arXiv:cond-mat/0102422v1 [cond-mat.supr-con] 23 Feb 2001
Metal-insulator crossover in superconducting cuprates in strong magnetic fields
P.A. Marchetti,aZhao-Bin Su,bLu Yuc,b
aDipartimento di Fisica “G. Galilei”,INFN, I–35131 Padova, Italy
dInstitute of Theoretical Physics, CAS, Beijing 100080, China
cAbdus Salam International Centre for Theoretical Physics, I-34100 Trieste, Italy
The metal-insulator crossover of the in-plane resistivity upon temperature decrease, recently ob-
served in several classes of cuprate superconductors, when a strong magnetic field suppresses the
superconductivity, is explained using the U(1) × SU(2) Chern-Simons gauge field theory. The ori-
gin of this crossover is the same as that for a similar phenomenon observed in heavily underdoped
cuprates without magnetic field. It is due to the interplay between the diffusive motion of the
charge carriers and the “peculiar” localization effect due to short-range antiferromagnetic order.
We also calculate the in-plane transverse magnetoresistance which is in a fairly good agreement
with available experimental data.
PACS Numbers: 71.10.Pm, 11.15.-q, 71.27.+a
The in-plane resistivity in heavily underdoped sam-
ples of cuprates (in particular LSCO) exhibits a mini-
mum and a crossover from metallic to insulating behav-
ior upon the temperature decrease . Recently a similar
crossover was observed in several classes of superconduct-
ing cuprates [2–5] when a strong magnetic field (up to 60
Tesla) suppresses the superconductivity. The “obvious”
interpretation in terms of two-dimensional (2D) localiza-
tion or 2D insulator-superconductor transition is ruled
out, as the sheet resistance, defined as ρsh≡ ρab/a, where
a is the interlayer distance, at the crossover point is be-
tween 1/25 to 1/12 in units of h/e2[2–4], or, using a free
electron model, the estimated product kFl, where kF is
the Fermi momentum, l the mean free path, is between 12
and 25! The insulating ground state persists up to opti-
mal doping in LSCO  and in electron-doped supercon-
ductors Pr2−xCexCuO4(PCCO) , while in newly stud-
ied Bi2Sr2−xLaxCuO6+δ(BSLCO, or La-doped Bi-2201)
it persists only up to 1/8 hole-doping without show-
ing any signature of stripe formation. Thus ascribing
this metal-insulator (MI) crossover to a quantum critical
point related to charge density instability  is open to
objections . There were several attempts to interpret
the insulating ground state in doped cuprates using var-
ious arguments on non-Fermi liquid behavior . How-
ever, the crossover phenomenon as temperature varies
has not been addressed up to now, to the best of our
knowledge. It was thought earlier that the MI crossover
in the absence of magnetic field  and that induced
by magnetic field is of different origin (the sheet resis-
tance in the first case was substantially higher) . This
is doubtful, because the recent experiments on YBCO
show a MI crossover in nonsuperconducting compounds
in the absence of magnetic field , and the same type
of MI crossover (with comparable sheet resistance at the
crossover)in the superconducting compounds of the same
series doped with Zn in the presence of magnetic field
. We believe the two crossover phenomena are of the
same origin. We have used the U(1) × SU(2) Chern–
Simons (C.S.) approach to the t − J model, proposed
by us earlier  to explain the MI crossover in heav-
ily underdoped cuprates in the absence of magnetic field
. In this Letter we generalize our formalism to in-
clude the effect of magnetic field and show that such a
MI crossover is a universal feature of doped cuprates,
and it is due to a “peculiar charge localization” effect
(using the wording of Ref. ), resulting from the inter-
play of the spin-excitation gap (corresponding to short-
range antiferromagnetic order (SRAFO)) and the holon
induced anomalous dissipation. Moreover, we will show
that the observed large positive in-plane transverse mag-
netoresistance (MR) at low temperatures [12,13] can be
semi-quantitatively explained within this formalism.
The U(1) × SU(2) C.S. gauge field approach is a par-
ticular scheme of slave-particle formalism to treat the
t − J model based on an exact identity , introduc-
ing a U(1) field gauging the global charge symmetry and
a SU(2) field gauging the global spin symmetry, both
with C.S. actions. Using an optimization procedure of
free energy , a careful mean field (MF) approxima-
tion gives the following results: The U(1) gauge field for
low doping δ develops a π-flux per plaquette converting
holons into Dirac fermions with a Fermi energy ǫF∼ tδ.
The holons induce a vortex structure in the MF con-
figurations of the SU(2) gauge field, with vortices cen-
tered at the holon positions. These dressed holons in
turn are seen as slowly moving impurities by spin waves
giving rise to a spinon mass ms∼
this feedback is self-consistent because for low δ we have
ǫF ∼ tδ << ǫs∼ J?δ|lnδ|. We use J = 0.1eV, t/J = 3
of “gauge fixing” (using the N´ eel gauge) the SU(2) gauge
field becomes physical, describing the spin fluctuations.
The spinon action is given by a nonlinear σ-model with
a mass term (spinon gap) which in the CP1represen-
tation yields a new self-generated U(1) gauge field A
coupling holons and spinons ( This field is analogous to
the U(1) gauge field in the standard slave-particle ap-
proaches [15,16]). Due to coupling to holons (fermions
in our approach), this gauge field acquires an anoma-
?δ|lnδ|. Notice that
in our numerical computations. Due to a special choice
lous dissipation term, the “Reizer singularity” , which
dominates the low-energy action for the transverse com-
ponent of the gauge field AT. For ω,|? q|,ω/|? q| ∼ 0 we have
?ATAT?(ω,? q) ∼ (χ|? q|2+ iκω
diamagnetic susceptibility and κ ∼ δ the Landau damp-
ing. The interplay of the two different energy scales, the
spinon gap and the holon induced anomalous dissipation,
is the key factor in our interpretation of the MI crossover
. For the temporal component in the same limit, we
have ?A0A0?(ω,? q) ∼ (γ + ωp)−1, where γ is the fermion
density of states and ωpthe plasmon gap.
Now we consider the introduction of a magnetic field
H perpendicular to the plane.
, R = Rs+ Rh, i.e., the observed resistivity is the
sum of the holon and spinon contributions, can be gen-
eralized to this case. The external electromagnetic po-
tential Ae.m., corresponding to the constant magnetic
field H can couple with coefficient −ε to spinons and
1 − ε to holons. In principle, 0 ≤ ε ≤ 1 is arbitrary.
However, to be consistent with the requirement that the
physical inverse magnetitic susceptibility should be the
sum of the inverse of that of holons and spinons, i.e.,
malized spinon and holon susceptibility, respectively (this
relation can be derived in the same way, as the Ioffe-
Larkin rule), we find ε = χ∗
was argued earlier using variational considerations .
Replacing these quantities by unrenormalized values, we
find 1 − ε ∼ χs/(χh+ χs) ∼
Coulomb gauge A0
e.m.= 0, the effective action for the
gauge field A can be written as:
|? q|)−1, where χ ∼ t/δ is the
The Ioffe–Larkin rule
h)−1, where χ∗
hare the renor-
s). This value
?δ/|lnδ| << 1. In the
+(AT+ (1 − ε)Ae.m.)Π⊥
h(AT+ (1 − ε)Ae.m.)
s(AT− εAe.m.)] +iσh(H)
bles due to holons and spinons, respectively, σh(H) is
the Hall conductivity due to holons. Note Ae.m.appears
in two places in this low energy effective action: one is
simply a shift of the transverse component of the gauge
field ATby (1 − ε)Ae.m.and −εAe.m.corresponding to
the minimal coupling to holons and spinons, respectively,
while the other is a C.S. term due to parity breaking in-
duced by H.
As remarked in , the leading effect of the integra-
tion over A0 is the renormalisation of the diamagnetic
susceptibility: χ → χ(H) = χ +
tive action. The holon contribution Rhcan be evaluated
using the Boltzmann equation, taking into account the
classical cyclotron effect, as in , obtaining
are unrenormalized polarization bub-
4π2γin the ATeffec-
h[1 + ((1 − ε)Hτ
where τ is the transport relaxation time, τimp the
impurity scattering time and mh ∼ δ/t the holon
mass.The spinon contribution Rs is evaluated here
using the Kubo formula for the spinon current: Rs =
polarization bubble at ? q = 0, renormalized by gauge
fluctuations.At large scales, for x0≫ |? x|, Π⊥
is approximately given by ?∂µG(x|A)∂µG(x|A)?, where
? ? denotes the A-expectation value and G(x|A) is the
spinon propagator.Using the Fradkin representation
[20,11] it can be transformed into a gauge invariant form
?∂µG(x|F)∂µG(x|F)?, where G(x|F) in terms of a path-
integral over 3-velocities, φµ(t) ≡ ˙ qµ(t),µ = 0,1,2, is
s(ω))−1, where Π⊥
sdenotes the transverse
G(x|F) ∼ i
·(φi(s′) − 2pi)(φj(s′′) − 2pj)[#]
Fij(p,s|s′,λ) = Fij(x + λ
Fij≡ ∂iAj− ∂jAi.
After integration over AT, using the effective action, and
integration over the 3-velocities in the gaussian approxi-
mation, and over p and s by saddle point for low T, one
obtains at large scales
ds′′φ(s′′) − 2ps),
0− |? x|2)
where α =
with the anomalous skin effect due to Reizer singular-
is monotonically increasing, vanishing quadratically near
the origin and g is monotonically decreasing vanishing
at large argument.(See  for explicit expressions.)
The integration over |? x| and x0appearing in the Fourier
2, q0 is the momentum scale associated
?1/3. f and g are functions describing the
effect of gauge fluctuations and for a real argument, f
transformation are evaluated by saddle point for |? x|, at
large x0, and with scale renormalization by principal
part evaluation for x0[11,21]. The integrals are domi-
nated by a complex saddle point at |? x| = 2q−1
s(T,H) = m2
with c = −if(eiπ/4), a constant with real part ∼ 3 and a
small imaginary part. For the range of physical parame-
ters considered here (H<∼100 Tesla), these bounds gave
a temperature range validity lying between a few tens
and a few hundreds degrees.
The saddle point produces the following effects: it in-
duces “renormalization” of the spinon mass yielding a T
and H dependent damping: m2
rise to an attraction between spinon and antispinon lead-
ing to the formation of a damped spin wave; it introduces
a multiplicative renormalization of the correlation func-
tions which for Rsis given by Z(T,H)[m2
Z(T,H) = (c′
stant ∼ f′′(eiπ/4).
The final result in the range of T described above is
s(T,H); it gives
0)1/2with c′a new con-
where ms(T,H) ≡ |ms(T,H)|e−iΘ(T,H). Basic features
of our formulas can be summarized as follows: for low
T, the effect of the spinon gap is dominating (Θ ց 0),
leading to an insulating behaviour; at higher tempera-
tures one finds a metallic behavior due to the dissipation
induced by gauge fluctuations, contained in |ms(T,H)|,
that becomes the dominant effect.
crossover is recovered, decreasing the temperature. The
minimum of R as a function of T, TMI(δ,H), is decreas-
ing with δ and increasing with H (see the MR curve
(R(H)−R(0))/R(0) in Fig. 1), in agreement with exper-
iment . In the absence of magnetic field the crossover
is determined by the interplay between m2
T/χ ∼ Tmh ∼ λ−2, with ξ the magnetic correlation
length, λ the thermal de Broglie wave length.
λ ≤ ξ, the “peculiar” localization effect due to SRAFO
is not “felt”, and a metallic behavior is observed. In the
opposite limit λ ≫ ξ the cuprate is insulating. The exter-
nal magnetic field effectively reduces the thermal energy,
or increases the thermal wavelength, so the crossover
temperature goes up.The resistivity is diverging at
, which approaches T = 0 as H vanishes.
This divergence is lying outside the region of validity of
our formulas and should be considered as an artifact.
Therefore a MI
FIG. 1. The calculated magnetoresistance for cases when
the quantum effects related to σh(H) are strong, for doping
δ = 0.05. It becomes negative near the minimum which itself
shifts to higher temperatures upon the field increase.
However, the shift in MI crossovertemperature leads to
a large positive (in–plane transverse) MR at low T which
is our main new result, and it was absent in the earlier
treatments . The derived MR scales quadratically
with H (See Fig. 2) in agreement, in particular, with
data on LSCO [12,13], away from the doping δ = 1/8
where the stripe effects dominate. As remarked in ,
the shift of χ induced by the C.S. term reduces the damp-
ing and the H2-term due to minimal coupling acts in the
same direction. In the region of T where dissipation dom-
inates this induces a reduction of resistivity, a tendency
contrasted by the classical cyclotron effect. One then has
two possible types of MR curves: one is always positive
but it exhibits a knee below the crossover temperature
between the mass gap and the dissipation dominated re-
gions (See Fig. 3). This behavior can be compared with
the one observed in LSCO reported in  and we find
a reasonably good agreement. If, on the contrast, the
quantum effects related to σh(H) are sufficiently strong,
FIG. 2. The calculated field dependence of the magnetore-
sistance for doping δ = 0.075, in comparison with experi-
mental data on La1.925Sr0.075CuO4+ǫ (inset), taken from Ref.
0 100200 300
FIG. 3. The calculated temperature dependence of the
magnetoresistance for doping δ = 0.075, in comparison with
experimental data on La1.925Sr0.075CuO4+ǫ (inset), taken
from Ref. .
a minimum develops, eventually leading to a negative
MR in some region around it. This is illustrated in Fig.1.
We should point out that the large positive MR at low
temperatures is foreseen in this theory for both cases.
A comment on the limit H ∼ 0 is in order, where we
recover the results of , in particular m2
icT/χ, Z(T,0) ∼
an inflection point at temperature T∗(δ) ∼ 200 − 300
K, (found also in the experimental curves), above which
the theoretical curves start to deviate strongly from the
experimental data. We propose to interpret this in-
flection point as a mid–point of a crossover to a new
“phase” where our MF treatment is not valid anymore.
If we identify our T∗(δ) with the crossover tempera-
ture T∗found in experiments, both MI crossover tem-
perature TMI(δ) ≡ TMI(δ,0) and T∗(δ) are found in
a reasonable agreement with experimental data (in the
of doping dependences:
in this limit can be written in terms of a dimension-
less variable x ≡ cT/χm2
?|lnδ|. Hence, if we neglect the contribution ∼ T4/3
˜R ≡ (R − R(TMI))/(R(T∗) − R(TMI)), this is a func-
tion only of x, thus exhibiting a “universal” behaviour,
as observed in YBCO .
As a final remark it might be worthwhile to notice
that the same U(1) × SU(2) approach is able to repro-
duce qualitatively [10,23] the behavior of the spin lattice
relaxation rate (1/T1T)63found in underdoped YBCO
 and a structure of Fermi arcs around (π
spectral density detected by ARPES .
We thank J.H. Dai for collaboration in the early stage
of this work, and Y. Ando and T. Xiang for very helpful
discussions. The work of P. M. is partially supported
by RTN Programme HPRN-CT2000-00131, while L.Y.
acknowledges the partial support by INTAS Georgia 97-
s(T,0) = m2
√δ. In this limit the resistivity exhibits
< 0.08), due to a delicate cancellation
sapart from an overall factor
due to holons and define a “normalised resistivity” by
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