Tunneling of correlated electrons in ultra high magnetic ﬁeld
Shan-Wen Tsai1 , D. L. Maslov1 and L. I. Glazman2
1)

arXiv:cond-mat/0203034v1 [cond-mat.str-el] 2 Mar 2002

Institute for Fundamental Theory and
Department of Physics, University of Florida, Gainesville, FL 32611
2)
Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455
(June 4, 2018)
Eﬀects of the electron-electron interaction on tunneling into a metal in ultra-high magnetic ﬁeld
(ultra-quantum limit) are studied. The range of the interaction is found to have a decisive eﬀect
both on the nature of the ﬁeld-induced instability of the ground state and on the properties of the
system at energies above the corresponding gap. For a short-range repulsive interaction, tunneling
is dominated by the renormalization of the coupling constant, which leads eventually to the chargedensity wave instability. For a long-range interaction, there exists an intermediate energy range in
which the conductance obeys a power-law scaling form, similar to that of a 1D Luttinger liquid.
The exponent is magnetic-ﬁeld dependent, and more surprisingly, may be positive or negative, i.
e., interactions may either suppress or enhance the tunneling conductance compared to its noninteracting value. At energies near the gap, scaling breaks down and tunneling is again dominated
by the instability, which in this case is an (anisotropic) Wigner crystal instability.
PACS numbers: 71.10Pm, 72.15Gd, 72.15

law scaling with energy, whereas the renormalization of
the vertex is not yet important. In this interval, the system behaves as a Luttinger liquid and the results of Ref.
[ 5 ] are applicable. At energies close to the gap, the
power-law scaling breaks down and tunneling is dominated by the electron properties in the vicinity of the
critical point. For a short-range interaction, the critical
point dominates tunneling at all energies, and there is
no Luttinger-liquid behavior. We also ﬁnd that above a
certain magnetic ﬁeld, when the lowest Landau level is
strongly depopulated, the interaction enhances the tunneling conductance of the Luttinger-liquid regime above
its free-electron value. Such an unusual behavior receives
a natural explanation in terms of electron scattering from
the Friedel oscillation near the tunneling barrier.
First we review the procedure of ﬁnding the eﬀect of
the electron-electron interaction on the tunneling conductance of a metal in the UQL5 . Let the magnetic ﬁeld
be perpendicular to the contact plane (z = 0) that separates two metallic sides. We consider both the symmetric conﬁguration, where both sides are in the UQL, and
the asymmetric one, where one of the sides is made of a
high-carrier concentration and/or dirty metal so that, in
the ﬁrst approximation, the magnetic ﬁeld does not aﬀect
that side. The transmission and reﬂection amplitudes for
non-interacting electrons, t0 and r0 , are assumed to be
known. We choose the Landau basis for the free electron
wavefunctions (non-interacting electrons in the UQL and
in the presence of the barrier):

Low-dimensional systems exhibit “zero-bias anomalies” in tunneling (non-linearities of the current-voltage
characteristics at small biases), which reﬂect the renormalization of the density of states by the electronelectron interaction. In particular, tunneling into a onedimensional (1D) metal (Luttinger liquid) is characterized by a power-law suppression of the tunneling conductance, gT . A three-dimensional metal placed in a strong
magnetic ﬁeld that depopulates all but one Landau levels (ultra-quantum limit, UQL) provides an example of
a very special quasi-1D system. It is well-known1–3 that
repulsive interactions in the UQL lead to a charge-density
wave (CDW) or a Wigner crystal instability of the ground
state. This has been conﬁrmed, for example, by experiments on graphite in high magnetic ﬁelds4 . The most
complete analysis of this instability for the case of a shortrange interaction was performed in Ref. [ 3 ] by solving
the renormalization-group (RG) equation for the interaction vertex. On the other hand, it has recently been
shown that for the case of a long-range (Coulomb) interaction gT exhibits a power-law, Luttinger-liquid-like behavior at suﬃciently high energies5 . This result was obtained in two ways – via a perturbative Hartree-Fock procedure, which results in the RG equation for the transmission amplitude, and via bosonization in the coherent state basis – neither of which took into account the
renormalization of the interaction vertex. In this paper,
we combine the approaches of Refs. [ 1,3 ] and [ 5 ] to
study the behavior of the tunneling conductance in the
whole energy interval from the Fermi energy down to the
energy associated with the instability. This is accomplished by solving a coupled system of the RG equations
for both the interaction vertex and transmission amplitude. We ﬁnd that for a long-range interaction there is a
parametrically wide energy interval in which the ﬂow of
the transmission amplitude already results in its power-

(0)
ψpz ,px (r) = ψpz (z)χpx (x, y),

where
ψpz (z) = θ(z)t0 eipz z + θ(−z)(eipz z + r0 e−ipz z )
ψ−pz (z) = θ(z)(e−ipz z + r0 eipz z ) + θ(−z)t0 e−ipz z .
1

(1)

that exchange contribution involves integration over the
transverse momentum k⊥ , whereas the Hartree contribution enters Eq.(4) at q⊥ = 0. This is due to a simple fact
that the Friedel oscillation in charge density, and hence
in the Hartree potential, is translationally-invariant along
the barrier, whereas the exchange potential involves the
density matrix and is thus non-local. If the interaction
potential is peaked strongly at q = 0 (e.g., Coulomb
potential) the Hartree contribution is enhanced due to
q⊥ = 0 and may in fact dominate over the exchange one.
In this case Γ0 in Eq.(3) is negative and thus the transmisson amplitude is enhanced by the interaction. Such an
unusual behavior should be contrasted, e.g., to a strictly
1D case6 , in which Γ0 = U0 (0)−U0 (2kF ) and is thus positive for any “realistic” interaction potential, i.e., such
that U0 (q) is a monotonically decreasing function of q.
In our case, due to the 3D nature of the problem, the
exhange and Hartree contributions are not simply given
by the corresponding forward and backscattering amplitudes but also involve averaging over the transverse direction. As a result, Γ0 can take either sign.
At the next order, Vex and VH are re-calculated using
the corrected wavefunctions and are substituted back into
Eq.(2). The higher order corrections to δΨ(r) are higher
powers of logs. Similarly to tunneling through a barrier
of weakly interacting electrons in 1D,6 summation of the
most divergent corrections to t in all orders of the perturbation theory can be performed via an RG equation

(Here and thereafter, we set ¯ = 1 and also deﬁne the
h
h
magnetic length as our unit of length: ¯ c/eB ≡ 1).
Backscattering of magnetically quantized electrons at
the barrier gives rise to a Friedel oscillation in the charge
density, whose amplitude decays away from the barrier
as z −1 . Correspondingly, the exchange, Vex (r, r′ ), and
Hartree, VH (r), potentials calculated using exact (in the
presence of the barrier) but otherwise free wavefunctions,
also exhibit rapid 2kF -oscillations and decay as z −1 away
from the barrier. The ﬁrst-order correction to the wavefunction due to interaction is given by
δΨpz ,px (r) =

dr′

dr′′ G(r, r′ ; E) [Vex (r′ , r′′ )+

δ(r′ − r′′ )VH (r′′ )] Ψ(0) x (r′′ ),
pz ,p

(2)

where G(r, r′ ; E) is the Green’s function in the presence
of the magnetic ﬁeld. When the momentum transfer
(2pz ) matches the wavevector of the Friedel oscillation
(2kF ), a slow (z −1 ) decay of the oscillations in Vex and
VH leads to a logarithmic singularity in δΨ(r), and consequently, in δt. For electron’s energy close to the Fermi
energy, we have
t = t0 1 − |r0 |2 cΓ0 ln (W/E)

(3)

where c = 1(1/2) for symmetric (asymmetric) conﬁguration, W ∼ EF is the eﬀective bandwidth, Γ0 ≡ Γ(q⊥ =
0), and Γ(q⊥ ) is expressed via the Fourier transform of
the interaction potential U0 (qz , q⊥ ):
Γ(q⊥ ) =

1
(2π)2 vF

dt
= −cΓ0 t (1 − |t|2 ),
dξ

2
d2 k⊥ iq⊥ ·k⊥ −k⊥ /2
e
U0 (0, k⊥ )
2π
2

−U0 (2kF , q⊥ )e−q⊥ /2 .

(5)

where ξ ≡ ln (W/E) and t(ξ = 0) = t0 . The solution of
(5) is
(4)
t=

The interaction potential, U0 , can be both of long- and
short-range. The ﬁrst situation corresponds to a singleband metal, in which there are no more charge carriers besides those already in the UQL. In this case,
U0 (qz , q⊥ ) = 4πe2 /(q 2 + κ2 ) (for q ≪ kF ), where κ =
ωp /vF and ωp is the plasma frequency at B = 0. The
perturbation theory works if e2 /vF ≃ κ2 ≪ 1 , which
means that the interaction is long range. As the magnetic ﬁeld reduces the phase-space available for the motion along the ﬁeld, kF and vF decrease with the ﬁeld (in
the UQL, kF , vF ∝ B −1 ). Therefore, the perturbation
theory breaks down at suﬃciently strong ﬁeld so that
κ ≃ 1. The case of a short-range interaction may correspond to a situation when a small pocket of the Fermi
surface is in the UQL, whereas other parts of the Fermi
surface still contain many Landau levels. Screening of
the interaction among the magnetically quantized carriers is then mostly due to the “external” carriers, and the
interaction may be of short-range.
The ﬁrst and second terms in Eq.(4) are the exchange
and Hartree contributions, respectively. Due to the Pauli
principle, they enter Eq.(4) with opposite signs. Note

t0 (E/W )cΓ0
|r0 |2 + |t0 |2 (E/W )cΓ0

.

(6)

For small values of |t0 | (high barrier), Eq.(6) reduces to
cΓ
a power-law form t = t0 (E/W ) 0 . The tunneling conductance is related to the transmission amplitude via the
Landauer formula.
The power-law solution for t is obtained using the bare
interaction vertex. However, the vertex is also subject to
renormalization. The ﬂow of Γ(q⊥ , ξ) is described by
the integro-diﬀerential RG equation3 , which has a more
transparent meaning when written for the Fourier transform of γ(r⊥ , ξ) = 1/(2π)2 d2 q⊥ Γ(q⊥ , ξ)e−iq⊥ ·r⊥ :
dγ(r⊥ , ξ)
=
dξ

′
dr⊥ γ(r′ , ξ)γ(r⊥ − r′ , ξ)
⊥
⊥
′

× 1 − eir⊥ ∧r⊥ ,

(7)

where a∧b = ax by − ay bx . The initial condition for Eq.
(7) is given by Γ(q⊥ , ξ = 0) = Γ(q⊥ ), where Γ(q⊥ ) is
deﬁned by Eq. (4). The ﬂow of the vertex aﬀects the RG
equation for t, in which now the bare vertex has to be
replaced by the renormalized one:
2

dt
= −cΓ(q⊥ = 0, ξ) t (1 − |t|2 ) .
dξ

We now turn to the case of a long-range interaction. It
is instructive to consider ﬁrst an example of an ultra-long
interaction: U0 (r) = const → γ(r⊥ , 0) ∝ δ(r⊥ ). To analyze the ﬂow of γ(r⊥ , ξ) for small ξ it suﬃces to substitute
γ(r⊥ , 0) into the right-hand side of Eq.(7), upon which
the latter vanishes. It means that the ultra-long range interaction is equivalent to a strictly 1D case, when γ(r⊥ , ξ)
is invariant under renormalization. The example just
given is certainly not realistic: even for the Coulomb interaction the ground state is unstable with regard to the
CDW formation2 . However, remnants of this behavior
survive for the Coulomb potential in a sense that γ(r⊥ , ξ)
ﬂows slower than t(ξ). As a result, there exists an intermediate energy scale, EP L , (∆ ≪ EP L ≪ W ) such that
at E ≃ EP L the higher order logarithmic terms generated by Eq.(8) have already summed up into a power-law
form but the renormalization of Γ is not signiﬁcant yet.
EP L may be deﬁned by the condition that the lowestorder correction to t0 becomes equal to t0 :

(8)

Eqs. (4), (7) and (8) provide a full description of tunneling into a three-dimensional metal in the UQL.
Few words about the general features of Eq.(7) are
now in place. Physically, γ(r⊥ , ξ) plays the role of an interaction potential between two electrons whose guiding
centers are separated by a “distance” |r⊥ | across the magnetic ﬁeld. (Note however that γ(r⊥ , 0) is related to the
Fourier transform of the bare interaction, cf. Eq. (4).)
The ﬁrst and second terms in the brackets in Eq. (7) correspond to the Peierls and Cooper interaction channels,
respectively. A strictly 1D case is recovered by putting all
guiding centers onto the same line: r⊥ = r′ . In this case,
⊥
r⊥ ∧ r′ = 0 and the Peierls and Cooper channels cancel
⊥
each other exactly. This cancellation reﬂects the wellknown fact that there are no instabilties for 1D (spinless)
electrons and the system remains in the massless phase
(Luttinger liquid) down to the lowest energies. In the
UQL case r⊥ = r′ , Peierls and Cooper channels do not
⊥
cancel each other, and an instability becomes possible.
For a short-range interaction, the q⊥ -dependence of U0
can be neglected: U0 (0, q⊥ ) → u0 , U0 (2kF , q⊥ ) → u2kF
2
and Γ(q⊥ ) = (u0 − u2kF ) /(2π)2 vF e−q⊥ /2 . For repulsion,
the solution of Eq. (7) with the Peierls term alone is1,3
Γ(q⊥ , ξ) = Γ

−1

(q⊥ ) − ξ

−1

.

ξP L = ln (W/EP L ) = 1/cΓ0 .
For a screened Coulomb potential, Eq.(4) together with
the condition κ ≪ kF < 1 yields
∼
Γ0 = e2 /πvF

t0 f (E)
,
|r0 |2 + |t0 |2 f 2 (E)

(9)

(10)

where f (E) ≡ [Γ0 θ(E − ∆) ln E/∆]c . Near the gap
(E ≈ ∆), the tunneling conductance gT ∝ |t|2 assumes a
universal form
gT ∝ (E − ∆)2c θ(E − ∆).

−1

.

(12)

A simple estimate for EP L can be obtained if
the ﬁrst (exchange) term in Eq.(12) is larger than
the second (Hartree) one, in which case EP L ≃
W exp(−πvF /ce2 | ln κ|). At the same time, for kF ≃ 1,
the gap is estimated as ∆ = W exp(−avF /e2 ), where
a ≃ 1. We see that EP L ≫ ∆ due to the presence of a
large logarithm under the exponential.
As it has already been pointed out, Γ0 is not necessarily positive: if kF is suﬃciently small (the lowest Landau
level is strongly depleted), the Hartree (backscattering)
term dominates and Γ0 < 0. The dependence of Γ0 on kF
is shown in the inset of Fig. 1 for a range of κ. Recalling
that both kF and κ are magnetic-ﬁeld dependent, one can
express the ﬁeld B0 at which Γ0 changes sign in terms of
the nominal ﬁeld BQ , at which the UQL is achieved, and
−1
the familiar gas parameter, rs , as B0 ≈ 2BQ / ln1/3 Crs ,
where C ≃ 1. For rs ≃ 1 B0 ≈ 2BQ . The gap as a function of kF and κ was obtained by solving the RG equation (7) numerically with Eq.(4) as an initial condition.
The ratio ξc /ξP L ≡ (ln W/∆)/ ln(W/EP L ) is plotted in
Fig. 1. All numerical results presented here are for a symmetric contact, i. e., c = 1. For a suﬃciently long-range
interaction (cf. ﬁlled circles for κ = 0.1 in Fig. 1) there
is a wide range of kF (and thus of the magnetic ﬁeld) in
which ξc /ξP L > 1. Once this condition is satisﬁed, there
exists an energy interval ∆ ≪ E < EP L (ξP L < ξ < ξc )
∼ ∼
∼
in which the system eﬀectively behaves as a 1D Luttinger
liquid, exhibiting a characteristic power-law scaling of
the tunneling conductance. (For EP L ≪ E ≪ W the
power-law reduces to the ﬁrst log-correction.) Near the
gap (E ≃ ∆), the power-law behavior crosses over to
the threshold behavior [Eq. (11)]. If the interaction is

The vertex diverges most rapidly at q⊥ = 0 and the
equation for the pole ξ = Γ−1 (0) yields the gap ∆ =
W exp(−1/Γ0 ). As the electron density remains homogeneous across the ﬁeld, the instability correspond to
a CDW along the ﬁeld. Inclusion of the Cooper term
merely changes the magnitude of the gap3 . The RG equation for t(ξ) [Eq.(8)] with Γ(q⊥ , ξ) from Eq.(9) allows for
an analytic solution:
t=

| ln κ| − 4kF 2

(11)

Comparing t from Eq.(10) with Eq. (6), obtained without taking into account the ﬂow of the interaction vertex
Γ, we see that the two coincide only at the lowest order
in the interaction (ﬁrst log-correction to t0 ). In contrast
to Eq.(6), the full solution shows that the higher order
logarithmic terms do not sum up into a power-law. This
is already seen from the fact that the ﬁrst log-correction
to t0 [Eq.(3)] becomes of order t0 at E ≃ ∆, i.e., when
the renormalization of Γ cannot be neglected. In other
words, the CDW critical point dominates the physical
properties in the entire energy interval from W to ∆.
3

not suﬃciently long-ranged, there is no energy interval
in which ξc /ξP L > 1 is satisﬁed. This is illustrated by
the open circles in Fig. 1 (κ = 0.6). (We did not plot the
points corresponding to kF < κ, as it would have been
incosistent with our weak-coupling approximation.)

2

Γ0 / (e /πvF)

12
10

0

8

ξc/ξPL

a good agreement between the full solution (solid lines)
and the Luttinger-liquid solution (dashed lines). As E
approaches ∆ (ξ → ξc ), scaling breaks down and t(E)
vanishes at E = ∆. In the inset of Fig. 2, kF = 0.4. The
rise of t with ξ corresponds to the negative value of the
bare vertex, Γ0 . This corresponds to an enhancement of
gT above its non-interacting value. Had the vertex renormalization been absent, t would have reached unity in the
limit E → 0 (ξ → ∞). However, the vertex renormalization curbs the rise in t and eventually brings it down
to zero at E = ∆. At the instability, the system becomes an insulator. For long-range interaction, we ﬁnd
that the vertex diverges most rapidly at ﬁnite q⊥ (for
κ → 0, q⊥ →≈ 0.7). This is consistent with previous
work2,4 and suggests that in this case the instability is
of the Wigner-crystal type, with charge modulation both
along the ﬁeld and in the transverse directions.
In conclusion, we found that the range of the electronelectron interaction plays a crucial role both in the nature
of a ﬁeld-induced instability and in tunneling at energies
above the corresponding gap. For a short-range interaction, there is a CDW instability which dominates tunneling in the whole energy interval. For a long-range interaction, depending on the depletion of the lowest Landau
level, tunneling may exhibit two types of characteristic
behavior. For not too strong depletion, the tunneling
conductance decreases as a power-law of the relevant energy scale at higher energies and vanishes in a thresholdlike manner near the gap (which for the long-range case
is the Wigner-crystal gap). For a stronger depletion, the
conductance ﬁrst increases as a power of energy, then
goes through a maximum and ﬁnally vanishes at the gap.
These predictions are amenable to a direct experimental
veriﬁcation by tunneling into low-carrier-density materials, e.g., graphite.
We are grateful to E. V. Sukhoroukov and V. M.
Yakovenko for very instructive discussions. D. L. M. acknowledges support from the NHMFL In-House Research
Program, NSF Grant No. DMR-9703388 and Research
Corporation (RI0082). L. I. G. acknowledges support
from NSF Grant No. DMR-9731756.

0.2

0.4

6

0.8

0.6

kF

1

4
2
0

0.2

0.4

0.8

0.6

kF

1

FIG. 1. Ratio ξc /ξP L for κ = 0.1 (ﬁlled circles) and κ = 0.6
(open circles). The dashed line corresponds to the value 1.
Inset: Γ0 /(e2 /πvF ) as function of kF for various values of κ
(from top to bottom: κ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6).

0.1
κ=0.5
1

0.01

t

κ=0.1

t

0.1

0.001

0.01
0.001
0.1

0.0001

1

2

ξ/(e /πvF)
0.01

0.1

2

10

ξ/(e /πvF)

1

10

FIG. 2. Solid lines: solution of the full system of RG
equations Eqs. (8, 7, 4) for the screened Coulomb potential.
Dashed lines: solutions of Eq. (8) with Γ0 given by its bare
value. In the main plot t0 = 0.1, kF = 0.8 and the values of
κ are indicated in the ﬁgure. Inset: kF = 0.4. The curves,
from bottom to top, correspond to κ = 0.1, 0.2, 0.3 and 0.4.

1

S. A. Brazovskii, Zh. Eksp. Teor. Fiz. 61, 2401 (1971) [Sov.
Phys. JETP 34, 1286 (1972)].
2
H. Fukuyama, Solid State Comm. 26, 783 (1978).
3
V. M. Yakovenko, Phys. Rev. B 47, 8851 (1993).
4
Y. Iye, et al., Phys. Rev B 25, 5478 (1982); Y. Iye and G.
Dresselhaus, Phys. Rev. Lett. 54, 1182 (1985); H. Yaguchi
and J. Singleton, Phys. Rev. Lett. 81, 5193 (1998).
5
C. Biagini, D. L. Maslov, M. Yu. Reizer and L. I. Glazman,
Europhys. Lett. 55, 383 (2001).
6
D. Yue, L. I. Glazman and K. A. Matveev, Phys. Rev. B
49, 1966 (1994).

A full solution of Eqs. (4,7,8) illustrating various
regimes of the t(E)-dependence for a screened Coulomb
potential is represented by solid lines in Fig. 2 for t0 =
0.1. Dashed lines are solutions of Eq. (8) only with Γ0
given by its bare value [Eq. 4], i. e., without taking into
account the renormalization of the coupling constant. In
the main panel, kF = 0.8. The region of small ξ shows

4