Zeeman splitting of zero-bias anomaly in Luttinger liquids
A.V. Shytov1,2 , L.I. Glazman3 , and O. A. Starykh4
1

Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030
L.D. Landau Institute for Theoretical Physics, 2 Kosygina Str., Moscow, Russia 117940
3
Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455
4
Department of Physics and Astronomy, Hofstra University, Hempstead, NY 11549

arXiv:cond-mat/0211303v1 [cond-mat.mes-hall] 15 Nov 2002

2

Tunnelling density of states (DoS) in Luttinger liquid has a dip at zero energy, commonly known
as the zero-bias anomaly (ZBA). In the presence of a magnetic ﬁeld, in addition to the zero-bias
anomaly, the DoS develops two peaks separated from the origin by the Zeeman energy. We ﬁnd
the shape of these peaks at arbitrary strength of the electron-electron interaction. The developed
theory is applicable to various kinds of quantum wires, including carbon nanotubes.
PACS numbers: 73.63.-b, 73.21.Hb, 73.22.-f

Interaction between electrons in a conductor leads to
the formation of an anomaly in the tunnelling density of
states (DoS) at the Fermi level. The one-particle DoS is
directly related to the diﬀerential conductance of a tunnel
junction, and the anomaly in DoS translates to the zerobias anomaly (ZBA) of the tunneling conductance. This
anomaly gets stronger if the conductor is disordered, and
if the dimensionality of the electron system is reduced.
The perturbative treatment of the DoS anomaly in disordered conductors is well-developed [1]. In a disordered
wire or ﬁlm, the perturbation theory in the interaction
strength is divergent at the Fermi level, and therefore
a non-perturbative treatment is needed to describe the
DoS at low energies [2,3,4]. In one-dimensional conductors with one or a few propagating electron modes the
suppression of the DoS due to the electron-electron repulsion is strong even in the absence of disorder. The density of states in this case is adequately described within
the Luttinger liquid theory [5]. The ZBA was observed
in experiments with higher-dimensional disordered systems [6,7]. The recently measured [8,9] strong suppression of the tunnelling in a single-wall carbon nanotube at
low bias gave an evidence that electrons in a nanotube
indeed form a Luttinger liquid.

The perturbative theory of Zeeman splitting of ZBA
in a clean 1D system was developed in [11]. There,
the anomalies in the conductance were ascribed to the
physics of Bragg reﬂection of electrons oﬀ the Friedel oscillation. Magnetic ﬁeld B splits the standard Friedel
oscillation in two. The diﬀerence between the two corresponding wave vectors is proportional to B. Electron
scattering oﬀ the two components of the Friedel oscillation results in the conventional DoS anomaly at zero
energy and two additional peaks in the DoS at ±gµB B.
This single-electron picture is valid for a weak electronelectron interaction only, and is not applicable to the Luttinger liquid with a strong repulsion between electrons.
However, the strongest manifestation of the Luttinger
liquid behavior is found in carbon nanotubes, where the
interaction is not weak. Thus, one may question the existence of peaks in the DoS at Zeeman energy in such
systems.
In this paper, we demonstrate that the tunneling density of states in a Luttinger liquid is singular at energies
ε = ±g ∗ µB B. The eﬀective Land´ factor g ∗ here is renore
malized by the interaction. The overall magnitude of the
singular correction to DoS is proportional to the constant of electron-electron interaction in the triplet channel. The energy dependence of the DoS around the singularities is given by a power law, δν(ε) ∼ |ε ± g ∗µB B|γ .
We calculate the exponent γ in terms of the Luttinger
liquid parameters.
Consider the one dimensional Luttinger liquid ﬁlling
the half-line x > 0 and conﬁned by a barrier at x = 0.
†
We decompose the electron creation operator ψs (x) into
†
†
†
the left- and right- moving parts: ψs = ψ+,s + ψ−,s . Here
± denote left and right movers, and s = ±1 denote two
spin states. Then, we bosonize electrons [12]:

Zero bias anomaly thus provides important information about strongly correlated electron systems. However, ZBA is sensitive mostly to the dynamics of electron
charge but not to that of spin. To probe the spin physics,
one may study the eﬀect of a magnetic ﬁeld on the properties of the electron system. The perturbative calculation shows [1] that the application of a magnetic ﬁeld
modiﬁes the anomaly in the DoS. It acquires, in addition
to the zero-bias dip, two peaks at energies ε = ±gµB B,
where gµB B is the Zeeman energy. The peaks heights
are equal and proportional to the electron-electron interaction constant in the triplet channel [1], which is not accessible in a measurement of the conventional ZBA. The
described Zeeman splitting of the ZBA in disordered normal conductors was not observed yet, but a related phenomenon was studied in thin superconducting ﬁlms [10].

π
1
†
exp ±ipF x ± i
(1)
ψ±,s (x) = √
2
2πa
i
− √ [±(ϕρ (x) + sϕσ (x)) + ϑρ (x) + sϑσ (x)] .
2
Bosonic variables ϕi and ϑi describe charge (i = ρ) and
1

Kσ becomes essentially independent of ε, while g⊥ ﬂows
toward zero [13]. In this way, the non-linear term H ′
is not relevant at large times, and can be treated as a
perturbation.
We are interested in the DoS at ε → gµB B. This allows us to consider the Hamiltonian Hσ +H ′ acting on the
states within the energy band of the width D somewhat
exceeding 2gµB B. The constants Kσ (D) and g⊥ (D) in
the Hamiltonian (3) should be renormalized according to
Eq. (6).
Before developing a rigorous calculation, we provide a
hint, where the singularity in the DoS may come from.
Tunneling of an electron may be viewed as spreading
of charge and spin densities, which initially at t = 0
were formed near the barrier (x = 0). Because of the
spin-charge separation, the two density perturbations
propagate independently. The charge propagates freely,
since the corresponding Hamiltonian Hρ in is quadratic.
The propagation of the spin density is aﬀected by the
backscattering term Eq. (5). To demonstrate qualitatively its eﬀect, we expand H ′ in ϕσ to the second order
and then derive the linear equation of motion for the
ﬁeld ϕσ . The ﬁrst-order expansion term only shifts by a
small amount the solution of that equation. The secondorder term generates a contribution ∝ g⊥ cos(bx)ϕσ in
the equation of motion, and leads to the phenomenon of
Bragg reﬂection with wave vector b/2. As the result, the
backscattered component of ϕσ oscillates with frequency
ωz = uσ b/2 = Kσ gµB B. These oscillations give rise to
features in the DoS at energies ε = ±ωz .
We start with retarded Green’s function

spin (i = σ) ﬂuctuations, and a is the short-distance
cutoﬀ. The ﬁelds ϕi and ϑj are canonically conjugate:
[ϕi (x), ϑj (x′ )] = iδij θ(x − x′ ). Since the electron wave
function is zero at the barrier (x = 0), the ﬁelds ϕi satisfy boundary condition:
ϕρ (0) = ϕσ (0) = 0 .

(2)

The Hamiltonian can be divided into four parts:
H = Hρ + H σ + H ′ + H B ,

(3)

where the ﬁrst two terms include the density-density interactions:
Hi =

ui
2π

dx Ki (∂x ϑi )2 +

1
(∂x ϕi )2
Ki

,

(4)

and H ′ represents spin-ﬂip backscattering:
H′ =

2g⊥
(2πa)2

dx cos

√
8ϕσ .

The last term in Eq. (3) describes Zeeman splitting:
HB = −

gµB B
2

dx ρspin (x) =

gµB B
√
2π 2

dx∂x ϕσ ,

where ρspin (x) is spin density. The parameters ui , Ki
and g⊥ can be expressed in terms of interaction potential V (x), but here we treat them as phenomenological
constants. For free electrons, Kρ = Kσ = 1, while for
repulsive interaction Kρ < 1 and Kσ > 1. Also, the bare
parameters Kσ and g⊥ are not independent. For g⊥ ≪ 1,
they are related as Kσ ≈ 1 + g⊥ /2πuσ .
For convenience, we absorb the magnetic ﬁeld term HB
√
into quadratic part by shift ϕσ → ϕσ +gµB BxKσ / 2uσ .
Then, the backscattering term transforms into
2g⊥
H′ =
(2πa)2

√
dx cos( 8ϕσ + bx) ,

GR (x, x′ , ε)
s

(5)

with b = 2gµB BKσ /uσ .
The Hamiltonian (3) decouples into charge and spin
sectors. While the charge excitations do not interact
and their Hamiltonian Hρ is quadratic, the spin sector
is described by the sine-Gordon model Hσ + H ′ . In zero
magnetic ﬁeld the constant g⊥ is renormalized at low energies [12],
g⊥ (D) =

1+

g⊥ (W )
πvF

log W
D

,

= −i

dteiεt

†
ψs (x, t) , ψs (x′ , 0)

0

(here {. . .} is anticommutator) and compute the tunneling density of states as [14]
ν(ε) =

g⊥ (W )

∞

1
4π(ikF )2

Im
s

∂2
∂x∂x′

GR (x, x′ , ε) . (7)
s
x=x′ =0

For slowly varying ϕi (x) and ϑi (x), one may neglect their
derivatives, and diﬀerentiate only the factors e±ikF x in
the electron operators (1). Equation (7) can be rewritten as
1
ν(ε) =
Re
2π 2 a

(6)

∞

0

dt Gρ (t)eiεt + Gρ (−t)e−iεt
× [Gσ (t) + Gσ (−t)]

(8)

−iϑi (x = 0, 0)
iϑi (x = 0, t)
√
√
exp
2
2

(9)

where

where W is the initial, and D is the running bandwidths, and g⊥ (W ) is the “bare” interaction constant.
The renormalization group (RG) ﬂow occurs along the
∗
line Kσ ≈ 1 + g⊥ /2πvF towards the ﬁxed point Kσ = 1,
∗
g⊥ = 0. Finite magnetic ﬁeld does not aﬀect the RG
ﬂow for energies larger than gµB B. For smaller energies,

Gi (t) =

T exp

are time-ordered Green’s functions of charge and spin,
and T denotes time-ordering.
2

we use Eqs. (10) and (11), and arrive to

To compute these correlation functions at g⊥ = 0, we
express the ﬁelds ϕi and ϑi in terms of bosonic eigenmodes aq , a† of the Hamiltonian (4). Because of the
q
boundary condition (2), only odd modes contribute to
ϕi :
cq,i sin qx aq,i eiui qt + a† e−iui qt
q,i

ϕi (x, t) =

2ig⊥
δGσ (t) = −
(2πa)2

aq,i eiui qt − aq,i e−iui qt
cq,i
cos qx
.
Ki
i

δG± (x, t, t′ ) = ±

Here cq,i = e−qa/2 πKi /q, and the short-distance cutoﬀ
a = uσ /D is related to the reduced bandwidth D. The
summation in Eq. (10) involves wave vectors q > L−1 ,
where L is the length of the system. One can compute
the average in Eq. (9) using the relations [15]
1

eA eB = eA+B e 2 [A,B]

and

1

eA = e 2

A2

,

×

(0)
Gi (t)

ia
ui |t| + ia

=

(11)

.

δGσ (t)
(0)
Gσ (t)

(12)

C0
|ε|α ,
Γ(1 + α)

(13)

C0 =

1
πa

a
uρ

a
uσ

1
2Kσ

Gσ (t) =

iϑσ (0,t)
√
2

exp
S

−iϑσ (0,0)
√
2

0

,

S =1−i

−∞

dt′

(14)

ig⊥
2uσ

a
uσ |t|

2(Kσ −1)

θ(t) eiωz t ,

(17)

i π (α+1)
2

∞

cos εt dt δGσ (t)
,
(0)
tα+1 Gσ (t)

(18)

with the exponent γ = α + 2(Kσ − 1), i. e.
γ=

H ′ (t′ ) dt′ .

ω z = g ∗ µB B ;

i(ϑσ (t) − ϑσ (0))
√
dx T exp
2

√
√
× cos( 8φσ (x, t′ ) + bx) − cos( 8φσ (x, t′ ) + bx)

1
−1
−1
Kρ + Kσ − 2 + 2(Kσ − 1) .
2

(20)

Equation (19) is valid for arbitrarily strong interaction
in the charge channel, and conﬁrms the existence of singularity in DoS centered at energy

∞

0

=−

0

Computing the correction
∞

1
.
uσ t′ ∓ x + ia sgn(t′ )

γ

−∞

2ig⊥
δGσ (t) = −
(2πa)2

(16)

δν(ε)
ε − ωz
g⊥ (ωz ) 1
(19)
=−
(0) (ω )
4uσ sin πγ
ωz
ν
z
Γ(1 + α)
cos π (α − γ) for ε > ωz
2
×
×
,
cos π (α + γ)
ε < ωz
Γ(1 + γ)
2

where the averaging is performed over the non-perturbed
Hamiltonian Hσ . To the ﬁrst order in g⊥ , the S-matrix
is
∞

(15)

which is singular at ǫ = ±ωz . Since Eq. (18) is even in
ε, we consider further only ε ≈ ωz . Integrating over the
time domain t ∼ |ε − ωz |−1 , we ﬁnd for the singular part:

.

S

2xuσ t
∓ x + ia sgn(t − t′ )

D−1

To develop perturbation theory in g⊥ , we use the standard expression [16]
T exp

uσ (t −

t′ )

Kσ

0

δν(ε) = πC0 Re e

−1
−1
with the anomalous exponent α = (Kρ + Kσ )/2 − 1,
and the prefactor
1
2Kρ

−∞

a2
a2 + 4x2

with ωz = g ∗ µB B, and renormalized Land´ factor g ∗ =
e
Kσ g. Using Eqs. (8) and (12) at t > D−1 , one ﬁnds the
correction to the DoS

Substituting this expression into Eq. (8) and evaluating
the integral for ǫ ≪ D, one arrives to the well-known
formula [5],
ν (0) (ε) =

dx

Unlike Eq. (12), the correction δGσ (t) contains an oscillating part. We will retain only this part, since we are
interested in singularities at non-zero energies. The oscillation originates from the point of enhanced singularity in
Eq. (16), x = uσ t′ = uσ t/2. The oscillating contribution
to δGσ (t) is

valid for any operators A and B linear in aq and a† . At
q
zero temperature, the only non-zero average is aq a† =
q
1, and one ﬁnds
1
2Ki

∞

where

†

q

dt

′

(0)
× Gσ (t) δG+ (x, t, t′ ) eibx + δG− (x, t, t′ ) e−ibx ,

, (10)

q

ϑi (x, t) =

∞

0

0

g∗
g⊥ (D ∼ µB B)
.
= 1+
g
2πuσ

(21)

At small g⊥ , which implies Kσ ≈ 1, the main contribu, tion to the exponent in Eq. (19) comes from the charge
3

mode. Therefore, the exponents α and γ of the powerlaw singularities in the DoS at ε = 0 and ε = ωz respectively, are nearly identical, γ ≈ α. The contribution (19)
was found for zero temperature, and its energy dependence is non-analytic at any γ. However, it may be easily distinguished from the regular part of ν(ε) only at
γ < 1. Also, ﬁnite temperature T smears the singularity
at |ǫ − ωz | ≃ T .
The DoS anomaly (19) is directly related to the bias
dependence of the tunneling conductance G(V ) between
a conventional metal and a one-dimensional conductor.
The corresponding singular contributions are related as
δG(V )/G = δν(eV )/ν. Tunneling between the ends of
two identical one-dimensional conductors (such as an intramolecular junction between carbon nanotubes [9]) also
has a singularity at Zeeman energy. In this case the peak
at eV = ωz has a diﬀerent shape, because it is deﬁned
by the singularities in the DoS both at ε = ωz and ε = 0.
The conductance can be calculated as G(V ) = dI/dV ,
where the tunneling current I(V ) between the two conductors is proportional to the convolution of the two corresponding DoS:

γnt ≈ αnt ≈

0

dε ν(ε) ν(ε − eV ) .

To conclude, the application of a magnetic ﬁeld to
a Luttinger liquid creates additional singularities (two
peaks) in the tunneling density of states. We have
found the power law characterizing these singularities
and demonstrated that they are robust, i. e., they persist
at any interaction strength in the charge channel. The
magnitude of the peaks is determined by the short-range
interaction, which plays a minor role in the charge physics
of a Luttinger liquid and therefore is hardly accessible in
the measurements of the conventional zero-bias anomaly
of the tunneling conductance.
We are grateful to L.S. Levitov, P.B. Wiegmann,
K.A. Matveev, M.P.A. Fisher, A.G. Abanov and
I.L. Aleiner for useful discussions. This research was
supported by NSF grants PHY99-07949, DMR97-31756,
DMR02-37296, EIA02-10736, KITP Scholarship and by
an award from the Research Corporation.

(22)

Calculating this integral, one ﬁnds the singular contribution to the diﬀerential conductance
eV − ωz
1
g⊥
δG(V )
=−
π
(α + γ)
G(V )
4uσ sin 2
ωz
Γ(1 + 2α)
×
.
Γ(1 + α + γ)

α+γ

(23)

The exponent α + γ here coincides with 2γ up to a small
term of the order of Kσ − 1. This “exponent doubling”
at eV = ωz is similar to that occuring at zero bias [9].
It is interesting to analyse Eq. (23) in the limit of weak
interactions, in which [12]
Kσ − 1 ≈

U (2kF )
g⊥
≈
,
2πvF
2πvF

(24)

Thus, the exponent γnt again nearly coincides with the
ZBA exponent αnt . The reported values of αnt in the experiments with carbon nanotubes were αnt = 0.3 ÷ 0.6,
and therefore the peak at ǫ = ωz should be sharp and
easy to observe.

eV

I(V ) ∝

1
−1
Kρ − 1 .
4

Kρ ≈ Kσ −

U (0)
.
πvF

To the ﬁrst order in the interaction potential U (q),
Eq. (23) yields δG/G = (U (2kF )/4πuσ ) ln(|eV −ωz |/ωz ).
This result is in agreement with the ﬁrst-order expansion
of the tunneling conductance obtained in [11]. However,
beyond this order, there is a diﬀerence between Eq. (23)
and Eq. (47) in [11]. It stems apparently from the inapplicability of the RG approach developed in [11] for the
treatment of tunneling at energies close to ωz .
We derived Eq. (19) and (20) for the single-mode Luttinger liquid. In the case of carbon nanotubes, one has to
take into account the degeneracy between the two conic
points in the Brillouin zone [17]. Treating the interaction in the charge channel non-perturbatively, as we did
before, we ﬁnd the exponent of the singularity at ǫ = ωz ,

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4

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[14] The electron wave function is zero at the barrier, and one
should therefore consider tunneling in a small vicinity of
the barrier. Thus, the tunneling operator contains derivatives of the electron operator.

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5