(uc +us )θ/2(uc −us )1/2 ( 1+θ)]2/(1+θ)  with (z) theand LLc = [(2uc )2gamma function. The last term denotes the contribution froma simple pole at x = vF t −iη, seemingly describing an electron
moving with the velocity vF , but not quite a usual quasiparticle
in the Fermi liquid owing to the existence of the power-law
decay with the exponent μF (= 1), where μF = φF = −θ and(1+θ)/2(uc +vF )θ/2 (vF −us )1/2 ]1/(1+θ) . Thus weF = [(uc −vF )shall call it a pseudoelectron. Its decaying behavior persists
even at k → 0.
Casting iG(p,t) in Eq. (2) into the form
[ (p)t]−1+μ (p) e−i (p)t+iπφ (p)/2 , (3)iG(p,t) =Λ

HIDEAKI MAEBASHI AND YASUTAMI TAKADA

Δ

ω

ω

Λ

FIG. 1. (Color online) Full spectral function A(pF +k,ω) at k =0.1 for the spin- 21 Luttinger model with use of us = 0.4406pF /m,
uc = 1.582pF /m, and θ = 0.05 351 obtained by the Bethe ansatz for
the Yang-Gaudin model with λ0 = 1.0 (solid curve). The contribution
from a pseudoelectron excitation (ω = vF k), which has been omitted
in preceding studies [15,16], is explicitly shown by A(pF +k,ω)(dotted curve).

simplification occurs, leading to an analytic expression ofA(p,ω) in terms of the confluent hypergeometric functionof two variables. In this case, Voit found that A(p,ω) has anadditional peak with logarithmic singularity at ω = vF k besides
the spinon and holon ones, while n(p) remains the same as thenoninteracting one with a finite jump at |p| = pF [18]. But, hegave up calculations without the replacement of η by −1 forthe spin- 12 Luttinger model with θ = 0 because of the difficulty
in dealing with a double Fourier transform in A(p,ω). We haveovercome this difficulty by use of a mathematical trick virtually
identical to Feynman parameters [16,19]. By introducing two
Feynman parameters, we can analytically perform the double
Fourier transform and then resort to numerical computationof the remaining double Feynman integral. The result thus
computed with high accuracy is plotted in Fig. 1 and exhibitsa singular behavior precisely matched up with that suggested
in Eq. (4).
Now let us discuss the effect of nonlinear dispersion onA(p,ω) as well as the case of |p| far away from pF by adopting
the mobile impurity model which was previously introduced
for treating spinon and holon modes without spoiling the
rigor of the whole theory [5,7]. Our strategy is to employ
the same model to calculate the contribution to A(p,ω) fromthe pseudoelectron excitation. In the model, we first consider
an electron with either linear or nonlinear dispersion ξp asa mobile impurity in the LL with the velocity vp = ∂ξp /∂p.
(Here we take only the case of ξp  0, i.e., |p|  pF .) This
impurity electron is assumed to couple with spin and chargecollective excitations near the right (+) and left (−) Fermi
points with the interaction Ṽαν ≡ Ṽαν (p) (α = ±, ν = s,c) [20].
Then, we can calculate the Green’s function for the mobile
impurity electron by straightforwardly extending the method
developed by Dzyaloshinskiı̆ and Larkin [21,22] or in a moregeneral GW framework [23,24], from which we obtain an
integral equation to determine the Green’s function G(p+k,ω)201109-2

RAPID COMMUNICATIONSSTRUCTURAL EVOLUTION OF THE ONE-DIMENSIONAL . . .

PHYSICAL REVIEW B 89, 201109(R) (2014)

in momentum-frequency space as(ω − ξp − vp k)G(p + k,ω)
dq dε W (q,ε)
G(p + k − q,ω − ε)e−|q|/ ,=1+i(2π )2 ε − vp q

λ
Λ

(5)

ν=c,s

ω

−ie−ie (p+k)t,(δαν /2π)2 /2
α=± [1 + i(uν − αvp )t]λG(p + k,t) =λwith  the cutoff of momentum transfer q and W (q,ε) =(1/4) αβν Ṽαν χαβν (q,ε)Ṽβν with use of χαβν (q,ε) the exact
correlation functions for the Luttinger model.
Since Eq. (5) is very similar to the one known well in
the Luttinger model [10], we are familiar with its solution; bytransforming the variables from momentum and frequency into
real space and time, we can write down a differential equation
for G(x,t), from which we obtain(6)
for t > 0, where e (p+k), the dressed impurity energy, contains
an energy shift in proportion to the cutoff  ase (p + k) = ξp + vp k − (δαν /2π )2 (uν − αvp )/2. (7)ανIn Eqs. (6) and (7), phase shifts δαν ≡ δαν (p) are independentof  and related to Ṽαν and vp through√√Ṽ±c 1 + θ − Ṽ∓c θ
Ṽ±s
δ±c =
, δ±s =.
(8)uc ∓ vp
us ∓ vp
The results in Eqs.
√ that e (pF +k) =√ (7) and (8) indicate
vF k, δ+c (pF ) = 2π 1+θ , δ−c (pF ) = 2π θ, δ+s (pF ) = 2π , andδ−s (pF ) = 0. Thus at p = pF , we see that Eq. (6) is reduced tothe pole contribution in Eq. (1).
For illustration of the overall behavior of A(p > pF ,ω) withthe change of p and ω, we adopt the Yang-Gaudin model
analytic exin the weak-coupling region (λ0  0.1) in which √pressions√such as vF = pF /m,
u
=
(1−λ
)v
,
u
=
1+2λ0 vF ,
s
0
F
c
√
and θ = ( 1+2λ0 −1)2 /4 1+2λ0 are successfully checked
to reproduce the exact Bethe-ansatz results with sufficientaccuracy, indicating that we may well specify the mobile
impurity model by employing the weak-coupling results of thequadratic dispersion ξp = (p2 −pF2 )/2m and the interactions
Ṽ±c (p) = −Ṽ+s (p) = V0 and Ṽ−s (p) = V0 /(1−2λ0 ln[ξp /Ei ]),all of which are obtained by the poor man’s scaling with Ei ,an initial energy scale. Then, from G(p,t) in Eq. (6), we can
explicitly compute A(p,ω) which is insensitive to the choiceof Ei . In Fig. 2, A(p,ω) thus obtained at λ0 = 0.1 is displayedwith increasing p from pF to show its complete structural
evolution in the 1D weakly interacting electron gas with thequadratic dispersion. Since we focus on the region of ω in thevery vicinity of ξp , only the pseudoelectron mode appears asa singular structure in A(p,ω) in Fig. 2.
By comparing Eq. (6) in its long-time limit with Eq. (3),we can determine μe (p) and φe (p) as(δαν /2π )2 /2,(9)μe (p) = 1 −φe (p) = −


αν

αν

(δαν /2π )2 sgn(uν − αvp )/2.

(10)

ω ξ

Λ

FIG. 2. (Color online) Structural evolution of A(p,ω) as a function of ω in the very vicinity of ξp with p going away from pF (fromp/pF −1 = 0.01 to p/pF −1 = 0.2 by the interval 0.01) in the mobile
impurity model at λ0 = 0.1.

They are independent of  and universal, but they dependon p in the system with nonlinear dispersion [9], as plotted inFig. 3 in which μe (p) and |φe (p)|−1 are given by the solid anddotted curves, respectively. The structural feature of A(p,ω) is
specified by both μe (p) itself and its relative value with respect
to |φe (p)|−1.
two dimensionless constants,√ Thus we define √c1 = 1−1/ 2+O(λ0 ) and c2 = 3+O(λ0 ), corresponding to
the intersections of the solid and dotted curves. By virtueof Eq. (4), A(p,ω) at ω = e (p) has a peak with a cusp
for p/pF −1 < c1 λ0 (the spin-charge separated regime) anda divergent one for p/pF −1 > c2 λ0 except for the narrow
vicinity of p/pF −1 = c2 λ0 (the nearly-free-electron regime).
In the intermediate regime of c1 λ0 < p/pF −1 < c2 λ0 , on the

μ

μ
μ

φ

μ

λ
φ

λ

λ

FIG. 3. (Color online) Leading exponents of the spectral function
at the spinon, antiholon, and pseudoelectron modes, which aredenoted by μs (p), μc (p), and μe (p), respectively, versus p/pF −1
for the Yang-Gaudin model with λ0 = 0.1.

201109-3

RAPID COMMUNICATIONSHIDEAKI MAEBASHI AND YASUTAMI TAKADAPHYSICAL REVIEW B 89, 201109(R) (2014)

other hand, A(p,ω) shows only a broad peak around ω = ξp
due to the strong damping effect of δ+c → ±∞ for vp → uc in
Eq. (8) (the pseudoelectron damping regime).
In order to give an entire picture of A(p,ω) including spinon
and antiholon modes, we have also plotted the exponents μs (p)and μc (p) in Fig. 3, which are exact for the Yang-Gaudin modelwith λ0 = 0.1. (Accurate values for those exponents can bederived from the exact Bethe-ansatz results for s (p) and c (p)for Galilean invariant systems [7].) By comparing them withμe (p), we can identify the mode(s) to dominate A(p,ω): (i)For |p|/pF −1  λ0 , the situation is characterized by μs (p) ≈μc (p)  μe (p), implying the dominance of spinon and antiholon modes. Thus the physics here is well described by theconcept of spin-charge separation, but the pseudoelectron is
now found to appear as an additional singular cusp structure as
depicted in Fig. 1. (ii) For |p|/pF −1 ∼ λ0 , the pseudoelectron
excitation is overdamped as shown in Fig. 2 and sandwiched
between the spinon and antiholon divergent peaks, making
it difficult to be detected, although we do not expect thatits total contribution to the spectral weight is negligible.
(iii) For |p|/pF −1  λ0 , μe (p) becomes much larger than either μs (p) or μc (p), indicating that the long-time evolution ofG(p,t) is controlled by the single mode of pseudoelectron, welldefining the nearly-free-electron regime. In fact, according toour explicit calculation based on Ref. [7], μs (p) is a decreasing
function of |p| and becomes negative for |p|  2pF , so thatthe divergence at the edge of support of A(p,ω) disappears.
As for the antiholon, μc (p) slowly increases with increasing|p| and reaches 1/2 at |p| → ∞. In the high-|p| limit with
λ0 fixed, both e (p) and c (p) approach the free-electron
dispersion, but because μe (p) ≈ 2μc (p) ≈ 1, the antiholon peakgets absorbed into the pseudoelectron one. Thus A(p,ω) is
composed of a single divergent peak, reduced to a delta function at |p| → ∞ as A(p,ω)≈ (vp )−1 fγ ([ω−p2/2m]/vp )−1 ∞where fγ (y) ≡ (2π ) −∞ ds(1+s 2 )−γ eiys ≈ γ |y|−1+2γ with
γ = [1−μe (p)]/2 ∝ (pF /p)2 → 0+ .
Our discussion has been limited only for |p|  pF , but asimilar discussion can be made for |p| < pF in which electronhole asymmetry due to nonlinearity in the dispersion may be
a matter of interest. Our theory is applicable to lattice systems

by suitably choosing the dispersion ξp and an interaction
parameter λ0 . Then the pseudoelectron damping regime appears in general for |1−|vp |/vF | ∼ min(1,λ0 ) due to the strong
scattering effect for |vp | = |∂ξp /∂p| → uc ,us . According to thenumerical results for A(p,ω < 0) obtained by the dynamical
density-matrix renormalization group (DDMRG) method forthe 1D Hubbard model in the intermediate-coupling regionof U = 4.9t and the filling n = 0.6 (for which λ0 ∼ 1) [25],the peaks of spinon, holon, and its shadow band for |p| < pF ,which are predicted by the Bethe ansatz, are found, but there
appears no signature of an additional noticeable structure, atleast for |p| not close to pF . This absence of the pseudoelectron
structure in the DDMRG does not contradict our theory,
because the data without a strong artificial broadening effectare given only in the pseudoelectron damping regime of
1−|vp |/vF ∼ λ0 .
In summary, we have theoretically studied the overallbehavior of A(p,ω) in a 1D metal at zero temperature inorder to establish the concept of a “pseudoelectron,” describing
the behavior of an electron injected into the LL with either
linear or nonlinear dispersion. This pseudoelectron is found to
manifest itself in an entire range of p, though its importancein the whole structure of A(p,ω) depends on p; for |p| ≈ pF , itappears only as an additional cusp structure to the main peaksof spinon and (anti)holon, while for |p|  pF , it provides amain and divergent peak with the correct high-|p| limit ofa delta function. This pseudoelectron very much resembles
a quasiparticle in higher-dimensional Fermi-liquid systems(e.g., see Fig. 3 in Ref. [24]), although it is not quite
the same, reflecting the specialty of 1D physics. We hopethat this concept of a pesudoelectron will be confirmed in
the future through experiment and/or large-scale numericalcalculation with deliberately chosen parameters so as to avoid
its overdamping regime.

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H.M. thanks H. Fukuyama, M. Nakamura, and K. Penc
for valuable comments. This work is partially supported
by Innovative Area “Materials Design through Computics:
Complex Correlation and Non-Equilibrium Dynamics” (Grant
No. 22104011) from MEXT, Japan.

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RAPID COMMUNICATIONSSTRUCTURAL EVOLUTION OF THE ONE-DIMENSIONAL . . .

PHYSICAL REVIEW B 89, 201109(R) (2014)[12] In standard notations [10], we have taken g4 = 0 and g2 = g2⊥
by the Pauli principle and spin-SU(2) symmetry, respectively,
and defined g4 ≡ g4⊥ and g2 ≡ g2 = g2⊥ .
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201109-5